Stealing the Gold: A Celebration of the Pioneering Physics of Sam Edwards

in a useful form and to this end a simple model will now be discussed in detail which will suggest the right approach.

4. A simple model The non-linear term 'ZMuu causes the turbulence and the simplest way to think of it is as a force which as far as one is concerned is roughly random. This suggests examining the problem of the response of a linear system to a random force in detail and though all the results of this section are well known it is useful to be reminded of them and to present them in an appropriate way. Consider then the motion governed by

where & is statistically specified as in (2.17). (The label k and the vector character of U are dropped in the model; J > 0.) Then Liouville's equation is

or, more usefully, the Green function of Liouville's equation satisfies

This function has the property that it propagates F, i.e. if F(u, t') is the value of F at the time (', then at a subsequent time t

Now if one starts at a time t' with a definite F(u, t') and the force & has a distribution (defined from t' on) of ^([<^]) then the average of F at t, (F(u, t)) say, is propagated by the average of G, say, defined by

This follows by multiplying (4,4) by 3P and integrating over the function !F. The mean Green function is well known in the theory of Brownian motion so the result of the integration (4.5) will be quoted. (A derivation is given in Appendix 1 since its generalization is needed in Appendix 2.) It is

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where

and g is defined by (2.19). Two properties of / need to be noted. First and secondly

Thus (G) loses its dependence upon u' as t — t' tends to infinity and this implies that whatever F(u, t') is, (Fy settles down to the Gaussian

In particular, if g has an exponential form then

and Since ((?} propagates (_F) the equations for (F) and (G) can be written down from the exact expression (4.6)

It follows that as t — t' tends to infinity, the steady value of <-F) satisfies

which has the solution (4.9), i.e.

while (G) settles down to the solution of

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A straightforward generalization is to the case of several ^ characterized by constants ya, w^. Then (4.17) has the factor y(J + ra)"1 replaced by

Another way of stating these results, which will turn out to be the basis for subsequent work, is to observe that if the equation for G is written

where and this is expanded in terms of the solution of (4.13), which for the moment will be called P rather than<<3>, then

In this expansion terms of order yn and of order S'Zn (and mixed terms like yre-m^-an-zmj are collected together. Upon averaging the expression in each brace vanishes, as do all terms odd in &', and <(?) = P. Clearly the second-order brace can be considered as denning y,

and hence P, i.e. ((?> itself. This of course is just what one would expect from a Gaussian distribution: it is specified by its second moment, which in this case is then related to the externally given 7, w, and J. A straightforward extension of this property will now be developed to get an intuitive solution of (2,24). 5. A simple derivative of the turbulent distribution function In this section the model discussed in §4 will be used to derive the equations for the mean distribution function, using an intuitive argument. Consider then the equation

(The mean value signs <...) will now be dropped since these alone are referred to when discussing F and G.) This equation describes the physical situation of energy entering the system due to the fluctuating forces, at a rate h^, and leaving due to viscosity vk2. The term ~LMuu neither creates nor destroys energy but mixes it up amongst the various uk. To some extent, as far as one particular uk is concerned, 2L3fuu must appear as a random force of the type discussed in §4. But in addition the force acting upon u, say will contain a term Mu^u_t+t so that the gain of energy by the component Uj will depend on the magnitude of

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uk and hence the loss of energy by uk will depend upon its magnitude. One can expect then that as far as one component uk is concerned the effect of the rest of the turbulent fluid will be to produce a diffusive term represented as in §4 by a second derivative, and also a term which represents a dissipative force proportional to uk, i.e. to use the nomenclature of the Fokker-Planck equation, a dynamical friction. Thus one may expect the change of just one component uk (and u_k for they always appear together) to be described by

In the same way the steady distribution function will be governed by

The diffusion constant rfk, and the total viscosity «k can be written

i.e. total input into components uk (and u_k) = external input+input from all other components; (5.4) total output = viscous loss (i.e. external output) + output into all other components. (5.5) It must be emphasized that this section is only introductory to the next, so the derivation of $k and Rk which will now follow will employ assumptions that are unnecessary and sometimes incorrect, but it will turn out that the forms of /Sk and .Rk so obtained are correct, so that the consequences of assuming (5.2) are better founded than (5.2) itself. With this proviso the solution of (5.3) is clearly

where (There is always implicit in/ the restriction divu = 0 so that the normalization is appropriate to two degrees of freedom rather than three. The^k's in (5.2) and (5.3) ensure that the constraint is not violated by the motion, but in practice it is most convenient to keep the constraint as a subsidiary condition and in

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effect drop^ from the equations.) Then the mean of J7Jf7^k can be expressed in terms of gk,

If (5.2) is multiplied by M| on the left and by fw^k on the right one obtains upon integration over all u the relation

From these results and those of §4 one may obtain Sk and R±. First, consider *S^ which is the analogue of the expression

The present o^ is the analogue of J, the lifetime associated with the one component k, and the w^ is the lifetime of the fluctuating force, i.e. of

The expression 7 is the analogue of the mean square of the force, i.e. of

This is readily evaluated if it is assumed that in the first approximation the components are independent and their distribution function

(It must be understood that terms are not counted twice in the exponent.) One then has the integral

(The possibility of j = 1 = j' = 1' gives a contribution smaller by a factor A than those quoted and is disregarded. Of the three terms the first gives zero, a property of M"0v, and the answer stems from the remaining two which have j + j' = 0, 1 + 1' = 0 and 1+j' = 0, j +1' = 0 respectively. This then implies that the is represented by
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or in the limit of infinitely large volume

The expression for L is derived in Appendix 3 from (5.15) and is most conveniently written

(where it will be noted that the second part of L is antisymmetric under the interchange of 1 and j, so it contributes nothing to the integral for S^; but the notation is very convenient when .Rk is evaluated below). To obtain -Rk one may argue this way. The coefficient can be derived by expanding the complete G about n(?k and averaging the. analogue of the series (4.20) on the assumption that/ 3 HA- Thus Sk arises from the term k

But there is another term with a non-vanishing average when all but u±,u_k are averaged; it is

Now it is a tricky matter to average away the uv (p =)= k, — k) but the correct procedure is to consider the djdu',. acting upon//giving

Just as with Sk this now gives

where

and

and when calculated agrees with the definition of (5.17). This derivation of JJk is of course little more than an argument, but it is not worth rigorizing it by this method for a proper derivation is given in the next section. That the loss of energy should appear as a dynamical friction term was originally postulated by Heisenberg and the customary term 'turbulent viscosity' can be adopted for R^ (Hinze 1959) which will be seen to have precisely the same role as viscosity in (5.2), (5.3).

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The essential point of this section lies in the need for using two functions 8^ and JJk to describe the turbulent state. This has already been noted by Kraichnan (1959) in a paper which though differing in detail, has a very similar point of view to that of this section.

6. The general method of expansion The basic assumption of this paper is that there exists a degree of randomness in the steady state of turbulence which permits the calculation of the turbulent viscosity. In this section a systematic expansion will be given for these two quantities, the expansion parameter being in effect the degree of randomness of the system. That of course is rather an inprecise statement, but a full discussion of what precisely one is expanding in terms of cannot be given without more experience of the expansion and its consequences, so it will be deferred to § 10. The time independent case is the simpler so it will be dealt with first. One needs then to find the solution of (2.22), in particular one may expect the solution to be simpler in an infinite volume than in a finite volume. In such a large volume the argument of Maxwell in kinetic theory will apply: that if in some region T^ the probability distribution of the u is FVi, and in another adjacent region Vz it is Fv , then taken together in the region V1 + V2 the distribution function will be and hence where Z is a functional of the u which contains no reference to the boundaries. For example, the case of a random external force which leads to (2.22) can be solved in the absence of the mixing term M to give

In general, however, Z cannot be determined exactly and even if it could it would still leave a most intractable functional integral to be performed to obtain the various moments. Indeed the only functionals which are known to be integrable are polynomials multiplied by the exponentials of quadratic forms. Now the eigenfunctions of the kernel

are the (complex) Hermite polynomials multiplied by the exponential

Since the kernel is expected to play a central role in any expansion, this suggests expanding F as a series in the Hermite polynomials

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defined by

where -F0 is given by (5.14).-f Thus one writes where n is a vector in the Hilbert space of all the Hermite polynomials The eigenvalue associated with the label n is It will be seen that the «j,, w_kth polynomial is a tensor of rank wk + w_k. (The tensor indices are not written in explicitly.) Well-known relations exist between the polynomials such as

but in practice only a few of the polynomials will be needed in an infinite system

These polynomials are orthogonal to one another against the Gaussian -F0, but are not normalized to unity but to (gk/A)"k+w-fe. Consider the differential equation for F rearranged in the form

Now, following the ideas of §§4 and 5, ascribe to S and R the (superficial) order M2 and consider F expanded as a series in M. Then one has where, if the operator is denoted by %]

f The suggestion that Hermite polynomials should be used for the expansion has also been made by Hopf (1962). I am grateful to Dr Kraichnan for this reference.

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If the right-hand side of each equation is now rearranged in terms of the Hermite functions, one has then to solve equations of the type

which have the immediate solution

To complete the specification of the expansion one wishes to give correctly the mean value of •MJ:t«/Lk from F0 alone, so that

Proceeding now to calculate Flt

The right-hand side contains H^H^H^ so that

or in continuous variables

The expression for Fz is much more involved,

The resolution of the right-hand side into Hennite polynomials is straightforward but tedious and leads ultimately to the form

where, writing and

(k...l' all different), (a,b,c,d all different, selected appropriately from k...l'), (a,c,d,e,f all different as befpre),

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It is implied in the summations that all the u are different Fourier components, all products of the same component having been resolved into the polynomials. Now of all these terms only the last contributes to (6.19), and *S^ and R± are therefore chosen to make this vanish for each k. By comparing the coefficients of 5k one sees that S± and Rt are precisely those of the previous section. The remaining terms will give the values of to this order of approximation. It is to be noted that the four-w correlation cannot be factorized into two (MM) correlations, and the six u cannot be factorized into 1i(uu x uuuuy. Now in view of the complexity of F% it might be supposed that the higher approximations become unbearably complicated. This, however, is not the case when one passes to the infinitely large volume and a general procedure for writing down the nth term of the series is given in Appendix 4. At each stage new terms appear which give corrections to S and R in order that (6.19) be fulfilled. Some general results can be stated about this series: (i) The number of positive terms to any order equals the number of negative terms, (ii) The nth term of S can be written symbolically as and the nth term in R as For example, the expressions (5.16) and (5.21), and typical terms of the next order which are (Appendix 4)

and

for 8 and R, respectively. In the symbolic notation these are

the M containing ^-functions which remove three integrations.

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This completes the discussion of the steady distribution, and in the next section the time-dependent case is discussed. 7. The general expansion in the generalized phase space To resolve the time-dependent case the same method as that of § 6 will be employed. There is no point now in averaging out the random input force beforehand, so the expansion will be made in & as well as M, they being considered of the same order in as much as both are approximately random as was implicit in §6. At this stage it is worth noting that there is no need for ^ to have the Gaussian distribution of (2.17) and independent values can be given for say (&&&&}. These will however affect only the corrections to 9" and ^ being corrections to the basic assumption of randomness. The equation

will be rearranged as

where The quantities ^^ 9tk, Q.k, Dk will appear analogues of Sk, E^, wk, d^. Expanding as before it will now be required that

where the averaging is over the distribution of the force ^ which will still be given by (2.17) and (2.20). The analysis goes through exactly as before, for example,

so that only the results will be quoted. Since it is still true that the mean rate of input of energy is given by one may define Then if

and

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There is one great simplification to be noted in these equations. Since it follows that If one introduces Qk by the definition then from the definitions of J and q it follows that

If one now tries &lk = J?k as a solution of (7.9) it does indeed satisfy it. But R± can be taken as known since the equations for it are independent of 6^. and #k. It follows that one can now write a closed equation for Qk

in which q±, w^, £ift, L^ all have the same meaning as before. This can usefully be written in time-dependent form by Fourier transformation

where and £ is the kernel of the integral in (7.15). The extension to higher approximation goes through exactly as before and in the time-dependent form the remark (i) still holds, as of course does remark (ii) when dskdkn replaces dPk. An expansion has now been obtained in the time-independent and timedependent cases for the distribution function. By analogy with other branches of theoretical physics it may be termed the generalized random-phase approximation. To understand its implications one needs to solve the equations in as many cases as one can, and it will turn out that it is possible to make considerable progress in spite of the complexity of the equations. It is important to emphasize that the expansion developed here is quite different from those obtained by truncating the infinite set of equations got by taking moments of the Navier-Stokes equations, which have a structure similar to the equations developed in quantum-field theory. These approaches in effect try to make R±, @k do the work of both (Sk, ^.,), (R±, ^ and the form of the solution suggested can be got from the forms above by taking, in say the timeindependent case, and dividing the numerator into the denominator

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Now expand the integral for _Rk

If one now writes and puts further

then

which is a form often studied, and typical of the kind of expression obtained by manipulating the Navier-Stokes equations directly. There is no underlying physical plausibility for this form, however, and though rather complicated mathematical manoeuvres have been performed above, they follow as closely as possible the intuitive models of earlier sections. The relation of the equations derived here with the work of Kolmogoroff and Kraichnan will be discussed at the end of the next section.

8. Properties and solutions of the equations Before attempting to solve the equations derived in §§ 6 and 7 one must verify that the expansion of P is in accord with the original Navier-Stokes equations from which the whole analysis stemmed. To see that this is the case multiply the original Navier-Stokes equation (2.8) taken at the time t, by Uk at the time t'', and average. This gives

or in the four-dimensional Fourier transform

Using the expansion for P, the two terms on the right are evaluated from Pt (7.5), and give ^/Qj and £fk—B^^k respectively, so that the original equation implies that which is indeed (7.15). A similar result applies to the time-independent case and there represents a discussion of the flow of energy. The total energy is

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hence, at any time, the ratio of change of energy is given by

Now the term in M vanishes by symmetry, but it can also be written from F1 and gives But the definitions of S and R ensure that this expression vanishes for

It follows that the total external input and output balance and also the total internal input and output, as indeed must be the case since no work is done by the inertial terms. Whereas 8 and Rq being rates at which energy is absorbed or emitted are familiar concepts, the conservation properties are also true of &1 and B<2, which refer to action. Turning now to the solution of the equations, the simplest case is clearly that of § 6, so one may ask whether there are any conditions, however remote from physical attainability, under which an exact solution can be obtained. It is a property of M that, bearing in mind that divu = 0, one may rewrite

in the form It follows that if
If h and v both tend to zero, their ratio is arbitrary and one can write q = (2/cT7)-1, since this is the case of thermal equilibrium, but in general q is well defined. For constant q the actual integrals for S and E are divergent which is scarcely surprising with an input rising like k2. But there still stems from this analysis the useful comment made earlier that in the corrections to S and S the number of positive terms to any order equals the number of negative terms, for it is only

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by this means that every order vanishes when q is constant (both q, (a are of course positive definite). To consider more realistic cases one can simplify the equations by assuming that the input is more concentrated near small k so that the viscosity can be ignored in the first approximation. This is equivalent to the statement that the Reynolds number of the turbulence may be considered infinite. Of course one cannot balance input and output in the absence of viscosity, but this point can be resolved as will be shown. The simplest input is a power and it is possible to solve this case in the limit of infinite Reynolds number. So consider in which case the equation for _Bk contains no dimensional parameters, and therefore gk, Rk must be powers, and therefore also St. Define q, R by the equations

Then from (5.21)

Writing and using the explicit form of L\

one finds

where A2 is a numerical constant, independent of k. Therefore and

In the same way from (5.16), From the definition of q± one now has and it follows that

i.e. and that

i.e This result is not restricted to the approximations to S± and Rk of equations (5.16) and (5.19) but is true to all orders, as is seen by considering the symbolic expressions for the higher terms quoted in § 6. The effect of the higher terms is to alter

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the coefficients which occur. It follows that there exist constants KX, pa, trx such that the solution is given exactly by

(In the literature it is customary to use the distribution per unit |k|, i.e. so the above result gives A particular case is that of white noise a = 0 (white that is with respect to wavelength; the definition of h* has already assumed whiteness with respect to frequency), for which

The validity of these solutions depends on the integrals A and B, and the integrals appearing in higher approximations, all converging. It is R± which is the critical function for if it exists it is easily confirmed that 8t and all the higher functions also exist. Tor small | j | so that the Rt integral goes like

which implies that a < 2. For large j, L contains terms like k. j and ka but allowing for the angular integration both give the expression

implying that a > — 1. The solution then is valid for For a, < — 1, a solution cannot be obtained without invoking the viscosity which then affects the solution for all |k|, as in the first example of this discussion. For a > 2 the precise nature of the input at small [k| can be expected to affect the entire solution, and the input itself will of course be modified at small |k| in order that the total input be finite. This case will be discussed further shortly. One has then a picture of energy entering the kth component of the system at a rate Aft(|k|/&i)~0!, gradually being transported to higher and higher |k| till finally the viscosity can no longer be ignored, with the result that gk falls away very much faster. Finally, a region is reached in which -Rk is negligible and only external input and output matter giving

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The apparent paradox that one may investigate the small |kj region by dropping the viscosity, even though the latter is required for overall energy balance, is resolved by observing that in the absence of viscosity the integrals over $k and -Rk do not converge. Large |k| values act as an infinite capacity well for energy, and the formal identity of the integrals over jk| of $k and Bt is not meaningful. The inclusion of the viscosity restores the precision of the identity. There is in the model of turbulence with source Adkl/fej)"", a localization of influence of one |k value upon another. Thus if one takes a region 8 of k space near to k2 say and asks how much this region contributes to 8%, the answer is an amount which tends to zero as k2 + kx increases. To be precise which tends to zero as k-t + k2 tends to infinity. A similar statement holds for -Rk; both remarks are essentially contained in the statements that the integrals over all |k| space for S^ and J?k converge. Consequently one can regard the situation as a cascade of energy from small |k| to larger |k|, supplemented by an external input which, per component, decreases as |k| increases. Finally, for large k viscosity comes in and kills the flow of energy. Of course it is clear from the definitions of-Bk and S^ that energy flows in and out of all components to all components, of larger and smaller k; but on an average it flows from small to large k, the imbalance being taken up by the input. The above model is still not satisfactory, however, since though the input per component decreases, the input per unit wave-number does not, the latter incorporating a weight factor |k|2 from |k|2<M;, which overcomes the k|~«. A. physically realistic model will decrease much faster than a power and one should expect \ to be zero for |k| > K^ say. (This remark need not apply in magnetohydrodynamie turbulence where white noise from electromagnetic sources is quite feasible and the previous model therefore significant.) Consider then such an input,

Integrate the equation up to a value K, where K > Kv If one introduces

then Since L contains a term 8(k.+j +1) one may write and so, by writing k + j for — k in the second term obtain

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where Sis the region between the two spheres k| = K, |k+j| = K. Now suppose tentatively that the solution to this problem is again a power law, with Then writing |k| = -BT|k'| and |j| = K\}'\ one sees that where Y is the value of the integral, a dimensionless constant. But 3f is a constant, independent of A, so one has which when combined with the equation for S^, i.e. with (8.21), gives

This means that or in terms of the distribution per unit wave-number

the Kolmogoroff spectrum (see, for example, Batchelor 1959). Unfortunately the problem cannot be resolved so simply for the dimensional arguments only apply if all the integrals converge, and they do not. The relation between gk and R± is still (5.19) and still only converges if J|j| 2 2j
where p^ tends to zero with j j j. Suppose that for |j| < J the value of q^ directly mirrors the input, and remove this region from the integral. Near j = 0, Rj is small so one may write giving If again J?k and q± are taken tentatively to be powers, the integral as before can be transformed into

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This integral will not be independent of k unless it converges as J -> 0. This is the case for the part depending upon pa but not for the first term which again gives a contribution like k2^1. So altogether one may write This now mntrfl.riinta thfi original assnniTvHnn of n, mirfi nnwpr la.w ainnn it, sncr
that B lies between and

Assuming the former, since 8 = 2m + n, one has This law has been obtained by Kraichnan in the paper mentioned earlier. Kraichnan gives a discussion of the experimental situation in this paper. If one adopts the other extreme, one obtains the Kolmoeroroff result Presumably the complete solution will lie between these two extremes which are in fact very close to one another. These remarks again apply to all higher order terms which have the effect of modifying the coefficients 3$ and #. The Kolmogoroff hypothesis hi the present context is that the coefficient 3S is negligible compared to <€, but there seems no reason in the present analysis for this to be the case. The surprising point is that if one makes the exact opposite of the Kolmogoroff assumption: that the energy input into a component of large k is directly dependent on the behaviour of the system in the external input region, i.e. US ^> *$, one only changes the |k| dependence of gk from |k|~~^ to |k|-V. There now remains the time dependence to be investigated and this will be done in the next section.

9. The time correlation of the velocity correlation functions The basic equation may be integrated over all k0, and in order that it will reproduce the equations of § 6 one must have

i.e.

In time-dependent form one has then to solve

with the boundary conditions

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which will assure that and so wkgk = ht+Sk. The time dependence of <^. is highly involved since it has a highly convoluted structure. If one tentatively associated a time dependence of exp ( — d\t) with Qi(t) then <$£ would behave as an average over In different regions of k space (Wfc +W-t-j)"1 will be larger or smaller than w^1. In particular, if 6^ is a power, ((uk + w_k_j)~1 will be a maximum for j = Jk. Though it is now clear that Qk cannot have a simple exponential decay one can still argue that .$£ will consist of some part decaying slower than <2k and some part faster, so a crude assessment of the situation will be to write where Wk decays faster than $k, Vt slower. For short times, the system moves slowly because St — 0 at t = 0. This means that initially i.e.

After a while Wk will become small, V^ will still be slowly varying, so the behaviour at intermediate times will follow the solution of

i.e.

where p^(t) is some slowly varying function. Finally, at very long times only the most slowly varying components remain, and following the suggestion above a crude model will be to consider and since Jt is small, therefore This equation has the solution (v^./^) t -?*' where y is an arbitrary constant chosen to fit on to the intermediate solution. It is very crude of course to assume that all Jj decaying more slowly than ^k do so at the minimum rate, but a more elaborate argument allowing for the variation of Jj leads to a time dependence which is not very different on a logarithmic scale. These arguments can again be applied to higher terms in the expansion and rather surprisingly still go through. For example, one finds corrections to (9.13) of the type ^|k and so on, which again leads to the final power law, so the general picture of an initial Gaussian form (9.9), followed by a main exponential region (9.11) characterized by 6^, then finally a power law tail is independent of the order of the approximation. There is some correlation between any one n± and any other uit and between any v^ and the input. For very long times in the behaviour of every MJ will be found the residue of the behaviour of the most slowly varying parts of the system, either of those MJ for j ~ 0, or of the input should there be slowly varying components in it (which have not been considered here).

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10. The accuracy of the expansion In much of the work of §§ 8 and 9 it has been possible to state that the higher terms of the expansion do not alter the functional form of the solution but only the constants which appear. There still remains the question of how well these constants are represented by the simplest approximations to S and R. In the case of an input A(|k|/fc1)~a it has been shown that the solution can be written in terms of the two constants pa,
(The precise statement of the approximation is given in Appendix 4 with the discussion of higher order terms.) The expansion in this approximation is then in the ratio of the external input squared to the internal input squared, and the accuracy of the expansion is measured by the smallness of the external input required to keep the system steady. This argument is of course limited to the power input case and does not mean that if h^ = 0 anywhere the method is exact in that region of k space. Since these purely mathematical arguments are not very convincing it is worth considering the situation from the physical point of view and asking what phenomena are considered small by stopping at the first approximations for S± and R±. The distribution function

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predicts the correlation of three velocities

but it also predicts correlations of four and six velocities which cannot be expressed as simple factors (terms which vanish if all k, j, 1, m are unequal)

[terms which vanish if k... p are not all unequal)

To ignore F6, Ft, etc., is to assume that the five U correlation can be expressed in terms of the three and the two, and the eight U correlation can be expressed in terms of the six and the two, and the extent to which they are not is a measure of the inaccuracy of the theory. This is rather a remote physical effect. A much simpler one is to consider the probability that the velocity at a point x is U. This is given by and will be independent of x hi the homogeneous case. Clearly if the expansion for F is used it develops/ in a series of Hermite polynomials

The basic Gaussian/0 is chosen so that Hz never appears. Working to second order, since by symmetry <4re) is zero, one has fourth- and sixth-order polynomials alone

The residual terms in Fz are the cause of the subsequent terms of the expansion, so one may now say that the expansion should be good if / is a Gaussian with

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small additional terms in £T4, Zf8 and no further corrections such as the Hs, Hla, H1!L which arise from F&. A more detailed analysis shows that the effect of the F2 corrections is to make the Gaussian rather more peaked at the origin. More elaborate correlation functions such as the joint probability of finding H! at Xj while U2 at X2 can be defined

and discussed in a similar fashion. Similar arguments can be applied to the time-dependent case. It will be noted that gk, .Rk and \ form a soluble set of functions without discussing time dependence so that if they could be obtained from experiment, the equation (7.16) governing time dependence could be viewed as one for <2k alone with -^k- \ Pu* m as externally defined function

11. Conclusions Many problems in theoretical physics can be expressed in terms of functional differential equations, but turbulence is an exceptional problem in that there is in the limit of large Reynolds number no external parameter which can be used as a basis of an expansion technique. In the language of quantum field theory it is a problem of infinitely strong coupling constant. It follows that an expansion must be based on the internal properties of the system and with one's present limited knowledge of non-trivial mathematical operations in Hilbert space the only substantial fact is that since the probability of finding a particular velocity at a particular point in a fluid is quite close to a Gaussian (Batchelor 1959, ch. 8), the system is substantially random and the generalized random phase approximation should be applicable. This method appears to be the simplest which stems directly from Liouville's equation or the generalized phase space equation, and is entirely consistent to any order in the sense that it contains no features which contradict the original equations from which it was derived. The situations discussed in this paper are all highly idealized, being, it is believed, as simple as can be whilst still containing the mathematical essence of the problem. For this reason no detailed comparisons with experiments are offered, though it is hoped that the method of attack will prove a sound basis for the discussion of real situations and further calculations to this end are in hand. The author would like to thank Dr Batchelor for a helpful discussion which led to an expansion of §8, Dr Rraichnan for a correspondence which eliminated several errors from the first draft, and caused the author to invent the work of §3. The first draft was an account of lectures given by the author at the Culham Laboratory in 1961, and the author is grateful for the opportunity to visit the laboratory on that and many other occasions and for many discussions there, particularly with Prof. W.B.Thompson. He would like also to thank Prof. Flowers for many helpful discussions concerning the problems of turbulence and the writing of this paper.

60

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S. F. Edwards REFERENCES

BATCHBLOB, G. K. 1959 Homogeneous Turbulence. Cambridge University Press. GEL'IAITD, I. M. & YAOLOM, A. M. 1960 J. Math. Phys. 1, 48. HINZE, J. O. 1959 Turbulence. New York: McGraw-Hill. Horr, E. 1952 J. Rat. Mech. Anal. 1, 87. HOPF, E. 1962 Proc. Symp. App. Math. 13. KBAICHNAK, R. 1959 J. Fluid Mech. 5, 497.

Appendix 1 The equation of motion can be solved directly with the boundary condition U(t') = u', giving

The Green function for the motion is then

It follows that the mean Green function is given by

Write 6? parametrically by using the integral representation of the 8 function

By substituting in (2.17) the integration over !F is performed by completing the square, i.e. by changing variables to

which leaves

This is finally evaluated by again completing the square to give

where

61

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269

Appendix 2 If the non-linear term is added one may still write the solution of

as and proceeding as before eventually obtain

with the condition divu = 0. This of course is of no use as it stands since the unknown interaction term remains, but upon explicitly differentiating <$>, as in (4.13) above, the interaction term only appears at the time t, when U is just u, so one still has the simple form

and the similar equation for <J).

Appendix 3 One needs to evaluate the expression to obtain the coefficient in the integral for S. Now S™' must have the structure ^"'St so to simplify the analysis since one may write the coefficient as In a compressed notation this is

Similarly for R one has the expression

which when written out becomes

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270

This can be written in terms of a symmetric part under the interchange of k and j

and an antisymmetric part The symmetric part is just the coefficient in 8 with 1 and k interchanged, whilst the antisymmetric part gives In all then, since one may leave the antisymmetric part in L^ for convenience in writing, one has

Appendix 4 The series is obtained by successively solving equations of the type

To obtain a picture of the right-hand side one must invent a graphical notation (as in the virial cluster expansion of a gas or the Feynman diagrams) for algebraically it becomes very complicated. The diagrams however are quite different from the examples mentioned and are constructed this way. For M write a dot, for M a full line, for 3/9« a dotted line. Then Mj$r wf u( d/Sul^ is written

where the arrows give a vector sense so that Similarly one can define

It will be assumed that dotted lines will always be drawn to the left, whilst full lines have arbitrary directions. Now to solve (A 4.1) one needs the righthand side in Hermite polynomials where upon the inverse of the right-hand differential operator is (S^w,,)"1. It will turn out that these factors can be k

easily inferred from the diagrams and need not appear explicitly. That being

63

The statistical dynamics of homogeneous turbulence

271

the case Fn+% will consist of combinations of the diagrams (A 4.1,2, 3) running across the paper from right to left as they would in an algebraic expansion. For example F% consists of

and Ft of

and similar terms in JB, and in S and R. Now the definition of 8 and R is that

If one starts to perform the integrals it is clear that a 8/9itj must meet a MJ to its left or, by parts, the integral vanishes. Li other words the dotted lines in the diagrams either meet and annihilate a full line, or else meet one of the 'external' %, w_ k of the integral (A 4.7), or else give zero. Clearly they all act to their left. Now consider the remaining full lines. They also give zero unless they can pair with another, i.e. a Uj must find another M_J to be non-vanishing. (This amounts to the same as replacing WjW_j by H0 and Hlil_j). In order that a contribution be made to A 4.7, it must follow that all the lines but two in the diagram must link up, two full lines giving a q, a full and a dotted giving unity, or more precisely a Qi factor. The two emerging lines consist of the dotted line emerging from the first subdiagram on the left, and then either a dotted line (i.e. S like) or a full line (S like). Both emerging lines are labelled k. It is possible for say Uj M_J MJ M_J to appear, i.e. higher Hermite polynomials, but their contribution always turns out to be of order A relative to the terms already noted and so the volume will now be assumed so large that this possibility can be ignored. There now remains the terms in (S?ikwk)~1. It is clear that the % are either zero or unity and they can be characterized this way: the diagrams are well ordered from right to left. So if a vertical line is drawn between each junction it will cut lines of the diagram and for every cut of a line marked y an «,• is added to the sum and such a factor

64

S. F, Edwards

272 1

(Sw)" is inserted between each junction. Some examples should make this clear. Consider working to order Ft. The diagrams are

A vertical line gives the factor in each of (A4.8):

The equations of § 5 are got by equating the S and R like parts of A 4.8, 9 to zero,

where the two full lines in the 'bubble' in (A4.10) give <7|g_k_|, i.e. (5.16), and in (A 4.11) the one full line gives q$, the mixed line J®, i.e. (5.19). If one now goes to fourth order one gets many diagrams. Use the first approximation to 8 and B, i.e. (A 4.10, 11), some sets of diagrams already completely cancel, for example

cancels exactly with

Some partially cancel e.g.

65

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273

and

These have the same M and q factors but the (Sw) factors are different. For the former, one has whilst for the latter, recalling the integral for R, one has differing by a single term in the central factor. (The topologically similar terms in the perturbation expansion of electrodynamics do cancel exactly.) Finally, there are terms which do not contain any subdiagram equivalent to a lower order and are topologically irreducible. Such a term is

and with the residue of the partially cancelled terms, these terms give rise to the corrections to B and 8 in this order. By counting lines and (Sw)"1 factors, the formal expressions quoted in § 6 are now readily obtained. The crude evaluation quoted in §10, is obtained by ignoring the partially cancelled diagrams and assuming the value of the irreducible diagram can be approximated by distorting them into cancelling diagrams, for example

Diagrams of the latter type are readily evaluated, the one shown being ^|/wk. Adding all the types up with due regard to sign one obtains the estimate quoted in §10. These diagrammatics can be extended to cover the case in which^([^]) has a general distribution which is only approximately Gaussian, but since the generalization is quite straightforward it will not be given.

4

SAM EDWARDS AND THE TURBULENCE THEORY Katepalli R. Sreenivasan1 and Gregory L. Eyink2 1

Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34100 Trieste, Italy 2 Department of Applied Mathematics and Statistics, The Johns Hopkins University, 3400 North Charles Street, Baltimore, Maryland 21218-2682, U.S.A. Abstract

We provide a brief assessment of the contributions of Sir Sam Edwards to field-theoretic methods in the statistical theory of turbulent fluid dynamics, and connect those contributions to later developments in the subject. 4.1

Introduction

The closure problem in hydrodynamic turbulence is notorious for its difficulty (see, e.g., Monin and Yaglom 1971). Expansions of high-order moments in terms of powers of Reynolds number do not converge; truncation of the hierarchy of moments or cumulants beyond a certain order yield unrealizable results such as negative energy. Sam Edwards was immersed in this problem in the mid-sixties. In his first paper on the subject, 'Theoretical dynamics of homogeneous turbulence,' J. Fluid Mech. 18, 239 (1964) he already stated its essential difficulty: 'Many problems in theoretical physics can be expressed in terms of functional differential equations, but turbulence is an exceptional problem in that there is in the limit of large Reynolds number no external parameter which can be used as a basis of an expansion technique. In the language of quantum field theory it is a problem of infinitely strong coupling constant.' The turbulence problem perhaps no longer appears to be as 'exceptional' as it once did, for other important strong-coupling problems have since been faced in theoretical physics. Some of these, such as color confinement in quantum chromodynamics, are still with us; others, such as critical phenomena in three space dimensions, have been successfully solved.1 Experience has taught us that each 1 We have used the word 'solved' to indicate that critical scaling exponents have been calculated by several methods, such as Borel-resummed e-expansion, high-temperature series, and Monte Carlo simulation, and that the results agree to several significant digits [for recent discussions, see Guida and Zinn-Justin (1998), and Pelissetto and Vicari (2002)]. Mathematical

Contributions of Edwardsh

67

such strong-coupling problem stands on its own, and no general and encompassing method is available, or perhaps likely to be found, to solve them all. In the case of critical phenomena, it was discovered that there were 'hidden' small parameters, such as the deviation of space dimension d from an upper-critical dimension dc (often four), namely e = dc — d, or an inverse number of components of the order parameter, 1/N. These parameters were made the basis of successful perturbative calculations even for e = 1 or N = 1, especially when augmented with Fade or Borel resummation techniques (Wilson and Fisher 1972; Ma 1976; Fisher 1998). It has also been possible to develop successful nonperturbative numerical schemes to calculate critical scaling exponents, e.g., by fast Monte Carlo algorithms. [See Guida and Zinn-Justin (1998), and Pelissetto and Vicari (2002) for a survey of current results.] None of these methods that enabled break-through successes in the theory of critical phenomena has yielded results of comparable significance in understanding or predicting turbulent flows. Nevertheless, considerable progress has been made. Edwards himself was a pioneer in the application of quantum field-theory tools to turbulence. In his paper cited above (Edwards 1964), he developed a self-consistent expansion method which, in his own words, was 'based on the internal properties of the system.' His methods, as well as the related earlier work of Kraichnan (1959, 1961) and Wyld (1961), have yielded some important insights. Recently, perturbative techniques have scored a very significant success in calculating turbulent scaling exponents in a simplified model of a white-noise advected passive scalar (for a review, see Falkovich et al. 2001). It is our purpose in this chapter to review the contribution of Sam Edwards to developing field-theoretic methods in turbulence theory, and to summarize some recent progress and hopes for the future. In Section 4.2 we shall take a quick tour through Edwards' classic paper and point out some of its significant results that have played a role in later developments. In particular, in Section 4.3, we shall discuss the recent progress in calculating anomalous scaling exponents in the Kraichnan model for passive scalars by perturbative field-theoretic methods. In Section 4.4 we offer some prognosis for the much harder problem of NavierStokes turbulence. The paper concludes with a brief summary and perspective in Section 4.5. 4.2

Contributions of Edwards

Edwards (1964) contains many noteworthy aspects but we shall focus on those that seem to us most important in view of our present understanding of the subject, and on those that have played some part in later developments. Other physicists regard aspects of the problem as open: e.g., no one has yet constructed by rigorous mathematics a strong-coupling fixed point of the renormalization group for a realistic, short-ranged model in three dimensions, although the numerical evidence is that it exists and the scaling properties in its vicinity are understood by the e-expansion. Various non-universal quantities of significant interest, such as critical temperatures, cannot yet be readily calculated for real physical systems found in nature or realized in laboratories. Engineers might in general regard a problem as open if it is understood only to this degree of detail.

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5am Edwards and the Turbulence Theory by Sreenivasan and Eyink

scientists revisiting the paper will certainly uncover riches left uncounted and undescribed in our summary. Everyone interested in the subject of turbulence is thus warmly encouraged to read the paper for himself and discover hidden vistas unremarked upon by us. The problem that Edwards considers is the idealized case of homogeneous isotropic turbulence maintained statistically steady by a stochastic fluctuating force, which acts as the random source of energy:

Here U is the fluid velocity, with pressure p determined to enforce incompressibility V • U =0. The stirring force F chosen by Edwards is a Gaussian random field with zero mean and covariance given by

Because the energy input is stochastic, it is only possible to seek statistical information about the system. Edwards thus focuses on the multivariate probability density for Fourier amplitudes. The first step in his paper is the derivation of the Liouville equation for the probability density. It resembles Hopf's (1952) functional equation with which it shares the property of linearity. On the way to deriving this equation, Edwards obtains the now well-known relation for the mean energy input of a Gaussian white-noise force, for which the spatial Fourier transform of the noise covariance gk(t — t') is equal to hkd(t — t'). The mean energy input is just the integral over wavenumber of the forcing spectrum h^ [see Edwards' eqn. (2.23) and the formula below it]. This result was obtained independently by Novikov (1964) at about the same time, and is usually attributed to him. A difficulty with the Liouville equation for dissipative systems is that it is impossible to write down its stationary solution in analytical form, as one can write down the Gibbs distribution for thermodynamic equilibrium. However, Edwards realized that an effective substitute is to write down an analytical expression for the distribution over histories, or a path-integral. This is his formula (3.7), which provides an exact non-Gaussian distribution over spacetime histories of the turbulent velocity. An advantage of this approach is that it allows a calculation also of multi-time statistics, such as the two-time correlation functions, which are of independent interest. From his path-integral formula, Edwards derived a set of statistical field equations (3.5), by a standard method (e.g. section 10-1-1 of Itzykson and Zuber 1980). These equations are the main focus of his later analysis. The modernity of Edwards' approach is quite striking: while similar path-integral formulas had been introduced before for linear statistical dynamics by Onsager and Machlup (1953), this may be the first introduction of such a formula for classical nonlinear dynamics. It is a small step to transform Edwards' path-integral expression into the now-standard one for the Martin-Siggia-Rose field-theory with an extra 'response field' (Martin

Contributions of Edwards

69

et al. 1973, Janssen 1976, DeDominicis and Peliti 1977). Such path- integrals for stochastically forced Navier-Stokes equations have proved useful in more recent work— e.g. the evaluation of tails of probability density functions (PDF's) using instanton methods (Falkovich and Lebedev 1997). As Edwards notes, his path-integral reformulation of the problem is in principle 'a solution to the problem of turbulence, [but] as it stands it is quite useless in practice.' It can only be useful in conjunction with some method of calculating approximately the integrals of non-Gaussian densities over high-dimensional spaces. To this end, Edwards develops a 'self-consistent' perturbative expansion based upon a physical analogy of turbulence dynamics to stochastic Langevin dynamics. That is, Edwards proposes that the effects of averaging over turbulent fluctuations in his field equations can be subsumed into two dynamical contributions: an 'eddy viscosity' augmenting the damping from molecular viscosity and an effective stochastic force, or 'eddy noise', chaotically generated by the nonlinear dynamics. To develop the necessary formalism for his expansion, Edwards derives in Section 4 of his paper the Gaussian path-integral for a linear Langevin model. His procedure in the following two sections is then to postulate such a Gaussian expression as the leading-order approximation to the path-integral for Fourier coefficients of the turbulent velocity. However, this expression contains two unknown functions: an 'eddy-damping coefficient' R^ and an effective 'eddy-noise covariance' S^ acting in each wavenumber k. To determine these functions, Edwards expands his non-Gaussian path-integral density in Hermite polynomials orthogonal with respect to his reference Gaussian (so that the polynomials themselves depend upon the unknowns). For each order, Edwards thereby obtains a set of closed "self-consistent" equations for the functions R and S. To first order, they are given by expressions (5.16) and (5.21) in his Section 5, namely,

Here Lyt are known coefficients, determined from the Navier-Stokes nonlinearity, and
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5am Edwards and the Turbulence Theory by Sreenivasan and Eyink

more akin to the 'quasi-particle' procedures used in condensed matter physics. This is particularly clear in Edwards' use of formal expansions in Hermite polynomials, which are the eigenfunctions of the self-consistent evolution operator [his eqn. (6.16)]. However, Edwards' idea is based on the same physical intuition as that of the other authors. In fact, his Sections 4-6 can be regarded as a precursor to the works of Kraichnan (1970) and Leith (1971), who several years later discovered that Kraichnan's direct interaction approximation (DIA) closure is realized by a self-consistent Langevin model. A number of later simplifications, such as Orszag's (1977) eddy-damped quasi-normal Markovian (EDQNM) closure also have a Langevin realization. Such a model is the essential content of Edwards' Sections 4-6 regarding time-independent statistics, and of Section 7 regarding time-dependent ones. In Section 8 of the paper Edwards applies his general formalism to a number of concrete problems. In particular, he discusses the approximate eqns. (8.1)-(8.8), for the energy spectrum, namely,

where R and S are given by (4.3). In the text near his eqn. (8.13), Edwards shows that the above equation has the correct thermal equilibrium 'equipartition solution' when the random force has the k"2 spectrum required by the fluctuation-dissipation relation. Also, as a consequence of (4.3), he notes in his eqn. (8.6) that

which corresponds to the conservation of energy by the nonlinear terms in Navier-Stokes equations. However, this cancellation is only formal if the integrals diverge. At the top of p. 261, Edwards has an interesting discussion about this 'apparent paradox,' namely that there can still be finite energy dissipation even when the viscosity, i/, vanishes (e.g., Frisch 1995). Edwards points out that the apparent cancellation of integrals in his (8.6) is not 'meaningful' when they are separately divergent. This is very much related to earlier remarks of Onsager (1949) about the impossibility to reorder Fourier series which are not absolutely summable but only conditionally convergent. However, Edwards does not quite reach Onsager's sharp conclusion that energy dissipation is possible without viscosity. Rather, he only concludes that adding a little viscosity makes the integrals convergent and permits their exact cancellation. More recently, Polyakov (1993) has pointed out an analogy of 'inviscid dissipation' in turbulence in two dimensions to conservation-law anomalies in quantum field theory, such as the axial anomaly in quantum electrodynamics. The case considered by Edwards at some length in his eqns. (8.15)-(8.35) is random forcing with a power-law spectrum. This same problem was later also considered by DeDominicis and Martin (1973) and Yakhot and Orszag (1987)

Contributions of Edwards

71

in any space dimension d (whereas Edwards considered only d = 3). The parameter a of Edwards [see his eqn. (8.15)] is the same as the parameter y used by these later authors. Edwards does not consider an expansion in the parameter c = 4: + y — d = I + y (for d = 3). However, his self-consistent equations yield similar results as those obtained by the later authors using e-expansion renormalization group methods. His treatment is perhaps more similar to that of Mou and Weichman (1995), who analyzed the same problem using Kraichnan's DIA equations or the mode-coupling approach. In particular, Edwards obtains a power-law energy spectrum through his eqn. (8.30), this result being equivalent to the later authors' result E(k) ~ k~x with

(Note there is a typographical error in the first of Edwards' equations, which is missing a minus sign in the exponent.) It is interesting that Edwards explicitly states that the validity of this solution is limited to — 1 < y < 2, or, equivalently, when d = 3, to 0 < c < 3. There is an important qualitative change in the problem at y = 2 or c = 3, at which the spectrum goes through E(k) ~ 1/k, and there begins to be more energy at low k than at high k. That is exactly where 'energy cascade' begins. This limitation was also recognized also by DeDominicis and Martin, whereas Yakhot and Orszag imagined that the e-expansion result would be valid all the way to e = 4. In fact, it appears likely, from both physical arguments and work to be described later on a model problem, that intermittency corrections start to appear for e > 3. The restriction of the dimensional analysis results to 0 < e < 3 was understood by Edwards, and was probably pointed out in this paper for the first time. Within the confines of the randomly forced Navier-Stokes equation, the problem of 'true turbulence' corresponds to the case of a compact force in Fourier space, supported at low wavenumbers. Edwards considers this important problem on pp. 261-263 of his paper. In certain limiting situations, he obtains the spectral exponents of Kolmogorov (1941) and Kraichnan (1959). Recognizing that there are potential infrared divergences, Edwards argues that the energy spectral exponent must lie between limits set by the Kolmogorov value of —5/3 and the Kraichnan value of —3/2. From the point of view of potential intermittency corrections, it is interesting that he only allows spectra that roll off less steeply than Kolmogorov's, whereas we now believe the roll-off to be steeper (e.g. Kaneda et al. 2002). Finally, Edwards (1964) makes detailed calculations for two-time correlations of the velocity Fourier amplitudes, on pp. 263-264. He predicts that they will fall off as a Gaussian for small times, as an exponential for intermediate times, and as a power-law for long times. The decay rates that he calculates are all wavenumber dependent. Before this work of Edwards, and the somewhat earlier work of Kraichnan (1959, 1961), the interesting subject of time-correlation functions in turbulence had been neglected. There is no discussion in Edwards of Lagrangian vs. Eulerian time correlations, and it is apparent that his predictions must be

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understood to be for Eulerian time-correlations. However, sweeping effects, which Edwards does not discuss, must come into play in that case, as discussed by Kraichnan (1964) and later by Kraichnan and Chen (1989). Before concluding this section, we should also point out that Edwards has returned to studies of turbulence several times since his first 1964 paper, even as recently as 2002. We will not make extensive comments on these later forays but make only brief comments on their scope and point out some important subsequent developments. Following the framework of the 1964 paper, Edwards and McComb (1969) calculated the Kolmogorov constant for the energy spectral density. A notable feature of this paper was the use of a maximum entropy argument to develop a second relationship, in addition to (4.3), between the two functions R^ and oo. For such a limit the existence of Onsager's negative temperature states has been proved within the microcanonical ensemble, by Eyink and Spohn (1993) and Kiessling and Lebowitz (1997). A number of issues discussed by Edwards and Taylor (1974)—such as the critical energy for appearance of negative temperatures and the possible nonequivalence of canonical and microcanonical ensembles—have now been definitively resolved. The most important development in the field has been the application, independently by

The White-noise Passive Scalar Model

73

Robert (1990) and Miller (1990), of Gibbsian statistical mechanics directly to the continuum Euler equations without the point-vortex approximation. In a paper written to commemorate Kubo's sixtieth birthday, Edwards (1980) obtained working expressions for two basic features of turbulence: the single-point probability density function for the velocity fluctuation and the small-scale intermittency. Using polymers with freely hinged chains as the analogy, Edwards closed the problem and showed that the PDF of the single-point velocity has a Gaussian core with stretched exponential tails, which he explicitly obtained. In contrast, experience from simulations and measurement seems to favor slightly sub-Gaussian tails. For the intermittency problem, Edwards used a different analogy to the localized states in disordered semiconductors, and discussed the energy growth of eddies in the inertial range. The jury is still out on these ideas. Most recently, Edwards and Schwartz (2002) have considered turbulence and surface growth models (particularly focusing, for growth phenomena, on the Kardar-Parisi-Zhang model) and, using path-integral methods, discussed theoretical developments needed for determining two-time correlation functions. This is a direct outgrowth of the first paper, Edwards (1964). A key result is an approximate Markovian equation, i.e., local in time, for the two-time correlation. This approach also yields a natural alternative to the maximum entropy constraint of Edwards and McComb (1969). One of the more characteristic results of the new approach is the stretched-exponential decay of time correlations, rather than a purely exponential form. It is thus clear that turbulence has gripped the interest of Sam Edwards for many years. His body of work contains many nuggets of technical mastery and intuitive notions, some of which have seen fruition in different ways. The subject has advanced significantly in several directions since Edwards made his entry. In the next two sections, we provide a brief overview of recent developments in the application of field-theoretic methods to the problem of turbulence. Although we will not attempt to trace accurately the influence of Edwards on these recent developments, we shall briefly highlight their relation to his earlier ideas.

4.3

The white-noise passive scalar model

As mentioned earlier, the perturbative methods that Edwards helped pioneer have so far not been crowned with absolute success for the problem of Navier-Stokes turbulence. However, there has been recent noteworthy progress on another problem: the advection of a passive scalar by 'synthetic turbulence,' a Gaussian random velocity field which is white-noise in time. This model was introduced by Kraichnan (1968) and, for this reason, is called the Kraichnan model. Since an authoritative review of the subject is now available (Falkovich et al. 2001), our survey below of the recent work on this model will be brief. The Kraichnan model considers the concentration 0(r,t) of a passive scalar such as a dye or temperature field injected into developed turbulence. The

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dynamical equation it satisfies is Here, F(r, t) is a Gaussian white-noise source term of a passive scalar, similar to that considered by Edwards for the velocity, with zero mean and covariance F(r — r')(5(t — t'). The advecting incompressible velocity field U(r,t) is also a Gaussian random field with zero mean and covariance The cases of greatest interest are those for which the source F is supported at low-wavenumbers, while the velocity field U has a power-law spectrum ~ k~(1+^ at high wavenumbers, for any 0 < £ < 2. This corresponds to a scaling

for r —s- 0 in space dimension d [see eqn. (48) of Falkovich et al. (2001)]. As was first realized by Kraichnan himself (see Kraichnan 1968), the special feature of the model is that there is no closure problem, independently of the precise form of the function ^(r). The single-time scalar correlation functions C n ( r i , . . . ,r n ;t) = (0(ri,t) • • • 0(r n ,t)} satisfy the equation

Here the summation is over all pairs / < m of integers l,m = 1 , . . . ,n while the hats ^ over the position vectors indicate their omission from the correlation function. Since this equation for Cn involves only the lower-order correlation function Cn_2, it is possible, in principle, to mathematically solve this hierarchy of equations inductively for all of orders of scalar correlations. An important problem, as in the case of critical phenomena, is the prediction of the scaling exponents for the small-scale scalar field. The phenomenology in this case is quite similar to that for the turbulent velocity itself. One imagines that there is a cascade of the scalar 'energy' or intensity, ^ j d d r$ 2 (r, t), from the injection scale L to small scales. When the scalar is weakly diffusive, there is a large range of scales, the so-called inertial-convective range, over which the flux of scalar intensity is constant. However, this is only true in an average sense, and fluctuations in individual realizations of the cascade do develop. These fluctuations become large as one considers increasingly smaller scales. This property of 'intermittency' is the origin of the anomalous scaling for the scalar structure functions defined by

The White-noise Passive Scalar Model

75

where $2 = {$2} is the mean-square scalar fluctuation and £p is a scaling exponent. Classical dimensional analysis of Obukhov (1949) and Corrsin (1950), itself an outgrowth of Kolmogorov (1941), predicts that £p = p/3. In fact, it is known from experiments on real turbulent scalars that £p is a concave, nonlinear function of the order p of the structure function (Antonia et al. 1985, Meneveau et al. 1991, Chen and Kao 1997, Moisy et al. 2001, Skrbek et al. 2002). This type of anomalous scaling law is now called multifractal, after an intuitive interpretation by Parisi and Frisch (1985); see also Mandelbrot (1974). Multifractality involves an infinite concave family of exponents and is fundamentally different from the scaling observed in classical critical phenomena.2 Returning to the Kraichnan model, the velocity field in the model does not possess anomalous scaling, since it is Gaussian and unifractal. Surprisingly, however, the scalar field advected by this Gaussian velocity field does show multifractal scaling. Although the input velocity is self-similar, it induces a scalar cascade through which intermittency develops in successive steps from large scales to small. In fact, it is possible to guess from Edwards' work the spectral exponent £ of the velocity [see eqn. (8)] at which multifractality begins to develop. As was noted by Edwards for the Navier-Stokes fluid stirred by a random force with power-law spectrum, the energy spectrum has the form A; 1 ^ 26 / 3 for 0 < c < 3. In this range, there is more energy at high wavenumbers than at low wavenumbers, and no energy cascade occurs. Precisely at e = 3 the velocity spectrum makes the transition from most energy at high wavenumbers to most at low wavenumbers. Edwards realized that dimensional reasoning breaks down for e > 3 and that possible corrections to the scaling laws can occur there. Since the energy spectrum of the velocity field in the Kraichnan model has the powerlaw form ~ fc~(1+£), there is the formal identity £ = 2(e — 3)/3. Hence, it is likely that anomalous scaling begins in the Kraichnan model precisely at £ = 0. In analogy with e-expansions used in critical phenomenon, it is suggested that the natural perturbation parameter is not e, as considered by DeDominicis and Martin (1979), and Yakhot and Orszag (1986), but instead e - 3, or £. The first to realize this fact for the Kraichnan model were Gawedzki and Kupiainen (1995), who carried out the corresponding expansion. Fortunately, the Kraichnan model is exactly solvable and the scalar statistics are Gaussian for £ = 0. Indeed, setting £ = 0 in (4.8) and substituting into (4.9) gives

2

Our point is not that multifractality cannot appear in ordinary critical systems, but that it is different from the 'classical critical scaling' in which only a finite number of exponents are readily apparent. It is, in fact, possible (see, e.g., Fourcade and Tremblay 1995) for more 'exotic' operators than those considered in 'classical' critical phenomena to possess multifractal properties.

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The very rough velocity field for £ = 0 acting at high wavenumbers is seen to mimic exactly a molecular diffusion. This equation for translation-invariant correlation functions is just a multi-dimensional heat equation, and it is not hard to show that Gaussian correlation functions, satisfying Wick's theorem, are the solution. Prom this point Gawedzki and Kupiainen were able to develop an expansion for the anomalous exponents of the scalar (see also Gawedzki and Kupiainen 1995; Bernard et al. 1996), with the result that

At nearly the same time it was realized by Chertkov et al. (1995) and Chertkov and Falkovich (1996) that the scalar statistics also become Gaussian for d = oo. They worked out a corresponding expansion in 1/d, yielding a result consistent with the above.3 The Kraichnan model also simplifies for a smooth velocity at £ = 2, the so-called Batchelor regime of the passive scalar. In that case, all Cn = 0, in the sense that the correlation functions are logarithmic, instead of being power-laws. In this limit, an expansion was worked out by Shraiman and Siggia (1995, 1997). This is technically a more difficult limit than the other two: it is a singular perturbation problem with a boundary-layer, for which the relevant expansion parameter turns out to be (2 — ^) 1 / 2 . The problem has thus been worked out only for the triple correlation n = 3. All of these authors realized that, in the Kraichnan model, so-called 'zero modes,' that is, stationary solutions of the homogeneous version of equation (4.9), play the key role in the development of anomalous scaling. This is due to the fact that equations formulated for the scalar correlation functions are linear with known (singular diffusion) operators. This linearity property is essentially unrelated to the linearity of the advectiondiffusion equation. However, the fact of closure, i.e., the existence of closed linear equations, does depend upon the linearity of advection-diffusion and also the white-noise character of the velocity; as already implied, it is this combination that truncates the hierarchy in the Kraichnan model. All these perturbative results have now been checked by clever Lagrangian numerical methods, for the £ and (2 — S,)1/"2 expansion by Frisch et al. (1998, 1999), and Gat et al. (1998), and for the 1/d expansion by Mazzino and Muratore-Ginanneschi (2001). To know the scalar field at position x and time t, it is enough to track the corresponding tracer particle back to its (Lagrangian) initial position. It follows that the evolution operator for the n-point average of the scalar coincides with that for the probability that n tracer particles from given positions reach new positions x^, k = 1 , . . . , n after time t. For the n-point function, it is only necessary to focus on the relative evolution of n particles simultaneously. This reduces the problem to a study of the evolution of the geometry 3 The 1/d expansion for equilibrium lattice systems was introduced by Fisher and Gaunt (1964).

Navier-Stokes turbulence

77

of polyhedra with n vertices. Deviations from dimensional estimates (i.e. anomalies) are traceable to the nontrivial evolution of these geometric objects as advected by the flow. Among all the geometric figures that grow in time according to dimensional estimates, the ones that matter are those whose shapes are preserved: these statistically conserved objects are the ones that dominate the behaviour of scales in the inertial range and the anomaly of the exponents. This study also reveals the role of 'hidden' statistical integrals of motion. Indeed, the 'zero modes' are statistical integrals of motion of the Lagrangian fluid particles. The perturbation theories originally worked out for the Kraichnan model did not use a Martin-Siggia-Rose field-theory formulation of the sort that Edwards helped to pioneer. Nor, for that matter, did they use the renormalization group (RG) as a basis to organize the expansions. However, in later works, this has been done, first by Gawedzki (1997) and later in an extensive series of work by the St. Petersburg school in Russia (e.g. see Adzhemyan et al. 2001, 2002). As might be expected, RG is a more important tool when working out higher terms of the expansions: with its aid the ^-expansion has now been carried out to third-order (Adzhemyan et al. 2001). Even more interestingly, the same perturbative RG approach has been extended to a generalized model, still with a Gaussian random advecting velocity field, but now with correlations exponentially decaying in time rather than delta-correlated (Adzhemyan et al. 2002). This is a step in the direction of greater realism of the models. Some previous conjectures for 'additive' operator product expansions as the basis of turbulent multifractality (Eyink 1993; Lebedev and L'vov 1994) have been confirmed in the models by using perturbative RG techniques. Thus, for a nontrivial model problem and some generalizations, the state of our understanding of anomalous scaling is now essentially as good as for threedimensional critical phenomena, where there are Borel-resummed e-expansion results, high-temperature series expansions, and numerical Monte Carlo results, which agree to several decimal places. In both cases, rigorous mathematical proofs are still lacking, but it is quite clear that scaling exponents exist and what their numerical values are. Needless to say, we have far from adequately reviewed all the results that have been obtained by now for the Kraichnan model of a passive scalar. For example, new insight has been obtained into dissipative anomalies in turbulence related to the non-uniqueness and stochasticity of Lagrangian particle paths for a non-differentiable advecting velocity. White-noise advection models have also been fruitfully investigated for related problems such as compressible turbulence, magnetohydrodynamics and the coil-stretch transition of polymers in turbulence. We refer to Falkovich et al. (2001) for a fuller discussion of these many aspects. 4.4

Navier Stokes turbulence

The results we have described for the Kraichnan model have already had a significant impact on our understanding of Navier-Stokes turbulence. There is now rather general agreement that velocity structure functions will show anomalous,

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multifractal scaling. There was already good empirical evidence for this, both experimental (Anselmet et al. 1985; Sreenivasan and Dhruva 1998) and numerical (Cao et al. 1996). It is hard to imagine that anomalous scaling would arise in the cascade for the passive scalar and not for the nonlinearly self-interacting velocity field. In this respect, the Kraichnan model results have served much the same purpose as Onsager's exact solution of the two-dimensional Ising model, which eventually convinced most statistical physicists that Landau's mean field theory needed to be replaced. In the same way, the fact that Kolmogorov-style dimensional analysis fails in the Kraichnan model lessens one's faith that these same arguments will succeed when applied to Navier-Stokes turbulence. A better theory is clearly needed. Technically, it has not proved possible to carry over readily the formal methods successfully employed in the Kraichnan model to Navier-Stokes equations. This, also, is very much as for the two-dimensional Ising model, whose specific methods of solution (such as infinite-dimensional Lie algebras, spinors, and Toeplitz determinants) played no direct role in the apparatus of the successful general methods such as the renormalization group and the e-expansion. The only method used in Onsager's solution that has subsequently shown general applicability to a wide class of systems is the transfer matrix. Likewise, it may be that there are certain features of the solution of the Kraichnan model that are more general and can be carried over to Navier-Stokes turbulence and to other similar nonequilibrium scaling problems, such as random surface growth. The concept of a 'zero mode' seems a plausible candidate. Zero modes can also be considered in nonlinear dynamics (e.g. in shell models or Navier-Stokes turbulence) by working with the linear Hopf equation or linear Liouville equation or, equivalently, with the infinite linear hierarchy of equations for multipoint correlation functions. The linearity of the scalar advection-diffusion equation is not a prerequisite for the existence of zero modes, and it is likely that they play an important role in anomalous scaling more generally. The perturbative expansion techniques employed successfully in the Kraichnan model have not so far found any success in Navier-Stokes turbulence. It is quite likely, as we discussed earlier, that randomly stirred fluids with power-law forcing spectrum first develop intermittency at e = 3. However, the statistics do not become Gaussian in that limit and there is no other obvious analytical simplification at e = 3. So, an expansion in e — 3 does not look very feasible. It is also a frequent speculation that Navier-Stokes turbulence should simplify in infinite-dimensional space (Frisch and Fournier 1978), but this has not yet been demonstrated. Yakhot (2001) has proposed an expansion in d — dc, with 2 < dc < 3 a critical dimension where the energy cascade changes from inverse to direct, and made some progress by attributing a simple behaviour for pressure terms. One likely hope for a successful perturbative treatment is the 1/N expansion for Kraichnan's 'random coupling' model (Kraichnan 1961), in which N copies of the Navier-Stokes equation are coupled together with quenched random parameters. It is known for this and related models (Kraichnan 1961; Eyink 1994a; Mou and Weichman 1995) that Kraichnan's DIA closure becomes exact

Conclusion

79

and the statistics become Gaussian at N = oo. Furthermore, in an TV-component version of a shell model it has been shown numerically that the anomalous scaling corrections to Kolmogorov's (1941) arguments vanish proportionally to l/N (Pierotti 1997). Thus, the anomalous exponents should be perturbatively accessible in the shell model by a l/N expansion. Unfortunately, for Navier-Stokes turbulence, the DIA closure is not consistent with Kolmogorov (1941) scaling and a Lagrangian formulation of the random-coupling model would need to be devised (Kraichnan 1964). None of these steps appears to be trivial. 4.5

Conclusion

The turbulence problem remains a hard nut to crack. Anomalous scaling almost certainly occurs, but no controlled approximation to the exponents exists. Furthermore, there are many other flow properties in turbulence that we would like to calculate, not just scaling exponents. More practical, and also physically very interesting, are quantities such as drag coefficients, mixing efficiency, spread rates, single-point probability density functions, and mean velocity profiles. So far, none of these aspects of the problem can be reliably calculated at high Reynolds numbers. We do not, however, wish to leave the impression that the situation is hopeless. Despite the lack of adequate analytical tools, quite a lot of understanding has been gained, through the work of many people, Sam Edwards among them. Much of the lore is reasonably well established on the basis of experiment, simulation, and theoretical arguments. As we have argued above, multifractal scaling of the turbulent velocity field is now doubted by few, although occasional papers still appear claiming the contrary. It is clear in flow experiments (Sreenivasan and Dhruva 1998) and simulations (Cao et al. 1996), and has been established theoretically for the Kraichnan model. Likewise, there is good evidence for a dissipative anomaly in the conservation of energy at zero viscosity. It is observed in experiments (Sreenivasan 1984) and simulations (Kaneda et al. 2003), and there are known theoretical mechanisms to produce it (Onsager 1949; Eyink 1994b). Concepts such as fractality of isosurfaces (Sreenivasan 1991), fusion rules for powers of velocity-gradients (Fairhall et al. 1997), Kolmogorov's refined similarity hypothesis connecting scaling of velocity-increments and dissipation (Stolovitzky and Sreenivasan 1994), stochasticity of Lagrangian particle trajectories (Bernard et al. 1998; Yeung 2002), and many other key ideas, seem well-founded and likely to survive into the future. Our ability to calculate with the Navier-Stokes equations continues to extend to higher Reynolds numbers because of advances in computing power, and new experimental techniques (Donnelly and Sreenivasan 1998) and modeling techniques (Meneveau and Katz 2000) extend that capacity. The present status for the 'turbulence problem' seems to us really not so different from that for other strong-coupling problems in field theory, e.g., the color confinement problem in QCD. In the confinement problem there are also heuristic ideas [the QCD vacuum is a color magnetic monopole condensate, a Type II chromomagnetic superconductor with confined chromoelectric flux tubes acting

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as 'strings' between color charges; see Mandelstam (1978), and 't Hooft (1978, 1981)], but there is great difficulty in making quantitative calculations. Direct numerical simulations, i.e., lattice QCD (Wilson 1974) allow us to calculate, in principle, anything we wish (e.g. the hadron spectrum), but in practice computer limitations confine us to modest results for the foreseeable future. Nevertheless, many aspects of QCD are regarded as reasonably well-established (e.g. a mass gap, color confinement, chiral symmetry-breaking), even though they are not rigorously proved. Turbulence is very similar. It is no accident that both these problems ended up as Clay Institute Millenium Prize Problems: proving the mass gap for four-dimensional quantum non-abelian gauge theory, and proving regularity of Navier-Stokes solutions at high Reynolds numbers. They are both strong-coupling problems and those are mathematically hard. However, in both cases we have a lot of very good insights and ideas that can spur intuitively new developments. The difficulties that remain should not lead us to despair, but, instead, ought to inspire us to greater imagination and ingenuity—qualities that Edwards has always had in generous quantities.

Acknowledgments

It is a pleasure to be a part of the celebration honoring Sir Sam Edwards. In this chapter, we have attempted to outline his contributions to the important topic of turbulence. It is not likely that we have done full justice to Sam's effulgent spirit, which the subject clearly did not contain entirely. Asked how one would know if one has arrived at the right answer to the problem, KRS well remembers Sam saying, 'A little angel will whisper in your ears that you are right.' We wish to thank Michael E. Fisher for his many helpful comments on this chapter. The work was supported by a grant from the U.S. National Science Foundation, CTS-0121007. References

Adzhemyan, L. Ts., Antonov, N. V., Barinov, V. A., Kabrits, Yu. S. and Vasil'ev, A. N. (2001). Calculation of the anomalous exponents in the rapidchange model of passive scalar advection to order e3. Phys. Rev. E 64, art. no. 056306. Adzhemyan, L. Ts., Antonov, N. V. and Honkonen, J. (2002). Anomalous scaling of a passive scalar advected by the turbulent velocity field with finite correlation time: Two-loop approximation. Phys. Rev. E 66, art. no. 036313. Anselmet, F., Gagne, Y., Hopfinger, E. J. and Antonia, R. A. (1984). High-order velocity structure functions in turbulent shear flows. J. Fluid Mech. 140, 63-89. Antonia, R. A., Hopfinger, E., Gagne, Y. and Ciliberto, S. (1984). Temperature structure functions in turbulent shear flows. Phys. Rev. A 30, 2704-2707.

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Eyink, G. L. and Spohn, H. (1993). Negative-temperature states and largescale, long-lived vortices in two-dimensional turbulence. J. Stat. Phys. 70, 833-86. Fairhall, A. L., Dhruva, B., L'vov, V. S., Procaccia, I. and Sreenivasan, K. R. (1997). Fusion rules in Navier-Stokes turbulence: First experimental tests. Phys. Rev. Lett. 79, 3174-3177. Falkovich, G. and Lebedev, V. (1997). Single-point velocity distribution in turbulence. Phys. Rev. Lett. 79, 4159-4161. Falkovich, G., Gawedzki, K. and Vergassola, M. (2001). Particles and fields in fluid turbulence. Rev. Mod. Phys. 73, 913-975. Fisher, M. E. (1998). Renormalization group theory: its basis and formulation in statistical physics. Rev. Mod. Phys. 70, 653-681. Fisher, M. E. and Gaunt, D. S. (1964). Ising model and self-avoiding walks on hypercubical lattices. Phys. Rev. A 133, 224-239. Forster, D., Nelson, D. R. and Stephen, M. J. (1977). Large distance and long time properties of a randomly stirred fluid. Phys. Rev. A 16, 732-749. Fourcade, B. and Tremblay, A.-M. S. (1995) Field theory and second renormalization group for multifractals in percolation. Phys. Rev. E 51, 4095-4104. Frisch, U. (1995). Turbulence: The Legacy of A. N. Kolmogorov. (Cambridge University Press, Cambridge, England). Frisch, U. and Fournier, J. D. (1978) d-dimensional turbulence. Phys. Rev. A 17, 747-762. Frisch, U., Mazzino, A. and Vergassola, M. (1998). Intermittency in passive scalar advection Phys. Rev. Lett. 80, 5532-5535. Frisch, U., Mazzino, A., Noullez, A. and Vergassola, M. (1999). Lagrangian method for multiple correlations in passive sealer advection Phys. Fluids 11, 2178-2186. Gat, O., Procaccia, I. and Zeitak, R. (1998). Anomalous scaling in passive scalar advection: Monte Carlo Lagrangian trajectories. Phys. Rev. Lett. 80, 55365539. Gawedzki, K. and Kupiainen, A. (1995). Anomalous scaling of the passive scalar. Phys. Rev. Lett. 75, 3834-3837. Gawedzki, K. (1997). Turbulence under the magnifying glass. In: Quantum Fields and Quantum Space Time (NATO ASI Series, Series B, Physics, 364). 't Hooft, G., Jaffe, A., Mack, G., Mitter, P. K. and Stora, R. (eds.). Plenum, New York. Guida, R. and Zinn-Justin, J. (1998). Critical exponents of the N-vector model. J. Phys. A 31 8103-8121. 't Hooft, G. (1978). On the phase transition towards permanent quark confinement. Nucl. Phys. B 138, 1-25. 't Hooft, G. (1981). Topology of the gauge condition and new confinement phases in nonabelian gauge theories. Nucl. Phys. B 190, 455-478. Hopf, E. (1952). Statistical hydrodynamics and functional calculus. J. Rat. Mech. Anal. 1, 87-123.

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5

REPRINT THE STATISTICAL MECHANICS OF POLYMERS WITH EXCLUDED VOLUME by S. F. Edwards Proceedings of the Physical Society, 85, 613—624 (1965).

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PBOC. FHVS. S O C . , 1 9 6 5 , V O L . 85

The statistical mechanics of polymers with excluded volume S. F. EDWARDS Department of Theoretical Physics, The University of Manchester MS. received \8th November 1964 Abstract. The probability distribution of the configurations of a polymer consisting of freely hinged links of length I and excluded volume v is studied. It is shown that the interaction of the polymer with itself can be represented by considering the polymer under the influence of a self-consistent field which reduces the problem to an equation like the Hartree equation for an atom. This can be solved asymptotically, giving the probability of the «th link of the polymer passing through the point r to be

where L = nl is the length along the polymer and Jf(L) Thus the mean square of r, , is

the normalization.

The theory is extended to polymers of finite length, to the excluded random walk problem and to n dimensions. 1. Introduction The effect of finite thickness on the configurational statistical mechanics of polymers is an important problem in polymer science and biophysics, since it has long been believed that the probability of finding the «th link at r in such a polymer (assumed for the present to consist of freely hinged links of length /) will not be the random walk distribution

but a broader distribution. This will have the effect of making the mean of r2 greater than the Einstein value nl2, and important physical results stem from the failure of the Einstein law. This problem has been extensively studied in the model in which the links of the chain are restricted to joining neighbouring points of a perfect lattice. Though it was for some time believed that the asymptotic form of ~ «y where 0-22 > y > 0-18. Phenomenological arguments implying this law are also given by Flory (1953). This paper will give a general derivation of this law and the probability distribution nom which it stems. The basic fact which the analysis relies on is as follows. Consider 1 613

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the probability of finding the polymer at the point r knowing only that it starts at the origin. Use the Gaussian distribution for a start and consider the polymer of infinite length. Since the probability that the «th link lands at r is

where L — nl is the length along the chain, the probability required is

Compare this with a rigid polymer pointing in an arbitrary direction. Since the surface of a sphere is 4w2 one has

One can expect that a real polymer with excluded volume will have a law lying between these extremes, and it is found that the realistic answer tends to

where v is the excluded volume, defined by

where U is the potential between two segments separated by a distance r. For a discussion of the excluded volume see Flory (1953) and Volkenstein (1963). All these functions decrease quite fast, but it will turn out that/ will play the role of a potential in subsequent calculations, and it will be recalled that there is a great difference between potentials of 1/r and 1/r2, the former being rather pathological in spite of its familiarity, whilst the latter is rather harmless. The realistic function r~*13 needs to be treated with care but is not as bad as r"1. Since this probability is decreasing fast the following physical picture is proposed. The polymer starts at the origin and the point L moves slowly outwards as L increases, so that an 'average view' from far outside gives a 'polymer density' ofp(r). Think of this as established, and now again move out along the polymer from the origin. The current point at L will be deflected from the random walk path by encounters with parts of the polymer having an L' quite different from the current point, and one can think of this current point being deflected by the 'polym" density'. So one may argue that the motion will be like a random walk in the presence of a potential, and this potential will have to be calculated from the complete solution itself. Here is a strong similarity to Hartree's treatment of the atomic problem. Hartree replaced the many-electron problem by the problem of solving the motion of one electron in a certain self-consistent field. From this solution Hartree then went back and calculated the self-consistent field itself. That problem is impossible analytically, but it turns out that in the polymer problem the asymptotic forms can be obtained, though the complete solution is a problem comparable with the atomic problem.

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This self-consistent field approximation will be derived in the next section by a straightforward argument; that it is rigorously correct is shown in the appendix. The equation will be solved in § 3 and some generalizations offered in § 4. 2. The self-consistent approximation Consider briefly the Einstein case. One may argue that the entire problem is described by the probability that the point, a distance L from the origin along the polymer, is at the point r in space, ps(r, L). Starting at r, L, consider the probability of finding the polymer at r, L +1. This will stem from adding the contributions from all the points r+ Sr, L, where clearly Sr is li, i being a unit vector. Thus

and expanding

or

This is, of course, a completely standard problem, and has the solution (1.1). This has the property of a Markov process, that one can always break up the interval (o, r), (0, L) in as many places as one likes: and write

If one takes the ends of all the links of the chain as the Rt, one can write down the probability of the entire configuration

(The L, labels now need not appear since L, — Li +1 = I.) Now consider the effect of an interaction potential V between points of the chain (7 is Ul~2 of 1-6) labelled as R(L). It will have the effect of multiplying PE, which expresses just the length of the links, by a Boltzmann factor which for an infinite chain will be

1 he constant C allows for the change in normalization caused by the Boltzmann factor.

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This factor will include the case of hard rods of radius a, but in the following analysis V will be considered soft and the appropriate generalization made in due course. Clearly P([R]) cannot be broken up as was PE in (2.5). But let us now make the self-consistent field approximation and argue that one may replace the configuration R(L") by the probability distribution p(i, L"), i.e. let us write

where p is to be determined. It is now possible to break up P into a Markov chain, for writing

one can split

Hence it follows that if Px is the probability of finding a configuration in the presence of the potential W instead of V, the approximation (2.8) gives

where P^R], r, L) is the probability of finding a chain [R] starting at o length 0, ending at r length L, and Pi([R] ;r,L; tm, £„) is the probability of finding a chain [R] starting at r length Land ending at r „, Lx, and C(L) is inserted because again normalization •will be required. Having made the approximation one can split up P± into a set of pi, just as in (2.4):

and now thep! satisfy a differential equation. For consider again the derivation (2.1, l-->)When one allows a change in the weight factor between L and L + l of

and a change of normalization, one finds

As before, this must equal

so that

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617

Returning to the definition of W, since V is short range one may write

One may recognize in (l/2KT)$V(s)d3s the virial coefficient in the soft potential approximation. It is well known that for hard potentials one should replace this by the excluded volume

This replacement will be assumed here, further details being given by Volkenstein (1963). Thus one may write vp(r)l~z for W, and also one may define vp*(L)l~2 = C'/C, so that finally

This is the basic self-consistent field equation which will be solved in the next section. It will be assumed from now on that v is positive; the solution for 11 negative is quite different and it is hoped to discuss this in another paper. 3. The solution The term j>(L) can be removed from the differential equation by extracting a term exp(J\ij>/-2^L'), and this should be done, leaving the question of normalization right to the end. Strictly speaking, (2.18) should have a source on the right, since as L tends to zero^j mdp should tend to S(r). One can expect a solution of the type^ = Q(r, L)lr, where

Let us introduce the Fourier transform

and write This gives

This will be solved in the WBKJ approximation (Jeffreys and Jeffreys 1961) which expands about the solution

" can be verified from the solution which will be produced that the 82(f>/8r2 term is

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negligible in the asymptotic region. Moreover, since p^ is in fact a Green function it can always be written in terms of the eigenfunctions of the differential equation, which will exist in pairs

As L tends to infinity this sum will be dominated by En ~ 0, which suggests that a valid further approximation will be to expand <j> in E:

At this point one should recall that/) differs from^ by the functional1, rx; L, Lx). Clearly the only survivor in the sum (3.5) will be En = 0, and within multiplicative functions of L to the order of (3.6) this is just

Thus

where/(L) is the normalization. This now yields, putting in normalization, and ignoring factors like r"1 which are effectively constant in the asymptotic region relative to the distribution below,

where

and

Now it will be argued that this distribution is completely dominated by the region

so that effectively, for the purpose of calculating fi,

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619

and

Putting

then

Hence

or

so that

and

It is convenient to rewrite this as a distribution around r, i.e. to put

when in terms of r', to the same accuracy as has been used so far,

This gives

where a, p are coefficients which, are not accurate since corrections of the same order come from the approximations in (3.4), (3.6), (3.19); and the transition from (3.9) to (3.24) also gives corrections of the same order. The serious expansion of the whole method is in fact the parameter i,-i/iVly5J7/10 and the condition for the present analysis to be valid is that

or

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To put in another way, for chains whose total length is less than O(77/&2), the Einstein law (r* > = LI should be valid, but for chains longer than O(/7/^2) the law (3.25) should hold The present numerical data on lattices are insufficient to comment upon this criterion 4. Some generalizations The discussion above concerns the probability that a polymer of infinite length starting at the origin will pass through the point r, at length L. One obvious generalization is to consider a polymer of finite length Lx. This leads to a straightforward change, replacing p~(r) by p~(r, LJ where

and

Clearly

and As functions of r and L respectively, j>(1> and^'1' are more complicated thaa^ andj> and a solution has not been attempted, though, it would be a comparatively easy matter to give an expansion of j>(1),j>(1) in terms of L^1, and, since it has been argued that p is effectively a 8 function anyway, this should converge rapidly. Another obvious generalization is to the random walk with excluded volume. This problem is equivalent to the polymer at r, L only seeing that part of the polymer of length less than L. This is equivalent to replacing/", j> by j><2), j>m where

The integro-differential equation now becomes much more difficult, the only obvious comment being that as L tends to infinity j><2:>, j>i2> will tend top and J>. 5. Conclusion The self-consistent field approach has been shown to give the law <>2 > ~ £6 • The number of dimensions in which the problem resides is essential to this answer. The value of j>E comes out to be r"1 in three dimensions, r~2 in four dimensions, and, ffl general, rz~n in n dimensions (n ^ 3). In two dimensions pE does not exist. Now one can treat r~2, r~3 etc. by perturbation theory. It follows that there is only a coefficient change in the Einstein law (r2 > oc L in four dimensions, and not even that in five and higher. In two dimensions, although j>B does not exist, one may still calculate the settconsistent field and| becomes r-2'3, the final law being ~ L3;2. In one dimension,

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The statistical mechanics of polymers with excluded volume obviously <

621

when there is volume exclusion, so for

for It is a property of the self-consistent field for atoms that, although the first approximation is straightforward though involving heavy computing, the higher approximations are virtually impossible. The author suspects the same situation here. It should be possible to evaluate the full self-consistent field of (2.19) numerically, but to improve (2.19) is probably very hard. Another interesting and indeed much more drastic change from the Einstein law appears when v is negative, for the possibility then arises that the polymer can collapse under suitable circumstances. It is hoped to discuss this in detail in a subsequent publication. Acknowledgments This problem was suggested to the author by Professor G. Gee and the author has had the benefit of many helpful discussions with him. He would also like to thank Professor C. Domb for very useful discussions and suggestions concerning the writing of this paper. Appendix A formal assessment of the problem In this section the chain will be taken as continuous. The problem is to evaluate

taken over all paths which go through J?(0) = 0 and R(L) = r. (These path integrals are discussed, for example, by Gel'fand and Yaglom (1960) as are the x integrals below.) The functional integral can be parameterized by writing

where V l is the inverse operator to V (assumed positive definite for simplicity) and the mtegral is taken over all functions x- The identity is proved by completing the square

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Since V is always short range one may replace K.TV ^ r — $ ) b y e 1 S(r—s), and this will also work when the excluded volume stems from a hard potential rather than a soft, in the usual way. Thus

(The symbol J/~ will always be used for the various normalizations.) Now one has rigorously obtained the Markov type process under the x integral sign. It follows now that

where

or

This Green function must have sets of solutions i/>n(y) exp(£nZ,) and $n +(r) exp (-.£„£) (i/r, i[i+ are the (at most) two solutions for given E; in Hermitian systems they are complex conjugates, but cannot be so identified since x is a variable of integration): Cleraly as l

tends to infinity only En =0 winll survice, ie.

Now G will be obtained by the method of steepest descent, writing

The next terms in the series for 6, due to V2^, have been investigated by Jeffreys and Jeffreys (1961) who give a systematic expansion, but attention here will be restricted to the leading terms only. Thus

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The statistical mechanics of polymers with excluded volume

623

and

and

As has been noted, large L corresponds to small E, so one may expand in E to give

whereupon

Now one has the x integration to perform, and this also is done by steepest descent. The function of steepest descent xs is dominated by the first two terms of the exponent, since the third vanishes at the maximum of the function. Thus one finds since

which is real and is indeed vj>l~* of (3.17). As before, one can now write p in the forms (3.9) and (3.24). The other cases are obtained by considering

for the finite chain, giving just G, but the random walk problem

a

ppears to have no simple parameterization. The functional integral (A5) cannot be evaluated more generally. Further terms can oe taken in the solution of G, and the steepest descent equation can be studied in the

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exact form

but it appears these are beyond analytic treatment.

References DOME, C., 1963, /. Chem. Phys., 38, 2957. FLOSY, P. J., 1953, Principles of Polymer Chemistry (Ithaca: Cornell University Press), p. 523. GEL'FAND, I. M., and YACLOM, A. M., 1960, J. Math. Phys., 1, 48. JEFFREYS, H., and JEFFREYS, B. S., 1961, Methods of Mathematical Physics (London: Cambridge University Press). VOLKENSTEIX, M. V., 1963, Configurational Statistics of Polymer Chains (New York: John Wiley), p. 368. WALL, F. T., and ERPENBECK, J. J., 1959, J. Chem. Phys., 30, 634.

6 THE ENTRY OF FIELD THEORY INTO POLYMER SCIENCE R. C. Ball1 Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom With thanks to Sam, for all his help and support over many years. Abstract Two papers by Sam Edwards in 1965-6 changed the face of polymer theory with the introduction of field theoretical methods. Using these he identified what we now know as universal regimes, and opened polymer science up to theoretical physics. 6.1

Background

Sam began his research career renormalising Quantum Electrodynamics with J. Schwinger, his PhD supervisor: Renormalisation in those days simply meant extracting the finite predictions from a theory in which the bare quantities (particularly, in QED, the electron mass) are divergent—an almost miraculous achievement in its time. What I believe Sam was less satisfied with was the method: perturbation theory seemed the only workable option [and Sam did make some progress here (Edwards 1953)], but it is not an appealing outlet for such a creative mind as Sam's. The underlying Quantum aspect of QED was also a central obstacle to progress, as Feynman weights are complex exponentials which almost totally cancel when computing the sum (strictly path integral) over notional classical paths. This makes it extremely hard to control approximations, large answers not necessarily being good ones. Thus it was that Sam was very interested in field theoretical problems where one might need a Feynman Path Integral over real positive exponentials, which should open the door to variational techniques. Using these, where the biggest answer is the best, one should be able to rank and hence systematically to improve methods, and there was always the hope that something might develop which would be of use back in the likes of QED. Polymers were a particularly attractive theoretical target because of the very literal applicability of Feynman Diagrams, which Sam introduced in his papers of 1965-6. 1

E-mail: [email protected]; website: http://www.phys.warwick.ac.uk/theory

100 6.2

Field Theory in Polymer Science by R. C. Ball The new polymer theory, and excluded volume

The first paper in question (Edwards 1965) appeared in Proc. Phys. Soc. in 1965 and in the main text presents itself as an analysis of the single chain excluded volume problem. A substantial appendix presents the material from a more mathematical point of view and makes plain the field theoretical roots of Sam's approach. Based on a cryptically worded acknowledgement to Cyril Domb, it is tempting to guess that the appendix is more the original and the main text an explanation in response to reader request! As an attack on the excluded volume problem the paper is a very substantial success. To tackle it Sam developed what we now know as the Self-Consistent Field approximation (SCF), which moved the field on from Flory's two line argument for the swelling exponent into a full quantitative theory. With this Sam could calculate all properties in principle. Naturally, he recognized that one would run up against limits of analytic tractability, and of course the SCF is only an approximation— its answers are not guaranteed to be correct. Concerning the swelling exponent the answers are interesting, as Sam obtained not only Flory's old result for what we would now call the swelling exponent v in three dimensions, but also noted how key was the dependence of this result on space dimension d:

We should really call this Edwards' Formula— it is eqn. (5.1) in Edwards (1965) — despite the entrenched usage 'Flory Formula' and subsequent rediscovery by Fisher (1969)! Sam's discussion includes the observation that (using more recent language) the upper critical dimension is dc = 4, and the way he stated the validity of the above formula as d < 3 reflects how consideration of non-physical dimensionality was yet to come to the study of Critical Phenomena. Sam's generalisation of Flory's exponent was clearly exact for d = 1 and d = 4, albeit logarithmic corrections were not mentioned in the latter case. I suspect the logarithmic corrections were a casualty of the challenge to extract analytical results from the full SCF; they are more accessible by simple perturbation theory, and re-summing these is one route to what we now know as the Renormalization Group and the expansion of critical exponents in the quantity c = 4 — d. Historically, this was brought to bear on the excluded volume problem indirectly, through de Gennes' mapping to the n = 0 component 4 field theory (de Gennes 1972). It was only later that des Cloizeaux (1980) showed how to directly renormalize the polymer perturbation theory, leading to various universal ratios (such as characterize the comparison of end-to-end radius to radius of gyration) in addition to the critical exponents. The Renormalization Group and expansion in e = 4 — d established that Sam's/Flory's result for v was only approximate as d is lowered below 4. However, it took an immense amount of work to establish that the the value 3/5 was too high by of order 0.01 in d = 3, Sam having noted that the earlier numerics

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101

by Domb (1963) could not discriminate against it. Nienhuis (1982) first showed that the result v = 3/4 is (or happens to be) exact for d = 2, and Conformal Field Theory [e.g. Cardy (2001)] has since given much more detail. The present author conjectures that the SCF exponent result may be correct over the full range 1 < d < 2, where perturbation theory for the free energy lacks a leading short-range divergence. The Self-Consistent Field approximation was rightly presented by Sam as a general technique, modelled, as he indicated, on the Hartree approximation in electronic structure, and constitutes a landmark contribution to Polymer Theory in its own right. Along with the full SCF formulation also came the eigenfunction decomposition of the propagator, and the exploitation of ground-state dominance for long chain spans. Sam recognised the potential for SCF methods to treat adsorbed polymer layers, which he worked out with research student Dolan (Dolan and Edwards 1975). In considering both the single chain and the adsorbed layer, Sam's strength in analysis enabled him to obtain results without recourse to numerics. He came slightly to regret this when Scheutens and Fleer (1979) took the numerical lead, and subsequently much of the founding credit, for SCF analysis of adsorbed polymer profiles: all that work (whose application continues today) stems from Sam's analytic formulation. There is one slight contradiction in the single-chain story above, in that SCF is not a variational technique, or if judged as such then it gives a very poor free energy estimate. Here lies a story of a lost paper: Sam did indeed seek to build a systematic variational theory of the polymer coil, based on a general Gaussian trial form. He initially thought this led again to the Edwards/Flory result, until J. des Cloizeaux spotted that a divergent term had been missed in the analysis. This led to the publication by the latter of a different (and less encouraging) result for the swelling exponent (des Cloizeaux 1970).

6.3

Polymer solutions

A year later appeared the second paper (Edwards 1966) addressing polymer solutions, where the key issue is what happens when polymer coils are forced to interpenetrate. Sam's analysis led to the quantitative identification of a screening length £, beyond which length-scale the chains are Gaussian and their correlations weak. This idea plays a central role in all the subsequent theory of polymer solutions. Right in the abstract of the paper the coil overlap concentration is identified, c* in the later notation of the French school, and Sam also excises from discussion the melt-like regime where his focus on the monomer-monomer virial coefficient would cease to be appropriate for the interactions. His intermediate concentrations regime is now recognised as dividing into two regimes. In the semiconcentrated regime, screening largely prevents excluded volume from swelling the chains, and Sam's headline result for the osmotic presssure applies (he terms it partial pressure).

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The lower end of Sam's intermediate concentrations is ruled by what (by 1972 at least) the French school termed the semi-dilute regime (Cotton et al. 1972), where there is significant coil swelling at scales below the screening length, which in turn was renamed the blob size (de Gennes 1979). Sam could not carry his direct analysis all the way through for this regime, and it was left for de Gennes to capture it by arguments of cross-over scaling (de Gennes 1979), underpinned by des Cloizeaux's calculation for a case where chain length is only restricted on the average (des Cloizeaux 1975). Nevertheless, Sam did clearly recognise that there was a problem connecting the semi-concentrated and dilute regimes in the discussion of Section 4 of his 1966 paper. Finally, Sam gave an outline of how formally to approach dilute solutions at the level of a virial expansion in chain concentration. Although these expansions are of limited practical use, they are of considerable theoretical interest as the chain-chain virial coefficient is diagrammatically the easiest quantity to compute to a given order of perturbation theory in the excluded volume. This makes it a natural parameter of choice for renormalization calculations such as those of des Cloizeaux (1980). 6.4

Mathematical aspects

Sam's favoured notation and direct mathematical approach shows through in his 1965 Appendix: A formal assessment of the problem (Edwards 1965). This is an unfettered application of functional analysis, with a presentation that exposes the essentials in a most direct way, and (despite Sam's heading) stripped of any unnecessary formalities. It establishes what we now know as the Edwards Hamiltonian,

(where I have compromised somewhat towards Sam's later notation), with the backbone path of the polymer parametrized in terms of a continuous arc-length variable. The first term in Edwards' Hamiltonian, when exponentiated, constitutes the Wiener measure, and comes straight from Sam's acknowledged field theory sources. This alone leads to pure Brownian walk statistics for the polymer on all length-scales. The mathematics is then that of diffusion equations, and hence a whole armory of mathematical techniques came into play. Having come from Quantum Mechanics, Sam took care to emphasise this contact with classical mathematical physics in his text, but of course he had opened the door for others to bring related expertise from Quantum Mechanics to bear on polymer problems, as acknowledged by de Gennes (1969). Another key feature of the Wiener measure is that its structure is scalefree (indeed formally scale- invariant). The notions of arc length and persistence length enter exclusively through the combination (.L, as adopted by Sam's longstanding friend des Cloizeaux, but Sam has always kept the parameters

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103

separate—perhaps a lasting inheritance from his polymer chemistry colleagues in Manchester. The second, excluded volume, term is not written so explicitly in Sam's 1965 paper, notwithstanding this being a standard citation for the whole Hamiltonian! What Sam does is essentially to establish its basis, via what would now in other fields be described as a Bogoliubov transformation. In his terminology that is 'parametrizing' the full monomer-monomer2 interaction in terms of an integral over an auxiliary field (see below). All of the Hamiltonian is summed up by Sam's innovation of Feynman diagrams for polymers, in which the noninteracting chain corresponds to a single (free) particle propagator and the excluded volume to a two-particle interaction. It should be noted that Sam also drew attention to the residual three-point interaction (from the third monomeric virial) in these diagrammatic terms, providing the foundation stone for later appreciation of nontrivial scaling at the polymer theta-point, where the excluded volume itself cancels. These diagrams provide key clues to de Gennes' mapping from polymer statistics to 4 field theory (de Gennes 1972), and later to the tricritical 6 theory for the theta-point. The auxiliary field % furthers the analysis by separating the analysis of chain propagation from the details of the interaction. At fixed % the polymer chain is mathematically a Markov process and, most importantly for Sam, this conforms to a differential equation. He can then bring to bear much machinery from Schrodinger quantum mechanics, including the whole development of the SCF. The details of interaction come in the 'bare' propagation of the auxiliary field (in an inverse form, which does require some assumptions for existence), and here lies the key to the delta function approximation. The main text states this directly in terms of the inverse interaction, but when the auxiliary field is expressed in Fourier representation (as in Sam's 1966 paper) the issue becomes clear. As long as the calculations of interest are dominated by long enough wavelength components of the auxiliary field, the long wavlength limit (or virial coefficient) suffices for the Fourier transform of the interaction. In that approximation, integrating out the auxiliary field leads back to the delta function interaction. The auxiliary field became the central feature in the 1966 paper (Edwards 1966) on polymer solutions. Sam showed how the decoupling of n chains through X immediately led to expressions involving just the n th power of the single chain partition function,

and hence the extraction of the thermodynamic limit was clear. The analytical complication remained that the partition function F in question depended functionally on the realisation of %, and this is the point where approximations are necessary. 2 Sam

consistently uses molecule for monomer, and chain for macromolecule.

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For the main calculation, Sam adopted the most tractable approximation of expanding InF up to second order in % (a second order truncation of a cumulant expansion). This judiciously preserves the leading divergent term, which is a simple expectation value of the excluded volume interaction. Sam went to the trouble of showing how this term was truly finite when reverting to the full form of the monomer-monomer interaction (as oppposed to just the delta-function limit), in gratifying contrast to QED, where one did not have a direct basis to limit the divergence. The fate of the divergent term does, however, mirror what happens in QED: it drops out of the physically relevant laws, such as the osmotic pressure. The discussion of chain screening makes the simple physics clear, particularly in the theorists' terms of Feyman diagrams. Inevitably, it also confronts the issue of how the behaviour at intermediate concentrations can match on to the dilute regime. Sam's central difficulty was that the Gaussian truncations served well to capture the extensive aspects of the polymer solution, whilst his earlier SCF approach captured something of the internal correlation leading to coil swelling. To deal with (what we now term) the semi-dilute regime he needed the best of both, and this he could only carry a limited distance. There has recently been some interest in bypassing the challenge of computing analytically chain partition functions in an auxiliary field, in favour of doing this numerically for many samples of that field. This would have been unthinkable with the resources of the 1960s, but lately F. Bates has attempted such results on block co-polymer microstructures (Bates 1999). 6.5

Future directions

The role of Theoretical Physics has penetrated throughout Polymer Science since Sam's landmark papers and, inevitably, the more the field matures the more it is led by theory. Whilst that wider heritage is the greatest impact of Sam's work, I should signal the Self-Consistent Field as the particular tool introduced in the 1965 paper that remains most immediately active today. It is the front-line tool in understanding the rich variety of block co-polymer phases (Bates and Frederickson 1999). What about that original agenda, as to what polymer theory could contribute back to Quantum Mechanics? When Sam left this subject the key challenge was the need for renormalisation and the gap in understanding this implied about high energies. After thirty years of the Standard Model, coupling constants running with energy and Grand Unification in sight, encouraged by Supersymmetry and developments beyond, particle theorists would probably claim some prospect of success on that front. However there remains another difficulty from a scientific generation earlier: the nonlocality inherent in quantum theory. John Bell in his later years convinced Sam that these problems were quite sharp, but it remains probably the best move on the part of Sam's generation to have got on with Theoretical Physics, leaving the worries to Einstein and Bell.

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105

The advent of Quantum Cryptography and widespread efforts to realise Feynman's dream of Quantum Computing mean we can no longer set the issue of nonlocality to one side. Either we give up on the idea that the Laws of Physics are both Causal and Local, or we accept (interfering!) Multiverses, or we come up with some new interpretations. I propose to indulge in a little of the last, aided by Sam's favourite technique of parametrisation through an auxiliary field. A simple but pertinent example is the Electromagnetic Field (in simple flat space-time). In a Schrodinger representation, we have the wave function ^[A;t] which at particular time t depends functionally on the 4-vector potential A as a function of 3-position r. The Green Function to propagate ^ forwards in time can be formulated as a Feynman Path Integral,

where as usual Fap = Afj^a—Aa^ = daAp—dpAa and the functional integration is over all (classical) histories of the field A(r, t) which could possibly connect between specified initial and final configurations. Following Sam, we can parametrise the quadratic form in the integrand by coupling to an antisymmetric auxiliary field x<x/3 = —Xfia, leading to

Now the outer functional integral is just a classical average over a Gaussian distribution for the components of %. Inasmuch as the inner functional integral still represents quantum mechanics (with a strange phase on the time) it is a totally factorisable quantum mechanics because we can integrate its action by parts, using

where the right hand side is all completely local in A(r,t). Quite generally, provided the derivative terms in the Lagrangian density are quadratic we can always remove them in such a way. More complicated local terms are no obstacle to the procedure, and indeed we need to invoke some in the electromagnetic case, as otherwise the resulting integral over the local value of the field A is divergent. Let us review where this has got us. For a given realisation of the auxiliary field the quantum mechanics (of the electromagnetic field) has been rendered completely trivial, with each point in space-time decoupled from every other, but in a x-dependent manner. If you like, there is still quantum mechanics at each space-time point, but in this case it happens to be rather trivial. It is when we average over realisations of the auxiliary field, using a simple uncorrelated classical distribution, that we recover conventional quantum mechanics globally for the resulting average state function.

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Two extreme costs have to be noted. Quantum mechanics requires us to compute observables from the bilinear combinations ^ • • • 'f, and the only way we can get the average of these combinations to factorise is to insist that the random field % we use to propagate ^tbe independent of % used to propagate fy. Thus, as well as strange phases on the time variable, rendering the evolution of ^ at fixed (history of) % no longer conventionally hermitian, we also lose linear interdependence between ^ and ^. The conventional structure is only recovered upon taking the averages over (independent) % and %. The crucial question (which I cannot answer definitively here) is whether full averages over % and % are required to match the statistics of one-off experiments, because, if so, we would be in at least as much difficulty of interpretation as Multiverse ideas. However, truly pointwise particle detection measurements would imply sensitivity to particles of all energies and momenta, and such measurements would always be swamped by vacuum fluctuations. For measurements at finite energy there must be some local averaging over time and space, and interestingly this is at the level of state functions not probabilities—so it is naturally an independent averaging of ^ and ^. We need these local space-time averages to do enough to damp out the fluctuations in ^ and 'ft to avoid conflict in prediction of experiments. Of course QM is more discriminatingly checked by its prediction of the complete probability distribution of outcomes of a 'classical' measurement, but the more times one repeats an experiment to refine the observed distribution, the more a true average is obtained of the auxiliary field. That is where this whole idea originated: the sharper tests of nonlocality in QM require the rather nonlocal notion of many repeated measurements. I should note that even if you entertain this new interpretation, there remains a problem with fermion fields. Here, decoupling the Lagrangian (or Hamiltonian) over different points in space-time is not enough, because fermion operators are still left with their anticommutators: because of these, there appears no easy way to achieve a factorisation of the state function over localities that is preserved by the motion. The only obvious hope is to express the fermion fields (nonlinearly) in terms of boson fields, which can be done elegantly in 2 +1 dimensions but not (elegantly) in 3 + 1 dimensions. Acknowledgments

The author would like to thank the organizers of this volume for their initiative and for offering the opportunity to contribute. References

Bates, F. S. and Frederickson, G. H. (1999). Physics Today, 32, 32-8. Bates, F. S. (2001). Invited talk at Principles of Soft Matter. 21st CNLS Annual Conference, Santa Fe (May 2001). Cardy, J. (2001). J. Phys. A 34, L665-72 . des Cloizeaux, J. (1970). J. de Physique I 31, 715-36.

References

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des Cloizeaux, J. (1975). J. Phys. (Paris) 36, 281. des Cloizeaux, J. (1980). J. de Physique 41, L151-5. Cotton, J. P., Farnoux, B. and Jannink, G. (1972). J. Chem. Phys. 57, 290-4. Dolan, A. K. and Edwards, S. F. (1975). Proc. Roy. Soc. A 343, 427-42. Domb, C. (1963). J. Chem. Phys. 38, 2957. Edwards, S. F. (1953). Phys. Rev. 90, 284-91. Edwards, S. F. (1965). Proc. Phys. Soc. 85, 613-24. Reprinted in this volume. Edwards, S. F. (1966). Proc. Phys. Soc. 88, 265-80. Reprinted in this volume. Fisher, M. E. (1969). J. Phys. Soc. (Japan) 26 (suppl.) 44; de Gennes (1979) is surely wrong in attributing this reference priority, de Gennes, P.-G. (1969). Rep. Prog. Phys. 32, 187. de Gennes, P.-G. (1972). Phys. Lett. A 38, 339. de Gennes, P.-G. (1979). Scaling Concepts in Polymer Physics. Cornell University Press. Nienhuis, B. (1982). Phys. Rev. Lett. 49, 1062. Scheutjens, J. M. H. M. and Fleer, G. E. (1979). J. Phys. Chem. 83, 1619.

7

REPRINT THE THEORY OF POLYMER SOLUTIONS AT INTERMEDIATE CONCENTRATION by S. F. Edwards Proceedings of the Physical Society, 88, 265-280 (1966).

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PSOC. PHYS. soc., 1966, VOL. 88

The theory of polymer solutions at intermediate concentration S. F. EDWARDS Department of Theoretical Physics, The University, Manchester MS. received 12th January 1966 Abstract. It is argued that polymer solutions can be classified into three broad types which may be characterized in terms of N the total number of (micro) molecules, n the number of polymer chains, I the effective length of a micromolecule, c the excluded volume per micromolecule and V the total volume. The types are: 0) dense solutions in which VIN Vjn>v or (Z,/)1/2> Vjn>v according to the magnitude of«, i.e. L-niOp-i/spiio greater than 1 or less than 1, and (Hi) dilute solutions in which V/n
A discussion is given of the way this expression fails as one enters regions (i) and (lii). 1. Introduction Although a fully realistic theory of polymer solutions will involve considerable technical complexity in such matters as the precise flexibility of the molecular linkages and the molecular forces, and the nature of the interaction of the molecules with those of the solvent, there remains a core of general functional relationships in, for example, the equation of state, which can be reduced to problems which are easily posed, but which ran only be resolved by fairly powerful mathematical tools. In this paper it is hoped t uerive this skeleton theory in which the dependence of the thermodynamic functions upon the parameters specifying the solution will be demonstrated. Firstly one must specify the quantities on which the system will depend. These will be as follows: V the volume of the system, T the temperature, n the number of polymer chains present, which w iU be assumed to be all of the same length and constructed of m molecules each of •ength I, so that m = N [n the number of molecules per chain, where N is the number of (micro) molecules, and L = ml the length of a single chain. It will be assumed that the molecules are freely hinged; if they are not / is to be regarded as an effective length, i.e. parameter in the diffusion equation for the configurational probability of the chain, appearing where the step length would appear for a freely hinged chain. Finally the excluded volume of the molecule, defined by

where

U is the molecule-molecule interaction, and orientation is averaged away. The

viri 1e ° Ca-n ^e rePresented as the effective size of a molecule as in the usual second coefficient. It will be assumed positive throughout this paper. 1 265

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These parameters are sufficient to describe several regimes of polymer solutions, s these are now enumerated. Firstly, one can have a dense solution and almost every molecule of every chain is in contact with several others. It is clear that the excluded volume -parameter described above is quite inadequate to describe this system, and i will not be attempted in this paper. It is clear from the virial expansion of a dense gas that this dense state corresponds to having a volume of polymer comparable with tie volume of solvent, i.e. V ~ Nv. But at lower densities of polymer one can discern tw regimes which are amenable to treatment. One can reach the condition V ^> jVoandyet have a rather homogeneous system because of the length of the chains, i.e. as in th sketch in figure 1. In figure 1 the system is dilute but the size of a chain greatly exceeds

Figure 1.

Figure 2.

the spacing of a chain. Supposing the configuration of a chain to be that of a random walk, it will, roughly speaking, occupy a region of space of the order of (LI)1'2 to ^ meter. The mean spacing of chains is (V/n)113, so one has the regime of figure 1 provided that (i/)1'2 > (VjnY13. This regime will be characterized by a mean density with fluctuations. But if the density of chains decreases further the density will becoffl6 increasingly lumpy till the chains separate (it should be remembered mean repulsr?6 forces are operating) and figure 2 is a good description of this.

Ill The theory of polymer solutions at intermediate concentration

267

This is an intermediate state to that of complete separation when the chains behave as molecular balls (figure 3). To give criteria for the validity of the different pictures it is useful to reduce all the physical quantities to lengths. The four basic quantities are then 1B = z> i;3 ,1, r<, = (V/n)113, r: = (LI)1'2; and it will prove useful to add two derived quantities £ = (UnvLjVl3)-112 and q = (wl//)1/s,
Figure 3. Then the dense regime is q ~ r 0 . The dilute regime of figure 3 is r^ < r0l with a further classification according to whether Ir-f is greater or less than /03, a criterion which decides if the chain follows the random walk or excluded volume laws. The intermediate region will be shown to exist provided that

Since it is consistent that all these parameters can be experimentally adjusted to this criterion, this is a perfectly feasible regime and happens to be amenable to calculation, leading to a soluble equation of state. A discussion will also be given on how this regime tends, via figure 2, to the dilute regime, figure 3, but although the qualitative picture is clear an exact description does not appear possible. 2. The formulation of the problem The reason that there is a problem is that, when two chains interact at one point A, there is an enhanced probability that they will interact at another B, simply because they are chains (figure 4). This is quite different from the multiple collisions giving rise in

Figure 4. gas theory to the second virial coefficient, which are here equivalent to the situation in "gure 5. It will always be assumed that these triple encounters have negligible probability. It follows that, if one defines the excluded volume parameter v as in equation (1.1), t™ can be thought of as a pseudo-potential which will describe an interaction such as at

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A of figure 4 by perturbation theory. What must not be treated as a perturbation is th combined effects of interactions at A, B and so on. One expects then that the free energy of the system will be a function of v, often quite a subtle function of v, but will not involve

Figure S. 'higher virial' type coefficients. If one defines

then it will often be adequate to think of v(i] as

but circumstances will arise when the form (2.1) is needed. Although molecules have a definite shape or orientation they will be considered as points so that the chain is described by a function R(Z/)> 0 ^ L' ^ L giving the coordinates of the point a distance L' along the chain. The Boltzmann factor for the potential energy is then

where the sum includes a = b. Now, as has been discussed, this replaces the true potential UficT and will be treated as a pseudo-potential unless circumstances warrant any improvement. The fact that the polymer is formed of chains is expressed by wnting the probability of the chain configurations as Wiener integrals, so that if -^([RJ' W'"' is the functional probability of finding configurations Ra(i'), R6(i") ...

The free energy associated with the configurations is then

where 8Ra means integration over all configurations of the chain Ra(L')> and g is *' weight associated with each step. In the limit of a freely hinged continuous chain, I has the nature of a differential, but is a constant. In actual fact one can expect g to K

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less than this value because of non-free hinging, which to some extent should be present in the Boltzmann factor, for neighbouring links will not be correctly treated by the pseudopotential, for example they cannot meet with arbitrary relative orientation. Since in the present solutions variation of the total volume of the system cannot effect these linkages, (2.14) will be taken to represent the system truly and g taken to be a constant. Equation (2.6) should strictly be an integration over the coordinates and orientation of each molecule in the chain, but it is much more convenient to run it into the continuum. At this point a transformation used previously in a polymer problem by the author {Edwards 1965, 1966) is adopted. It is well known that the exponential of two-body forces can be written as the Gaussian average of a potential associated formally with the fields, i.e.

where Z

l

is an operator such that

For example, if Z(j8, y) is ^-'"-^/('Hp-Yl) then

so that (2.8) is the well-known equation

It will be noted that the left-hand sum over a, b includes the case of a = b. Applying this result to (2.5) one has

| his trick is equivalent to the situation in electrostatics where one can discuss the interaction of point charges by having a (Coulomb) potential between them, or by saying they Interact with an electric field which satisfies Poisson's equation. The first point of view B that of (2.6). Since one can evaluate (2.7) by completing the square, i.e. 'where

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clearly this latter equation is the analogue of Poisson's equation and xi of the electrostatic potential. If v(t) is taken to be »S(r) its inverse is c~1S(r), but this must be treated with caution when Ra(I/) = Rj,(£") (the equivalent of y = p in the model) and a more refined c(r) employed near the point. The transformation has now decoupled the chains. On can now consider each chain moving in a field x, the averaging over x being equivalent to stating that x is caused by the other chain. Having decoupled the chains, it is -well known that the functional average over SRa is equivalent to a differential equation. Suppose the chain Ra starts at r a (0) and ends at r0(i), then

where G satisfies

In this way the free energy integral is reduced to

All the integrals over the G, are equal, so if one writes

The two forms (2.15) and (2.18) are those to be used in the next sections.

3. Polymer solutions at intermediate densities Although formulae (2.16) and (2.18) are equivalent, their form suggests that they will be useful in different circumstances. In (2.18) the fact that there were n chains only appears in the factor n log F. The exponent contains only an integrated out form. This suggests that it is a good starting point when the system itself is rather homogeneous, i.e. as in figure 1. From this point of view one may forget all about chains and just think of fields %. There will be a mean field corresponding to the mean polymer density, and then fluctuations about this mean. The mean field must be a constant in space, so writing

where V — A3 and mlt m2, m3 are positive integers, one has

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The theory of polymer solutions at intermediate concentration

271

and, assuming that an expansion in x is permissible,

where Then

where

Tor values of jmj upwards of order (3A2L/V2/3)1/2, f m is given by the first term, but for |mj smaller than this £m tends to \Lzl~z. Finally then

The sum over m (the positive integers) will tend to the integral

«s integral does not converge at large [mj, the cause being traceable to the fact that e sum ination in (2.4) or (2.5), the term with a = b, i.e. the self-interaction of the chain,

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is included. This term is one for which the pseudo-potential term is inadequate, for by expansion one finds that

If •»(*) is treated as a 8 function for large L, one obtains the term

being the first term in the expansion of the logarithm in (3.11). The correct form of (3.11) is then obtained by allowing an improved accuracy in v(t) for single chains, whereupon the pseudo-potential can be successfully used, i.e. (3.11) should be replaced by

where t>(m) is the true Fourier transform of a(r). In effect the trouble comes from allowing the chain to interact with itself at a point, and a precise treatment of the interaction of a (micro) molecule with its nearest neighbours in the chain is already sufficient to make the integral convergent. The self-interaction of the chain is independent of F as one would expect and therefore does not alter the equation of state ; so it will be called ne. In the remaining expression it is adequate to replace £n by 3A2L/m27r2Z3 (the error corresponding to an error in the free energy of order KTv2n2LS!2l~nizV~l). One ha then

But

Thus the integral (3.15) gives

so that finally

and

if one introduces M, the total volume of polymers, and uses the length

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273

fjjjs can be written as

which can be physically interpreted this way. The ni
4. The behaviour of a chain in the solution The formal analysis for a single chain is simple, for clearly the probability of the chain starting at r' and ending at r at length L is given by

or if

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When xo is extracted from G(r, r'; L; [x]}, it will not influencep, so our problem becomes that of evaluating

Now consider the series expansion of p, derived from the series for G:

If we use ( ... > to mean averaged as in (4.4), the value of <x(rMr/)) is readily obtained by using the transforms and gives

If we use the value of f m as in (3.IS), this gives

which clearly stems from the original definition of v as a pseudo-potential (and is indeed c(r—r') in general) and a term resulting from the interference due to the chain nature of the polymer. Though it has an unusual form, the effect of the other polymers is a kind of screening, for the integral of <x(r)x(r')> of (4.8) is now zero or more exactly, using the exact expression (3.8) for f m ,

a strong reduction. One can see the origin of this behaviour in this way. Suppose one represents the polymer by a full line, and the interaction by a dotted line. Then the interaction of a chain with itself to order, i.e. a single interaction, is as shown infigure6.

Figure 6.

But other chains will get in the way resulting in a long-range effect building up (figure 'I' The sum of all these effects is the effective potential (4.8). One can understand the fact that the correction is an attraction by remembering that it is the fluctuations which are under discussion. The mean interactions have been allowed for in ^oj and the effect o

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The theory of polymer solutions at intermediate concentration

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the other chains is to try to bring the system back to the mean and counter the direct t;(r-r') term. The effect of the other chains is therefore to weaken the self-interaction of the chain and make its configuration close to that of a random walk. The % transformation shows that the self-interaction of the chain can be represented by the diffusion of a

Figure 7. particle in a (complex) random potential. Now the motion of a particle in a random potential is described by the Boltzmann equation for the Lorentz gas, which suggests that an equation of this form is also appropriate in the present problem. Since the derivation of this equation is standard and it will not be used further in this paper, it will simply be quoted. Let the mean <(G) have Fourier transformg^L), then

Since

one can write for

where the new terms ensure the conservation of probability

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understood in polymer problems and is further noted below.) Consider then the other extreme of an isolated chain. This has been treated elsewhere by the author (Edwards 1965, 1966) and the results will be quoted here. Consider one end of the chain at th origin and consider adding the last link on to the chain. Suppose for the moment that the chain is Gaussian. It is known that it starts at the origin and of course that it ends at the given end point which will be called the current point. Since it went through the end point there must be a considerably enhanced probability of rinding the chain in the neighbourhood of the end point, but this enhanced probability will give rise to th interaction term of (4.11) and be absorbed into changes in entropy per link, effective step length and so on. But it also started at the origin. There is, therefore, an enhanced probability of rinding it near the origin, an enhanced probability which will die away the further one gets from the origin. This probability will be

which for a very long Gaussian chain is T>\Ttrl. This term will produce an effective repulsive potential at the origin which is sensibly thought of in those terms. It is no sensible to consider an effective potential around the current point since one would be always at its origin as the chain builds up, and this term is better thought of as a scattering term. Now a Coulomb potential is well known to have dominant behaviour at large distances so it is clear that the assumption that the chain is, even in its first approximation, Gaussian must be inadequate and a self-consistent argument developed. This is done in the references above where it is shown that one can expect

which in turn leads to a relation

-where the c are numerical constants (for this latter result compare also Flory 1953, chap. XII). To translate this verbal argument into mathematics consider the Green function for the single chain:

A straightforward attack is to write

and evaluate by steepest descent

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The theory of polymer solutions at intermediate concentration

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where

and the expression in the curly brackets is shorthand for the logarithm of the Fredholm determinant of the kernel

If one wished to obtain the partition function, i.e.

one could alternatively consider, as in § 3, working directly upon

which would, of course, give xi a constant, and the complicated effects of the excluded volume are transferred to the fluctuations and to higher terms. It is clear that one gets different answers by integrating
n we light of the discussion above it seems adequate to consider only one point of the chain asfixedand treat the rest in the fluctuation term. Thus to understand the transition rom constant ^i to the peaked xi of a very dilute solution consider the form

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In this, each chain is pictured as a blur surrounding the points rt. Then

If the chains strongly overlap one can approximate %i by integrating over the r, and recover the %o of § 3. If they are very far apart one can write where

which, according to the references cited, tends to

In intermediate cases write so that, in an abbreviated notation,

Now write G; for G([rj;]) and expand the right-hand side

Thus one can approximately identify

say, or Through approximation this form now spans the two limits discussed here, being tw cases of g, rj, f constant in space, and g small. Like all intermediate problems there seems no easy way to solve this further, and some final comments will now be made on the very dilute case. 5. Very dilute solutions The conclusion of this section is that heavy numerical work appears necessary to solve the very dilute case, so the account given will be brief. Clearly a cluster expansi°B

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The theory of polymer solutions at intermediate concentration

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is appropriate corresponding to the interaction of one, two, three chains and so on. Thus using the definition

and generally

Thus

Clearly the first term corresponds to the non-interacting chain, the second two chains interacting, and so on. It will therefore lead to a series of the type

and to get a^ one has to evaluate

Formally this leads to an expansion about

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and the answer

and log(l + c/T) represents the Fredholm determinant as before. It is also clear physically that Xia represents the mean 'polymer density" of a pair of chains starting at r± and r2. Far apart, for very long chains, one can expect where

But as t-i -> r2 the two regions of high density coalesce; this process cannot be considered in any sense a perturbation of the separated chains since the whole problem is that of a self-consistent field which is virtually unrelated to the random flight unperturbed form. The calculation of the virial series has been considered by several authors and has been recently reviewed by Stockmayer (1960). The method here has the advantage o being systematic and allowing full weight to the correlations within the chains, so that it appears doubtful if the virial coefficients can be obtained without heavy numerical work. A closer study of (5.11)-(5.12) suggests that the coefficient aa will be proportional to vslsL9ls for long chains but the numerical coefficient will require a calculation comparable with a molecular self-consistent field. 6. Conclusion It would appear that between the very dense region (which is as yet functionally intractable) and the very dilute region (which is functionally tractable, but exceedingly hard work) there exists a region which is theoretically tractable and the author hopes to b able to check the validity of the equation of state suggested here with experimental studies taking place in the polymer chemistry group at Manchester. Acknowledgments This problem evolved through discussions with Professor G. Gee and Professor G. Allen and the author thanks them for their suggestions and encouragement. He would also like to thank Mr. Martin Oliver for his help in checking the analysis. References EDWABDS, S. F., 1963, Proc. Phys. Sac., 85, 613-24. 1966, Proc. Nat. Bur. Stand. Con/, on Critical Points, April 1965, (Washington: National Bureau of Standards), in the press. FJLORY, P. ]., 1953, Principles of Polymer Chemistry (Ithaca: Cornell University Press). STOCKMAYER, W. H., I960, Makromol. Chem., 55, 54-74.

8 THE COARSE GRAINED APPROACH IN POLYMER PHYSICS Y. Oono1 and T. Ohta2 1

Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois 61801-3080, U.S.A. 2 Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan 8.1

Introduction

In 1965 Edwards introduced the Edwards model of the self-avoiding walk o polymer chain in a good solvent (Edwards 1965). Its relation to the 4-model, found by de Gennes (1972), became the starting point of the modernization of the statistical physics of polymer systems. In 1966 Edwards went beyond the single-chain problem and studied non-dilute polymer solutions, pointing out that polymer solutions could have a regime between dilute and concentrated solutions (Edwards 1966). The many-chain version of the Edwards model used in this study was later adopted by des Cloizeaux (1975) to establish the concept of the semidilute solution. The revolutionary nature of Edwards' work may now be hard to recognize, particularly in view of the widespread adoption of renormalization group concepts and approaches during the 1970s and 1980s. In this chapter we wish to highlight the revolutionary nature of Edwards' solution theory and to discuss some spin-offs. In Section 8.2 the prehistory—the Edwards model of the self-avoiding walk—is critically discussed. In Section 8.3 Edwards' theory beyond dilute solutions is outlined by closely following the original paper. In Section 8.4 the renormalization group theory of semidilute solutions is discussed that is a direct offspring of Edwards' theory. Section 8.5 discusses the outcome of the Edwards type model for diblock copolymer melts as an example of spin-offs. In Section 8.6 we critically review the real nature of the model proposed by Edwards and its power. In a brook where Edwards found nuggets even the writers of this chapter could still find nuggets. It was simply because he had left the brook quickly for higher-quality gold mines. 8.2

Edwards model and cutoff

We theoretical physicists are interested in the large-scale picture of polymer systems; we are not very much interested in the chemical or local conformational

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details of a polymer chain. We are interested in the features of polymer chains that survive coarse-graining. What is, then, a good model that can capture the features of a self-avoiding walk after coarse-graining? Edwards' answer (Edwards 1965) is: it is the Wiener process with the 6-function-like repulsive penalty for each binary contact. Although there is already an exposition of the Edwards model elsewhere in this volume (Ball 2004), let us summarize its spirit, because we see the same spirit working behind the non-dilute solution theory as well. Furthermore, a need for renormalization is already in the single-chain model, and this necessity was recognized in Edwards' solution theory by Edwards himself. The spirit of the model is intuitively very appealing, so a lot of (mean-field type) work ensued. We know, however, that these mean-field-like approaches are not really meaningful, because they do not recognize the most important aspect of the problem. In short, a lesson to be learned is that sometimes mathematical difficulty (e.g. the ill-defined nature of the model) can imply something physically novel, which triggers further development of the theory. As proved by Donsker (1951), in his celebrated invariance principle (see also Billingsley 1999), the ordinary lattice random walk, which may be a good model of a polymer chain without any self-interaction as viewed by a macroscopic observer, may be described by sample paths of the Wiener process. Therefore, it is a natural idea to use it to describe the polymer conformations. To realize the self-avoiding nature of the chain, a short-ranged binary repulsive interaction described by a delta-function-like potential should be added. This interaction is traditional in polymer dilute solution theory (the so-called two-parameter theory); see, e.g., Yamakawa (1971). The reason why mean-field type approaches do not work is probably that the mean field that is supposedly proportional to the average monomer density cannot capture large fluctuations in the monomer density; since monomers are in a string, their spatial distribution strongly deviates from its average. For a given sample path of a d-dimensional Wiener process, the total interaction energy may be formally expressed as a contour integral along the path. However, with probability one the three-dimensional Wiener process has a double point (Dvoretsky et al. 1957). This implies that any small chunk of a sample path almost surely has a double point. Consequently, without a small scale cutoff for monomer-monomer interactions the conformational energy cannot be defined. The importance of the cutoff can be understood simply and clearly by means of dimensional analysis. Let the monomer (or unit step) size of the polymer be i and the total length (the stretched length) of the polymer be L. If there is no self-interaction then the mean square end-to-end distance (R2) of the chain is proportional to IL. Therefore, dimensional-analytically, we may write

where / is a supposedly well-behaved function. Since the model is not welldefined in the i —> 0 limit, it is very likely that / is, in reality, not well-defined in this limit. That is, the function / in the above formula is not expected to

Edwards Goes Beyond Dilute Solutions

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converge in the L/t —> oo limit. In other words, we cannot apply the 'classical dimensional analytical wisdom' asserting that dimensionless quantities that are very large or small may be ignored. This is the case of Barenblatt's intermediate asymptotics of the second kind (Barenblatt 1996), and we may guess that in the L/t —s- oo limit:

where q is an exponent describing the deviation from the non-interacting case, and is called the anomalous dimension. This cannot be determined by a naive dimensional analysis. As was discussed in Goldenfeld et al. (1989), renormalization group theory can derive the functional form of / under a mild condition, so we need not assume that / has the form f(x) ~ x~q. Perhaps, a lesson we have learned is that occasionally mathematical difficulties imply genuinely new physics. Another such example in polymer physics, though rather minor, may be the correct interpretation of the theta point. Needless to say, neither the Edwards model nor the model used in the two-parameter theory has a hard core in its interaction potential, so the system becomes thermodynamically unstable, if the so-called excluded-volume parameter v becomes negative. Therefore, the zero excluded-volume limit, whose importance has been stressed since Flory as the theta point, is a genuine singularity even if v\fL is bounded in the L —> oo limit (Oono 1975). This is a physically unnatural result. Therefore, there must be many-body (or at least the three-body) repulsive interactions even at the so-called theta point (de Gennes 1975; Oono 1976). That is, the theta point was clearly recognized as a result of a many-body effect, and turned out to be, in a sense, more difficult (Duplantier 1980, 1982) than the good solvent case contrary to the claim of Flory. In retrospect, in the conventional theory of polymer solutions, including the so-called two-parameter theory, there was a fundamental misunderstanding of the nature of the ideal state of a polymer chain. In a certain sense, the delta-function-like singular interaction misguided polymer physics. The ideal state of gas is not realized through turning off the interactions but through dilution. If dilution is impossible, then the most natural ideal state is probably the hard core interaction limit as in liquids (Barker and Henderson 1982; Talbot et al. 1986). Therefore, in this sense the self-avoiding walk should be regarded as the ideal state of polymer chains (Oyama and Oono 1977). Another important view point arises from the need for the cutoff: the Edwards model is actually not a single model but a collection of models with various cut-offs in the self-interaction contour range, similar to the situation for any renormalizable field theory. [This point is clearly and pedagogically explained in Aoki (2000).] Therefore, the model can only predict universal quantities, independent of microscopic details. 8.3

Edwards goes beyond dilute solutions

After the Edwards model of the self-avoiding walk or a single polymer chain in a good solvent, naturally, the next target was to understand the polymer solution

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Coarse Grained Approach in Polymer Physics by Oono and Ohta

with nonzero concentration. Edwards was the first to recognize that between the dilute and the concentrated solution regimes is another regime. He writes in the original paper, 'One can reach the condition V ^> Nv and yet have a rather homogeneous system because of the length of the chain, i.e., as in the sketch in figure 1. In figure 1 the system is dilute but the size of a chain greatly exceeds the spacing of a chain.' Here, V is the system volume, N the total number of monomers in the solution and v is the monomer volume (= excluded volume). He continues, 'The mean spacing of chains is (V/n)1/3, so one has the regime of figure 1 provided that (L^) 1 / 2 > (V/n)1/3. This regime will be characterized by a mean density with fluctuations.' Here, n is the number of polymer chains in the system. The difference between this regime and a more concentrated regime is in the intensity of interactions among monomers; 'one can have a dense solution and almost every molecule of every chain is in contact with several others. It is clear that the excluded volume parameter described above is quite inadequate to describe this system, and it will not be attempted in this paper.' As is mentioned above, the starting point of the Edwards solution theory is a minimal model (a model that captures all the universal features of the system being modeled with the number of parameters exactly the same as that in its phenomenological description), i.e., the many-chain version of the Edwards model. The canonical partition function of the chains is given by

with

where N$ is the (bare) polymer 'length' of the a th chain, ca(r) (r € [0, TV"]) is the conformation of the a th chain with r being the contour variable, VQ is the (bare) excluded-volume parameter, and n is the total number of chains in the solution. D is the 'uniform measure' on the (continuous) conformation space, and should be understood meaningful only when it appears together with the first 'spring' term of the Hamiltonian [these equations correspond to (2.5) and (2.6) in the original paper]. We understand that self-interaction is removed by placing a cutoff in the contour variable. Now, Edwards expects that individual chains are not good descriptive units but that the monomer concentration fluctuations have to be good collective coordinates for such systems. Therefore, the many-body version of the Edwards model defined above is rewritten in terms of a new field ?/> as

Edwards Goes Beyond Dilute Solutions

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with

where {• • • )^ is the expectation value with respect to the Gaussian random field ip with mean zero and covariance given by

In rewriting (8.3) to (8.5) an infinite constant is removed by shifting the origin of the free energy [(2.11) in the original paper corresponds to (8.5) above]. 'This trick is equivalent to the situation in electrostatics where one can discuss the interaction of point charges by having a (Coulomb) potential between them, or by saying they interact with an electric field which satisfies Poisson's equation.' The above formula may be rewritten as

where PQ is the (bare) polymer length density distribution function describing the polydispersity of the solution, and the density distribution of the Gaussian white random field 1(1 is explicitly written [this corresponds to (2.18) in the original paper]. Splitting 1(1 into the spatially uniform part ifiQ and the rest, ^\ = 1(1 — ifiQ, and performing the average over ?/>o, we maY further rewrite this as

Here, V is the system volume as in the above. If the concentration fluctuation is not large then we may expand G(No, V>i) in powers of ^i as

where { { • • • } } implies an average over polydispersity, and SQ is the static form factor of the unperturbed chain (Wiener chain), given by

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Coarse Grained Approach in Polymer Physics by Oono and Ohta

Now, we can perform the Gaussian average over ?/>i in (8.9) to obtain the free energy as

This is the starting point of all the calculations and is essentially (3.10) in the original paper. Let us continue to follow the original paper. Edwards points out that the term containing the integral over q does not converge at large |q|; 'the cause being traceable to the fact that... the self-interaction of the chain is included. This term is one for which the pseudo-potential term is inadequate,' and the correct form of this divergent term is 'obtained by allowing an improved accuracy in v ( r ) for single chains', as seen from the average (denoted as {• • • }o) of the binary interaction over the unperturbed chain that can be written as

where is the Fourier transform of the binary interaction potential. This is a constant VQ for the pseudo-potential and the q integral is divergent, as noted, but for the real potential due to the short length cutoff, the integral converges. Thus, the divergent term in the free energy should be correctly written as

The first integral converges, and the regularized last term does not alter the equation of state, because 'the self-interaction of the chain is independent of V as one would expect. . . '. This way, Edwards completed his computation of the free energy. Differentiating the free energy with respect to V, we can obtain the osmotic pressure TT in three dimensions as [(3.18) in the original paper]:

where c is the polymer number density. 'The ckpT represents the pressure of c units, in this case the chains.' The second term is the contribution expected from 'TV units whose total effective volume is' v^cN. (The notations are changed from the original.) 'The fact that there are chains appears in the last term.' This term is proportional to (ro/£) 3 , where TO is the average distance between the centers of mass of polymers, and £ is the correlation length of the monomer

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concentration fluctuation (see below) . The last term is due to the connectedness of the chain, but this 'is modified by the interference of other chains getting in the way, and £ represents that distance over which' the enhanced presence of monomers due to connectedness would be expected. Thus, the idea of the screening of the monomer-monomer interaction is introduced, as is explicitly computed in Section 8.4 of the original paper. We can read off the screened interaction (its Fourier transform) result from (8.9) and (8.10) as

We can accurately approximate the unperturbed form factor (8.11) as

where the overline implies the number-averaging, and /x = Mn/Mw (Mn: the number-averaged molecular weight; Mw: the weight-averaged molecular weight) describing the polydispersity. Therefore, the effective interaction (8.16) reads (for the monodisperse case)

implying £ ~ (cvoN^1/2. The last term in (8.15) is indeed proportional to

8.4

Semidilute solutions

About ten years later, after de Gennes realized that the Edwards model is the spin-dimension-tends-to-zero limit of the 4-model, des Cloizeaux translated the Edwards solution theory in the grand canonical ensemble formulation into the 4-model (des Cloizeaux 1975). He could obtain the scaling law from the renormalization-group theoretical result of his formulation:

where v is the exponent appearing in the end-to-end distance of a self-avoiding chain (R"2} ~ N"2" and F is a well-behaved function. When the variable X oc cNvd was large, he found the asymptotic relation:

X is a parameter called the overlap parameter, and is proportional to the number of polymers that overlap with a given polymer (or the number of centers of mass of polymers within the range of a given polymer spread).

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Des Cloizeaux remarks that Edwards' result is incompatible with the scaling law given above. If we assume that Tv/kpTc is a universal function of X, and that for large X the osmotic pressure should not be dependent on the molecular weight (dependent only on the monomer density p = cN), we must conclude (8.20), so there is no doubt about the scaling law. The field theoretical approach is very powerful, but, generally speaking, if one wishes to understand the properties of a single chain in the solution, or if one wishes to consider the polydispersity effect, it is rather awkward (the molecular weight distribution of the grand canonical ensemble of polymers is exponential). In contrast, as is clear from the formulas in the previous section, any molecular weight distribution can be studied in the Edwards formulation. Also contourvariable-dependent interactions that appear in, e.g., copolymers can easily be incorporated. Thus, the formulation of renormalization group approach using the real chain picture (the so-called conformation space formalism) becomes indispensable. [However, it should be noted that the motivation for devising the formalism (Oono 1979) was to study polymer dynamics, because the field theoretical formalism is incapable of treating dynamical quantities; a related, systematic formalism by des Cloizeaux (1980, 1981) was, we guess, with the same motivation.] Also we should not forget a subtlety that in the n-component spin model one has to deal with the apparent singularity due to the Nambu-Goldstone mode before taking the n —> 0 limit. It is clearly recognized by Edwards that if the monomer concentration is too large, then his theory does not apply. Up to how large a monomer concentration can we use his formalism? He says the upper bound of the concentration is given by Ncv < 1, where v is the monomer volume. However, if the monomer concentration is not ignorable, i.e., unless Ncv
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growing into a unified theory of singular perturbation methods for differential equations (Chen et al. f996; Oono 2000). For the first author, the methodology of renormalized semidilute solution theory is the direct source of this unification. To avoid the difficulty of singular perturbation, we must look for the state around which the perturbation parameter becomes dimensionless. Consequently, we must perform a double expansion in powers of VQ and c = 4 — d. That is, the so-called e-expansion is a logical necessity of perturbative calculation. Our conformation space renormalization approach (Ohta and Oono 1982) starts from Edwards' result (8.12) with d = 4 —e. From this, the osmotic pressure becomes

where ((So(k)}} may be approximated as (8.17) to obtain analytical formulas explicitly. In the (4 — e)-space without cutoff the integral does not exist. In the actual calculation, dimensional regularization is convenient, but if explict cutoff is used in the calculation, the fundamental idea of renormalization may be made explicit. The essence is that the resultant equation of state is sensitive to the cutoff (i.e., microscopic details), but this sensitivity may be absorbed into the parameters in the model, A^o and VQ. We must clearly recognize that these 'bare' parameters are not definable unambiguously, because the concept of 'monomer' is ambiguous even from the polymer chemistry point of view; after all polyethylene could be prepared from diazomethane, or any cycloalkane, in principle. The renormalized counterparts of TVo and VQ are the parameters appearing in the macroscopic description, i.e., the equation of state. The resultant equation of state is a function of X and if the exponent v is replaced with the known value, it agrees with experimental results without any adjustable parameter, as shown in Fig. 8.1. Michael Fisher has remarked that quantum mechanics was not needed to do cutting edge research, quoting our result, so 'one feels that if some of the giants of the past, like Boltzmann or Gibbs or Rayleigh, were able to rejoin us today, they would be able to engage in research at the cutting edges of condensed matter physics...' (Fisher 1988). The conformation space renormalization procedure (Oono 1985) enables us to evaluate other quantities accessible experimentally. The correlation length of monomer density fluctuations has been obtained in the semidilute regime (Nakanishi and Ohta 1985), which is in good agreement with experiments. The scattering function has also been calculated up to order e = 4 — d (Nakanishi and Ohta 1985). However, its agreement with experiment (Noda et al. 1983) is not satisfactory for large values of the wave number. The origin of this discrepancy has not been clarified for two decades.

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FIG. 8.1. The osmotic compressibility of the semidilute solution as a universal function of the overlap parameter X (Ohta and Oono 1982). Notice that the relation between X and the polymer concentration is given by a universal proportionality factor that can be computed with the aid of the RG theory. Therefore, there is no adjustable parameter in this comparison of the theoretical result and the experimental result (Noda et al. 1981). 8.5

Block copolymer melts

A diblock copolymer is a polymer consisting of a sequence of one type of monomers, A, joined chemically to a block of another type of monomers, B. When these two blocks are incompatible, they tend to segregate each other at low temperatures. However, because the A and B blocks are joined together covalently, macrophase separation is impossible. Instead, various mesophases are formed, depending on the block length ratio and on the temperature. Microphase separation of block copolymers has been studied extensively for more than two decades (e.g., Matsen and Bates 1996). Edwards investigated statistical properties of a single diblock copolymer with the mean field quite similar to the one employed in his polymer non-dilute solution theory discussed in the preceding section (Edwards 1974). He showed that there would be a phase transition (microphase segregation) that makes the polymer look like a dumbell. Helfand (1975) developed a similar approach for copolymer melts, which is now called the self-consistent density functional theory. Although its dynamical version is also possible, we believe the approach along the line explained below is more flexible and easier to use than the self-consistent method as is illustrated in the following. Here, we are interested in the melt consisting of block copolymers. As mentioned in the preceding section, when we deal with concentrated solutions (solutions with a nonzero monomer concentration), we have to take into account the real excluded-volume effect. In this sense, the delta-function expression of the

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135

monomer interaction in the original Edwards Hamiltonian is inappropriate for this problem. However, we must correctly understand the essence of the Edwards model. It is the energetic Boltzmann factor e~E, where E is the conformational energy, averaged over the measure on the phase space. In Section 8.2 we have discussed that the ideal state of a polymer chain should be the self-avoiding chain. The same spirit may conveniently be followed here. The measure that averages the above Bolzmann factor should be interpreted as the configurational probability measure of the hard-core liquid. Thus, the energy E has only to take the attractive interactions and the harmonic spring describing the connectivity of the chains into account. Following this spirit, we can write down the model Hamiltonian (8.4) for A — B type diblock copolymers as

where is the difference between the monomer density of A and that of B and W is the (free) energy density due to the monomer composition (but the hard-core exclusion effects are assumed to be in the fundamental measure of the configuration space). We assume that the system is a polymer melt, so the system is incompressible and the total monomer density is almost constant throughout the system. Therefore, only the monomer composition, or , matters. The relation between the conformation variables Q and may be formally written as

where

with / being the monomer fraction of A. To proceed further, we wish to construct a field theory for only. Perhaps the best way is to follow the Edwards strategy used in his solution theory, already discussed above. We obtain an effective free energy functional for micro-phase separation (Ohta and Kawasaki 1986, 1990):

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Coarse Grained Approach in Polymer Physics by Oono and Ohta

where a is a positive constant, stands for the spatial average of , and

with a vanishing boundary condition at infinity, where A is the Laplacian. The function W has been redefined in (8.26) such that all the local entropic contribution from a single chain has been absorbed into W. The functional form of W is determined by the microphase separation diagram. At higher temperatures the system is uniform, but at lower temperatures, the A and B monomers tend to phase-segregate. Therefore, W must exhibit a pitchfork type bifurcation (the cusp singularity). Thus, if we assume that the phase behaviour is structurally stable (i.e. that small modifications of the phase boundaries in the phase diagram do not alter the phase diagram qualitatively) then we may assume that W is homeomorphic to

The coefficient g is a positive constant, and r is negative for the disordered state and is positive for low temperature micro-phase separated states. A crucial approximation in the free energy (8.26) is that the nonlocality is considered only in the bilinear terms of . However, we emphasize that the long range interaction in the last term in (8.26) is the most characteristic feature of diblock copolymers that originates from the osmotic incompressibility and causes microphase separation. This type of effective Hamiltonian was derived first by Leibler (1980), but he then approximated the long-range interaction described by G and reduced the effective Hamiltonian further to a short-ranged Brazovsky-type form. Ohta and Kawasaki (1986, 1990) were the first to realize the importance of competition between this type of long range interactions and the short range interaction (V) 2 in (8.26) as the fundamental cause of various mesoscopic structures (see Seul and Andelman 1995 for a summary). Although such a model is often derived from a microscopic model by a formal reduction method, it is very difficult to justify the procedure in a controlled fashion. Therefore, it would be desirable to have an alternative justification as well. This will be discussed in the next section. A fairly quantitative description of the phase diagram and the phase transition dynamics can be accomplished often with the polynomial free energy, if the segregation is weak. We may expect a similar thing here: the naive 4-model can even be quantitative, if the segregation is weak. The double gyroid structure in the block copolymer system is realized under the weak segregation condition (Matsen and Bates 1996), so we wish to apply the above model to this structure.1 A crucial approximation in the free energy (8.26) is that the nonlocality is considered only in the bilinear terms of . However, we emphasize that the 1 However, we should note that even if the segregation is not weak, carefully choosing the coefficients as functions of temperature could make a model quantitative, both statically and dynamically; see, e.g., Goldstein et al. (1992). Therefore, we believe the modelling with the aid of W above is quite legitimate.

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137

long-range interaction in the last term in (8.26) is the most characteristic feature of diblock copolymers: it originates from the osmotic incompressibility and causes microphase separation. It has been shown that competition of this type of long-range interactions and the short-range interaction (V) 2 in (8.26) is the fundamental origin of various mesoscopic structures (Seul and Andelman 1995). A similar free energy function can be reached via a very different route (Oono and Shiwa 1987), so we believe the above free energy is a good mathematical expression of the physics of (at least not too strongly segregated phases of) block copolymers. It is well known that the segregation of A and B forms a lamellar structure when / ~ 1/2, because the covalent bond between the A portion of the block copolymer and that of B prohibits indefinite segregation. If there is no connectedness, then the phase segregation dynamics is well modelled by the Cahn-Hilliard equation

The form factor is known to be very insensitive to W (i.e. universal) (Oono 1996). Since we know that the chain connectedness prohibits formation of large uniform domains with excess A or B, the polymer effect must be to encourage = . Mathematically, the simplest means to accomplish this is to add a linear damping term — a( — ) to (8.29):

where a is a small positive constant. Indeed, this can be obtained with F given by (8.26). The resultant equation (8.30) reasonably models the block copolymer pattern formation dynamics (Bahiana and Oono 1990). The mode expansion method, valid in the weak segregation limit, has been applied to investigate the equilibrium structures and the kinetics of morphological transitions in three dimensions (Qi and Wang 1997; Nonomura et al. 2003). A unique feature of the free energy (8.26) or the evolution equation (8.30) is that it admits a stable double gyroid structure, as has been shown numerically by Teramoto and Nishiura (2002). It is well known that a double gyroid can be expressed approximately by a level surface defined by

where s and t are the parameters. We expand in terms of its fundamental modes as

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Coarse Grained Approach in Polymer Physics by Oono and Ohta

where a» and 6j are real amplitudes and c.c. means the complex conjugate. By comparing (8.31) with (8.32), the fundamental reciprocal vectors qi and Pj for a double gyroid are readily identified. For instance, q1 = CQ(I, — 1,1) and pl = CP(2, 2,0) with CQ = Q/A/6, CP = P/(2 v / 2) and Q2 = fP 2 . Substituting (8.32) into (8.30) and ignoring the higher harmonics, we obtain a coupled set of time-evolution equations for the amplitudes supplemented by the equation for the wave number:

where h is a positive constant and f a mp is obtained from (8.26) and (8.32) as a function of the amplitudes and P

We do not write the explicit form since it is lengthy. Note that the expression (8.32) includes other relevant morphologies, as a special case, such as lamellar, hexagonal and body-centered cubic structures. The equilibrium structures can be obtained by solving the amplitude equations numerically and evaluating the free energy Famp. The phase diagram obtained in this way is displayed in Fig. 8.2, where the phase boundaries derived analytically (Nonomura and Ohta 2001) without taking the double gyroid phase into account is also entered for comparison. Figure 8.3 exhibits the formation

FIG. 8.2. Phase diagram in the (horizontal axis) and r (vertical axis) plane. The regions indicated by white circles, black triangles, white squares, and black circles are the stable phases of lamellae, hexagons, double gyroid, and body-centered cubic, respectively. The phase boundaries are obtained analytically by a one-mode approximation without considering the double gyroid.

Some Reflections on Models

139

FIG. 8.3. Formation of double gyroid for T = 2.2 and = —0.1 starting from a disordered state. of a double gyroid for a = g = h = 1, T = 2.2 and = —0.1, starting from the high temperature disordered state. The present theory enables us to study morphological transitions, such as from a double gyroid to a lamellar structure or to a hexagonal structure (Nonomura et al. 2003). The reader may question that the expansion (8.32) cannot properly describe the phase transition dynamics, because neither the spatial variation nor the imaginary parts of the amplitudes are taken into account. Therefore, spatially highly localized processes such as nucleation and growth of localized new structures cannot be investigated. The study described above is thus preliminary, but our main concern here is the domain evolution such as breakup and reconnection of domains in order to explore the possible intermediate states that appear in the course of the transitions. Note that even such preliminary studies did not exist. 8.6

Some reflections on models

The study of the block copolymer system illustrated in the preceding section is about the general structure of the phase diagram and the characterization of various possible phases. As is clearly seen, no very accurate free energy is needed. This study is essentially in the spirit of Landau. However, if we wish to understand the osmotic pressure of a polymer solution with a finite monomer concentration, the theoretical framework used above is not enough, as already mentioned. Thus, to proceed from here, we cannot avoid serious consideration about the nature of models.

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As can be realized explicitly from the correspondence to the spin-dimensionzero Landau (or 4) model, the spirit of the Edwards model is close to that of Landau's: to capture the key physics in order to understand universal (or general) features of the system being modeled. Thus, with the aid of the renormalization group ideas the Edwards model produced far-reaching results, i.e., the quantitative universal features of polymer solutions with infinitesimal monomer concentrations without any adjustable parameters. Therefore, we wish to conclude this article with a discussion of two points about Landau's approach. One is its justification and the other its practical improvement. Let us discuss the second point first. The Landau free energy is of a polynomial form as is W. Needless to say, this type of free energy with only a few adjustable parameters cannot quantitatively describe all the possible phase diagrams and phases of all the materials systems. In metallurgy, however, the polynomial Landau free energy has successfully been used to produce (semi)quantitatively reliable results with careful choice of coefficients. Then, we may expect that a small modification of Edwards' original model with the delta-function-like interaction would allow us to go beyond the semidilution limit even quantitatively. The effect of finite monomer concentration should not drastically alter the semidilute solution theory result. Perhaps the most natural way is to take the entropic effect of the real excluded volume in the k = 0 component ?/>o m the Edwards theory. The rest could be treated as in our semidilute renormalization treatment. We need one adjustable parameter that relates the monomer concentration and the monomer volume fraction. The result (Shiwa et al. 1990) is a sort of a hybrid between the FloryHuggins theory and semidilute solution theory, and seems to work; e.g., it correctly describes the polymer molecular weight dependence of the osmotic pressure. Now, let us consider the fundamental aspect. What is the nature of the Edwards model? Landau wrote down his free energy, assuming the smoothness of the free energy as a function of the order parameter near the phase transition point. However, we now know very well that such an expansion is impossible, because the free energy is not smooth. Most textbooks on phase transitions still introduce the Landau free energy by expansion in the original fashion, and then tell the reader that actually there is a fundamental flaw and that we cannot do this. Thus, fluctuation effects are taken into account. Perhaps, the method of the stochastic field used by Edwards (which is usually called the HubbardStratonovich transformation, but originally due to Kac) may be used to 'derive' the 4-theory by expanding (and truncating) the formal result. Of course, the level of this justification is almost the same as Landau's, except for its seemingly (or deceptively) exact formal manipulation. In any case, no justification of the use of a Landau type free energy is ever given, except perhaps for empirical justification. Among mathematical physicists the belief is that Landau theory will be eventually be justified by renormalization transformation of a microscopic model. If

Acknowledgments

141

we accept this, then a natural strategy to justify the use of the Landau free energy can be as follows. The bulk phases correspond to the (low codimension) fixed points with a vanishing correlation length of the renormalization group transformation; see, e.g., Goldenfeld (1992), Section 9.3.3. Therefore, mean field theory for an appropriate free energy or potential function must be able to capture all the information about the bulk phase. Consequently, such a free energy function or potential can correctly determine the relative locations of phases on the phase diagram, although it cannot reliably predict whether the phase transition is first order or second order. Suppose the phase diagram of a system of our interest is fc-dimensional. According to Landau's assumption, the minimum of the free energy function specifies the phase, so the function must have k control parameters. Let us assume (without much loss of generality) that the phase diagram is structurally stable, i.e., small modifications of the phase boundaries do not alter the relations among the phases (e.g. there is no 'four corners' if k = 2). Then, we have only to look for a structurally stable potential function. As the reader might have anticipated, this is exactly the topic of Thorn's theory of classification of singularities of gradient models (cf. Arnold 1984). k is the number of unfolding parameters. If k < 4 there are only 7 diffeomorphically different potentials. Since we are studying a thermodynamically stable system, the potential function must increase indefinitely for large variables. This implies that k must be 2 or 4. The former requires the 4-potential, and the latter (f>6 uniquely (modulo C'00-diffeomorphism). The model of the block copolymer discussed in Section 8.5 may be (best?) justified with the logic outlined here. Since the so-called applied catastrophe theory is regarded as an almost trivial abuse of mathematics, at least for phase transitions, the reader may well frown on the above argument, but note that the above argument never tries to understand the phase transition itself, in contrast to the mentioned 'abuses! Donsker's invariance theorem (Donsker 1951) is nothing but the universality claim of the renormalization group theory applied to the lattice random walk. Therefore, just as a deep justification of Landau theory is expected, the Edwards model should some day be fully derivable from the lattice self-avoiding walk in a controlled way. Needless to say, however, the most important message of the Edwards model is that correct physical insight with good taste can produce very useful models that can eventually be justified even rigorously. We clearly observe in Edwards' track record that having good taste is crucial for being a good physicist. We wish Sir Sam to enjoy extending his track record further into the future. Acknowledgments

The crucial phase of our work on semidilute solution theory took place in Pittsburgh. We wish to record our gratitude, long overdue, to David Jasnow.

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This work was supported by the Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture of Japan. References Aoki, K.-I. (2000). Int. J. Mod. Phys. B 14, 1363. Arnold, V. I. (1984). Catastrophe Theory. Springer, Berlin. Bahiana, M. and Oono, Y. (1990). Phys. Rev. A 41, 6763. Ball, R. C. (2004). Contribution to this volume. Barenblatt,

G. I.

(1996).

Similarity,

Self-Similarity,

and Intermediate

Asymptotics. Cambridge University Press, New York. Barker, J. A. and Henderson, D. (1982). Rev. Mod. Phys. 48, 587. Billingsley, P. (1999). Convergence of Probability Measures. 2nd ed., Wiley. Chen, L. Y., Goldenfeld, N. D. and Oono, Y. (1996). Phys. Rev. E 54, 376-394. des Cloizeaux, J. (1975). J. Phys. (France) 36, 281-291. des Cloizeaux, J. (1980). J. Phys. Lettres (France) 41, L151-5. des Cloizeaux, J. (1981). J. Phys. (France) I 42, 635-52. Dvoretsky, A., Erdos, P., Kakutani, S. and Taylor, S. J. (1957). Proc. Camb. Phil. Soc., 53, 856. Donsker, M. D. (1951). An invariance principle for certain probability limit theorems. Mem. Amer. Math. Soc., No. 6. Duplantier, B. (1980). J. Phys. Lett., 41, L409. Duplantier, B. (1982). J. Phys. (France) 43, 991. Edwards, S. F. (1965). Proc. Roy. Soc. 85, 613-624. Reprinted in this volume. Edwards, S. F. (1966). Proc. Phys. Soc. 88, 265-280. Reprinted in this volume. Edwards, S. F. (1974). J. Phys. A 7, 332. Fisher, M. E. (1988). Condensed Matter Physics: Does Quantum Mechanics Matter? In: Niels Bohr: Physics and the World. Proc. of the Niels Bohr Centennial Symposium (Boston, MA, USA November 12-14, 1985), p. 65. Feshbach, H., Matsui, T. and Oleson, A. (eds.). Harwood Academic Publishers, Chur, Switzerland. Flory, P. J. (1942). J. Chem. Phys. 10, 51. de Gennes, P.-G. (1972). Phys. Lett. 38A, 339. de Gennes, P. G. (1975). J. Phys. Lett., 36, L55. Goldenfeld, N. (1992). Lectures on Phase Transitions and the Renormalization Group. Addison-Wesley. Goldenfeld, N. D., Martin, O. and Oono, Y. (1989). J. Scientific Comp. 4, 355-72. Goldstein, R., Zimmer, M. and Oono, Y. (1992). Modeling mesoscale dynamics of formation of J'-phase in AlsLi alloys. In Kinetics of Ordering Transformations in Metals. Symposium in honor of Prof. Jerome B. Cohen, p. 255. Chen, H. and Vasudevan, V. K. (eds.). TMS. Helfand, E. (1975). J. Chem. Phys. 62, 999.

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Muggins, M. L. (1941). J. Chem. Phys., 9, 440. Muggins, M. L. (1942). J. Phys. Chem., 46, 151. Leibler, L. (1980). Macro-molecules 13, 1602. Matsen, M. W. and Bates, F. S. (1996). Macro-molecules 29, 1091. Nakanishi, A. and Ohta, T. (1985). J. Phys. A: Math. Gen. 18, 127. Noda, I., Kato, N., Kitano, T. and Nagasawa, M. (1981). Macro-molecules 14, 668. Noda, I., Imai, M., Kitano, T. and Nagasawa, M. (1983). Macro-molecules 16, 425. Nonomura, M. and Ohta, T. (2001). J. Phys. Condens. Matt. 13, 9089. Nonomura, M., Yamada, K. and Ohta, T. (2003). J. Phys. Condens. Matter 15, L423. Ohta, T. and Kawasaki, K. (1986). Macro-molecules 19, 2621. Ohta, T. and Kawasaki, K. (1990). Macro-molecules 23, 2413. Ohta, T. and Oono, Y. (1982). Phys. Lett. 89A, 460-4. Oono, Y. (1975). J. Phys. Soc. Jpn. 40, 917. Oono, Y. (1976). J. Phys. Soc. Jpn., 41, 787-93. Oono, Y. (1979). J. Phys. Soc. Jpn., 47, 683-4. Oono, Y. (1985). Adv. Chem. Phys., 61, Chap. 5 (i.e. 301-437). Oono, Y. (1996). Structural stability of spinodal decomposition. In Mathematics of Micro structure Evolution. Chen, L.-Q., Fultz, B., Cahn, J. W., Manning, J. R., Morral, J. E. and Simmons, J., eds. TMS SIAM. Oono, Y. (2000). Int. J. Mod. Phys., B 14, 1327. Oono, Y. and Shiwa, Y. (1987). Mod. Phys. Lett. B 1, 49. Oyama, T. and Oono, Y. (1977). J. Phys. Soc. Jpn., 42, 1348. Qi, S. and Wang, Z. G. (1997). Phys. Rev. E 55, 1682. Seul M. and Andelman, D. (1995). Science 267, 476. Shiwa, Y., Oono, Y. and Baldwin, P. R. (1990). Mod. Phys. Lett. B 4, 1421-8. Talbot, J., Lebowitz, J. L., Waisman, E. M., Levesque, D. and Weis, J.-J. (1986). J. Chem. Phys. 85, 2187-2192. Teramoto, T. and Nishiura, Y. (2002). J. Phys. Soc. Jpn. 71, 1611. Yamakawa, H. (1971). Modern Theory of Polymer Solutions. Harper & Row, New York.

9

REPRINT STATISTICAL MECHANICS WITH TOPOLOGICAL CONSTRAINTS: II by S. F. Edwards Journal of Physics A: General Physics, 1, 15-28 (1968).

145

J. PHYS. A (PKOC. PHYS. SOC.), 1968, SER. 2, VOL. 1. PRINTED IN GREAT BRITAIN

Statistical mechanics with topological constraints: II S. F. EDWARDS Department of'Theoretical Physics, University of Manchester MS. received 13th October 1967 Abstract. It is shown that the full specification of an assembly of long flexible molecules, needed for a statistical-mechanical study, requires an infinite set of topological invariants, and the first two of these are derived in detail. It is argued that these invariants provide a better description of the topology of the system than a more intuitively obvious one, for example, to state the condition that a molecule contains a single knot is very complicated requiring an infinite number of invariants, just as the specification of a function at a point requires an infinite number of Fourier coefficients. It is shown that the probability of molecules taking up configurations with given values for the invariants is a problem in quantum field theory, and that for example the first invariant leads to a formalism isomorphic with the electrodynamics of scalar bosons, and the governing differential equations for one and two molecules are derived. The transition from a real polymer to its representation by a continuous curve leads to divergences, but these can be absorbed by renormalizing the step length and entropy per monomer; within these two changes the topological properties are independent of monomer structure.

1. Introduction In the previous paper (Edwards 1967 a, to be referred to as I) a discussion was given of the effect of topological constraints on the statistical mechanics of long chain molecules and the discussion was illustrated by examples where the constraint was caused by the configuration of the molecule relative to some given curve in space. This is an incomplete discussion since a path in space can have an invariant topology relative to itself, and also to obtain topological invariance of one curve relative to two or more others requires a knowledge of the ordering of the entanglements. This also leads to a more complicated set of differential equations when the curves in question are Brownian motion paths. In this paper we shall try to produce a set of invariants in terms of the intrinsic equations of the curves which serve to define topological characters. Then the probabilities of these characters being taken up will be shown to satisfy certain differential equations. The situation is much more complicated than that of I, but, nevertheless, it appears that it must be resolved before any really complete theory of polymers can be attempted. 2. The specification of knots To have knots at all, that is configurations of a curve falling into a certain class which cannot be transformed into other classes (i.e. different knots, including no knot), one must have infinite or closed curves. To start the discussion consider two unknotted closed curves. Two possible configurations are

Figure 1.

15

146

16

S. F. Edwards

Let the curves be i^s) and t2(s) where ^(Lj) = r^O) and r(L2) = r 2 (0), and let ia = ri —r2- Let us consider

r

which, using Stokes' theorem,

But curl curl = grad div — V 2 >

an

d from Green's lemma

whereas

Now

algebraic number of times the curve r 2 passes through the surface S^ with i^ as perimeter, n, say.

Therefore

For example, figure l(a) has n — 0, figure l(i) has « = + 1, according to the sign ascribed tor 1; r 2 , i.e.

Figure 2.

have opposite signs of n. When one has an explicit integral constraint one can start asking for the probability, total entropy, etc., with the constraint integral having a particular n, Now, say one had a closed curve in the configurations

Figure 3.

If one evaluates

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essentially a self-integral version of the previous form, one obtains different values for (a) and (6). For 3(a) one can deform it into a circle when r(%) x r(s2) is at right angles to r(sx) —r(s 2 ), so the value is zero. 3(6) can be deformed into figure 4, and the two strips 13, 24 can

Figure 4.

be brought together to cancel, giving the final value 47r. But it is also possible to have the value — 47T, since if we consider the two curves

Figure 5. these give equal and opposite values. If both these knots are on one curve

Figure 6.

the integral gives zero, although it is not possible to unknot the curve back to 3(a). A double knot can then give the values 877, 0, —8-rr to the integral, so it is clear that, whils figure 3(«) implies zero for the integral (2.6), the converse is not the case. To understand this, let us return to the case of separate closed curves which have no self-knots. For three such, consider the equivalent configurations

Figure 7.

These are known as Borrowmean rings. The integral (2.6) vanishes for each pair, and if any one of the curves is removed the remaining pair are indeed free. But the three together are clearly locked together, so some new integral has to be found to express this in addition

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S. F. Edwards to the previous pair integral. The situation here is clarified by considering an equivalent two-dimensional problem. Let us consider two of the rings expanded infinitely to be straight lines which are twisted until parallel. The third ring (which is now topologically a square lashing) can now be shown in projection on the plane perpendicular to the lines thus:

Figure 8. The angle swept out by the whole curve around either point is zero, but the curve is clearly entangled with the points, i.e. cannot be removed without one of the points passing through the curve. Clearly the order in which the points are circumnavigated is giving rise to a new topological invariant. A similar situation can arise in which a non-zero angle is swept about the one point, whilst zero about the other, with entanglement:

Figure 9. To pursue this study further in two dimensions it is evident that the appropriate tool is the study of functions defined over Riemann surfaces joined along cuts linking the various points about which the topology is defined. For example, figure 8 leads to the study of elliptic functions (see Ito and McKean 1965). This approach does not appear useful for three-dimensional problems such as are encountered in physics, so the rest of the present discussion will aim to develop invariants in forms analogous to (2.6) and (2.7). Let us consider then a version of figure 7 in which surfaces SI and Sz containing points R! and R2 have boundaries rlt r2

Figure 10. Let r 3 meet 5j at y.lt ^ and 52 at x2, j82> and also draw on S2 a curve from a 2 to /82. Roughly speaking, one wants to be able to get the sense of r3(s) to change as it passes through S2, so that if one starts, say, at e, denoting by Bj the integral

the value

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is roughly equal to Sw. So one wants an operator placing the minus sign in front of Jjs! Bj.ffrg, and also to add 2 J^aa
where 0 denotes an arbitrary origin, say the point e. It is readily checked that this is indeed an invariant for the change in its value on choosing a different surface S2, or a non-topological change in rs(s), can be evaluated by Stokes' theorem and is zero. The expression

has the value unity from e to j32, but changes to — 1 as one passes through the surface, staying — 1 until x 2 , when it returns to unity. But to be precise one has to add the second term, which clearly leads to two circuits. It is also true that for figure 10 the value of jE1.drs is zero, so only the surface integrals remain. The problem is now to transform (2.9) into an integral solely containing rs, rlt r 2 , dr3, dr1, di,2. It is shown in the appendix that the final form is

There are, of course, many ways of transforming this expression: in particular, integration by parts will interchange the roles of 1 and 2, and in the particular case under consideration of figure 10, i.e. figure 7, 712 3 can be recast in a form with symmetry. The invariant 712,3 having a non-vanishing value, whilst 713, /23 vanish, tells one that the curve ts, whilst sweeping out a net angle zero around curve 1, is still entangled around curve 1 by virtue of its topological relation with curve 2. We shall not attempt to evaluate higher terms, since we are assured that a systematic classification of entanglements and knots does not exist, but it is clear, given any configurations, an invariant characterizing it can be found. It is also clear that such an invariant will not specify the particular situation; it needs an infinite number of these invariants to do this. The situation is like defining a function in terms of its Fourier series: one needs all the Fourier coefficients to specify it. Knots are similarly described. By considering the formula for I12 s with r1 = r2 = r3, a new invariant for a single closed curve is obtained which will distinguish between the configurations of figure 6 and figure 3(a), both of which have the simple invariant (2.7) equal to zero. But the vanishing of the first invariant and the non-vanishing of the second does not imply the configuration of figure 6. These circumstances are necessary but not sufficient. 3. The application to statistical problems For the thermodynamic properties of a system defined to be created in a completely random fashion, the specification of the topology of the system in terms which are common to one's experience is quite unnecessary. For example, suppose one is told that an ensemble of strings exists, these strings taking up random flight configurations, and are all closed. One can ask for the probability that a string contains a knot. But one could equally ask for the probability that a string has a first invariant equal to 4?r. As has been noted in the previous paper (I), the entropy change of the system under distortion is

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where pT is the probability of_a string having a topological specification T, GT the total number of configurations, GT the total number in the new circumstances (usually (3T/GT = ffjp-f since the total number of configurations is often not altered in distortions). Now, one clearly can never specify all the properties labelled T, so rather than attempt the impossible, which is to label knots and entanglements in the simple intuitive way, one should do so by using the set of invariants developed here. Thus if the first invariant is used it will forbid a pair of infinite molecules going from

Figure 11. but will not forbid the change to

Figure 12. The former ought to be the most important effect, but should the latter also matter, then the second invariant can be used, and so on. Another reason for preferring the system of classification by invariants is that treating a polymer, for example, as a random path clearly must fail at small distances when the precise molecular structure dominates. It appears from the calculations of I and those below that the invariants permit one to separate the short-range behaviour from the long, just as the renormalization programme in quantum field theory separates self-energy effects from the dynamic behaviour of the electron in an electromagnetic field. It is not clear, however, whether the question of whether a random path contains a knot is at all meaningful in the mathematical idealization of infinitesimal steps. One would guess that such questions are not meaningful, getting into unresolved, perhaps unresolvable, questions of measure, i.e. the probability of a single knot is always zero since a random path permitting infinitesimal steps will be 'unfinitely knotted". The invariants, however, appear to be meaningful by explicit calculation. Presumably very high-order invariants will depend more and more on effects over very small path lengths, and these of course will be blotted out by physical requirements. We have no proof, however, of these statements. The following explicit calculations will discuss how equations can be deduced to give, for example, the probability that two random paths become entangled in the sense that the invariants take on specified values. The problem of one random path with topology relative to fixed given curves was discussed in I for the first invariant and was found to reduce to the solution, of a differential equation. The problem of the topology of a single chain relative to itself amounts to the constraint

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being added to the probability of the configuration. Thus

where Af is the normalization of the Wiener integral. The values taken by &l4r? will be integral for closed curves and differ by integers for paths going off to infinity. One can bring (3.4) into the form of a single s integral by a parametric representation. Let us consider a vector field variable !Jk such that k.tj k = 0, then it is an identity that

where JS£ = JIT d^k represents the integral over all the functions £h. The identity is proved by writing 2Jk = ? h '— e ^ _ h x k & ~ 2 , and invoking k.£ k = 0. In terms of the Fourier transforms of ?k andef> k , div £ = 0 and

If one takes one has

Under the Sc integral one has a Markov process, one integral in ds, so the path integral becomes a differential equation as in I. Thus

where satisfies

and W(£) is the weight factor exp( —z'AJtJ. curl Z, dsr), normalized as in (3.8). At this point it is convenient to look at the series for G;, in terms of £. This appears linearly and quadratically in the differential equation, so a convenient graphical repre-

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sentation is to represent by a full line the solution G 0 of

to represent z'A [V, ?] by a dotted line emerging from a full line, i.e. is represented by

and to represent A2?2 by two dotted lines emerging from a full line: A 2 G 0 £ 2 G 0 is represented as

The effect of integrating over !J is to link up the dotted lines into closed dotted lines starting and ending in full lines

The series for G^([Q) is then represented by

and upon integration over ij by

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Written out explicitly one has

Readers familiar with quantum field theory will recognize the graphical series as that encountered in the quantum field theory of charged scalar mesons. They will also not be surprised to find that the first diagrams are divergent. In electrodynamics the dotted line represents as against our and the full lines stem from the Klein-Gordon operator where m is the mass of the meson. Since the length here represented by s may be replaced by Fourier transform, i.e. let

This propagator is very close to that of the meson. However, quantum electrodynamics works in four dimensions, space and time, whilst here there are three dimensions. Now, in electrodynamics the divergences are not understood, but their influence has been circumscribed by the renormalization theorems. The idea of these theorems lies in the fact that the basic quantities appearing in G0 will not be those observed in physical situations. Thus the pole of G0 comes at &2 = m2. The physical mass is defined from the pole of the complete G, the equivalent of the present GA. Thus if one formally writes the solution in meson theorv as and defines

then and the pole of G' comes now at m' and there is no contribution to the residue from £'. The change m to m' renormalizes the mass, from G to G' the wave functions, and there is also in meson theory the need to renormalize the charge, here X. The apparent nonlinearity of transformation (3.18) is rather deceptive; in practice, one just has to throw away all divergences and this proves an easy matter in the perturbation expansion of S. In the present problem it all goes through. The renormalization of m is mainly a change in effective step length, and of G amounts to a change of the entropy per unit length of the chain. One does not have to renormalize \ since graphs of the typ

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which do appear in electrodynamics, do not here. (Something like them appears in polymerized material, but it is hoped to consider that in a later paper.) This is as well since A is intimately associated with integers as can be explicitly shown (see I), but not here. The proof of these statements in electrodynamics was first given by Salam, the key point being to order the diagrams and study the order of divergence of the diagrams. Now, for every Fourier integration in a diagram there will be a dotted line and thus a d3kkKk~2 times factors from the G's. In electrodynamics there is d^kk~'2 times the same factors from G's. Thus, although the present D is different and the dimensionality is different, the contribution from each dotted line is still k dk and, since the diagrams are the same, Salam's proof will hold. There are, of course, no divergences in the study of a real polymer system's topology. The divergences found here are a consequence of assuming the monomers being infinitesimal, and by cutting off the offending integrals at a distance of the order of a monomer size one can hope to make it all physically sensible. But the theorem goes much further than this. It states that the effect of finite monomer size appears in an effective step length and an effective entropy per link. There are no further effects on the asymptotic behaviour of the topological properties of the polymer. Although one is now in a position to go on to calculate the probability of a closed Brownian path to have a certain value for its invariant, the problem of two paths interlocking seems of more interest, and so will be considered in a little more detail in the next section.

4. The entanglement of a pair of molecules The diagrammatic analysis of the previous section can be used also to discuss two or more molecules. The simplest diagrams are (excluding those already given above)

and interferences

The order of the diagrams is shown by A, but these now must carry indices. Diagrams (a) ... (/) refer to simple entanglements where the second molecule does not get involved in knots in the first and vice versa; this arises in diagrams (g), (fi) and higher order diagrams, whilst knots not involving the other molecules are given in the series in the previous section. For simplicity it will be assumed that the knottedness of the separate molecules has been absorbed as before, and the 'Lamb shift" diagrams (e), (/), etc., will be ignored. So one is now discussing the probability of the entanglement of molecules which are not selfentangled. The literature of electrodynamics carries an extensive discussion of these diagrams since they occur in a study of the relativistic theory of energy levels of, say, an electron-positron bound state, positronium. Following the work of Schwinger, and Bethe and Salpeter (1956), one can rearrange the diagrams so that there are no repetitions in the sense that (b) is an iterate of (a). So, let us introduce & satisfying

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where

and ^(rlt r t '; fj, r2") 's chosen to give the diagrams

taking it to order A2. The first two diagrams will contain 8(1^ — r/^Sfi^ — r2"), etc. This rearrangement is such that, should one of the molecules be turned into a definite rather than Brownian path, for example, if G0 is replaced by the function appropriate to a straight line, the exact differential equation of I is obtained. But because both paths are Brownian the present infinite series is needed. In electrodynamics A plays the role of charge, and it is shown in the references cited that the problem is resolved in weak coupling by basing the calculation on 'S derived from the simplest approximation to J-. In the present problem one has to integrate over A, so that there appears no very good reason why the series for J should converge. Nevertheless, if one asks for the probability of high entanglement, i.e. large•&, small A, and imposes boundary conditions on ^(i^, r^; r2, r 2 ') that r^, r r are far from r2, t2', for that is compared with the natural size (L/)1 '2, then all the higher terms are small compared with («) and (/3), since they all have dimension r~ 4 , but the higher terms all contain G0's which yield exponential factors at large separation. Outside this condition we have not made much progress. Assuming then that the series for Jf is good, one can further simplify by working at a particular problem. Writing 3? in terms of one can introduce

Further, one can take an average over lengths equivalent to studying the Laplace transform

Under these circumstances diagrams (x), (y), (8) and (e) all vanish, for example (a) is

whereas (/3) is A 3 5~ 4 , so that (absorbing constants like IJ6, etc.) This equation describes the mean behaviour of the simple entanglement of two chains, provided that their separation is not small. This is the simplest situation we have found, and from now on the treatment of T follows as in I, except that being a fourth-order differential equation it cannot be expressed in explicit terms, though there is no singularity in the equation and approximate solutions are readily obtained at small and large S. It is hoped to return to quantitative solutions in a later publication. It is perhaps worth noting finally that, if the higher invariants were required, as would be noticeably the case when three or more molecules were highly entangled, the corresponding interactions would lie outside electrodynamics and enter a class which have only

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been speculated upon in meson theory. For example, one would have things like

5. Conclusion This paper has done little more than set up the problem of the entanglement of random walks. But, given a technique for constructing invariants, and given the probabilities expressed in perturbation theory and in the various summations explored already in quantum theory, one is in a position to assess the difficulties involved in accurate solutions to entanglement problems. Problems of real polymers have other questions as well as these, and cruder methods will probably be needed to handle them (e.g. Edwards 1967 b) ; nevertheless, it is important to have an exact background against which to judge the accuracy of any approach. Acknowledgments I should like to thank Professor J. F. Adams for a conversation on the present state of the classification problem, Professor M. Gordon and Professor S. Prager for an interesting discussion of this work and Professor Prager's own approach (Frisch and Prager 1967), and to Professors Allen and Gee for helpful discussions. I should also like to thank Dr. M. A. Oliver for a detailed reading of the manuscript. Appendix The invariant was obtained in (2.9) in the form

where

and the curve r 3 intersects the surface S2, whose perimeter is r2 at the points y,2, /32. Points in S2 are denoted R2. To write this in a manifestly invariant way, it should not refer to Sz at all. To do this we note that

Then

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Now, by Stokes' theorem

and if

whilst

where r 30 is the origin of the di3 integral and, being a constant, is removed since jBi.dts = 0. This can now be rearranged to

Returning now to j"s d'R2.B1 one can always arrange that the surface S2 does not intersect some surface Slt which can be taken as making single valued a scalar potential representation for Bj. Thus, if one writes

Using this representation also in (A10), and also changing the coordinate system of rs to r 3 + R 2) the two integrals become

If we now invoke the vector identity we may rewrite this expression as

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But B! = Vi, so, returning to the original coordinate system, one has

which by Stokes' theorem yields

Finally, one obtains

since in our example $B1.drs = 0. This form can be put in a more symmetrical form by integration by parts, and also by constructing the other invariants by permuting 12, 3 and taking the various combinations of them. Clearly the same method can be used to construct invariants of an arbitrarily high order, corresponding to more complicated topological situations. References BETHE,H. A., andSALPETER, E., 1936,Handb.Phys.,XXXV/18 (Berlin: Springer-Verlag),p. 175 et seq. EDWARDS, S. F., 1967 a, Proc. Phys. Soc., 91, 513-9. 1967 b, Proc. Phys. Soc., 92, 9-16. FRISCH, H. L., and PHAGER, S., 1967, J. Chem. Phys., 46, 1475. ITO, K., and McKeAN, H. P., 1965, Diffusion Processes and their Simple. Paths (Berlin: SpringerVerlag).

10 NOTES ON 'STATISTICAL MECHANICS WITH TOPOLOGICAL CONSTRAINTS: I & IF Edward Witten School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey 08540, U.S.A. Sam Edwards pioneered the application of quantum field theory methods to condensed matter physics. He helped introduce many important techniques that shape how we approach the subject today. The resulting interaction between particle physics and condensed matter physics has been important for particle physics as well as for condensed matter physics. It has enhanced our understanding of Quantum Chromodynamics as well as our approach to other strongly coupled field theories whose study might shed some light on nature. The relations between condensed matter physics and particle physics seem obvious today, but that was not so when Edwards started. Here, however, I will concentrate on his two papers, 'Statistical mechanics with topological constraints: I & II] (Edwards 1967, 1968) and especially the second in which he considers the possible knottedness and entanglement of long molecules. I will approach these papers just from one vantage point, which is to relate them to subsequent developments involving the physics and mathematics of knots. Edwards proposed that invariants characterizing knottedness would give approximately conserved quantities in the dynamics of long molecules. Given two disjoint knots in three-space, which I will label as C\ and Ci, the simplest invariant describing how they may be intertwined is the Gauss linking number I(Ci, GI\ which Edwards in paper II [eqn. (2.1)] writes as follows:

The integral runs over the product C\ x C?. By a knot, I mean simply a closed loop in three-space with no self-intersection. It may in fact be unknotted, in which case we call it the trivial knot. As Edwards explained, this is only the simplest measure of entanglement of two knots. Many others have been discovered, and in the last couple of decades the study of such invariants has accelerated as a result of the discovery of new invariants (associated with the Jones polynomial of knots and its generalizations)

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that have roots in statistical mechanics and quantum field theory. The fact that statistical mechanics is useful for the mathematical study of knots is surely in keeping with the philosophy of much of Sam Edwards's work! Edwards goes on to consider the case of an invariant characterizing a single knot rather than a pair - here he considers the Gauss linking integral written above for the case that C\ = C^. The integral needs some regularization because of the singularity at ri = r2, and Edwards proposes some natural values of I(C, C) for knots with certain types of projection to the plane. This singularity of the self-linking number has played an important role in subsequent mathematical and physical work, and I will briefly mention how various authors have approached it. What has become the standard mathematical approach is to define the selflinking number not for a 'bare' knot C but for a 'framed' knot. A framed knot is a knot C together with a normal vector field to the knot which can be considered as a recipe for displacing C slightly to a new knot C' that does not intersect C. Let us call the framing /. Then, writing If(C) for the self-linking number of the knot C together with the framing /, we define If(C) = I(C,C'). This makes it explicit that the self-linking number is not a true topological invariant (in constrast to the linking number of distinct knots C and C1) but depends on the choice of framing. I believe that Edwards' discussion in paper II is in fact closely related to this mathematical notion of framed knots, since he presents his knots via diagrams that indicate a possible projection of a knot to the plane. Such a projection comes with a choice of framing; in fact, a vertical displacement of the knot gives a framing that is not a topological invariant of the knot but is natural once a knot diagram (such as those in paper II) is given. In general a knot, presented in the abstract, does not have a natural framing. If one framing of a knot C is given, any other framing can be obtained by rotating the normal vector field that defines the framing by an angle that varies from 0 to lim (for some integer n) as one goes around C. Thus, although there is no natural choice of framing, there is a natural notion of changing the framing by one unit; this is the case n = I . More recently, Polyakov (1988) offered an alternative regularization of the self-linking integral. Polyakov's main idea was to apply a 2 + 1-dimensional quantum gauge theory to condensed matter problems such as high temperature superconductivity and the quantum Hall effect. Of course, the very idea of doing something like this depends on the notion, introduced earlier by Edwards and others, of applying quantum field theory methods in condensed matter physics. Anyway, Polyakov considered a 2 + 1-dimensional gauge theory in which the Lagrangian (or at least the most important part of the Lagrangian) for a gauge field A was the Chern-Simons form

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(eJJ'fc is the Levi-Civita antisymmetric tensor. A; is a coupling parameter, which can be scaled out of the classical problem if the gauge group is abelian and only the gauge field is considered, but becomes significant when charged particles are included or the gauge group is non-abelian. Roughly, l/k corresponds to e2/he in electrodynamics.) Then, given two knots C\ and C% in space, he showed that the expectation value (§c A • c A) V He regularized the integral by a sort of pointsplitting method, and obtained an answer that is not a topological invariant but varies in a simple way when the geometry of C is changed. It would be interesting to know whether for knots that have been flattened out as in the diagrams considered by Edwards, Polyakov's regularization would agree with the results of Edwards. I suspect it might. Apart from its influence in condensed matter physics, Polyakov's paper was an important clue for my own work (Witten 1989) in which I used quantum gauge theory to offer a new approach to the Jones polynomial of knots. Basically, I considered the same Chern-Simons Lagrangian that Polyakov had used, but for a nonabelian gauge group such as SU(1). In this case, the Lagrangian has an additional A3 term and the theory appears non-linear:

But the quantum theory still turns out to be exactly soluble because of its topological invariance and relation to various soluble problems in 1 + 1-dimensional quantum field theory and statistical mechanics. For a nonabelian group, instead of the linear expression
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As Edwards noted, for applications to long molecules, the standard Gauss linking number is not enough - knotting and linking phenomena in three dimensions can be extremely complicated. In paper II, Edwards considers some concrete examples, such as the Borromean rings, that are not adequately described by the usual linking number. He then goes on to try to construct higher order topological invariants of knots and links by formulas involving higher order integrals rather than the double integral over C\ x GI in the definition of the Gauss linking number l(C\1 GI). Chern-Simons gauge theory actually sheds a new light on the construction of such invariants. From this point of view, higher order invariants such as the ones Edwards discussed are constructed by expanding the correlation functions of Wilson loop operators Wn(C) (for various choices of R and C) in powers of l/k, the effective gauge coupling of the ChernSimons gauge theory. The main novelty compared to the formulas considered by Edwards is that, because of the A3 term in the Chern-Simons action, there are interaction vertices in the bulk of three-space. From this point of view, Edwards considered multiple gauge boson exchange between different points on the knots Ci, but without gauge boson vertices on R 3 . The knot invariants that can be constructed by perturbation theory of Chern-Simons gauge theory have become the subject of a large mathematical theory. On the one hand, they give the expansion of the Jones polynomial of knots near q = 1. They are also closely related to a theory by Vasiliev of knot invariants of finite type - basically, knot invariants that can be characterized by a suitably simple transformation law under elementary moves. An approach to these invariants that follows rather closely the physical point of view about the quantum field theory is that of Axelrod and Singer (1991, 1994). An elegant and brief introduction to the gauge theory approach to knots and three-manifolds from a mathematical point of view has been given by Atiyah (1990). Finally, Rozansky and I somewhat more recently described (Rozansky and Witten 1997) another 2+ 1-dimensional quantum field theory that can be used to construct knot invariants and three-manifold invariants that are somewhat similar to those that arise from the perturbation expansion of Chern-Simons theory.

References

Atiyah, M. F. (1990). The Geometry and Physics of Knots. Cambridge University Press. Axelrod, S. and Singer, I. M. (1991). Chern-Simons Perturbation Theory. In: Differential Geometric Methods in Physics, hep-th/9110056. Axelrod, S. and Singer, I. M. (1994). Chern-Simons Perturbation Theory II. In: J. Diff. Geom. 39, 173, hep-th/9304087. Edwards, S. F. (1967). Statistical Mechanics with Topological Constraints I. Proc. Phys. Soc. 91, 513-519. Edwards, S. F. (1968). Statistical Mechanics with Topological Constraints II. J. Phys. Al, 15-28. Reprinted in this volume.

References

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Polyakov, A. M. (1988). Fermi-Bose Transmutations Induced by Gauge Fields. Mod. Phys. Lett. A3, 325. Rozansky, L. and Witten, E. (1997). Hyper-Kahler Geometry and Invariants of Three-Manifolds. Selecta Math. 3, 401, hep-th/9612216. Witten, E. (1989). Quantum Field Theory and the Jones Polynomial. Commun Math. Phys. 121, 351.

11 REPRINT THEORY OF SPIN GLASSES by S. F. Edwards and P. W. Anderson Journal of Physics F: Metal Physics, 5, 965-974 (1975).

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J. Phys. F: Metal Phys.. Vol. 5, May 1975. Printed in Great Britain. © 1975.

Theory of spin glasses S F Edwardst and P W Anderson:}: Cavendish Laboratory, Cambridge. UK

Received 14 October 1974. in final form 13 February 1975

Abstract. A new theory of the class of dilute magnetic alloys, called the spin glasses, is proposed which offers a simple explanation of the cusp found experimentally in the susceptibility. The argument is that because the interaction between the spins dissolved in the matrix oscillates in sign according to distance, there will be no mean Ferroor antiferromagnetism, but there will be a ground state with the spins aligned in definite directions, even if these directions appear to be at random. At the critical temperature, the existence of these preferred directions affects the orientation of the spins, leading to a cusp in the susceptibility. This cusp is smoothed by an external field. If the potential between spins on sites i.j is Jtis,.sj then it is shown that

where c{l is unity or zero according to whether sites i and j are occupied. Although the behaviour at low T needs a quantum mechanical treatment, it is interesting to complete the classical calculations down to T — 0. Classically the susceptibility tends to a constant value at T—>0, and the specific heat to a constant value.

1. Introduction

A dilute solution of say Mn in Cu can be modelled by an array of spins on the Mn arranged at random in the matrix of Cu, interacting with a potential which oscillates as a function of the separation of the spins. To simplify our analysis we consider the spins as classical dipoles pointing in direction s,, so the interaction energy is ./,•/>';.»> Now if, when the probability of finding a pair at points j, j is e ;j , it happens that Uyfiy =t 0 the system can show residual ferromagnetism or antiferromagnetism at sufficiently low temperatures. If U./y = 0, for the whole alloy, but still has domains in which it is nonzero, one may still construct a theory in which there are thermodynamic consequences, in particular in the susceptibility, a kind of macroscopic antiferromagne (Adkins and Rivier 1974). In this paper however we argue that there is a much simpler and overriding model, in which it can be assumed that "LJ(j :1 = 0 on any scale, and that the mere existence of a ground state is sufficient to cause a transition and a consequent cusp in the susceptibility, which is found experimentally (Canella and Mydosh 1972). There are many such states each of which is a local minimum and inaccessible from each other. This question is irrelevant to our argument. t Present address: Science Research Council. State House, High Holborn, London WC1R 4TA. * Present address: Bell Telephone Laboratories, Murray Hill, New Jersey. USA. M.P 5,5 ,

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The argument is that there will be some orientation of the spins which gives the minimum of potential energy. This orientation is such that <s,-> = 0 so the system is neither ferro- nor antiferromagnetic on any scale, nor need it be unique. Nevertheless there comes a critical temperature Tc at which the spins notice the existence of this state, and as T—»0 the system settles into the state. This physical picture is simple enough, but it requires some new formalism to express. The problem has a resemblance to problems of gellation in polymer science. When a solution of very long molecules becomes dense there comes a density at which the mobility of a molecule falls essentially to zero and the system gels. Such a molecule will still appear as a random coil, but if viewed later will be the same random coil. Thus what we must argue is that if on one observation a particular spin is s[" then if it is studied again a long time later, there is a nonvanishing probability that sp1 will point in the same direction, ie Recent observations by A T Fiory and co-workers using \i meson polarization have strikingly confirmed this qualitative change in behaviour. Above Tc there appears to be no mean magnetic field at the site of a stopped n meson, below Tc there is. At T = 0 one expects q = 1, at T ^ Tc, q = 0. The parameter q then takes the role of the mean field of the Curie-Weiss theory and we now construct the theory at the level of accuracy of the Curie-Weiss theory. 2. The mean correlation theory

To illustrate the basis of the phase change we firstly consider a single spin. The probability of finding orientation st is The joint probability of rinding $•" at one time and sj 2 ' at an infinitely remote time will be The fields that spins sj 1 1 and sj 21 find themselves in are

and

If a spin, as a function of position, is completely random, and

where « is the probability of finding a spin at a given that there is one at;'.

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Also

However if

Now reconsider (2.2) from the point of view of one spin, s^ say. Suppose that all the other spins are bundled into the fields c, so that

where & is the probability of finding the Ci's independently of any correlation caused by coupling to s ll) . which we assume obeys equations (2.10)-(2.12), and ^K a normalization. If there are a large number of s's arranged at random the £ variables can be expected to have a gaussian distribution so that

But

1. Hence

where

Finally, therefore,

as usual. As

so that q = 0, or, writing

correctly. Expanding near q = 0

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The structure is similar to the standard Curie-Weiss theory, with the proviso that, as T —» 0, q—> + 1 not - 1, whereas either root is permitted in ferromagnetism. So far we have considered the life of a single spin, and find an abrupt change in its behaviour at a Tc. Clearly if we considered the single spin on three separate occasions we would get correlation s (11 . s(2}. sl2). s(3), and more complex correlations. So far we have not related these functions to thermodynamics, and it is not clear that the q above is directly related to the free energy. In the next section a development of disordered thermodynamics will be given following the method used in rubber elasticity (Edwards 1970, 1971) which permits the calculation of the free energy. A new definition of q will be given which will be directly related to the thermodynamic functions. 3. The formulation of the thermodynamic functions

Consider a particular spin glass specified by a set of occupation numbers. We can absorb these into the definition of Jti so that there is a probability of finding a particular interaction operative, ie let P(/} is then the probability offindinga /. A particular spin glass will have a free energy &(/) defined by

where the integration allows for the probability of occupation. The ensemble free energy is then

In order to be able to perform the integrals over / and s it appears to be essential to be able to alter the order of the integration. A way to do this is to consider m systems and define an F(m) by

We consider that (3.6) can truly be evaluated for all m and continue the integral to a small m value and expand it

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Hence If, then, one can evaluate the 3miV dimensional integral (3.5) one can obtain the free energy of the system. The integral can be evaluated by the method of the previous section. Firstly, we note that whereas in the s(1), sm discussion above one had 2 21 s ji) _ S j2)- _ i (sj.^.sj ) = q and the argument works at the level of approximation in which s ; . s ( = 0. We now will have when we now employ the same symbol q, but it will follow a different definition. We take the simplest possible probability distribution for the / ft, ie where

and PO is the density of occupation.

Then

where To do the integral over the 5 we have to resort to a variational principle of the Feynman type, replacing the quartic form by a best quadratic. The replacement is then to write

So one chooses C so that

Thereupon one performs the final integral over s* and minimizes

to determine i\. Note that a term in s* .s* is not required since it is unity, and terms in st.Sj need not be included to the order of accuracy of this paper and without, of course, violating the extremal propertv of the solution. To evaluate we note that

and that

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Thus

Likewise

Let

Then

If q is now denned to be (si-' 1 . sf') then

and

Collecting the terms together

f will be a minimum with respect to variations in r\ and since q is defined solely in terms of tj, it follows that

or

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971

where higher order terms in m have now been neglected. The definition of q in terms of r\ now yields (again keeping only terms in m)

Note that dF/dq = 0 from the definition of q, and in accordance with the variational property of F. The equation for q and hence for F is not simple but we can solve it in the limits T —> Tc and T —» 0. For T —» T c , one has since

firstly the identity at q = 0 of

and then

Near 7 = 0, putting

From these results we can now calculate the specific heat and susceptibility. 4. The specific heat The free energy has the form F = — Tf, so the internal energy is

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Since F is stationary with respect to i ie q, only p0 need be differentiated in/to yield

Putting we have

This implies a cusp in C\. There is a little experimental data which neither excludes nor entirely supports this (de Nobel and du Chatenier 1959, Zimmerman and Hoare 1960, Zimmerman and Crane 1961). We are specially interested near T ~~ 0 where

Using this form one gets

which tends to a constant as T—>0. We are grateful to Dr K Fischer of (4.10). Dynamics near T = 0 always are really be properly discussed in the by Anderson (1973).

This is of course a classical result. for pointing out an error in the first calculation of the utmost importance and this region cannot present theory. A possible description is given

5. The susceptibility Theargument isagain straightforward, and if/ c is the normal paramagnetic susceptibility, one finds by adding the magnetic field to the energy and differentiating F in the usual way: Thus 7 = '/.c above T = Tc. Below T = Tc one finds, since

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and a = usual Curie constant that

The cusp is thus linear on one side but quadratic on the lower T side. From one's experience with the Bragg-Williams theory one can expect this lack of symmetry on either side of the cusp to be an artifact of the molecular field approximation employed here. The true structure will probably be more symmetric. At low temperatures one has from (3.38)

so

and is therefore independent of T as T—> 0. Note that the cusp at T = Tc is destroyed by an external magnetic field. This has the effect of altering r in sinhr[(p/p 0 )q]' 2 [(p/pu)qV 2 ' ' ] ' to in; + /J?| where \i = dipole moment/'/cT. This ensures that q > 0 for all T, being of order B2/T2 as T —> x. However since the cusp has a strong theory dependent shape we do not pursue the algebra of the cusp form as 6 increases. It will be noted that the cusp as calculated here is not symmetric unlike the experimental finding. This situation is analogous to the use of the mean field theory with thermodynamics of ordering in alloys (or the Ising model) after Bragg and Williams. The simple 'on or off theories give an asymmetry. Improvements like the Bethe-Peierls or Rushbrooke expansions redress the asymmetry to a certain extent, but exact theories find exact symmetry at the critical point. The purpose of this paper is simply to uncover the effect however and we do not attempt to apply the well known improvements to the mean field type of theory.

6. Conclusion

In this paper we have applied the simplest theory available to elucidate a new effect in disordered system physics. There are, apart from the obvious improvements required and possible in the present treatment of the phase change, several new avenues of studyopened up. Firstly the methods should be made quantum mechanical in order to give a reliable treatment near T = 0. Also, just as the mean field theory has many applications in ordered physics, the present theory will have many other applications in disordered state physics. One has already been mentioned, that of rubber elasticity which antedates

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the present work. But more generally the present approach permits the use of second quantization methods in problems which have hitherto been studied only as first quantization problems. It is hoped to return to these questions in later papers. Acknowledgments The manuscript has been critically discussed with Dr Conyers Herring and Dr David Sherrington who made several helpful suggestions. A letter from Dr K H Fischer corrected a serious error in (4.10) and made helpful comments. P W Anderson thanks the Air Force Office of Scientific Research for support under grant AFOSR-73-2449.

References Adkins K and Rivier N 1974 J. Phys.. Paris 35 C4-237 Anderson P W 1973 Amorphous Magnetism ed H O Hooper and A M de Graaf (New York: Plenum Press) p 1 Canella V and Mydosh J A 1972 Phys. Rec. B 6 4220 Edwards S F 1970 Statistical Mechanics of Polymerised Materials in 4th Int. Conf. on Amorphous Materials ed R W Douglas and B Ellis (New York: Wiley) 1971 Statistical Mechanics of Rubber, in Polymer Networks ed A J Chompff and S Newman (NewYork: Plenum Press) de Nobel J and du Chatenier F J 1959 Physica 25 969 Zimmerman J E and Hoare F E 1960 J. Phys. Chem. Solids 17 52 Zimmerman J E and Crane L T 1961 J. Phvs. Chem. Solids 21 310

12 REMARKS ON THE EDWARDS-ANDERSON PAPER P. W. Anderson Department of Physics, Princeton University, Princeton, New Jersey 08544, U.S.A. I first became aware that there was a spin-glass transition at the Amorphous Magnetism conference held in Detroit in August 1972. This first conference was not long after the center of Detroit had experienced riots, and the attendees were advised not to wander in the surrounding city; I remarked elsewhere that this conference, and a successor in Troy, New York, were in stark contrast to the sunny beaches and Alpine meadows of more favored specialties in physics. At this conference Canella reported on the work that he, Mydosh and Budnick had been doing (Canella et al. 1971), measuring the susceptibility of the canonical spin glass alloys such an CuMn and AuFe with a low-magnetic-field technique, which revealed a remarkably sharp cusp at a definite transition temperature Tc. This cusp was very nonlinear in field, rounding off very considerably at as little as 0.01 Tesla. In the discussion it was pointed out that Borg and co-workers had observed a sharp transition as early as 1970 in Mossbauer studies, and much later I found that I had actually refereed a paper by John Wheatley in my journal Physics in 1965 that showed the sharp transition in dilute CuMn via SQUID measurements. (Wheatley's paper also showed the characteristically constant magnetic susceptibility below Tc for field-cooled measurements, in contrast to Mydosh et al.'s cusp in the a.c. susceptibility.) I realized that this sharp phase transition was in complete disagreement with standard theories of what had already come to be known as spin glass, and felt that it was unreasonable to ascribe it to ordering, which was the common assumption. I puzzled over it on and off for two years, and acquired a couple of very rough hypotheses about it, but nothing publishable. This was the situation when Sam was elected to a professorship in the Cavendish Laboratory in 1974. He could not disentangle himself from the chairmanship of the Science Research Council for another year, so he mostly came into the laboratory on Saturdays, commuting down to London during the week. I came to look forward to meeting him for Saturday morning coffee in the Theory of Condensed Matter group and discussing a wide range of physics topics. He said that he needed a problem to work on during the monotonous train journeys, and asked me to talk out whatever was on my mind. When I got around to the spin glass transition his ears pricked up, and the very next week, I think, he came

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back with a sketch of the theory which is in the introductory paragraphs of our joint paper. It was my conviction that details of the interaction didn't matter much, so long as the sign was random—we didn't use the technical term 'universality' much in those days, but the concept had been around for quite a while; and I already had a notion of 'frustration' as fundamental to the cause of the phenomenon. Another thing I remember contributing was the thought that condensation would take place in the smallest extended eigenvalue of the random inverse susceptibility matrix after renormalization. His very first notes that he came back with contained the new idea of nonergodicity, of self-consistently calculating q, the average correlation between a spin now and the same spin in the indefinite future (now called our 'order parameter'). The clever way he did this, via a series of gaussian integrals over auxiliary variables, is pure Sam, and all I could do was check it—it's not my style, at all. There are other ways, as Palmer, Thouless, and I showed later. I mercifully don't remember whether the idea of nonergodicity was there in our discussions or not; certainly no formal statement was written down before those first notes. Most probably he already had the idea in his mind in some other context, and merely checked with me that it might apply to spin glass. Now we had the temperature-dependent order parameter
References

177

As I remarked, Sam had brought the replica technique from the subject of gels, and we were quite aware of its general interest in the context of disordered and amorphous materials, although actually few other cases of actual phase transitions appear. But the first inklings of a more general applicability of the Ed wards-Anderson ideas did not surface until 1977, at least to my knowledge, after David Sherrington and Scott Kirkpatrick had challenged us by working out a specific model where Edwards-Anderson (1975) should have been exact, but failed and gave nonsensical results in the low temperature phase (Sherrington and Kirkpatrick 1975). In order to investigate this paradox, Richard Palmer and I—with a crucial contribution from David Thouless—invented a different way of doing the problem (Thouless et al. 1977), the 'cavity method' or TAP, a revival of the old methods of Onsager and Bethe-Peierls in statistical physics which are reliable on a 'Bethe lattice' or Cayley tree, which the Sherrington-Kirkpatrick model effectively is. This method gave satisfactory answers for the phase transition, but in order to get the low-temperature behaviour it was necessary to work from an exact ground state at T = 0, which we thought was 'merely' a computer exercise, to be carried out on some randomly chosen instance. After Richard had struggled with this computer exercise for a while, he started beginning with our equations near Tc and gradually 'annealing' by lowering the temperature to zero. Scott Kirkpatrick (he and David Sherrington paid us a visit during this period, which was very helpful to us) also played with this method, and I think it was he who realized that our 'simple' computer exercise was actually an example of the horrendous 'NP complete' class of computer optimization theory, likely to be 'exponentially hard' in the sense of requiring time exponential in the size of the lattice. Generalizing from this work, soon thereafter he developed the method of 'simulated annealing' as a general way of attacking such problems (Kirkpatrick et al. 1983). More recently, he and others have shown that hard instances of computational problems often reflect the existence of spin-glass like phase transitions for equivalent Hamiltonians (Monasson et al. 1999). Parisi introduced the method of 'replica symmetry-breaking' which solves the low-temperature difficulties (see, e.g., Mezard et al. 1987). Armed with Parisi's methods, Yao-Tian Fu and I made the first foray of the replica method into computer science proper by obtaining new results on the graph partition problem (Fu and Anderson 1986), later generalized and improved by Wuwell Liao (1987). But at this point we have come very far from Sam's original replica method, and it may be time to turn the story over to others. References

Edwards, S. F. and Anderson, P. W. (1975). J. Phys. F5, 965. Reprinted in this volume. Fu, Y. and Anderson, P. W. (1986). J. Phys. A19, 1605. Kirkpatrick, S., Gelatt, Jr., C. D. and Vecchi, M. P. (1983). Science 220, 671. Monasson, R., Zecchina, R., Kirkpatrick, S., Selman, B. and Troyansky, L. (1999). Nature 400, 133.

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Paper by P. W. Anderson

Liao, W. (1987). Phys. Rev. Lett. 59, 1625. Cannella, V., Mydosh, J. A. and Budnick, J. L. (1971). J. Appl. Phys. 42, 1689. Mezard, M., Parisi, G. and Virasoro, M. A. (1987). Spin Glass Theory and Beyond. World Scientific, Singapore. Sherrington, D. and Kirkpatrick, S. (1975). Phys. Rev. Lett. 35, 1792. Thouless, D. J., Anderson, P. W. and Palmer, R. G. (1977). Phil. Mag. 35, 593.

13 EDWARDS-ANDERSON: OPENING UP THE WORLD OF COMPLEXITY David Sherrington Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, United Kingdom Abstract

A brief review is given of some of the riches exposed by and others germinated by the seminal spin glass work of Edwards and Anderson. 13.1

Introduction

The first spin glass paper of Edwards and Anderson (EA, 1975) showed th way to understand the existence of an unusual amorphously locally-oriented frozen magnetism in some alloys. But, more importantly, it introduced several novel concepts and new techniques whose comprehension, deep investigation and application have led to a revolution which has been almost boundless, permeating many areas of condensed matter physics, biology, computer science, economics and social science. It opened a passage through the mountains of the thenunknown to the 'El Dorado' of the modern science of complex systems, in which complex cooperative behaviour arises in many-body situations through conflicts between the desires of individual entities or small units thereof, even when those individuals or their few-body interactions are simple; the world of Complexity rather than that of complication. The authors' address on their famous paper reads simply 'Cavendish Laboratory, Cambridge, UK' A footnote gives a little more away, indicating Sam's (then) 'present address' as 'Science Research Council, State House, High Holborn, London' and Phil's as 'Bell Telephone Laboratories.' In fact, at the time this most influential work was done Sam was the Head of the Science Research Council, a full-time position with major responsibilities for British fundamental scientific research policy and funding. His seminal work was done between his Science Research Council commitments, on the train and during his 'leisure time' (and I suspect occasionally thought about during boring parts of meetings). Happily for me, he sometimes needed a sounding board in London on which to try his ideas. I was at Imperial College, we had overlapping interests and we had earlier established a good rapport when I was his student in Manchester. So it transpired that one day in the second half of 1974 he came to discuss his ideas

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for a theory and methodology for spin glasses with me. That discussion was a revelation. There had been a few earlier attempts to formulate theories to explain the unusual low temperature behaviour of metallic alloys such as AuFe and CuMn, christened 'spin glasses' by Bryan Coles, e.g. those of Klein and Brout (1963) and of Adkins and Rivier (1974), but the story that unfolded before my wonderfilled eyes and eager ears was extremely original, entrancing and convincing. I realised immediately that here was a watershed breakthrough and my mission thereafter was to understand it better, extend it and apply it. I did not, however, at that time appreciate how wide its influence would be; I suspect neither did Sam, but I have so often observed his incredible insight and forward vision that I cannot presume that. EA made several important 'quantum leaps' in their paper. They argued that a system whose Hamiltonian lacked periodicity, even over short distances on average, could nevertheless exhibit a phase transition to a state with long-lived spin freezing, albeit without periodic order. To demonstrate this they devised novel techniques that have had major impact on statistical physics, both in the stimulation of deep and consequential considerations of their implication and in application (Mezard et al. 1987; Sherrington 1992, 1999; Nishimori 2001; Mezard 2004; Parisi 2004; and below). In their new description of a phase transition without periodicity they had to introduce a new order parameter and in fact they did so in three ways, each of which led to new research directions: (i) in terms of correlations between spin orientations at different times; (ii) as the average of squares of local magnetizations; and (iii) via correlations between replicas of the system. In his chapter in this volume Giorgio Parisi (2004) explores the subsequent developments concerning overlap order parameters that these ideas engendered. Of particular note among the techniques EA introduced is their demonstration of the powerful use of a replication technique to discuss the typical behaviour of disordered systems, and especially to investigate the novel phase transition and determine its order parameter self-consistently.1 This paper changed the direction of my research. Many other scientists also soon recognized its interest and opportunities, analysed and developed them, and activity rapidly grew to an avalanche with consequences far and wide. I shall try to give a selected overview of some of these developments [see also the companion chapters in this volume of Mezard (Ch. 15) and Parisi (Ch. 14)]. The chapter by Phil Anderson (Ch. 12) also expands briefly on the history, whilst his series of Physics Today articles provides a valuable and extensive complementation (Anderson 1988, 1989, 1990).

1 Some aspects of the replica concept had been introduced earlier without self-consistency by others and it had also been considered self-consistently by Sam in the problem of rubber elasticity, but never so elegantly and with such power, nor in such a readily accessible journal.

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13.2.1 Edwards-Anderson In their approximate solution of the thermodynamics of spin glasses EA introduced almost the simplest model Hamiltonian2 that captures the essential ingredients of quenched disorder and frustration,

where the sum is over nearest neighbours on a lattice, the Jij are quenched independently distributed random interactions chosen from a characteristic distribution, and the S are classical vector spins.3 This model is now universally known as the Edwards-Anderson model and is the paradigm for spin glass theory. The model is not generally soluble exactly and EA used approximate mean field and variational methods in their analysis to obtain a new type of phase transition. Their model was beautifully minimal but also their analysis was very novel and sophisticated, employed radical new concepts and involved several new ansatze and several different approaches. 13.2.2 Exactly soluble model So I felt it would be helpful to find a modification of the model that was exactly soluble, and preferably for which their ansatze gave the correct solution.4 The part of their analysis that I found most intriguing was the use of the replica trick, firstly to average the physical observable In Z over exchange disorder, yielding an effective non-disordered system with higher-order spin interactions from which the averaged physical free energy of the original system is formally given by the linear terms in an expansion of the effective partition function in the number of replicas, and secondly to introduce a new order parameter for a self-consistent approximation. I recalled something I had learned when Sam's graduate student, exploring with him a new method of expansion in many-body theory where there was no small expansion parameter (Edwards and Sherrington 1967), itself built on his work in turbulence (Edwards 1964). I had been led to consider the use of auxiliary fields (Stratonovich 1957; Hubbard 1959; Edwards 1965) to replace interactions (Sherrington 1967, 1971) and had learned that these could easily be utilised 2

The simplest model, which has been the subject of most subsequent work, is the Ising analogue of the model. 3 EA chose the distribution P(J) to be Gaussian centered at zero, presumably to have just a single characteristic scale (the standard deviation), to exclude any possibility of periodic order, and to take advantage of the simplicity of Gaussian integrals. 4 In fact, my own first inclinations were threefold: to try to find a simpler way to express EA's observations (Sherrington 1975), to extend their methodology to include the possibility of periodic as well as spin glass order (Sherrington and Southern 1975) to accommodate the experimental observation of a concentration-dependent transition between a ferromagnet and a spin glass, and to find a soluble model for which their mean field theory would be exact (Sherrington and Kirkpatrick 1975).

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to demonstrate mean-field exactness for the infinite-ranged Ising ferromagnet [as exemplified by the work of Miihlschlegel and Zittartz (1963)], with physical correspondence provided that the exchange interaction scales inversely with the number of spins. It seemed to me that there should be a natural extension to a soluble Edwards-Anderson spin glass Hamiltonian with quenched random exchange, provided that the variance of the exchange distribution also scales inversely with the number of spins. It also seemed to me that the key conceptual ingredients would be captured by an Ising version. So I devised what is now usually referred to as the SK model (Sherrington and Kirkpatrick 1975),

where the Jij are independently distributed quenched random parameters, with mean and variance as indicated, the overbar indicating the average and the i and j running over all the spins. It was straightforward to apply the replica trick to this model and to show that if one inverts the order of limits for the number of spins, N —s- oo, and number of replicas, n —> 0, then the problem is extremally dominated and a mean field theory should be exact.5 The natural order parameters in the effective averaged system carry replica labels, and if one follows EA and assumes the natural symmetry between replicas (all quantities with only a single replica label independent of the label, all quantities with two replica labels the same for any non-identical pair) then the variationally-obtained selfconsistently determined result of EA follows (appropriately modified for Ising spins and non-zero mean exchange).6 Scott Kirkpatrick and I calculated its consequences. The resultant phase diagram looked reasonable in comparison with experiment, as also did the derived susceptibility. So at first sight everything seemed fine, with just corrections beyond mean field theory to consider further. But there was a catch; the derived low temperature entropy was negative and this could not be true for an Ising model. This observation was very serendipitous, since if we had followed EA directly and employed classical vector spins of dimension m > 1 then the resultant negative entropy would not have caused us concern and not been the clue to rich subtleties. Further work on stability in replica space (de Almeida and Thouless 1978) demonstrated that the problem lay in the assumption of replica symmetry. The first attempts at improvement allowed for one step of replica symmetry breaking (Bray and Moore 1978; Blandin 1978), albeit without obvious physical interpretation at that time. But this was not enough for the EA mean field theory/SK model. To go further Parisi devised a remarkable hierarchy for replica symmetry breaking (RSB) (Parisi 1979a,b), 5

This inversion of limits is itself not trivially justifiable, and became a source of concern, but is now believed. 6 Another novelty is that this procedure requires the maximizing of the free energy expression with respect to the spin glass order parameter, rather than the usual minimization familiar in conventional problems.

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which has proven stable against all tested fluctuations and has recently been demonstrated with mathematicians' rigour (Talagrand 2003a). Since replica space is a mathematical artefact, originally RSB was difficult to conceptualise physically, although now it has a clear interpretation (Parisi f 983) expanded upon in Parisi's companion chapter in this book (Ch. 14), that has led to much further extrapolation, both conceptual and technical. A system with a single pure thermodynamic state is replica-symmetric whereas an RSB system has several different equally acceptable pure states that are not simple global transforms of one another and that evolve chaotically as control parameters are varied. Experiments can yield either single pure-state expectation values or involve also correlations between different pure states, depending upon the the sequence of events (e.g. of cooling and application of a field). An important ingredient in yielding the complexity of RSB is 'frustration' (Toulouse 1977), the existence of conflicts between ordering inclinations, a clear feature of the EA model. Although these subtleties were fully appreciated only after years of investigation, already in their original paper EA recognised that 'there are many (ground) states each of which is a local minimum and inaccessible from each other'. A few years later, Edwards and Tanaka (1980a, b, c) calculated the degeneracy of states and demonstrated entropic extensivity over a range of energies,7 paving the way for the considerable recent interest in the configurational entropy of pure states, now often called the 'complexity.'8 The mean field thermodynamic equilibrium solution (as defined either by the mean field solution of the EA model allowing RSB or as the correct solution of the SK model) has many other subtleties, such as ultrametricity9 and non-self-averaging,10 to mention but two. For further details the reader is referred to Mezard et al. (1987), Parisi (Ch. 14) or Sherrington (1999). 13.2.3 Beyond spin glasses These concepts have been recognized as relevant to many other problems involving complex cooperative behaviour arising from the combination of frustration and quenched (real or quasi-) disorder. Some of these are, like the 7

Almost simultaneously, in a paper submitted a month later than the three papers of Edwards and Tanaka and apparently independently, Bray and Moore (1980) made a similar calculation as a function of temperature. 8 In the title of this article 'Complexity' is used more generally to describe situations in which the cooperative behaviour of a many-body system is much richer than that of the individual units and their few-body 'rules of engagement.' 9 A space is referred to as ultrametric if distances in that space obey the property that the two largest distances between pairs of points chosen from among three points are equal. In this context distance is a measure of the differences between 'pure' states, such as the Hamming distance. In the mean field theory of the EA or SK models the ultrametricity condition for three different states involves an inequality, the smallest of the three distances having the possibility of being smaller than the other two, implying a hierarchical state structure. 10 A disordered system is considered self-averaging if site-averaging and ensemble-averaging over the disorder distribution yield the same result. Conventional systems are self-averaging.

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original classic metallic spin glass alloys, within solid state physics, but the modern diaspora is much wider; covering computer science, e.g., in the guise of hard optimization studies involving best compromises with conflicts; biological sciences, e.g., in neural networks involving the simultaneous operation of combinations of excitatory and inhibitory synapses; or in social and economic sciences involving different preferences or mutually incompletely attainable goals. In these various analogous problems different microscopic entities play the role of 'spins' and others of their quenched 'interactions', whilst 'temperature' has yet other stochastic analogues; e.g., in neural networks the analogues of the spins are the neural firing rates, the analogues of the exchange interactions are the synaptic efficacies, whilst the temperature finds its analogue in rounding of the synaptic response function; in the optimization problems the analogues of the spins are the tunable variables, the exchange interactions are the conditions to be obeyed and the temperature is an artifice introduced to enable barrier-crossing, as in the now-famous 'simulated annealing' of Kirkpatrick et al. (f 983), itself an outgrowth of attempts to overcome barriers around metastable states in simulating EA and SK (Kirkpatrick and Sherrington 1978). In many cases the range-free character of SK, and therefore the mean field theory of EA, is probably more justified than in the solid state systems that originally inspired EA. In practice, however, often the effective connectivity is intensive (but random or quasi-random), more analogous to the Viana-Bray (1985) spin glass model characterised by

where the Cj are quenched randomly 0 or 1 on a dilute random network of finite connectivity c.11 This model maintains the range-free character and the need for replica symmetry breaking, but requires more than the original EA pair-wise order parameters. However, as it still has the low connectivity of a finite-range spin glass it is often considered an intermediate between the EA and SK models.12 I believe that in their original paper EA chose the distribution of exchanges to be symmetric around zero precisely to avoid any possibility of any periodic order. The possibility of periodic order can easily be restored by choosing an appropriate nonzero mean (Sherrington and Southern 1975; Sherrington and Kirkpatrick 1975). An analogue underlies several of the other complex systems of current interest. For example, in one form of data transfer a binary message is stored and transmitted as the ground state of an unfrustrated Hamiltonian [this is the case for the spin-glass model of Sourlas (1989) or for the recently re-discovered Low Density Parity Check codes of Gallagher (1962)] but transmission leads to distortion so that retrieval must be achieved from an effective 11

In the original Viana-Bray model the network connections were chosen randomly with probability c/N, so that the average connectivity is c, whereas in several other situations the individual valences of each vertex are maintained at c [cf. e.g., Banavar et al. (1987)]. 12 For finite connectivities the distribution of the J does not need an TV-dependence for normal thermodynamic behaviour.

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spin glass Hamiltonian with a displaced exchange mean that corresponds to the transmitted message; see Nishimori (2001). Another example is believed to exist in protein folding, where the bias is such as to favour the correctly folded tertiary structure.13 In these cases there is a single underlying dominant macroscopic state (albeit with possibly many locally stable microstates), but there are also examples with many retrievable macro-equilibria. One with a large number of analogues of ferromagnetic states with also frustration and effective quenched disorder is found in neural networks storing memory in global attractors, as pointed out by Hopfield (1982) and exhibited by the model with which he is associated:

where the
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recognised the relevance of temporal correlations in the first EA paper, where the first definition of a spin glass order parameter is given as

utilising his experience from polymer gels. In their second paper Edwards and Anderson (1976) also considered dynamics explicitly.18 Modern dynamical spin glass theory is usually formulated in terms of functional integrals, a methodology very close to Sam's heart and one he has utilised very innovatively and successfully in every field he has touched, since his earliest days as a graduate student with Schwinger (Edwards 1953). The first application of functional integral dynamical methods to spin glasses was by de Dominicis (1978), based on a method devised for non-disordered classical systems by Martin, Siggia and Rose (1978), itself closely related to earlier work of Edwards and Sherrington (ES, 1967) on strongly interacting quantum systems. De Dominicis was concerned with the use of dynamics in place of replicas and that has been a key theme in subsequent work, both for equilibratingand nonequilibrium systems.19 Sompolinsky and Zippelius (1982) pioneered the use of the method for stationary relaxational dynamics. A new important breakthrough was heralded when (Ted) Kirkpatrick, Thirumalai and Wolynes (1989) recognised and exploited its value and rich predictions for nonequilibrium nonstationary dynamics20 and related these to a revolutionary new understanding of structural glasses.21 It was further amplified when Cugliandolo and Kurchan (1993) showed how to solve exactly for important two-time macroscopic properties in a spherical p- spin- interacting analogue of the SK model in the limit of long times,22 18

As did SK in their second paper (Kirkpatrick and Sherrington 1978). Whereas in equilibrium statistical mechanics one employs as a generating functional the partition function, which weights microstates according to their Boltzmann factors, for dynamics one may employ a generating functional that ensures the satisfaction of microscopic dynamics. For example, schematically, 19

where the are the microscopic variables, the r\ are the stochastic noise, of distribution P ( r j ) , and the A are generating fields; the <5-function can be further exponentiated usefully in terms of auxiliary fields (f>(t), the analogues of ES's derivative fields S/S. Disorder averaging can be done directly on Z, without need of replicas as the physical limit has Z(\ = 0) = 1. Macroscopic order parameters involving multiple times may then be introduced (in lieu of replica overlaps) and the microscopic variables integrated out in their favour, albeit with non-trivial and temporally non-local self-consistency and with complex non-stationarity the analogue of replica symmetry breaking. 20 In fact they studied a Potts analogue of EA/SK. 21 In structural glasses there is no a priori quenched disorder; the system 'freezes' into a non-periodic state self-consistently and, it is believed, dynamically, although presumably not as the rigorously sharp transitions of infinite-ranged spin glasses. 22 In p-spin models the interactions have the form H =

EA Spin Glass and Beyond

187

demonstrated a breakdown of equilibrium and of the conventional fluctuationdissipation theorem beneath a critical temperature, and found its replacement by a non-equilibrating non-stationary relationship between two-time correlation and response in the lower temperature regime even at the longest times.23 The previously observed phenomenon of 'aging' was given firm mathematical flesh and the doors were opened to many further new concepts, theoretical methodologies and experimental observations. Here I shall only mention a few of these briefly, referring the reader for more detail to Parisi's (2004) and Mezard's (2004) accompanying chapters and to reviews such as that of Cugliandolo (2003). One is that in the new fluctuation-dissipation relation (FDR) for the non-equilibrating regime the ratio of integrated dynamical response to temporal correlation (which in a stationary equilibrium system is the inverse temperature) now both defines a new effective temperature and also relates to, and hence provides a means to measure, the equilibrium thermodynamic distribution of overlaps. Another is a whole new perspective on structural glasses, both theoretical and experimental, following the recognition of Kirkpatrick et al. (1989), mentioned earlier, that in suitable extensions of the original EA and SK models (in which there are local interactions with higher degeneracy of the preferred orientations of the 'spins' they connect) the dynamical 'freezing transition' occurs at a temperature higher than that at which a thermodynamic transition would occur, reminding one of the situation in structural glasses where the viscosity becomes immense at a 'glass temperature' 24 before the Kauzmann temperature at which the curve of excess entropy of solid over liquid would extrapolate to zero; this opened the door to investigate issues such as FDR and effective temperatures in glasses without imposed quenched disorder, as well as new ways to look for universality classes in aging. This situation (of a higher dynamical than thermodynamic transition temperature) is accompanied thermodynamically by a discontinuous onset of replica symmetry breaking, an observation which in turn fueled new and fruitful considerations of extensive complexity and its implications, in many contexts. Finally one might mention that these dynamical generating functional methods have now been extended much further, for example to studies of protein folding, to the dynamics of hard optimization algorithms and to idealised models in econophysics, and further application is a topic of much activity, but they present greater technical difficulties than are experienced in corresponding extensions of thermodynamic equilibrium. 13.2.5 Edwards-Anderson again The only spin glasses that are soluble, even with only theoretical physicists' rigour, have range-free interactions. But what is the real situation for short-range spin glasses such as epitomised by the EA model? It is generally believed that above critical dimensions (albeit possibly different for Ising and vector spins) 23 Shortly thereafter Cugliandolo and Kurchan (1994) extended their analysis to the softspin SK (p = 2) model, which is richer in some respects but less relevant to analogies with structural glasses. 24 Usually defined as where the viscosity reaches 1014 poise.

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true thermodynamic spin glass transitions do occur. It is more controversial as to whether the resultant spin glass states do or do not exhibit the curious features of replica symmetry breaking and ultrametricity.25 The sharp dynamical transitions of infinite-range systems are generally not expected to carry over as true transitions in the limit of infinite time, but many nonequilibrium finite-time features persist, even quantitatively, and many of the deductions, inter-relationships, analogies and alternative methodologies, developed with often-artificial models, are applicable to finite-range systems with appropriate interpretations. 13.2.6 Mathematics Although very fluent in mathematics and demonstrably able to use it very powerfully, Sam is a theoretical physicist who prefers physical insight and mathematical pragmatism to the rigour of a mathematical physicist.26 But through his work on spin glasses he has sparked a new school of probability theory and there is currently much interest in trying to justify (or correct) the sophisticated mean field theory that has grown out of EA and the work of theoretical physicists subsequently influenced by it. Rigour is difficult but significant progress has been made, particularly by Aizenman, Geurra, Frohlich, Talagrand and associates.27 So far, Parisi's ansatz is holding up. 13.3

Concluding Remarks

Sam and Phil certainly 'hit gold' with EA, in many interpretations of the expression. Firstly, literally; the experimental paper that inspired the search for a real new phase transition to an amorphously ordered magnet, as opposed to a smooth crossover to a slowed down paramagnet, was on (impure) gold, actually AuFe alloys (Cannella and Mydosh 1972). Secondly, they discovered a metaphorical gold-mine of unanticipated scientific riches of concepts and applications. Thirdly, in the replica method that Sam introduced they also found a new 'black gold'28 that has served as both a 'source of power' and a 'lubricant' for many further studies. Fourthly, research that has grown from the seed of their initiative has saved huge amounts of money in enabling more effective and less expensive methods in many application contexts.29 And the vein is by no means exhausted yet. 25

Both schools of thought have fervent disciples. In an analogy of challenging problems as mountains in our mental landscape, Sam's modus operandi is not to try to climb to the peaks but, as in the tradition of the great terrestrial explorers, to call on his unique panoply of skills, innovative vision and nous to discover passages to the lands beyond, opening up their riches. 27 See, e.g., Talagrand (2003b). 28 As oil has been called. 29 For example, through simulated annealing and other optimization analyses and algorithms, error-correcting codes, neural networks. 26

References

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Acknowledgments I would like to thank Sam Edwards for inspiration, interest and advice throughout all my scientific life; starting as a tutorial student of Sam's whilst an undergraduate, followed by supervision of my Ph.D. studies, support for my first academic appointment as an Assistant Lecturer in Theoretical Physics, all at the University of Manchester, and continuing after I moved to Imperial College and now Oxford. It will be clear from the above that Sam inspired me in the area of spin glasses, but he did so in many other topics as well. He has also been a great sounding board and a great encourager, always seeing potential sparkles even beneath the apparently dullest surfaces. And he has been a marvellous friend. I would also like to thank Phil Anderson for both overlapping and complementary support and advice, covering again a major part of my life, from much professional inspiration, advice and discussion, covering many aspects of condensed matter physics, to personal friendship and stimulating company exercised in many far-flung places (as well as ones closer to home). There are also many other colleagues I should thank for enriching my life in spin glasses and for the friendships that have grown beside our common scientific quest. There are too many to thank all, but I would particularly like to thank here my old collaborator Scott Kirkpatrick, my ex-collaborator and long-time friend Nicolas Sourlas, my two other co-contributors in commenting on EA, Giorgio Parisi and Marc Mezard, who have been constant sources of inspiration, Gerard Toulouse, whom I have known since our post-doc days and who played an important early role in the spin glass story, David Thouless, from whose book I learned my many-body theory as a graduate student and who was very influential in recognizing the problem with replica symmetry breaking and, with Phil and Richard Palmer, suggested the TAP alternative to the replica method, which was developed into the powerful 'cavity method' by Mezard, Parisi and Virasoro (whom I also thank), 30 Cyrano de Dominicis whose early papers with Roger Balian were among my main sources for functional methods in many-body theory during my doctoral studies, my many ex-students who can be represented by my co-editor Paul Goldbart, my many ex-postdocs who can be represented by Ton Coolen and Michael Wong, my many other collaborators and co-authors over the years, who might be represented by Hidetoshi Nishimori, John Hertz, Theo Nieuwenhuizen and Reinhold Opppermann, and last, but by no means least, my two long-time colleagues and friendly sparring partners, with whom I have never written a paper but regularly discussed very fruitfully over a quarter of a century, Mike Moore and Alan Bray. References Adkins, K. and Rivier, N. (1974). J. Physique 35, C4-237. Amit D. J., Gutfreund H. and Sompolinsky H. (1985). Phys. Rev. Lett. 55, 1530. 30

See Mezard et al. (1987).

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Anderson, P. W. (1988). Physics Today 41, (1) 9; ibid. (3) 9; (6) 9; (9) 9. Anderson, P. W. (1989). Physics Today 42, (7) 9; ibid. (9) 9. Anderson, P. W. (1990). Physics Today 43, (3) 9. Banavar, J. R., Sherrington, D. and Sourlas, N. (1987). J. Phys. A20, LI. Blandin, A. (1978). J. Phisique C6, 1568. Bray, A. J. and Moore, M. A. (1978). Phys. Rev. Lett. 41, 1068. Bray, A. J. and Moore, M. A. (1980). J. Phys. C13, L469. Cannella, V. and Mydosh, J. A. (1972). Phys. Rev. B6, 4220. Cover, T. M. (1965). IEE Trans. Elec. Comp. 14, 326. Cugiandolo, L. F. and Kurchan, J. (1993). Phys. Rev. Lett. 71, 173. Cugiandolo, L. F. and Kurchan, J. (1994). J. Phys. A27, 5749. Cugliandolo, L. F. (2003). Dynamics of Glassy Systems. In Slow relaxations and nonequihbrium dynamics in condensed matter. Barrat, J.-L., Feigelman, M. V., Kurchan, J. and Dalibard, J. (eds.). Springer-Verlag, New York, de Almeida, J. R. L. and Thouless, D. J. (1978). J. Phys. All, 983. De Dominicis, C. (1978). Phys. Rev. B18, 4913. Edwards, S. F. (1953). Phys. Rev. 90, 284. Edwards, S. F. (1964). J. Fluid Mech. 18, 239. Reprinted in this volume. Edwards, S. F. and Sherrington, D. (1967). Proc. Phys. Soc. 90, 3. Edwards, S. F. (1965). Proc. Phys. Soc. 85, 613. Reprinted in this volume. Edwards, S. F. and Anderson, P. W. (1975). J. Phys. F5, 965. Reprinted in this volume. Edwards, S. F. and Anderson, P. W. (1976). J. Phys. F6, 1927. Edwards, S. F. and Tanaka, F. (1980a). J. Phys. F10, 2471. Edwards, S. F. and Tanaka, F. (1980b). J. Phys. F10, 2769. Edwards, S. F. and Tanaka, F. (1980c). J. Phys. F10, 2779. Gallagher, R. (1962). IRE Trans. Inform. Theory 8, 21. Gardner, E. (1988). J. Phys. A21, 257. Hopfield, J. (1982). Proc. Natl. Acad. Sci. 79, 2554. Hubbard, J. (1959). Phys. Rev. Lett. 3, 77. Kirkpatrick, S., Gelatt, Jr., C. D. and Vecchi, M. P. (1983). Science 220, 671. Kirkpatrick, S. and Sherrington, D. (1978). Phys. Rev. 17, 4384. Kirkpatrick, T., Thirumalai, D. and Wolynes, P. G. (1989). Phys. Rev. A40, 1045. Klein, M. W. and Brout, R. (1963). Phys. Rev. 132, 2412. Martin, P. C., Siggia, E. D. and Rose, H. A. (1978). Phys. Rev. A8, 423. Mezard, M., Parisi, G. and Virasoro, M. A. (1987). Spin Glass Theory and Beyond. World Scientific, Singapore. Muhlschlegel, B. and Zittartz, J. (1963). Z. Phys. 175, 553. Nishimori, H. (2001). Statistical Physics of Spin Glasses and Information Processing. Oxford University Press, Oxford. Parisi, G. (1979a). Phys. Lett. A73, 203. Parisi, G. (1979b). Phys. Rev. Lett. 43, 1754. Parisi, G. (1983). Phys. Rev. Lett. 50, 1946. Sherrington, D. (1967). Proc. Phys. Soc. 91, 265.

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Sherrington, D. (1971). J. Phys. C4, 401. Sherrington, D. (1975). J.Phys. C8, L208. Sherrington, D. and Southern, B. W. (1975). J. Phys. F5, L49. Sherrington, D. and Kirkpatrick, S. (1975). Phys. Rev. Lett. 35, 1792. Sherrington, D. (1992). Spin Glasses. In Electronic Phase Transitions. Hanke, W. and Kopaev, Yu. V. (eds.), p. 79. Sherrington, D. (1993). Neural networks: The spin glass approach. In Mathematical Approaches to Neural Networks. Taylor, J. G. (ed.), 261—291. North-Holland, Amsterdam. Sherrington, D. (1999). Spin Glasses. In Physics of Novel Materials. Das, M. P. (ed.), 148-204. World-Scientific, Singapore. Sompolinsky, H. and Zippelius, A. (1982). Phys. Rev. B25, 6860. Sourlas, N. (1989). Nature 339, 693. Stratonovich, R. L. (1957). Dok. Akad. Nauk SSSR 115, 1097. Talagrand, M. (2003a). C. R. Acad. Sci. Parts Ser. 1337, 111. Talagrand, M. (2003b). Spin Glasses: A Challenge to Mathematicians. SpringerVerlag, New York. Toulouse, G. (1977). Comm. on Phys. 2, 115. Viana, L. and Bray, A. J. (1985). J. Phys. CIS, 3037.

14

THE OVERLAP IN GLASSY SYSTEMS Giorgio Parisi Dipartimento di Fisica, INFM, SMC and INFN, Universita di Roma La Sapienza, P. A. Moro 2, 00185 Rome, Italy Abstract

In this paper I shall consider many of the various definitions of the overlap and of its probability distribution that have been introduced in the literature, starting from the original papers of Edwards and Anderson; I shall also present some of the most recent results on the probability distribution of the local overlap in spin glasses. These quantities are related to the fluctuation-dissipation relations both in their local and in their global versions. 14.1

Introduction

A fundamental step forward in the history of spin glasses is represented by the papers of Edwards and Anderson (EA; 1975, 1976). There are many ideas in these papers that have formed the leitmotif of research during the subsequent years: the use of the replica method, the mean field equations, the overlap order parameter. In their approximate solution of the thermodynamics of spin glasses, EA introduced almost the simplest three-dimensional model Hamiltonian, the Edwards-Anderson model: it captures the essential ingredients of quenched disorder and frustration. In spin glasses, the Edwards-Anderson model (and its later modifications) plays the same central role as the Ising model does in the study of ferromagnetism: this model has become a standard both for theoretical analysis and numerical simulations. In this chapter I shall concentrate on the definition of the overlap and its subsequent evolution. I shall show how this simple and deep theoretical tool has been used in order to obtain information in quite diverse situations and how, during its evolution, it has acquired a multitude of facets. In Section 14.2 I shall recall the original definition of the overlap. I shall then show how it can be extended to the case where replica symmetry is broken and many finite-volume pure states are present. We will see how the probability distribution of the overlap controls the value of the free energy in the mean-field approximation. In Section 14.3 I shall discuss the issue of defining the overlap order parameter in a thermodynamic way.

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In Section 14.4 I shall show how the various definitions of the overlap are related to the various definitions of the magnetic susceptibility. The magnetic susceptibility may be defined in different ways, and different definitions lead to different values when replica symmetry is broken, as happens in experiments. In Section 14.5 I shall show how a thermodynamically stable definition of the probability distribution of the overlap is possible, i.e., how to define some functions P(q) such that they do not fluctuate from system to system and they coincide with the ensemble average of the sample-dependent probability distribution function of the overlap. Two different approaches may be used: (i) the study of the partition function in the presence of a small random magnetic field, and (ii) the introduction of generalized susceptibilities as the responses to many-spin perturbations. The generalization of this last approach leads to the introduction of the probability distribution of a site-dependent overlap. Finally, in Section 14.6, the last section before the conclusions, the generalized fluctuation-dissipation relations are introduced. Their relations with the overlap distribution are elucidated both for the global case and for the local case. These last relations are particularly interesting, as they lead to the prediction that in an aging system the effective temperature is site-independent. 14.2

The original definition of the overlap

14.2.1 Only one state Let us first recall some of the essential ideas in the Edwards-Anderson papers. We first introduce the Edwards-Anderson model: the spins are Ising variables (±1) defined on a regular lattice (e.g. in three dimensions on simple cubic lattice); their interaction is only between the nearest neighbours pairs:

where the sum on i and k runs over nearest-neighbour pairs. The J's are random quenched variables with zero average and unit variance; depending on the strain of the model, they may take the values ±1 with equal probability or they may have Gaussian distributions. The finite- volume free energy density is defined in the usual way:

where /?[= (k-sT)^1} measures the inverse of the temperature T, k^ is Boltzmann's constant, and V is the volume of the system, measured in units of the lattice cell volume (so that it is also the number of lattice sites or Ising variables). In the infinite- volume limit Fj becomes J-independent (with probability 1) and will be denoted by F. The value of F in the infinite- volume limit can

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also be computed by first averaging over the J (generally speaking such averaging will be denoted by an over-bar) and later sending the volume to infinity:

Now, we would like to define an order parameter analogously to that in the ferromagnetic case. Also, if we suppose that at low temperatures there is a spontaneous magnetization m^, it is difficult to mimic the steps that are used to define a global magnetization in the ferromagnetic case. In this case the magnetization density can be written as

where gi is the sign of the local magnetization in the ground (i.e. the lowest energy) state, which, in the ferromagnetic case, is given by
This equation can also be written as

where now we consider a system composed of two identical copies (replicas or clones) of the original system (
The quantity

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195

is the overlap of the two replicas, and the Edwards-Anderson order parameter can be written as

14.2.2 Many states If there are multiple states, as is usual when there is a spontaneous magnetization, one must be careful: in the previous formulae, the value of
where wa denotes the thermodynamic weight of state a. In this way each state may be characterized by the local magnetizations m(i)a such that

where G denotes the Gibbs average. We also define an overlap matrix

In principle, we could define an a-dependent Edwards-Anderson parameter,

Insofar as all the different pure states relevant under the control parameters must have the same free-energy density, one expects that, in the infinite-volume limit,
For each given sample (i.e. realization of disorder), it is convenient to introduce the function P j ( q ) , defined as the probability distribution of the overlap q[a, T] in a system of two coupled replicas with the same realization of J's 1

For the definition of pure finite-volume states, see (Marinari et al. 2000).

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(Mezard et al. 1987; Parisi 1992). This function is well defined, and can be used as the starting point for the theory without making reference to the decomposition into finite-volume pure states. However, if we assume that such a decomposition can be done, we find that

It is interesting to note that

In the usual case, i.e., when there is only one phase, the function P(q) has only one delta function. In ferromagnetic cases at zero magnetic field, where two states exist with opposite spontaneous magnetizations, the function P(q) is given by

In systems where there is a global symmetry that changes the sign of all the spins, one has P(q) = P(—q) and, in order to avoid this duplication of information, one usually uses a function P(q) that is restricted to only positive q: in this case, P(q) takes a value that is twice the original one. If we use this prescription in the ferromagnetic case, the function P(q) reduces to a simple delta function. Similar prescriptions are used in the presence of other global symmetries. In a nutshell, we can characterize the phase structure of the system by giving the function Pj(q) or the set fj of all the w's and
When the ultrametricity condition is satisfied (Mezard et al. 1987), i.e., when

if one uses the property of stochastic stability, knowledge of the function P(q) is sufficient to reconstruct the whole functional II [P\. In this case, the different states of a given sample have a taxonomic hierarchical classification, and one usually refers to this situation as the hierarchical or ultrametric approach (Mezard et al. 1987).

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14.2.3 A soluble model In order to see if the previous construction is not empty, it is convenient to consider a soluble model, where explicit computations can be done, i.e., the Sherrington-Kirkpatrick model (Sherrington and Kirkpatrick 1975), which can be formally obtained as the limit of the Ed wards-Anderson model when the number of dimensions of space goes to infinity. More precisely, this model contains V spins, the Hamiltonian has the same form as in Eq. (14.1), with the difference that the sum over i and k runs over all the possible pairs of different spins and the variables J have average zero and variance V~i. In this model it is possible to prove that the function P(q) cannot be a delta function at low enough temperature, because this hypothesis leads to contradictions (i.e. negative entropy at low temperature). The real advantage of the model is its solubility, i.e., the possibility of expressing the free energy in closed form. If one makes the usual ultrametric hypothesis, it is possible—by using standard manipulations—to compute the free energy as a functional of the function P(q). In this way, one finds (Mezard et al. 1987)

where f[/[-P] has a relatively simple form. In this way, one can find an explicit lower bound to the free energy (Guerra 2003), which has been recently proved to coincide with the exact result. (Talagrand 2003). However, the ultrametricity hypothesis has never been proved rigorously. In spite of its intricacies, it is essentially the simplest possible hypothesis, and more complicated ansatze have never been constructed. Quite recently, it was discovered that it is possible to write a simple formula for the free energy as

where /z[fj] is, as before, a probability measure over the w's and the q. This result (Aizenman et al. 2003) is very interesting: the explicit form of the function F[£l] is somewhat unusual (Aizenman et al. 2003) and will not be reported here. A detailed computation shows that if we consider the /x;/[fJ] that corresponds to the ultrametric case, we have

and we recover the well-known result (Guerra 2003) that the ultrametric construction provides a lower bound to the free energy. It is rather likely that ultrametricity is correct (although a rigorous proof is lacking) because it produces the correct value of the free energy. Although we know that a solution of the variational problem (14.22) is ultrametric, we have to prove that it is essentially unique. It is also possible that ultrametricity could be proved by a more direct approach (Franz et al. 1992).

198 14.3

The Overlap in Glassy Systems by Giorgio Parisi The thermodynamic definition of the overlap

The definitions of the overlap presented in the previous sections are nice, but they rely on the decomposition into states which, for finite-volume systems, is only approximate and is very difficult to perform in practice. The fact that the overlap enters in the explicit solution of the Sherrington-Kirkpatrick model hints that the overlap is a relevant quantity. It would be very important to define the overlap order parameter in a thermodynamic way, as we now discuss. In the ferromagnetic case this is by far the simplest approach. We add to the usual ferromagnetic interaction a constant magnetic field e, and we compute the e-dependent magnetization m(e) in the infinite-volume limit. Only after taking the infinite-volume limit do we send the magnetic field to zero; we then find that the spontaneous magnetization ms is given by

The homologous definition for spin glasses is given as follows. We consider a system of two weakly-coupled copies of the system, for which the Hamiltonian is

Let us define F(e) to be the free energy associated with this Hamiltonian. We can define the e-dependent overlap as the expectation of the overlap, computed with this Hamiltonian. It is given by

Exactly as in the ferromagnetic case, we obtain two order parameters:

We now face the problems of interpreting these two parameters and giving a physical meaning to the event q+ ^
where
On the contrary, for small negative c the e-dependent part of the partition function can be written as

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199

where
In the presence of a nonzero magnetic field there is no accidental degeneracy present.2 In usual systems, for which the equilibrium state is unique, we would have
The advantage of this overlap is that it is even in the spins and invariant under the transformation 14.4

The two susceptibilities

Using the previous definition, we can take two different replicas each in the Gibbs ensemble, at e = 0, and compute the probability distribution for the overlap. In the most interesting case, where
In zero magnetic field the configuration with r(i) = —
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In order to simplify the analysis, let us suppose that, after we average over the disorder, at different points i and k we have that

This property is valid in some models of spin glasses, e.g., in the EdwardsAnderson model in the limit of zero magnetic field. A simple computation shows that the disorder-averaged equilibrium magnetic susceptibility is just given by

Indeed, we have that

The terms with i ^ k do not contribute after disorder- averaging [as a consequence of eqn. (14.33)], and the only contribution comes from the terms with i = k. We finally obtain

It is interesting to note that we can also write

where Ma is the total magnetization in the state a:

The first term, Xss(= /?(! —
has a more complex physical origin: when we increase the magnetic field, the states with higher magnetization become more probable than the states with lower magnetization, and this effect contributes to the increase in the magnetization. However, the time to jump from one state to another is very high (it is

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201

strictly infinite in the infinite-volume limit if nonlinear effects are neglected), and thus produces the separation of time-scales relevant for Xss and XeqIf we look at real systems, both susceptibilities are experimentally observable. • The first susceptibility (%ss) is the susceptibility that one measures when one adds a very small magnetic field at low temperature. The field should be small enough in order to neglect nonlinear effects. In this situation, when one changes the magnetic field the system remains inside a given state and is not forced to jump from one state to another: we measure the ZFC (zero-field-cooled) susceptibility and we obtain the single-state susceptibility. • The second susceptibility (% eq) can be approximately measured by cooling in the presence of a small field: in this case, the system has the ability to choose the state that is most appropriate in the presence of the applied field. This susceptibility, the so-called FC (field-cooled) susceptibility, is nearly independent of the temperature, and corresponds to XeqTherefore one can identify Xss and Xeq with the ZFC susceptibility and the FC susceptibility, respectively. The experimental plot of the two susceptibilities is shown in Fig. 14.1. They are clearly equal in the high-temperature phase, whereas they differ in the low-temperature phase. The difference between the two susceptibilities is a crucial signature of replica symmetry breaking and, if it is an equilibrium phenomenon, can be explained

FIG. 14.1. Experimental results for the FC (field-cooled) and the ZFC (zero-fieldcooled) magnetization (higher and lower curves, respectively) vs. temperature in a spin glass sample (GusrMn^^) for a very small value of the magnetic field: H = 1 Oe (taken from Djurberg et al. 1999). For such a small field, nonlinear effects can be neglected and the magnetization is proportional to the susceptibility.

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only in this framework. Its physical origin is the fact that a small change in the magnetic field pushes the system into a slightly metastable state, which may decay only on a very long time-scale. This may happens only if there are many states that differ from one other by a very small amount in free energy. 14.5

Virtual probabilities

14.5.1 General considerations The previous arguments show that the average of the function Pj(q) over the different realizations J,

has a deep theoretical interest. Indeed, the previous formula for the equilibrium susceptibility, eqn. (14.34), reads:

The situation is rather strange. We have seen that we can define
Virtual Probabilities

203

The associated partition function Z[h] depends on the set of local magnetic fields hi. We define the generalized free energy via

for positive e, where d^i[h] is a Gaussian measure: different /i's are independent, have mean zero and variance 1. This definition is rather baroque: it tells us, in a convoluted way, something about the probability distribution of the response of the system to a random magnetic field, but it does have the advantage of being a bona fide thermodynamic quantity. We can define a generalized susceptibility

It is easy to see that, at x = 0, x(x) is the usual susceptibility: %(0) = Xec It can be argued that for 0 < x < I we have that

where the function q(x) is defined by the condition

Alternatively the function q(x) can be computed by the condition

Although eqn. (14.45) naturally arises in the replica formalism, its direct interpretation is not evident. By doing an explicit probabilistic computation, it was shown in Parisi and Virasoro (1989) that it is deeply related to the behaviour of the weights wa. Here, a discussion of the probabilistic derivation and/or of the probabilistic consequences of eqn. (14.45) would be out of place. It is important to stress that the function P(q), or equivalently x ( q ) , can be computed in a thermodynamic way, i.e., by differentiating the appropriate free energy. 14.5.3 Generalized susceptibilities Our aim is to prove that the moments of the distribution P(q) are related to the static (Guerra 1997; Aizenman and Contucci 1998; Ghirlanda and Guerra 1998; Parisi 1998) and dynamic (Cugliandolo and Kurchan 1993, 1994) behaviour of the system when one adds appropriate random perturbations. This approach has the advantage of being general. Moreover, it allows one to define all the relevant quantities in the case of a single large system (in the infinite-volume limit),

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in the same way as in the previous section (i.e., without disorder averaging). By contrast, in the original approach the function P(q) was defined as the probability distribution in an ensemble of different systems, characterized by different realizations of the disorder (i.e., P(q) = Pj(q)). This difference is crucial if we consider the case (as in structural glasses) where no imposed disorder is present.3 In the case of spin systems, an appropriate perturbation is given by

where the couplings Kilt___ti are independent Gaussian random variables with mean zero and variance K"2 • = p\/(2Vp~i}. One can easily see that the canonical average of Hp over a system with the perturbed Hamiltonian 61 ;• • • ; 6p

J.

/

\

/

for any e, satisfies a relation

irrespective of the specific form of Hj, where [• • • ]K denotes an average over the new couplings K at fixed Hj. Here, the function Pj(q, e) is the /^-averaged probability distribution of the overlap q in the presence of the perturbing term in the Hamiltonian. The derivation involves only an integration by parts in a finite system. The previous equation looks strange: the function Pj(
the RHS fluctuating with disorder but the LHS not. Note that P j ( q , 0 ) is the usual overlap probability distribution Pj(q), which depends on the instance J. By contrast, Pj(q) has been computed using Eq. (14.50), with the limit being evaluated outside the cross-over region, i.e., e ^> V^1/"2. In the presence of many equilibrium states, as happens when replica symmetry is broken, the situation is rather complex. Indeed, a random perturbation reshuffles the weights of the different ergodic components in the Gibbs measure. The principle of stochastic stability (Guerra f997; Aizenman and Contucci f998; 3 However, in a glass we always have the possibility of averaging over the total number of particles.

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Ghirlanda and Guerra 1998; Paris! 1998) assumes that, if one considers an appropriate ensemble for the initial random system, we have that

where P(q) is the average over instances of Pj(q), the latter being defined in Eq. (14.51). There are cases in which stochastic stability fails trivially, e.g., when the original Hamiltonian has an exact symmetry that is lifted by the perturbation. The simplest case is a spin glass with a Hamiltonian that is invariant under spin inversion. In this case, P(q) = P(—q) because in the unperturbed Gibbs measure each pure state appears with the same weight as its spin-reversed counterpart. On the other hand, if we consider Hp with odd p, this symmetry is lifted. This means that in the c —> 0 limit only half of the states are kept. If the reshuffling of their free energies is indeed random then we shall have P(q) = 20 (q) P(q) = P(q), where 0(q) is the Heaviside step function. The same type of reasoning applies whenever the overlap q transforms according to a representation of the symmetry group of the unperturbed Hamiltonian HQ. Once the effect of exact symmetries is taken into account, one may expect that, for a large class of systems, the limit function P(q), in the limit of small perturbations, tends to the order parameter function P(q) of the pure system, where the exact symmetries are lifted. Stochastic stability is nothing but a statement of the continuity of various properties of the system at small e. Ordinary systems with symmetry breaking, and mean-field spin glasses, are examples of stochastically stable systems. In symmetry-breaking systems (and in ergodic systems) the equality of P and P is immediate, both being delta functions. Thus, the problem of deriving the equality between P and P appears only when the coexisting phases are unrelated by symmetry. Unfortunately, we are not able to characterize the class of stochastically stable systems in general. In particular, we do not know for sure whether short-range spin glasses, for which stochastic stability is the most interesting, belong to this class. However, stochastic stability has been established rigorously only in mean field problems (Guerra 1997; Aizenman and Contucci 1998; see, e.g., Marinari et al. 2002; and Contucci 2003). If one studies the problem more carefully, one finds that stochastic stability has far-reaching consequences, e.g.,

These (and other) relations have been carefully verified in numerical simulations of three-dimensional spin glasses models (Marinari et al. 2000). 14.5.4

Local overlap

Here, we would like to extend the definition of the probability P(q) and to define a site-dependent overlap probability distribution Pi(q) with properties that recall the global definition.

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To this end, let us start from a spin glass sample and let us consider M identical copies of our sample: we introduce V x M variables af, where a = 1 , . . . , M (eventually we will send M to infinity) and V is the (large) size of our sample (i = 1 , . . . , V). The Hamiltonian in this Gibbs ensemble is just given by

where H(aa) is the Hamiltonian for a fixed choice of the couplings and Hp[u\ is a random Hamiltonian that couples the different copies of the system. A possible choice is

where the variables Kf are identically distributed independent random Gaussian variables with average zero and variance 1. In this way, if the original system were d-dimensional, the new system would be d+1-dimensional, where the planes are randomly coupled. We can consider other ways to couple the systems, e.g.,

or

where the variables K?'k are identically distributed independent random Gaussian variables with average zero and variance (VM)^1. As we shall see later, the form of HR is not important: its task is to weakly couple the different planes that correspond to different copies of our original system. The first choice, eqn. (14.55), is the simplest to visualize, and is the fastest for computer simulations; the last choice, eqn. (14.57), is the simplest one to analyze from the theoretical point of view. In the following, we do not need to assume a particular choice. Our central hypothesis is that all intensive, self-averaging quantities are smooth functions of e for small e. This hypothesis is a kind of generalization of stochastic stability. According to this hypothesis, the dynamical local correlation functions and response functions will go uniformly, in time, to the values they have at e = 0. We now consider, in the case of nonzero e, two equilibrium configurations,
Fluctuation-dissipation Relations

207

We define the /^-dependent probability distribution P?*(q) as the probability distribution of this site-dependent overlap. If we average over K at fixed e we can define

where this bar denotes the average over K. Finally, we define

where the limit e —> 0 is done after the limits M —> oo and V —> oo. (Alternatively, we keep cM and cV much larger than 1.) Consistency with the usual approach implies that if we make the definition

then the probability distribution Pt (
A detailed computation shows that this crucial relation is correct. 14.6

Fluctuation-dissipation relations

14.6.1 The global fluctuation-dissipation relations The usual equilibrium fluctuation-dissipation theorem can be formulated as follows. If we consider a pair of conjugate variables (e.g. the magnetic field and the magnetization), the response function and the spontaneous fluctuations of the magnetization are deeply related. Indeed, if Req(t) is the integrated response (i.e. the variation of the magnetization at time t when we add a magnetic field from time 0 on), and if (7eq(t) is the correlation amongst the magnetizations at time 0 and at time t, we have that Req(t) = /?((7 eq (0)—(7 eq (t)), where (3 = (kT)~l and 3/:/2 is the Boltzmann-Drude constant a (i.e. the average kinetic energy of an atom at unit absolute temperature in three dimensions). If we eliminate the time and plot Req parametrically as a function of (7eq we have that

The previous relation can be considered to be the definition of the temperature, and is a consequence of the zeroth law of thermodynamics. In an aging system, the generalized fluctuation-dissipation relations (FDR) can be formulated as follows. Let us suppose that the system is carried from

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high to low temperature at time 0 and is in an aging regime. We can define a response function R(tw,t) as the variation of the magnetization at time t when we add a magnetic field from time tw on; in the same way, C(tw,t) is the correlation amongst the magnetizations at time tw and at time t. We can define a function Rtm(C) if we plot R(tw,t) vs. C(tw,t) by eliminating the time t (in the region t > tw, where the response function is different from zero). The FDR states that for large tw the function Rtvj(C) converges to a limiting function R(C). We can define

where X(C) = I for C > Cx = lim^^ C eq (t) and X(C) < I for C < Cx. The shape of the function X(C) gives important information about the free-energy landscape of the problem, as discussed at length in the literature. Using arguments that generalize the stochastic stability arguments to the dynamics (Franz et al. 1998), it can be shown the function X(C) of the dynamics is related to a similar function defined in the statics. Indeed, let us consider the function x(q) (introduced in the previous section), defined as

Obviously, x(q) = 1 in the region where q >
This basic relation can be derived using the principle of stochastic stability, which asserts that the thermodynamic properties of the system do not change too much if we add a random perturbation to the Hamiltonian. 14.6.2 The local fluctuation-dissipation relations There are recent results that indicate that the FDR relation and the staticsdynamics connection can be generalized to local variables in systems for which quenched disorder is present and aging is heterogeneous. One can arrive at a local formulation of the fluctuation-dissipation theorem, where local dynamical quantities are related to the local overlap probability distribution. For one given sample, we can consider the local integrated response function R-i(tw,t), which is the variation of the magnetization at time t when we add a magnetic field at the point i starting at time tw. In a similar way, the local correlation function Ci(tw,t) is defined to be the correlation between the spin at the point i at different times (tw and t). Quite often in a system with quenched disorder, aging is very heterogenous: the functions Q and Ri change dramatically from one point to another.

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Local fluctuation-dissipation relations (LFDT)

with the function Xi(C) possibly having strong variations from site to site, have been derived analytically under the appropriate approximations (Castillo et al. 2002a, 2002b, 2003), and have been observed in simulations (Montanari and Ricci-Tersenghi 2003a, 2003b). It has also been suggested that, in spite of this strong heterogeneity, if we define an effective /3fs at time t at the site i to be

then the quantity /3fs would not depend on the site. In other words, thermometers coupled to distinct sites would measure (at a given time) a common temperature, independent of the site; different sites are thermometrically indistinguishable. One can show that these results are general consequences of stochastic stability in an appropriate context, and that there is a local relation between statics and dynamics (Parisi 2003, 2004). The result is the following: we start from the local probability distribution of the overlap for a given system at a point i [i.e. Pi(q)}', we define the function Xj(
and we show that the statics-dynamics connection for local variables is very similar to the one for global variables, and is given by:

The property of thermometric indistinguishability of the sites turns out to be a by-product of this approach: during an aging regime, all the sites are characterized by the same effective temperature. 14.7

Conclusions

We have seen that the overlap, introduced in the original papers by Edwards and Anderson (1975, 1976), plays a crucial role in the theory, especially if replica symmetry is spontaneously broken. The properties of the probability distribution of the overlap P(q) also have a fundamental role in the theory: they are the basis for a definition of the functional order parameter that enters into the computation of the free energy and of other thermodynamically relevant quantities. The principle of stochastic stability was introduced originally in order to explain some of the properties of the probability distribution of the overlap. It gradually became one of the most important guiding principles in the understanding of the behaviour of disordered systems.

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It is possible to give two different definitions of the overlap distribution function that are are well defined (i.e. they do not fluctuate) for a single system in the thermodynamic limit. The second definition is particularly interesting, because it is related to the dynamical behaviour of the system in off-equilibrium situations, and for this reason it is directly connected to the observed experimental violations of the fluctuation-dissipation theorem. It is amazing how a very fundamental idea (the overlap between two configurations) has been so useful and appears in the theory in so many different, but related, forms. References

Aizenman, M. and Contucci, P. (1998). J. Stat. Phys. 92, 765. Aizenman, M., Sims, R. and Starr, S. L. (2003). Phys. Rev. B 68, 214403. Castillo, H. E., Chamon, C., Cugliandolo, L. and Kennett, M. (2002a). Phys. Rev. Lett. 88, 237201. Castillo, H. E., Chamon, C., Cugliandolo, L. and Kennett, M. (2002b). Phys. Rev. Lett. 89, 217201. Castillo, H. E., Chamon, C., Cugliandolo, J., Iguain, J. L. and Kennett, M. (2003). Phys. Rev. B 68, 134442. Contucci, P. (2003). Replica equivalence in the Edwards-Anderson model. Preprint cond-mat/0302500 and references therein. Cugliandolo, L. F. and Kurchan, J. (1993). Phys. Rev. Lett. 71, 173. Cugliandolo, L. F. and Kurchan, J. (1994). J. Phys. A: Math. Gen. 27, 5749. Djurberg, C., Jonason, K. and Nordblad, P. (1999). Euro. Phys. J. B 10, 15. Edwards, S. F. and Anderson, P. W. (1975). J. Phys. F5 965. Reprinted in this volume. Edwards, S. F. and Anderson, P. W. (1976). J. Phys. F6, 1927. Franz, S., Mezard, M., Parisi, G. and Peliti, L. (1998). Phys. Rev. Lett. 81, 1758. Franz, S., Parisi, G. and Virasoro, M. (1992). Europhys. Lett. 17, 5. Ghirlanda, S. and Guerra, F. (1998). J. Phys. A: Math. Gen. 31 9149. Guerra, F. (1997). Int. J. Phys. B 10, 1675. Guerra, F. (2003). Comm. Math. Phys. 233, 1. Marinari, E., Parisi, G., Ricci-Tersenghi, F., Ruiz-Lorenzo, J. and Zuliani, F. (2000). J. Stat. Phys. 98, 973. Marinari, E., Parisi, G. and Ruiz-Lorenzo, J. J. (2002). J. Phys. A: Math. Gen. 35, 6805. Mezard, M., Parisi, G. and Virasoro, M. A. (1987). Spin Glass Theory and Beyond. World Scientific, Singapore. Montanari, A. and Ricci-Tersenghi, F. (2003a). Phys. Rev. Lett. 90, 017203. Montanari, A. and Ricci-Tersenghi, F. (2003b). Phys. Rev. B 68, 224429. Parisi, G. (1987). Physica Scripta 35, 123. Parisi, G. (1992). Field Theory, Disorder and Simulations. World Scientific, Singapore.

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Parisi, G. (1998). On the probabilistic formulation of the replica approach to spin glasses. Preprint cond-mat/9801081. Parisi, G. (2003). J. Phys. A36 10773. Parisi, G. (2004). Europhys. Lett. 65, 103. Parisi, G. and Virasoro, M. (1989). J. de Physique I 50, 3317. Sherrington, D. and Kirkpatrick, S. (1975). Phys. Rev. Lett. 35, 1792. Talagrand, M. (2003). Comte Rend. Acad. Sci. 337, 111.

15

THEORY OF RANDOM SOLID STATES M. Mezard Laboratoire de Physique Theorique et Modeles Statistiques, CNRS and Universite Paris Sud, Bat. 100, 91405 Orsay CEDEX, France Abstract

This chapter is a non-technical, elementary introduction to the theory of glassy phases and their ubiquity. The aim is to provide a guide, and some kind of coherent view, to the various topics that have been explored in recent years in this very diverse field, ranging from spin or structural glasses to protein folding, combinatorial optimization, neural networks, error correcting codes and game theory. 15.1

A few landmarks

15.1.1 Structural glasses Nature provides for us numerous examples of systems that may condense into an amorphous solid state. Probably the most common case is that of structural glasses, of which window glass has been known for several millennia; recent reviews can be found in Angell (1995) and De Benedetti (1996). Structural glasses consist of a phase of matter in which atoms or molecules are arranged in space in a structure that is frozen in time, apart from some small fluctuations. Yet, contrary to the case of crystalline solids, the arrangement of these atoms or molecules is not a periodic one. It is a 'random' arrangement: although the system exhibits some kind of regularity on small enough scales (in the range of a few inter-atomic distances), this regularity is lost on larger length scales, as attested by the absence of sharp peaks in the diffraction pattern. A random arrangement of the degrees of freedom, but one which is frozen and does not evolve in time: these are the basic ingredients of what we shall call the random, or amorphous, solid state, and what goes generally under the name of 'glass phase.' (I have preferred the former because the term 'glass' is more specialized and might lead to some misunderstanding when we shall move to the random solid states of some systems that are more remote from condensed matter physics.) Qualitatively this description is fine, yet the reader should be aware from the beginning of the difficulty of giving more precise definitions. We used the term 'phase of matter' but it may be (and has been) disputed whether this is a really new phase of matter. The glass state might not exist as a true separate phase, but just be describing a liquid with an extremely large viscosity,

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so that we do not see it flow in the limited time scale of our experiment. The fact that the structure does not evolve in time should not be thought of as implying that the position of each atom is frozen: because of vacancies, for instance, the atoms can actually drift, although very slowly if the system is at low temperatures, as they also do in a crystalline phase. The relative positions of the points in space around which an atom is located, these define this frozen structure. The definition of a 'random' arrangement is not a trivial one either: one could have some order that displays no Bragg peaks but can be described with a little amount of information, or else one could be obliged to describe the glass state by giving the average positions of all atoms, which requires an infinite amount of information (in the 'thermodynamic limit' of infinitely large systems). These are all important subtleties, and we shall partly address them below. Yet it is clear that, judging from its relaxation time, the glass state is at least a quantitatively different state of matter. Actually, one very peculiar aspect of glass-forming materials, and one which is so important in their manufacturing, is how rapidly this relaxation time, or the viscosity, varies with the external conditions. So-called 'fragile' glasses exhibit the strongest variations (Angell 1995; De Benedetti 1996) in the vicinity of the glass transition, with the relaxation time increasing by more than twelve orders of magnitude following a variation of temperature by only twenty percent. At temperatures well below the glass transition temperature their life-time is essentially infinite, and some million year old samples have been found. In the regimes where the experimental time is much smaller than the relaxation time the glass state is out of equilibrium and one observes aging phenomena. Inevitably, we shall thus need to face the time-dependent properties of these systems, which are even more difficult to describe than their equilibrium counterparts. Beside its special properties, the glass state is important because of its ubiquity. It can be reached in virtually all systems, by many different pathways. Cooling from a liquid phase is a common one. The cooling rate should then be fast enough for the system to be quenched into the glass state, thus avoiding crystallization (how fast one should quench depends enormously on the system at hand: as we all know, it is much easier to reach a glass state in liquid silica than in a metal). Probably in most systems, the crystalline state is the most stable one, although this has not been proven: at zero temperature, the famous conjecture of Kepler stating that the densest packing of hard spheres is a crystalline one (face centered cubic or hexagonal closed packed) was unresolved for four centuries, but has finally yielded to a computer-assisted proof (Hales 1998). Showing that the crystalline state is the most stable one at some finite temperature is thus likely to be a very hard task. The existence of a crystal state is annoying both for experimentalists who must 'beat the crystallization trap,' and for theorists, who must find a proper way of studying a metastable state. But this is not more troublesome than studying super-cooled water, or diamond. A more subtle point, to which we shall return, is the fact that it is extremely

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difficult to prepare a glass in one given 'glass state'. From the mathematical point of view the idea of a glass at thermal equilibrium is a useful concept, and it turns out to be a very useful starting point in order to start a study, but the last word will deal with out of equilibrium dynamics. As we shall see, there are some indications that these two approaches (thermodynamic equilibrium and out of equilibrium dynamics) are intimately related, but the deep reason for this is not so clear, and its search will be a major challenge for the near future. 15.1.2 From rubber to spin glass and proteins Another technologically important glassy material is rubber (Goldbart et al. 1996; Zippelius and Goldbart 1998). There, the basic microscopic constituents are long polymeric chains, and the amorphous solid state is obtained by adding cross-links which glue together permanently these chains—a process called vulcanization which was discovered by Goodyear one-and-a-half centuries ago. Thus, there exists a fundamental conceptual difference with the simpler structural glasses described above: vulcanization has created some permanent links between the polymers, which are located at random positions. Therefore the description of the vulcanized rubber involves some random variables—the positions of the cross-links. These random variables are given a priori, they depend on the sample which one is studying, and their number is extensive, i.e., it grows linearly with the volume of the sample. This is very different from our previous case. In simple structural glasses one can work with a system of N molecules interacting pairwise (higher order interactions can be added easily without modifying the argument) through a simple potential V(ri,rj). The energy function (the Hamiltonian) is very easily described, the potential energy being just the sum of the pair interactions. What is complicated to describe and study is the amorphous state adopted by the system under fast cooling. On the contrary in rubber, writing down the Hamiltonian for a given sample requires the knowledge of the positions of all the cross-links, a very long list which you cannot determine, nor store on your hard disk, and which will be different if you move to a new sample. This type of system, where the Hamiltonian depends on an extensive set of random variables, is said to have quenched disorder. The terminology comes from the fact that the cross-links are permanent (i.e. are not in thermal equilibrium), in contrast to the positions of the monomers constituting the polymers, which do fluctuate thermally. Quenched disorder is also present in some exotic magnetic alloys called spin glasses (Mezard et al. 1987; Fischer and Hertz 1991; Sherrington 2003). These systems are not present in every-day life; they can be found only in some specialised solid state physics laboratories, and only in small quantities. They surreptitiously appeared in various odd corners of materials science only a few decades ago, and nobody has been able to foresee any type of reasonable application in the near future, in spite of the strong evolutionary pressure of grant funding which pushes physicists to try and imagine some. Yet, during the last quarter of the twentieth century, there have been many thousands of articles dedicated to spin glasses, both experimental and theoretical, and the spin glass

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problem has been described as a cornucopia (Anderson 1988). The reason is that spin glasses provide a (relatively) simple laboratory for the study of glass phases, which themselves appear in many domains, in physics and beyond. The archetypical case of a spin glass is an alloy such as CuMn, with a concentration of a few per cent of magnetic manganese atoms diluted by a non-magnetic metal, here copper. The magnetic degrees of freedom are the localized magnetic moments of the Mn atoms. They interact with each other through a complicated process, an indirect exchange with the conduction electrons, but the net result is an interaction that either tends to align the magnetic moments (a ferromagnetic interaction) or to anti-align them (antiferromagnetic). Whether the interaction between two magnetic moments is ferromagnetic or antiferromagnetic depends on the distance between the manganese atoms: the coupling oscillates with distance. But the positions of these atoms are frozen in time, on all accessible time scales, and therefore the couplings between the magnetic moments form a set of quenched variables. Neglecting quantum-mechanical effects, a good approximation at the temperatures of study, and using anisotropy to reduce the spins to a set of Boolean degrees of freedom—the Ising spins, which describe the projection of the spin onto one axis—one soon arrives at a much simpler system indeed, a set of classical Ising spins interacting with random couplings. One can guess that this kind of generic problem of randomly interacting Boolean variables will provide useful insight into several domains of science, and indeed it does, as we shall see. But the richness and difficulty of this problem, which we shall briefly survey in the next section, will be a surprise to any newcomer to the field (Mezard et al. 1987; Fischer and Hertz 1991; Talagrand 2003b). Another example of an amorphous solid state, and one of the greatest importance, is offered by proteins (Garel et al. 1998). In its native form, a protein is a long polymer which is folded in such a way that the relative positions of the various atoms are frozen, apart from some small vibrations. In general, this structure is not a simple periodic one, although one may find some recurrent substructures, 'alpha helices' and 'beta sheets,' signalling a degree of local ordering. In a loose sense, proteins thus fall into our broad definition of amorphous solid states. Obviously, by including this very rich new field one is drifting from the purest mathematical definition of glass phases. One reason is the fact that proteins are finite-sized objects. Probably the proper level of description for describing protein folding is the one that considers the amino acid groups as basic entities, and the angles along the backbone as the relevant variables (as always when one chooses one level of description, there also exist some effects that require going to a smaller-scale description). So we typically face a problem of a few hundreds to a few thousands degrees of freedom. This is enough to justify a statistical-mechanical analysis, but it is not Avogadro's number. Of more fundamental importance is the fact that proteins generally have one conformation that is preferred, the native state. This is the shape that makes them function; this is the shape that they adopt in natural conditions, and into which they will refold if denaturated. Although they also possess many other metastable states, these seem to have rather higher free energies, so that the

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protein will be able to avoid these other metastable states and fold into its native shape, sometimes with the help of some auxiliary, 'chaperon' molecules. Sometimes the free energy gap must be rather precisely tailored in such a way that some change in the external conditions (e.g. concentration of other proteins) will lead to some change in shape and properties of the protein, as has been demonstrated in the case of protein-DNA interactions. This dominance of the native state is at odds with the situation of glasses or spin glasses where the systems can freeze into any of the possible meta-stable states. One reason for this difference is the fact that the proteins are not completely random objects. Although the primary sequence of amino acids constituting a protein often looks random, one should remember that the sequences used in nature constitute a very small subset of the very large number of possible sequences (20100 for proteins made of a hundred amino-acids), and a subset that has been carefully selected by evolution, precisely for the ability to fold into a given shape allowing for some function. A totally random sequence of amino acids, with uniform probability of having each of twenty possible ones at each point along the chain, has very little chance of being a useful protein, or even just a molecule able to fold into a well-defined native state. One needs some constraints on the sequence to achieve this, and the most obvious one is to have the right proportion of hydrophobic vs. hydrophilic amino-acids, in such a way that the molecule, in water, will tend to form a compact globule with the hydrophobic ones buried inside the globule so that they avoid the water. The type of correlations that are needed in the choice of the sequence, in order to have a good chance of building a protein from a random heteropolymer, is a very difficult and open problem. Proteins provide some type of glasses with quenched-in disorder (the primary sequence of aminoacids), but the nature of the probability distribution of this disorder, and how natural evolution selected it, is still unknown. We shall not attempt an exhaustive enumeration of glassy states of physical matter: numerous examples range from other biological polymers, such as DNA and RNA, to glasses of electric dipoles, or of vortex lines in high temperature superconductors (Blatter et al. 1994). A very rich class to which these vortex systems belong is that of elastic objects, lines, interfaces such as Bloch walls, modulated phases like charge density waves, which have some thermal fluctuations but are also pinned by some external impurities. The ubiquity of such situations in physics is well documented (as should be clear by now), but in addition glass states show up also in 'far out' contexts, further enlarging the domain of study. 15.1.3 Networks of interacting individuals: global equilibrium Imagine a group of N scientists, consider any two of them, and characterise their relationship at a very crude level by stating whether they are friends or not. These colleagues meet at a conference and the organiser, a very wise person, wishes to optimise their assignment to the two available hotels. He will thus make two groups and try as much as possible to have friends grouped in the same hotel and people who dislike one another separated. He first collects the data on who is

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217

friends with whom. For each pair of people i,j, he assigns a positive interaction constant Jij = +1 if they are friends; otherwise their interaction constant is negative, say J^ = —I. From this set of interaction constants, which builds up our sample, the organiser tries to optimise the assignment in the following way: he will allocate each person i either in the hotel uphill, in which case he denotes him in his files by the number Si = +1, or in the hotel downhill, labelled then by Si = —1. Obviously, considering two colleagues i and j, there are two optimal assignments for each situation of friendship, putting them in the same hotel if they are friends or in different hotels if they are not. These are described mathematically by finding the set of values Si, Sj that minimize the 'pair interaction energy' —JijSiSj. Of course, in a realistic case it is impossible to satisfy everybody: often the enemies of my enemies are not necessarily my friends, and the situation is then called frustrated, in a sense that it is not possible to satisfy simultaneously all pairs of people (the degree of frustration is measured by the fraction of triplets i,j,k such that the product JijJjkJki is negative). Finding the optimal hotel allocation in the set of 2N possible ones turns out to be a very difficult problem, intractable by the present computers even for such a small number as N = 200. This problem is a case of a combinatorial optimization problem that falls into the so-called NP-complete class: there are no known algorithms, so far, that are able to solve this optimization problem in a time that grows like a power of the size (N) of the problem. There may exist better algorithms than the enumeration of the 2N allocations, but they all require computer time growing exponentially with N. What is the relationship of this sociological problem with our glasses? As one can guess from the choice of notation, this is just an example of a spin glass problem, the famous 'SK model' (Sherrington and Kirkpatrick 1975; Kirkpatrick and Sherrington 1978). Assigning person i to the uphill hotel is equivalent to having the Ising spin Si pointing up (Si = +1); a person in the downhill hotel corresponds to the spin pointing down (Si = — 1); and the aim of the organiser is to find a spin configuration that minimizes the interaction energy E = — Si
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we can learn a few important facts about our original problem. The best assignment has a ('ground state') energy EQ behaving for large N as —0.7633 TV 3 / 2 , which is very far above what would happen in the simple unfrustrated world, where the energy scales as —N2: despite all the efforts of our organiser, and his spending a lot of computer time, most people will be rather unhappy and he will not do a much better job than a random assignment of people to the two hotels! The physicist looks at this problem not only at zero temperature (where the problem reduces to finding a ground state), but also at finite temperature, where the various assignments are given a probability defined by the Boltzmann weight exp(— E/T). Then he can get some information on the structure of the assignments of low energy. It turns out that there are many such metastable states, which can be very different one from another: typically one can find an assignment that has an energy E\ that is very close to EQ (the difference between the two remaining finite when N becomes large), but which is very different, half of the people having changed hotel. On top of this, the set of metastable states has a fascinating hierarchical structure, building what is called an ultrametric space (Mezard et al. 1984a, 1984b). A whole class of 'complex systems' can be studied similarly in the framework of equilibrium statistical mechanics. This class contains many combinatorial optimization problems, in which one seeks a globally optimal configuration (a ground state) in a very large set of allowed ones (Mezard et al. 1987). One new idea introduced by physics is precisely this generalisation of the problem to a finite temperature one: instead of asking for the ground state, one asks about the properties of the accessible configurations with a given energy, allowing for the introduction of useful notions such as entropy, free energy, phase transitions etc.... This turns out to be a fruitful strategy, both as an algorithmic device and as a theoretical tool. On the algorithmic side, the idea gave rise to the simulated annealing algorithm, which basically amounts to a Monte Carlo simulation of the problem in which one gradually reduces the temperature in order to try to find the ground state (Kirkpatrick et al. 1983). It is not a panacea, and it can probably be outperformed by more specialised algorithms on any given problem. But it is a very versatile strategy, and one that can be very useful for practical problems because of its flexibility. In particular, it allows one to add new constraints as penalties in the energy functions with a rather small effort, where a more dedicated algorithm would just require a new development from scratch. Practical applications range from chip positioning to garbage collection scheduling, to routing and on to financial market modeling! Apart from trying to get an algorithm in order to find the optimal configuration, one aim could be to get some analytic predictions about this ground state, without necessarily constructing it. This is what happened to our conference organiser, above: from spin glass theory he could get the optimal 'energy' of the best assignment of his colleagues to the two hotels (or, more precisely, its large N limit), without knowing how to construct it, and he could learn about the distribution of metastable states. This type of knowledge is the first step

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towards the elaboration of a phenomenology of the problem, where one would aim, for instance, to understand the importance of various type of correlations in the friendship distribution etc.... It also builds up an interesting class of problems in probability theory. These are the 'random' combinatorial problems in which one studies the properties of ground states of some random systems, given a certain probability distribution of samples. A famous example is the assignment problem: given N persons and N jobs, and a set of numbers giving the performance of each person for each of the possible jobs, find the best assignment of the jobs to the persons. The probabilist can ask the question of the performance of the best assignment for a given set of samples, for instance when the individual performances are independent, identically distributed, random variables taken from a given distribution. Very often, the large N limit is 'self-averaging,' meaning that this optimal length is the same for almost all samples in the set. The statistical mechanics approach has led to predictions concerning this optimal performance (Mezard and Parisi 1985), which have been confirmed recently by a rigorous approach (Aldous 2001). 15.1.4

Networks of interacting individuals: dynamics

Although the systems that we have just described already provide a large class of interesting problems, we are still very far from any real situation in sociology. Our use of equilibrium statistical mechanics is restrictive on at least two crucial points. One of them is the focus on an equilibrium situation; the other is the search for a global equilibrium. Keeping for a while more with our toy conference problem, you may have noticed that human activity is, in general, not organised in this totalitarian way of having an 'organizer' trying to optimise everybody's life (as we know, such attempts are catastrophic, not only because of the practical impossibility of finding the optimal configuration). The more realistic situation of individual strategies, where people have a large probability to change hotel if they are too unhappy leads to a dynamical problem, which could be described again as the relaxation towards some local equilibrium. We enter the world of dynamics, in a case which is still familiar, in the sense that we can think of relaxational dynamics (the situation can be described by a heat bath). Familiar does not mean easy: at low temperatures (i.e. when each individual insists a lot on changing when this is favorable for him), this is the dynamics of a spin glass, and the relaxation time will be very large. What is found in spin glasses is that such a system, starting from initial conditions, will not find an equilibrium state, but will wander for ever (Bouchaud 1992). However the more time that has elapsed, the longer the characteristic time-scale to diffuse further away: such a system is aging, meaning that its response to an external stress depends on its age. This property has been observed, for instance, in polyvinylchloride, or in spin glasses, and its study has turned out to be an extremely valuable tool (Bouchaud et al. 1998). One step further in complexity is dynamical evolution when there is no energy. At zero temperature the energy is a Lyapunov function, which keeps decreasing. Without such a Lyapunov function, all kinds of behaviours become

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possible. We are going away from the physics of systems close to equilibrium, into much more complicated situations, which are just beginning to be explored. Progress has been made in some cases (Challet et al. 2000a, 2000b; Dubois et al. 2001; Mezard et al. 2002), and I would particularly like to mention briefly one case, taken not from sociology but rather from biology. This is the study of neural networks, and particularly some attempts to build up a consistent theory of how memory can be organised in the brain (Amit 1989; Krogh et al. 1991). Elaborating on decades of experiments, it seems plausible that one important level of description of the brain, relevant for the treatment of information, is the level of activity of the neurons, measured as the number of spikes they emit per second (this is not obvious, and the information may be encoded in more subtle ways, such as, for instance, in spike correlations). Focusing on the spikes, one can take as the relevant elementary variables either the spiking rate in each neuron, averaged over some time window of some tens of milliseconds, or its instantaneous version, which is the Boolean variable: 0 if there is no spike; 1 if there is one. An active (spiking) neuron, through its synapses towards an other neuron, will either favor the spiking of this other one, if the synapses are excitatory, or it may inhibit the other neuron's activity. At a caricature level, the neural network might be considered as a highly interconnected network (there are of the order of 104 synapses per neuron) of variables, either continuous—if one models the activity through firing rates, or binary—if one uses spikes. The details of when the neuron decides to spike can be described by monitoring the membrane potential (the neuron fires when the potential exceeds some threshold) and, in the end, what such a network does is basically governed primarily by which are the excitatory synapses and which are the inhibitory ones. Fifteen years ago, in a typical physicist's approach, John Hopfield tried to understand if such a caricature network could be used as a memory (Hopfield 1982). He studied a network that was trained as follows: one shows it some external patterns and one reinforces a synapse whenever the two neurons it connects fire simultaneously. This process, known as Hebb's rule, builds a set of synapses such that the network memorizes the pattern: when presented with an initial configuration that is a corrupted version of the pattern, it will spontaneously evolve towards the pattern. This way of fixing the synapses actually builds a set of symmetric synapses: the influence of neuron i on neuron j is the same as that of j on i. Because of this equality of action and reaction, there exists an energy function for this problem, and the evolution of the system, taking into account the stochastic nature of firing, is just that of a spin glass, where the exchange couplings between spins are the strengths of the synapses. A spin glass that has been tailored in such a way that its metastable states are the memorised patterns. It is no surprise that such a physical spin system, when evolving from an initial configuration not too far from a metastable state (one pattern), will flow towards it, and thus recover the full information on the pattern. This spin glass problem has been studied in great details: one can show that if too many patterns are memorized then the system can no longer memorize them, one can

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compute memory capacities, one can degrade the network, destroying a sizeable fraction of neurons and/or synapses, without altering its memory, etc.... This was an extremely useful proof of the existence of associative memory effects in a highly simplified neural network, and it allowed for many interesting quantitative studies. Its starting point was very remote from reality on one crucial point: the assumption of symmetric synapses. Dropping this assumption forbids us from introducing an energy function, and immediately drives one away from any equilibrium statistical mechanics studies. Yet it has been shown subsequently that many of the key properties of the network still persist in the presence of some degree of asymmetry. Hopfield's daring assumption, which was once described by G. Toulouse as a 'clever step backwards,' allowed one to reduce the problem to a solvable one, which provided a solid background that one could elaborate upon in order to get a more realistic model. Several physicists started from this point and then introduced more realistic ingredients in order to get closer to biological reality. This is, of course, a very important elaboration, which is still moving ahead. One should remember that, even in the presence of asymmetric interactions, the statistical mechanics approach may be useful in various ways, whether it is providing a solvable limiting case as in Hopfield's model, or whether one uses some of the purely dynamical approaches that will be described in the next section. 15.2

Tools and concepts

15.2.1 Statistical description Let us also step backwards towards the 'easy' case of amorphous solid states: glasses. As soon as one tries to go beyond the crystal, or the crystal with defects, one faces the basic obstacle: how to describe an amorphous solid state? As we saw, it is out of the question to try and describe the glass by listing the equilibrium positions of all the atoms. The point is that, in a given glass state, and even after averaging over the thermal fluctuations, the environment of each atom differs from that of all the other ones. Furthermore there is a very large number of long-lived glass states, a number which scales exponentially with the size of the system and therefore gives a contribution to the entropy, called the configurational entropy. In systems with quenched disorder, each sample is different from all the other ones. All these facts call for a statistical description of the properties of amorphous solid states. We have to give up the idea of describing in detail the equilibrium positions of the atoms in a glass state. Instead, we shall give a statistical description of the relative equilibrium positions. The first step is to get rid of the thermal fluctuations, defining, in a given glass state, the density of particles at point x by the thermal average p(x) = '^/i(S(x — Xj)). Here, Xj is the position of particle i and the brackets stand for the average over thermal fluctuations in a given glass state at a given temperature. Whilst this would be just a constant in the liquid, it is a complicated function in the glass, with peaks at all the equilibrium positions of the atoms, a much too complicated object. Basically, what one can hope to compute are some correlations, such as

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the probability, given that p has a peak at a point x, that it will have another peak at some point x + r. This object in turn could depend on the glass state one is considering; in all cases studied so far it does not (a property of the large N limit called reproducibility), but if it would, one should again consider the probability distribution of the correlation when one changes the glass state. For systems with quenched disorder it could also depend on the sample, and one would play the same game, but again this situation has not been encountered: most properties of a disordered system, including all thermodynamic properties, are said to be 'self-averaging,' which means that they are the same for almost all samples (with probability one in the large N limit). Giving up the idea of deciphering one particular sample and moving to the study of generic properties of all samples is a big shift of focus, which has been described as a paradigmatic shift. It is comparable to what was done when people introduced statistical physics, giving up the idea of following the Newtonian trajectory of every particle and concentrating on probability distributions. In the study of glassy phases we have to take this step of statistical modelling twice: first, in order to deal with the thermal fluctuations (the usual statistical physics description); second, in order to describe the fluctuations in the local environments, which exist even after thermal averaging (I shall call it the second statistical level). Some of the first successful implementations of this idea appear in the pioneering works of Sam Edwards and collaborators, both in spin glasses (Edwards and Anderson 1975), and in cross-linked macromolecules (Beam and Edwards 1976). The reason for the introduction of statistical physics has its roots in the chaotic motion of particles, leading to sensitive dependence on initial conditions and forcing one to abandon the hope of following a trajectory. In our case, one reason for the statistical description is probably similar. In spin glasses it is well established that there exists some degree of chaos in the sense that changing the sample slightly (e.g. changing a small fraction of the coupling constants) leads to a system in which the metastable states are totally uncorrelated with the previous ones. In structural glasses the situation is less clear, but it seems plausible that by changing slightly the number of particles from N to N + 6N with 1
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necessary to encode the 1014 synapses is much larger than that contained in DNA). There is an amount of randomness in the wiring, and there also exist generic properties common to most brains, which one can hope to understand in this statistical sense, without having to care about all details of the wiring. In this respect the situation is very different from the study of a globally optimised device such as, for instance, a computer card. 15.2.2 Physics without symmetry: equilibrium The theoretical study of glassy phases is a notoriously difficult problem in physics, and one in which progress has been relatively slow. One key reason is the absence of symmetry. All the simple computations on crystalline solid states that you find in the first pages of textbooks (diffraction patterns, phonon spectra, band structures, etc.) rely completely on the existence of a symmetry group. Even the simplest of these computations cannot be done in the glass phase. To face this situation, theorists have invented a number of methods that all amount to using the second statistical level, and introducing some kind of auxiliary symmetry, as we shall explain below. In the pure problems, it is relatively easy to understand the type of phases that can be found, using simple mean field arguments. The only more subtle questions, which are not well captured by the mean field, usually refer to some special points of the phase diagram, where the vicinity of a second-order phase transition induces some long-range correlations. In glassy systems it turns out that understanding the gross features of the phase diagram is in itself a complicated task. The nature of the solid phase is much richer than usual. Mean field theory has naturally been applied to these problems, yielding a rather complicated but beautiful solution (Mezard et al. 1987). Again, the basic ideas are simpler to express in the case of Ising spin glasses, with N spins taking values ±1 and interacting with random exchange couplings. Detailed mean field computations have established the following picture. Above a critical temperature Tc the system is paramagnetic and the local magnetization vanishes in the absence of an external magnetic field: (Si) = 0, where {• • • } denotes an average over thermal fluctuations. Below Tc we enter the spin glass phase where an infinite spin glass will develop spontaneously a nonzero local magnetization: (Si) ^ 0. Compared to the more usual low temperature 'solid' phases, the spin glass phase possesses two distinctive properties: • The spontaneous magnetisation (S^) fluctuates widely from site to site; the global magnetisation vanishes, and in fact all its Fourier components also vanish. Mathematically, we face a breakdown of the lattice translational invariance to a random state, with no conserved symmetry subgroup of the translational group. A simple order parameter that characterizes the onset of the spin glass phase is the one introduced by Edwards and Anderson • There exists an infinity of glass states. In state a, the spontaneous magnetization on site i, (Si)a, varies from state to state. The idea of several

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states is familiar from the usual case of ferromagnetism: in an Ising ferromagnet there are two states, in which the magnetization points either up or down. Here, there exist many states, and they are not related one to the other by a symmetry. The order parameter should be written rather as laa = N^1 2i{^i)a; but ^ turns out to be a-independent. Working within one given state is very difficult: the spins polarise into 'random' directions, which one does not know how to deduce from the original exchange couplings of the system; so one cannot use a conjugate magnetic field to polarize the spin glass into a given state. Even the definition of states beyond mean field is an open mathematical problem. The best one can do, so far, is to postulate that the states exist and have properties similar to those found in mean field, and check if the simulation or experimental results can be analyzed in these terms. It turns out that this is the case. For instance a simple indicator consists of using two identical replicas of the system (with the same quenched disorder), weakly coupled through infinitesimal attractive interactions, such as the product of the local bond energies in each system. One lets the system size go to infinity first, and the coupling between replicas go to zero afterwards. If there remains a non trivial correlation between the two replicas in this double limit, the system is in a glass phase. Basically, in this game each system is playing the role of a small polarizing field for the other system. The same method can be applied to identify the glass phase in structural glasses (Mezard 2001). Taking, for notational simplicity, a glass composed only of N identical atoms, the microscopic degrees of freedom are now the positions Xi of these N particles. One can introduce a second replica of the same system, composed of N particles at positions y j . The x particles interact with each other, the y particles also. The x particles are nearly transparent to the y particles, except for a very small attraction, which is short range. The order parameter for the glass phase is then the cross correlation function between these two systems (i.e. the probability, given that there is an x particle at one point ri, that there be a y particle at a point r\ + r), in the limit where the cross attraction vanishes. In the liquid phase, the x and y particles just ignore each other in this limit, and there is no cross correlation. Instead, in the glass phase, the weak attraction ensures that the two systems polarize in the same glass state. They develop correlations because of the fact that they are in a solid phase, and these correlations still exist in the limit when the attraction vanishes. This provides a good mathematical definition of any solid phase. 15.2.3 Replicas For the theorist a choice method is the replica method (Mezard et al. 1987). It uses the idea of having identical replicas of the original problem, but their number is not limited to two, but can be any real number. The replica method is always presented as a trick to deal with quenched disorder: in disordered systems, the free energy is generally self-averaging in the thermodynamic limit,

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and therefore one can as well try to compute the average of the free energy over quenched disorder. This is rather difficult to compute, in general. A much easier task is to compute the average of the nth power, Zn, of the partition function, which is nothing but the partition function of n noninteracting replicas. Taking the n —s- 0 limit one gets the quenched average of the logarithm of the partition function, which is proportional to the free energy. This trick is certainly very old (Giorgio Parisi dates it back at least to the fourteenth century, when the Bishop of Lisieux, Nicolas d'Oresme, used a similar trick in order to define non integral powers!) and has been used many times in the literature. Its first nontrivial application to the statistical physics of magnetic systems with quenched randomness is the seminal work of Edwards and Anderson (1975). Going much beyond a simple mathematical trick, the replica method permits a study of the free energy landscape, and principally of the regions of low free energy. (The notion of a free energy landscape, in the very large dimensional space describing the configurations of a system in statistical mechanics, requires some thinking; however, it is well defined in mean field, and it helps in developing an intuitive picture, which is why I shall use it here for a simple presentation.) The replicated partition function, after averaging over disorder, becomes a partition function for n systems, without disorder but with an attractive interaction between the various replicas: the reason for this attraction is simple: Because they share the same Hamiltonian, with the same disorder, the various replicas will be attracted towards the same favorable regions of phase space, and repelled from the same unfavorable regions. Both effects tend to group the replicas together. If one has a simple phase space, with basically one large valley, then the replicas all fall into this valley, and the order parameter is a number, the typical distance between any two replicas, which gives directly the size of this valley. But in a system with several metastable states, the situation can be more complicated, with some replicas choosing to fall into one valley, whilst others fall into other valleys. This effect has been called 'replica symmetry breaking! Technically, it appears as a standard spontaneous breaking of a symmetry. This symmetry is the permutation symmetry Sn of the n replicas. The problem is that this symmetry is broken only when one considers some number of replicas n that is nonintegral and, in fact, smaller than one. Based on a remarkable intuition about the permutation group with zero replicas, at the end of the seventies Parisi proposed a scheme of breaking the symmetry which is consistent, and has been applied successfully to many problems (Parisi 1979, 1980). Basically, the order parameter turns out to be a function, which is the disorder-averaged probability density P(q) for picking, at random, two thermalised noninteracting replicas of the system and finding their distance to have the value 1 — q. This order parameter can be computed at the mean field level in a variety of systems. In some cases it has been checked against other analytic computations not involving the replica method; it can also be compared to simulations; for overviews, see, e.g., Mezard et al. (1987) and Young (1998). (A direct experimental measurement of P(q) is not possible, but recent developments on out of equilibrium dynamics, explained below, provide an indirect

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access to its measurement.) So far, it has always been found correct, although a rigorous mathematical status is still lacking. The cavity method (Mezard et al. 1985; Mezard et al. 1987) has been developed in order to write down explicitly the assumptions underlying Parisi's replica symmetry breaking scheme, and to develop a direct, self-consistent probabilistic approach, equivalent to the replica method, based on these assumptions. The recent proof of the validity of Parisi's solution for the SK model basically follows this kind of cavity approach (Talagrand 2003a; Guerra and Toninelli 2002; Guerra 2003). Fundamentally, three types of solid phases have been found so far with the replica method. Speaking in terms of an Ising spin glass system, with spins S^, and defining the overlap between two spin configurations, Si and S^, as

we can characterize them from the shape of the overlap distribution P(q). At high temperature, the system is not in a solid phase and one has P(q) = S(q): the thermal fluctuations win, there are no correlations between replicas. At low temperatures, in the presence of a small magnetic field, which breaks the global spin reversal symmetry, one can find either: • A replica symmetric phase with P(q) = S(q — qo). This happens for instance in a ferromagnet, where
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point, let me explain briefly how one can use a kind of replica method in the structural glass case. Let us assume that the free energy landscape of a structural glass is indeed made up of many valleys, such that the low-lying valleys point in uncorrelated directions of phase space. Assume further that the number of valleys at a given free energy / is exponentially large, so that the entropy of the system is the sum of an internal entropy measuring the size of each valley, and a configurational entropy Sc(f) measuring their number. Proving these assumptions, purely from the microscopic Hamiltonian, is a task that seems totally hopeless at the moment, but one accessible method of approach is to postulate this structure, work out its consequences, and compare them to what is observed in experiments and simulations. How can one use replicas in such a case? The technique is a simple generalization of the two replicas used in the previous section to define the order parameter. Take m identical replicas of our glass, with a small short range attraction. In the glass phase this small attraction will polarise the system into the same valley. It is easy to see that the free energy of the replicated system F(m), considered as a function of m, is the Legendre transform of Sc(f) (Monasson 1995). Whilst it is difficult to compute Sc(f) directly, one can develop approximation schemes for F(m), and this gives access to the thermodynamic properties of the glass phase (Mezard and Parisi 1999). 15.2.4

Physics without symmetry: dynamics

The glass phase is very difficult to observe at equilibrium. Experimentally, a glass is an out of equilibrium system, at least if the sample is large enough. The equilibrium properties that we have just discussed cannot be used in a direct quantitative comparison with experiments. They can be of direct relevance for other amorphous solid states, such as optimization problems or memory neural networks, which are evolving from an initial configuration close to one of the memorised patterns. They can be useful in interpreting some experimental findings, as is the case for the hierarchical structure of metastable states, but a direct comparison is difficult. The equilibrium studies provide the properties of the free energy landscape, focusing on to the low-lying states. It is doubtful whether experimentalists will ever come up with a system prepared in one glass state a (the equivalent of a ferromagnetic crystal, uniformly polarised, without domain walls). Instead their systems age for ever. The point may be illustrated by the dynamical definition of an order parameter, which we shall formulate again for simplicity in spin glass language. In its original formulation by Edwards and Anderson (1975), the order parameter was defined as the long-time limit of the spin autocorrelation: q = limt->oo limjv_ >00 {/S'j(t)/S'j(0)), where the brackets mean an average over the thermal noise (some underlying dynamics, for instance of a Langevin type, can be assumed for this classical spin system). This gives a correct definition only if the system is thermalized inside one glass state a at time t = 0. Then it is kind of tautological: the system remains inside the same state, the probability of the spin configurations decouple at large time, and we obviously get back to the equilibrium definition q = liniAr^oo N^1 '^/i(Si)a (Si}a. We are back

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to our problem: the system cannot be thermalised at time t = 0, so what should one do? Experiments provide the answer: the glass is aging. Somewhere it keeps a trace of the date at which it was born (Bouchaud et al. 1998). Let us call t = 0 this time, defined as the time at which the system was quenched below the glass transition temperature (if one cools a structural glass slowly, there are cooling-rate effects, which may tell us a lot, but we shall not discuss them here). The correlation function between times tw and tw + T is C(tw + r,tw) = limjv^oo N^1 J^(Si(t w )Si(t w + T ) ) . As the relaxation time is infinite, or in any case much larger than any experimental time scale, the system is never thermalized at time tw, whatever its age tw is. One must study the dependence of the correlation as a function of the two times: the age tw and the measurement time T. The correct definition of the order parameter becomes q = lim-r^oo lim^^oo C(tw + r,tw). This turns out to give the same result as the equilibrium definition, showing that the system in this sense comes arbitrarily close to equilibrium, but now this order parameter can be measured. One can realise the subtlety of the approach to equilibrium by noticing that, in the reverse order of limits, lim,--^ C(tw + T, tw) = 0, for any tw. This situation has been called weak ergodicity breaking (Bouchaud 1992), and seems to be present both in spin glasses and structural glasses. Experimental measurements, done on response functions rather than correlations, have found it for instance in such diverse systems as PVC (aging in the mechanical response: if I measure the response of your plastic ruler to a stress, I can deduce when the ruler was fabricated—provided I can perform a measurement on a time-scale of the order of its age!) and in spin glasses (aging in the relaxation of the thermo-remanent magnetization). Taking into account properly the aging effect implies thinking in the twotime plane: the effects one can then study are not just the very complicated and system-dependent transient effect, but they relate to what happens when both tw and T go to infinity, along various paths. It turns out that there seem to exist few universality classes for the behaviour of the two times response and correlation functions in this limit. This has been first found by Cugliandolo and Kurchan in mean field spin glasses (Cugliandolo and Kurchan 1993). Based on these relatively simple models, for which the dynamics can be solved explicitly, a generic scenario of glassy dynamics has been worked out, implying a well understood generalization of the fluctuation-dissipation theorem, where an effective temperature, measurable but distinct from the bath temperature, characterizes the proportionality between the time-derivative of the correlation and the instantaneous response, when these quantities are measured on timescales comparable to the age of the system. On these time-scales, the new relaxation processes that appear are 'thermalised' with an effective temperature that is close to that of the glass transition temperature rather than to that of the room. A proper account of these fascinating recent developments is beyond the scope of this chapter. What I just want to point out here is that the measurement of this new effective temperature appearing in the generalized fluctuation-dissipation

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theorem, which can be done by doing response and noise measurements, monitoring properly the age of the system, allows for an experimental determination of the type of glassy phase which one encounters, in the classification of Section 15.2.3 (Cugliandolo and Kurchan 1993; Franz and Mezard 1994a, 1994b; Cugliandolo and Kurchan 1994; Franz et al. 1998). Numerical simulations on spin glasses and structural glasses have confirmed that the P(q) order parameter can be measured either from a well equilibrated small system, or from the generalized fluctuation-dissipation theorem in the out of equilibrium dynamics of large systems (Parisi 1997; Kob and Barrat 1997); the two procedures give results which agree with each other, although this does not imply that the asymptotic regime has been reached. The results point in the direction of a one-step replica symmetry breaking in the structural glasses, and a full replica symmetry breaking in spin glasses. On the experimental side, a recent, beautiful experiment in a spin glass material has managed to measure the fluctuation-dissipation ratio, and finds a rather good qualitative agreement with the predictions of the full replica symmetry breaking scenario (Herisson and Ocio 2002), although again it is not clear if the 'true' asymptotic regime can be measured. At present, it seems that the mean field predictions provide at least good guidelines to the experimental systems, at least on the time-scales that can be obtained in the laboratory. Similar measurements have been attempted in structural glasses (Bellon and Ciliberto 2002) but the results seem to depend a lot on the observable, and the situation is not yet clear. 15.2.5

Simulations

As we have seen, the theory of amorphous solid states has been developed in close connection with the progress with numerical simulations, and will continue to do so. The collective behaviour of strongly interacting systems can display very complicated, and sometimes surprising, behaviour, for which simulations help to provide some intuition and to bridge the gap between theory and experiments. Reviewing the progress on the simulations goes beyond my abilities and beyond the scope of this chapter; I shall rather refer the reader to Marinari et al. (1998). But one should be aware that in this field, the simulations play a very important role, on equal footing with theory and experiments, and this three-fold strategy is necessary for progress. 15.3

Directions

Predicting what will be the important future developments is bound to fail. I will not risk doing this, but just state a few topics that I find interesting at the moment. Their importance, the stage of their development, and the time-scale of their study is totally uneven. The reader should just take them as discussion topics, such as arise more or less randomly in a chat with colleagues on a winter's evening, around the fireplace. As always, the most interesting developments will be those that I cannot think of at this moment.

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Theory of Random Solid States by M. Mezard Physical glasses

The theory of glasses is still in its early infancy. The idea that glasses may be experimental realization of systems with one-step replica symmetry breaking, although it is more than ten years old, has given shape to an actual microscopic model only very recently. The most obvious open questions concern the dynamics in the low temperature phase (we have no microscopic theory of aging in structural glasses so far), and the whole behaviour in the temperature-window above the glass transition temperature. Mean field models with one-step replica symmetry breaking have two transition temperatures. The thermodynamic transition temperature, which should be the ideal glass transition temperature (i.e. that of a glass cooled infinitely slowly), and a dynamical transition temperature which is larger, at which the system becomes non-ergodic, but where there is no thermodynamic singularity. This dynamical transition (which is also the one that is detected by mode-coupling theory) is presumably a mean-field artifact: the system gets trapped in metastable states which have a free energy higher, by an extensive amount, than the equilibrium state. One expects that this dynamical transition will be rounded in any real system by 'activated processes^ i.e., bubble nucleation. These are not understood at the moment, and their correct description is needed in order to understand the rapid increase of relaxation times upon cooling in glasses. Setting aside for a moment all the unsolved mathematical questions, which I shall discuss later, it is clear that the theory of spin glasses is more advanced. Yet we face two difficult problems concerning the extension of mean field theory to spin glasses in dimensions smaller than six. On the technical side, the standard field-theory expansion around mean field is extremely difficult. Progress has been steady but slow, and indeed some of its first predictions have been recently confirmed numerically. Getting further along this direction will require some better understanding of the mathematical structures underlying replica algebra. The physical picture is not crystal clear either. We certainly would like to understand better how the many states are realised in real space. The physical discussion that can be given now is at the more abstract level of phase space, and it has shown its value in the design and discussion of experiments, but a fuller understanding requires going to the level of spins. In spite of many attempts at defining length-scales in glasses, my feeling is that the situation is still rather unclear. Let me state a simple illustration: if one has only two states, the out of equilibrium dynamics is that of coarsening and, after gauge transforming the spins, one can think of it in terms of coarsening in an Ising ferromagnet. The generalization of the fluctuation-dissipation theorem then takes a simple form, which has a very intuitive interpretation. After a large waiting time tw, the system has developed some domains of each of the two phases, and the typical size of the domain is i(tw) [in a pure ferromagnet it would be i = i/t^; in presence of impurities the growth of the domains would be slower]. Then the dynamics after the time tw is very different, depending on whether one considers time tw + r with r tw. In the first case, a given spin, which is generically far away

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from the domain walls, sees an environment that is at equilibrium. One thus expects the usual fluctuation-dissipation theorem to be valid. On the contrary, when T
Random systems

We seem to be on the way towards a general classification and characterization of the behaviour of random systems, both in their equilibrium and nonequilibrium behaviour. The original schism between the systems with and without disorder (roughly speaking: spin glasses and glasses) has been partially bridged (Bouchaud and Mezard 1994; Marinari et al. 1994a, 1994b; Chandra et al. 1995): if a system without disorder has a glassy phase, this phase may look very much like that of a disordered system. This is kind of reminiscent of Wigner's successful step, when he substituted the complicated Hamiltonian of a nucleus by a random matrix with the same symmetries. In the framework of amorphous solid states, such a step has been carried through in the case of a few specific examples, but we do not have yet any systematic equivalence, and the symmetry classes are not known. Many of the ideas that I have presented here have a resonance with other problems of physics. A better characterization of the low-temperature thermodynamics of glasses involves the computation of spectrum and localisation properties of vibrations in random structures, which is a problem appearing in many areas of physics. The interplay of amorphous solid state ideas with the ones developed in electron localization could certainly also be a source of enrichment to both fields. Although I kept here within the scope of classical statistical mechanics, the quantum behaviour of amorphous solid states is also very interesting: quantum critical points appearing at zero temperature have very interesting properties, which have just began to be worked out but offer a wonderful playground for future developments. On top of all the examples I have mentioned so far, from protein folding to brain theory, some of the most active areas of glassy physics outside of physics involve problems in computer science and information theory (Mezard 2003) such as error correcting codes (Nishimori 2001) and the satisfiability problem (Dubois et al. 2001; Mezard et al. 2002), as well as its application to game theory and economic modelling (Challet et al. 2000a, 2000b; Bouchaud and Potters 2000). At a very basic level, the field that we have been studying over the last two decades is just that of the collective behaviour of interacting agents that are heterogeneous, whether this heterogeneity is there from the beginning or is generated by the system through its dynamical evolution. Obviously, this is a very general topic

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with many possible applications. I am thus confident that the spreading of these ideas will go on for a while. 15.3.3

The unreasonable inefficiency

of mathematics

In some sense, the equilibrium statistical mechanics of amorphous solid states is a branch of probability theory. A direct probabilistic solution of the mean field theory of spin glasses has been developed, at the mean field level, through the cavity method. After many years of study, and clever mathematical improvements, it now offers a rigorous solution of the SK model, and in optimization for 'simple' problems like the assignment or random link traveling salesman problem. Clearly, this is a very active line of research, and one can expect that new exact results will be obtained in this field over the forthcoming years. But by far the easiest approach, the most compact as far as actual computation are concerned, the first one that one would use on any new random problem, is the replica one. It is very strange that nobody has yet come up with a mathematical framework to study the permutation group with a real number of elements and provide a justification of Parisi's replica symmetry breaking scheme, or maybe generalize it. This is a perfectly well defined scheme, where the computations, as well as the underlying probabilistic structure (which is exactly the content of the cavity method) are completely understood. Amorphous solid states are the low-lying configurations of certain Hamiltonians. It is no surprise that these will be related to the theory of extreme event statistics. If the configurations of a glassy system have independent random energies then extreme event theory tells us the statistics of these energies: they are given by Gumbel's law, first obtained by Fisher and Tippett (1928), which is the one of relevance for us since we expect the energy distribution to be unbounded in the thermodynamic limit but to fall off rapidly enough, faster than a power law. It turns out to be exactly the statistics that is found by the replica symmetry breaking method at one-step replica symmetry breaking, as was found early on in the case of the random energy model. This provides some very encouraging connections between standard probability tools and physics. Of course, in any physical system the energy of the configurations are correlated random variables. But one may hope that, after grouping together the configurations that are near one other, one builds up valleys for which the free energies are uncorrelated (keeping to the low-lying valleys). These systems will form a universality class containing the systems where the amorphous solid state is of the 'one-step replica symmetry breaking' type. The present belief is that the glass phase of simple glasses (e.g. hard spheres or soft spheres) could be of this type. A better understanding of the random packings of spheres could help confirm this conjecture. But the spin glass offers us some other universality classes, in which the low-lying valleys are not uncorrelated but possess a very specific type of hierarchical correlations: these are the problems where the amorphous solid state is described by the full replica symmetry breaking scheme. Putting them in the framework of extreme events statistics is an interesting mathematical problem. (In this respect, one can draw an analogy with the universal behaviours of sums

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of random variables, rather than extremes, which is much easier. Everyone knows that if the variables are only weakly correlated, the sum is universally distributed as a Gaussian variable; phantom polymer chains offer a physical example. Now if correlations are stronger, which means here that they can couple very distant variables, then physics offers the new universality class of self avoiding polymers, where the typical size of the sum is known to scale as the number to a power v ^ 1/2, but which is much harder to describe mathematically.) The field of spin glasses, in particular, offers many examples of facts that every physicist believes is true, but one cannot prove rigorously. This is not unusual in other branches of physics, and one should not be too worried about it. However it would be very welcome to have a proof of the existence of a spin glass phase in a finite dimensional model with short range interactions, to just mention the most obvious of such facts. I would not be surprised if the study of random solid states, and the various tools that have been developed in physics for that purpose, would lead in the future to interesting new mathematics, maybe with connections to probabilistic arithmetics. 15.3.4 Consilience The statistical physics process of building a microscopic theory of amorphous solid states is a slow and difficult step in the development of physics. Many colleagues will just not want to make the intellectual investment of getting into it, and will argue that a phenomenological description is enough. While I understand that the investment is hard, and for most people it may be better to wait until the theory has been understood better so that it can be simplified, I do believe that microscopic modelling is an absolutely necessary step. We need phenomenological descriptions, trying to find out some description in terms of the smallest number of parameters. But we need to be able to relate them to the microscopic structure, and show the consistency of both. In this respect I think for instance that an elaboration of the scaling picture of spin glasses (McMillan 1984; Bray and Moore 1986; Fisher and Huse 1987, 1988), which would take into account the existence of many states, would be a very interesting achievement. As we saw, the field of amorphous solid states is full of connections with many other branches of science. This is because of the richness of these amorphous phases, and their ability to have many different states coexist. In this respect its theory is a part of the development of a theory of complex systems (in the very broad sense of many interacting agents exhibiting complex collective behaviour). This field is not well enough defined for there to be a unique theory of complex systems. There are various approaches to it, applying to various levels, and each will be judged both on its own results and on its consistency with the others. Of course, statistical physics is just about finding out the collective behaviour, starting from the microscopic description of the atoms. In this vague sense one could say it is central to the field. On the other hand, if one looks at what statistical physics is able to achieve, one would rather say that it is not (yet) central. The available techniques can be judged as rather

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efficient for dealing with systems in which dynamical evolution has the property of detailed balance, which means that they can be described by an energy function, and the evolution is just relaxation in some (free) energy landscape. This is a very strong restriction and, as we saw in the example of neural networks, most of the interesting problems in complex systems will not obey it. Although some attempts have been made to develop the statistical mechanics study of the dynamics of systems without detailed balance (in particular in asymmetric neural networks, or in random mappings of phase space), this is a very vast field that is much less understood. The virtue of the theory of the amorphous solid state is that it can provide some very detailed information on some specific and oversimplified problems, which can then serve as solid starting points for further elaboration. It might also be that some interesting problems, particularly in biology, have been so well selected by evolution that every single detail of the microscopic description is relevant: they are not generic at all, and the statistical description will have nothing to say about them. I feel reluctant to accept this as a general principle, mainly for philosophical reasons, which I will not bother the reader with. Basically, I feel that some level of statistical description, and therefore some degree of genericity, is unavoidable in order to build up a theory of many interacting elements, whatever they are (a simulation of tens of thousands of coupled differential equations reproducing some experimental behaviour is not what I would call a theory, although it may be a very useful step in the elaboration of a theory). Physics has a long tradition of oversimplifying the real world in order to achieve a correct description, and then re-incorporating the left-out details. (Think of the theory of gases, for instance.) This strategy, which is also the one that was followed for instance in the physical theory of neural networks, is probably the best one that can be followed in order to elaborate a theory. I understand that it may seem odd to our colleagues in other fields, particularly the fields that are very experimental ones, but I believe that one day or another their science will also benefit from such a strategy. Which field the statistical physics of amorphous solid states is able to help now, I leave the reader to decide, hoping that the above can provide a few guidelines.

Acknowledgments

Randomness may have some strange consequences. One of them is that I have not yet had the chance to talk with Sam Edwards. Nevertheless, I feel I know him well because of a long familiarity with his work. As this chapter shows, his ideas have had an enormous impact on the field of amorphous solid states, dealing with very fundamental issues, such as the order parameter and the second level of probabilistic description. After nearly thirty years, these concepts are totally central to the whole domain. It is thus a great pleasure for me to contribute to this volume in his honour.

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References Aldous, D. J. (2001). Random Structures Algorithms 18, 381. Amit, D. J. (1989). Modelling brain function. Cambridge University Press. Anderson, P. W. (1988). Reference Frame articles appearing in Physics Today from January 1988 to March 1990. Angell, C. A. (1995). Science 267, 1924. Bellon, L. and Ciliberto, S. (2002). Cond-mat/0201224. Blatter, G., Feigelman, M. V., Geshkenbein, V. B., Larkin, A. I. and Vinokur, V. M. (1994). Rev. Mod. Phys. 66, 1125. Bouchaud, J.-P. (1992). J. Phys. (France) I 2, 1705. Bouchaud, J.-P., Cugliandolo, L., Kurchan, J. and Mezard, M. (1998). In Young (1998). Bouchaud, J.-P. and Mezard, M. (1994). J. Physique I (France) 4, 1109. Bouchaud, J.-P. and Potters, M. (2000). Theory of Financial Risks. Cambridge University Press. Bray, A. J. and Moore, M. A. (1986). In Heidelberg Colloquium on Glassy Dynamics and Optimization, Van Hemmen, L. and Morgenstern, I. (eds.). Springer-Verlag, Heidelberg. Challet, D., Marsili, M. and Zhang, Y. C. (2000a). Physica A 276, 284. Challet, D., Marsili, M. and Zecchina, R. (2000b). Phys. Rev. Lett. 84, 1824. Chandra, P., loffe, L. B. and Sherrington, D. (1995). Phys. Rev. Lett. 75, 713. Cugliandolo, L. F. and Kurchan, J. (1993). Phys. Rev. Lett. 71, 173. Cugliandolo, L. F. and Kurchan, J. (1994). J. Phys. A 27, 5749. Beam, R. T. and Edwards, S. F. (1976). Phil. Trans. R. Soc. London 280 A, 317. Reprinted in this volume. De Benedetti, P. (1996). Metastable Liquids: Concepts and Principles. Princeton University Press. Dubois, O., Monasson, R., Selman, B. and Zecchina, R. (eds.) (2001). Theor. Comput. Sci. 265, special issue on NP Hardness and Phase Transitions. Edwards, S. F. and Anderson, P. W. (1975). J. Phys. F5, 965. Reprinted in this volume. Fischer, K. H. and Hertz, J. A. (1991). Spin Glasses. Cambridge University Press. Fisher, D. S. and Huse, D. A. (1987). J. Phys. A 20, L997. Fisher, D. S. and Huse, D. A. (1988). Phys. Rev. B 38, 386. Fisher, R. A. and Tippett, L. H. C. (1928). Proc. Camb. Phil. Soc. 24, 180. Franz, S. and Mezard, M. (1994a). Europhys. Lett. 26, 209. Franz, S. and Mezard, M. (1994b). Physica A210, 48. Franz, S., Mezard, M., Parisi, G. and Peliti, L. (1998). Phys. Rev. Lett. 81, 1758. Garel, T., Orland, H. and Pitard, E. (1998). In Young (1998). Goldbart, P. M., Castillo, H. E. and Zippelius, A. (1996). Adv. Phys. 45, 393. Guerra, F. and Toninelli, F. (2002). Commun. Math. Phys. 230, 71. Guerra, F. (2003). Comm. Math. Phys. 233, 1.

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Hales, T. (1998). http://www.arXiv.org/math.MG/9811071 Herisson, D. and Ocio, M. (2002). Phys. Rev. Lett. 88, 257202. Hopfield, J. J. (1982). Proc. Nat. Acad. Sci. USA 79, 2554. Kirkpatrick, S. and Sherrington, D. (1978). Phys. Rev. B 17, 4384. Kirkpatrick, S., Gelatt, Jr., C. D. and Vecchi, M. P. (1983). Science 220, 671. Kob, W. and Barrat, J.-L. (1997). Phys. Rev. Lett. 78, 4581. Krogh, A., Hertz, J. A. and Palmer, R. G. (1991). Introduction to the Theory of Neural Networks. Addison Wesley, Reading, MA. Krzakala, F. and Martin, O. C. (2000). Phys. Rev. Lett. 85, 3013. Marinari, E., Parisi, G. and Ritort, F. (1994a). J. Phys. A27, 7615. Marinari, E., Parisi, G. and Ritort, F. (1994b). J. Phys. A27, 7647. Marinari, E., Parisi, G. and Ruiz-Lorenzo, J. J. (1998). In Young (1998). McMillan, W. L. (1984). J. Phys. G 17, 3179. Mezard, M. (2001). First Steps in Glass Theory. In More is Different, Ong, N. P. and Bhatt, R. N. (eds.). Princeton University Press. Mezard, M. (2003). Science 301, 1685. Mezard, M. and Parisi, G. (1985). J. Phys. Lett. 46, L771. Mezard, M. and Parisi, G. (1999). Phys. Rev. Lett. 82, 747. Mezard, M., Parisi, G., Sourlas, N., Toulouse, G. and Virasoro, M. A. (1984a). Phys. Rev. Lett. 52, 1156. Mezard, M., Parisi, G., Sourlas, N., Toulouse, G. and Virasoro, M. A. (1984b). J. Physique 45, 843. Mezard, M., Parisi, G. and Virasoro, M. A. (1985). Europhys. Lett. 1, 77. Mezard, M., Parisi, G. and Virasoro, M. A. (1987). Spin Glass Theory and Beyond. World Scientific, Singapore. Mezard, M., Parisi, G. and Zecchina, R. (2002). Science 297, 812. Monasson, R. (1995). Phys. Rev. Lett. 75, 2847. Nishimori, H. (2001). Statistical Physics of Spin Glasses and Information Processing. Oxford University Press. Palassini, M. and Young, A. P. (2000). Phys. Rev. Lett. 85, 3017. Parisi, G. (1979). Phys. Rev. Lett. 43, 1754. Parisi, G. (1980). J. Phys. A 13, 1101, ibid. 1887, ibid. L115. Parisi, G. (1997). Phys. Rev. Lett. 79, 3660. Sherrington, D. (2003). Contribution to this volume. Sherrington, D. and Kirkpatrick, S. (1975). Phys. Rev. Lett. 35, 1792. Talagrand, M. (2003a). Comte Rend. Acad. Sci. 337, 111. Talagrand, M. (2003b). Spin Glasses: A Challenge for Mathematicians. SpringerVerlag, New York. Young, A. P. (ed.) (1998). Spin Glasses and Random Fields. World Scientific, Singapore. Zippelius, A. and Goldbart, P. M. (1998). In Young (1998).

16

REPRINT

THE THEORY OF RUBBER ELASTICITY by R. T. Beam and S. F. Edwards Philosophical Transactions of The Royal Society of London. Series A, Mathematical and Physical Sciences, 280, 317—353 (1976).

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THE THEORY OF RUBBER ELASTICITY BY R. T. DEAMf AND S. F. EDWARDS, F.R.S.J Cavendish Laboratory, Cambridge (Received 4 March 1975)

CONTENTS PAGE

1. INTRODUCTION

318

2. STATISTICAL MECHANICS OF AMORPHOUS MATERIALS

318

3. MATHEMATICAL MODELS OF RUBBERS (a) Mathematical preliminaries (b) The phantom chain network (c) The Flory assumption (
322 323 325 327 329 329

4. EXCLUDED VOLUME IN RUBBER

335

5. ENTANGLEMENTS IN RUBBER

342

6. CONCLUSIONS

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APPENDIX 1

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APPENDIX 2

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APPENDIX 3

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REFERENCES

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This paper attempts to improve several weaknesses in the classical theories of rubber elasticity. It develops a formulation of the statistical thermodynamics of amorphous materials analogous to the Gibbs formalism for conventional statistical mechanics. This then permits the replacement of 'phantom chains', i.e. long polymer molecules with the fictitious property that they experience no forces except at cross link points and are transparent to one another, by realistic molecules which do experience forces and which can become entangled. The crosslinked points are no longer assumed to deform affinely with the gross behaviour of the solid. Under the simplest conditions forms like the classical are recovered but with a different coefficient, and the term representing the degrees of freedom lost by crosslinking, over which the classical theories are in dispute, is found to lie between the previous values in a formula which can reproduce the classical results by making different assumptions. The entanglements give rise to more complicated forms than the classical sum of squares of strain ratios, which under certain circumstances can reproduce the Mooney-Rivlin term which when added empirically to the free energy usually improves the fit with experiment. The general expression is complicated, but is nevertheless an explicit function of the density of crosslinks, the density of the rubber and the interchain forces. f Present address: C.E.G.B. March wood Laboratories, Southampton. J Present address: Science Research Council, State House, High Holborn, London WG1R 4TA. Vol. 280. A. 1296.

27

[Published 8 January 1976

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1. I N T R O D U C T I O N This study was prompted by a need to improve the 'phantom chain' models of polymer networks already well known in the literature. Phantom chain models have two major defects. Firstly, phantom chains can pass freely through one another (and themselves); secondly, there are no molecular inter or intra chain forces. These effects have now been put into the model and the theory developed. The paper is divided into four sections. In the first an abstract exact formula is provided for the free energy for an amorphous system. Just as Gibbs's famous formula provides an abstract formula for the statistical mechanics of systems in which all states are accessible, the new formula extends to systems with frozen-in degrees of freedom. Although the basic ideas here are well known, the formulation is a practical one which proves a basis for calculation. The second section shows how the formalism fits the problem of rubber elasticity. The classical theories all seem to use part of the formalism, but not all of it, and it is believed that by writing down the whole problem albeit in the form of difficult expression a significant advance has been made. The classical theories are rederived in this section. In §3 the effect of excluded volume, i.e. of short-range forces, is included in the calculation of the free energy of a rubber, while in § 4 the effects of entanglements are included to complete the kinds offeree normally encountered. Entanglements produce complex structure into the free energy, but it is interesting that forms like the well known Mooney-Rivlin term appear quite naturally. They do not, however, appear in any unique or clear way as is seen in the formulae (5.46), (5.48), (5.52), (5.53) and (5.54). These last are the principal results of the paper. 2. STATISTICAL MECHANICS OF A M O R P H O U S MATERIALS The object of this section will be to show that the standard formulae of statistical mechanics as applied to gases, liquids and ordered solids need modification before they can deal effectively with disordered or amorphous solids. Since rubber is an amorphous solid it is well to tackle the statistical mechanics properly before settling on the microscopic model that is going to be used. The starting point of this treatment will be the need of the statistical formulation to produce a shear modulus for a solid. Solids resist change in shape, therefore any theory of solids must reproduce this property or their essential solidlike nature is lost. Crystalline solids are easily dealt with in this respect. A lattice is given, the problem can then be transformed into 'phonon gas' coordinates and the free energy calculated using the normal formula as applied to gases. The incorporation of shear is done by changing the shape of the lattice and recalculating the free energy. The change in free energy with change in shape is then known and the relevant modulus found. The substance resists change in shape (provided the correct lattice was chosen initially) and therefore is a solid. The same thing cannot be done for amorphous solids. No lattice is 'given' and thus no transformation to the phonon gas can be made. The structure of an amorphous solid such as a glass or rubber is more liquidlike than solid, yet the normal statistical methods of integrating over all phase space for each particle will not give us any shear property. The shear property of the crystal came from doing two separate calculations on different shapes of the same crystal lattice, so that the microscopic topology of the crystal was conserved. This is the clue for the treatment of amorphous solids. Some way of specifying the topology laid down at fabrication must be found,

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the topology being conserved in strain by the various microscopic constraints in the solid that have been put in when it was made. Thus, in principle, the same calculation could then be made as for the crystal. The free energy is calculated for two different shapes of the solid with the same topology. The shear property has been recovered. There is, however, an additional complication not encountered with the crystal. No one topology is 'given'. In fact a very large number of topologies may be possible for any given method of fabrication. So there is now the problem of calculating the free energy strained and unstrained for each topology and doing a weighted average of these at the end. (The 'weights' being determined by the method of fabrication of the solid.) If the number of possible topologies for any given method of fabrication is large enough, then what has been calculated will be the most probable modulus (or whatever property is being calculated) which by the usual statistical argument is equal to the average. Trouble comes when the number of competing conserved topologies is small each with a substantially different free energy state associated with it; such as might be found at the gel point for polymer systems. These ideas may be expressed concisely in mathematical form. Consider the Gibbs formula for the nartitinn fnnr.Hon 7,\

where ft = 1 \k T, His the system Hamiltonian and J dD represents integration over all accessible phase space. For a fluid the free energy is independent of the shape of the container but for a crystal all states in phase space are not accessible, the topology of the lattice is conserved if the crystal is strained. Thus where the m label restricts phase space to the crystal topology. The dominant contribution to the shear modulus is from the internal energy. In an amorphous solid such as rubber this is not the case, short range forces dominate as far as the bulk properties arc concerned and in shear the internal energy contribution is small. There is an entropic shear effect due to the conserved topology of the rubber. How the topology of a rubber is specified will be dealt with when specific microscopic models are considered. If entropic effects only are being considered then Boltzmann's formula can be used: where S is the entropy of the system and & the number of configurations in phase space available to the system. If phase space is restricted by constraints conserving topology then the formula is modified (as for the crystal) to where £?m is now the number of configurations in the restricted phase space with topology 'm'. Let this topology 'm' have a certain probability of being formed in fabrication pm. (Rarely does the fabricator have control over his material, so that he can put in a known topology.) Thus the entropy of such a system, the number of different topologies available being a very large number in a statistical mechanics sense, is

where

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A weighted average over all topologies has been taken. However, if the system is strained after fabrication the formula is where the tilde denotes the strained system. Note that

(only for an unstrained system S — S]. It is the fact that the pm does not have a tilde over it that means the system is capable of supporting shear. There are then three steps in the calculation of S, which is what most of the classical theories of rubber elasticity calculate: (1) calculate pm', (2) calculate Sm; (3) calculate S = 3jpmSm. m

There is an easier way of doing this by combining all three steps (Edwards 1970, 1971). The method calculates the free energy F from a generalized partition function. Before setting down the method it might be useful to show the thinking behind it. In the calculation of S above there are really two systems in which the statistical mechanics is done. Firstly, the fabrication of the amorphous solid in which the^ m are to be calculated; then the altered system (it may be strained or at a different temperature, magnetization, etc.) in which the free energy of that topology is to be found Fm. Pictorially state topology

For system A,pm (the probability of making topology 'm') can befound by allowing all topologies at TA, VA, etc. and using the Gibbs formula

where //A is the Hamiltonian of system A. J dflm means integration over all phase space with topologies 'm', alternatively this could have been written as Jtf(m—/(fl)) d,Q; i.e. an integration over all phase space with a delta function constraint picking out the desired topology 'm'. Given this topology Fm has to be found. This is the free energy of topology 'm' in the altered system (B). So using the Gibbs formula again

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The tilde over the dfl's represents the change in phase space due to going from system A to system B. If A is different from B only by shearing, then

Now all that remains to be done is to calculate P. Consider the following expression:

which gives and

whence If Z(») can be calculated then F can be found using (2.1). Z(n) can be expressed in terms of a generalized Gibbs formula as follows:

Or for integral values of n, Z(n) may be written

Thus there are n + 1 systems

The formula as written is only valid for n a positive integer, but by analytically continuing the result Z(B) = ^Zm(Zm)n would be recovered for all n. m

The advantage of the method is that the 'topology conservation' can be put in as a constraint and thus absorbed under the integral sign

The a, labels the system.

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Thus Z(n) contains all the information required, Z(« = 0) = Z, and equation (1.19) gives /''. A simple way of looking at the properties of Z(«) is to expand Z(n)/Z out in powers of K:

/^ = — AT^ In Z is the free energy of the original system A at fabrication without topology conservation. F is the free energy of the B systems with the topology set by A. F = F only when the B systems are unstrained and at the same thermodynamic point as A. The reason for shear properties being bestowed on the material by topological constraints can now be shown. Assume F = F (A and B under identical conditions). Then for any change A, AF = A (~£,p m Fm), whereas If A is infinitesimal But

thus

Differentiate again

Thus Thus second order and higher derivatives at F = F have additional contributions due to the fixing in of the topology in our system and not the other. The part that gives the shear modulus (and additional bulk modulus) is

The A term in the expansion of Z(n)/Z can be used to give an indication of phase transitions at critical points since [(^) 2 — C^)2] is the second moment of the free energy distribution in the solid. ((-F)2 — (f\)2)l(F)2 will normally be of order 1/./V (where JVis the number of crosslinks and is large). Thus ((F)^ — (Fzy)l(F)2is negligible compared to 1, only at the gel point might this be expected to become of order unity, in which case a critical phenomenon will have been encountered. 3. M A T H E M A T I C A L MODELS OF R U B B E R S The three models now about to be discussed will be reviewed only with reference to the assumptions inherent in each. The statistical mechanics will be done the same way for each using the theory as developed in the first section. The mathematical framework will be the same for each

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as well, that is using functional integrals to set up the models. In fact one model will be set up and the assumptions of each theory put into the final integrals, thus it is hoped it will be easier to compare the theories on an equivalent basis. Before setting up the mathematics it is worth emphasizing that the 'peculiar' properties of rubber it is hoped to gain an understanding of, come from the polymeric nature of the material, that is from the long range correlations of atoms in the rubber (greater than 2 nm say). A full understanding of the bulk properties (compressibility, etc.) will come from an understanding of liquid theory which is almost entirely due to short range effects. Generally short range effects are completely unimportant but the motivation for this study was brought about by a desire to understand why a rubber or gel (a dilute rubbery network) does not collapse into a very dense little ball and to understand this it is found that the short range forces between molecules are very necessary to prevent this collapse. Another point worth bearing in mind is that any chain model of a polymer, and indeed the real polymer itself under the Flory 0 conditions, will if the chain is long enough (and flexible) tend to obey Gaussian statistics for distances along the chain further than a certain length say /. This / can be thought of as being a measurable constant for experimental systems or theoretically derivable for any particular chain model used, e.g. freely hinged rods or bond rotation models (Flory 1969). The important point being that provided the chain is flexible to allow changes in conformation the details of the bonds will be unimportant for large correlation distances. It is these large distances which are important in rubber elasticity. With these ideas in mind the mathematics will now be set out in detail and the assumptions of the theories due to (a) Flory & Wall (1951), (b) James & Guth (1943), (c) the present model, will be put into the model and their answers rcderived and compared. (a) Mathematical preliminaries

A phantom chain will be taken to be the definition as introduced by Flory, namely an infinitely thin chain that can pass through itself and other chains. Let I? be a position vector in space and s be an arc length, 0 < s < L, along a phantom chain of length L. The function R(s) is then a 'phantom chain'. R(s) can now be used to set up the statistics of a chain. Let the probability of finding a particular R(s) be denoted by P[R(s)~\. IfR(s) is to be a Gaussian chain then it is true that P i s a functional of R(s), R (s) — dR (s) jZs. This is because

where j8U(.r) stands for the functional integral over all functions R(s) (Feynman & Hibbs 1965). This is the desired form of the two body distribution function. Thus, for example, a partition function may be set up for a chain with an interaction potential between points on the chain R (jx) and U(j 2 ) of V(R(s1) —R(s^)). If Zis the partition function then

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For a single chain in a box of volume F we can define the two body Green function

where the functional integral can be solved by writing down the equivalent 'Hamiltonian' and thus Schrodinger equation in the spirit of Feynman. G(ru r2; su £2) therefore satisfies Pick's equation in a box H The general development of the formulation for polymers is given by Edwards & Freed (1969, 1970). The diffusion equation in a box permits two classes of solution depending on the boundary conditions: (1) cyclic boundary conditions; (2) density of polymer zero outside the box. The solutions arc

where k = (2it/F^) (li + mj + nK) (note, at large s — s' the small k eigenfunctions dominate the expansion, i.e. when ir2/].^ —j 2 |/F$ |> 1). Pictorially

FIGURE 2 FIGURE 1

The two body correlation function C(r, r') can be calculated from Gj_:

thus where the first term comes from 'other chains' (cyclic condition) and the second is due to the extra correlation from monomers along the same chain. It is of interest to note that the LljVs < 1. condition corresponds to a polymer 'gas' of molecules which have internal degrees of freedom. So the perfect gas law would be obeyed at very low densities independent of the boundary conditions. However, if Llj V' J> 1 then the 'internal' degrees of freedom have become 'external', the volume of the box is now a limiting factor on the number of configurations the chain can take up. Thus if it is desired to describe a long 'phantom chain' in a box (Llj F» > 1), a choice between Gj (cyclic boundary conditions) and G2 (zero density outside the walls) must be made. If a phantom chain really did exist in nature then confining it to a box would produce a density distribution as described by G2 (i.e. bunched in the centre). Phantom chains are not realistic, intra and inter chain forces for instance favour an even density distribution. These can be put in

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for solutions of intermediate density quite easily, the main condition on the system being that the Ijr correlation effects are small compared with the effects of the mean density of the solutions, (that is fluctuations in polymer density must be small). So for realistic chain models with ILj V* J> 1 as in network problems, we are driven back to taking the Gt solution with the additional caution that the density fluctuations must be small compared with the mean density. Problems involving the choice of G2 (polymer density zero outside the wall) have been studied. This class of problems arise in the situation where one end of a chain is fixed to a wall so the density at the wall is fixed (Dolan & Edwards 1974, 1975), i.e.

FIGURE 3 The solution to this problem (and for other boundary shapes) is obtained by introducing a small length S in which the polymer is given 'extra room' in order to build up to its starting density at the wall

FIGURE 4 The reason for this is that the chain once it starts away from the wall can never cross it again so configurations that cross the wall are riot counted. This forces the maximum density of polymer to be away from the wall, also the mathematics insists that the density at the wall is zero, but the model must not be taken seriously at distances less than one step length (besides in nature the wall is not a sharp infinite potential). A finite density pa at the wall can be obtained by starting the polymer off at a fictitious wall S further back (where S w /). This might be thought of as the effective infinite potential for the wall. Thus problems of this type have solutions of the type G2, there is no even density part as in the Cj case (or in Fourier transform terms there is no k = 0 mode for G2, there is for GJ. The mathematics of phantom chains in a box will now be used in setting up the general partition function Z(n)f or a crosslinked phantom chain network. (4) The phantom chain network The model for the phantom chain network will be set up by considering a very long chain of length L crosslinked at N points in a box of volume V. (Four chain segments eminate from one crosslink point.) The difference between a very long chain crosslinked in this manner and 2N separate chains linked together by N crosslinks to form a network is only in construction. It is the topology of the network which is important for the thermodynamic properties of the rubber not the method of construction. The topology of the network is specified for the phantom chain case by the position of the crosslinks along the arc length, e.g. S(R(S1) -R(S2)) S(R(S3) -R(St)) specifies one crosslink at S1 and S2 another at S3 and St. 28

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Now the general partition function may be set up. First the n + 1 systems must be denned: system label thermodynamic parameters

where systems (l)-(n) have been strained by the extension ratio. A,,, in the # direction (A,,. = X^X^), where XaY0Z0 = V0, similarly for Aa and As. Put one chain of length L in each box, the chains will take the system labels, so that /?(j) is the chain in the ath box. Each system will have the same crosslinking topology. For one crosslink at ^ and ^2 this would be put in by the constraint

so that for a phantom chain Z(«) for one crosslink is

The advantage of introducing n+ 1 systems is now seen because the f dj± If d.s2 can be done Jo Jo first. Changing the order of integration we have

Thus for N crosslinks Z(n) becomes

This is the general partition function (namely equation 2.22) for a phantom chain network. This formulation of the problem will now be solved with the James & Guth, Flory & Wall and also the present assumptions. Before starting it is more convenient to rewrite the crosslinking constraint by a pole integration

A lower bound will now be found for Z(n). (Therefore an upper bound on F.) Consider therefore thus

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So in this approximation we may write

Therefore only

has to be evaluated, although a direct

evaluation of

is in principle possible. Now what is meant by the averaging { } must be decided, and this is where the difference between the James & Guth and the Flory theory comes. The Green function for the system is easily solved, since the n + 1 systems transform into a problem of a random walk in 3(re+ 1) dimensions

where {^^'GO} is a vector in 3(re + 1) dimensions. This is the solution of Fick's equation in a box of volume FW+1(AXA!/A,,)*1 in 3(n + 1) dimensions. As discussed previously, there are two types of solution appropriate, an even density (cyclic boundary conditions) and polymers with a fixed end at a wall. (c) The Flory assumption

Flory assumed that the crosslinks are fixed in space and are randomly distributed over the volume of the rubber. This is an even density assumption appropriate to cyclic boundary conditions with solutions of type Gx. Evaluation of

is simply

or diagrammatically

FIGURE 5

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G is given by

where has been replaced by the approximation

In order to evaluate these integrals the following transformation must be made where &**>($) is split up into the affine deformation plus the fluctuations about this deformation (the i/(j)'s). Thus

A minimum loop size is introduced, Therefore

where The resulting exp(^i/"<JjTr5(JRi — jRf)}) m the partition function should now be renormalized by exp ( — fiL2/Vn+1(Xx \v A2)re) since this alone would give the perfect gas answer. Thus the general partition function is

which by formula (2.1) gives

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THE THEORY OF RUBBER E L A S T I C I T Y (d)

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The James & Guth assumption

James & Guth assume that there are two types of crosslink, one type fixed at the edges of the rubber the other free to move inside the rubber. The appropriate solution here is a G2 type because polymer is fixed at the wall of the box. The calculation is the same as before except without the 1/F™+1(A,!,A!/AS)K terms from the mean density term. Thus

In Fourier terms Flory includes a k = 0 mode whereas James & Guth do not. Thus the James & Guth result is The logarithmic term is no longer present because the even density assumption does not apply. James & Guth assume the system is held in equilibrium at A.,. = A,, = Az = 1, by 'internal pressure', that is short range repulsive forces. This was unnecessary in Flory's result. In both the above treatments it is interesting to calculate and <(JR(0)(>S) - K^(S))^. If this is done (see appendix 1) it is found:

and

(where the JFJ dominates so that the density is uniform). Thus the crosslinks fluctuate about a mean position (given by the affine deformation) by a standard deviation of ILfiN. This contradicts Flory's assumption of crosslinks fixed in space. James & Guth's idea is to fix some crosslinks at the edges and to allow the rest to fluctuate and unless a constraint of constant density is used, this will give rise to a density distribution with the polymer all piled up at the centre of the box.

FIGURE 6

(e) The present model

This model incorporates the uniform density assumption but allows the crosslinks to fluctuate about their affmely deformed positions. The starting point is again (3.14)

as in the previous analysis Z(n) is going to be approximated by

But this time a more realistic Green function will be used to do the averaging.

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The idea can be outlined as follows. It is required to calculate

This has been approximated by

Now define

i.e. thus where an upper bound is obtained for the free energy F, as before. If, however, Gu is a better approximation to reality an upper bound closer to the free energy will result. Also Gu can be varied so as to give the lowest free energy, thus a variational procedure will yield a closer upper bound on the free energy. U(R) may be thought of as a trial potential for which a best fit to V(R] is found. V[K\ for the case considered is

a good choice for U in this case is

where The reason for this is that crosslinks must be localized about their affinely deformed position. An harmonic well will do this, wt will be a variational parameter. The Green function is then the solution to

& is now the product of n+i Green functions, for turning the problem into the differentia' equation form, n +1 separable differential equations are obtained

with Pick's equation for the R^ system. However, before proceeding it is useful to introduce a transformation T/t that has the following properties the X

(sy& being such that

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(a linear combination of f/ w (j)'s only); also

so that

and T is therefore merely a rotation in the n + 1 system space. T has the properties of a rotation matrix, detT = 1 and TJf = [ 7^]~7. See appendix 2 for an example of T. Special significance is associated with the 'centre of mass' coordinate, which will be Jf|0)(5):

The other Xf(s) may be chosen quite arbitrarily provided the rotation condition onTis met and it is found then that X£(s) for /? ^ 0 is just a linear combination of the ij^'s. Thus

so that where The system is in a box of volume VII (1 +«Af)i and f

which, when lu^s^ — s%\ is large, is dominated by the lowest eigenfunction

Thus From this Green function we obtain (F— £/}

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which was transformed to

Now

the lowest eigenfunction contribution being completely dominant for the /? =£ 0 systems, leads to

where a minimum loop size has been introduced to mimic chain stiffness over short arc lengths. Then

(because I/it < 1/lj, I/Li has been neglected), and

Thus the general partition function is given by (3.4):

Now applying (2.1) to find F gives

where Since a> appears in Z only in as much as it appears in F we can find its value by making F stationary,

we find

Thus the As dependent terms are

The 1/(1 +cjp) is the wasted loops correction factor, also the answer is a factor of \ smaller than the classical theories. The short range forces are expected to give the required free energy correction to keep the network from collapsing; thus Ax\t/A.s = 1, the incompressibility condition,

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should be used with this formula. The fact that u>t is independent of the strain condition means that the — £ |7Vln (cuJSn) term does not contribute to the elastic free energy. The important i

difference between this and the Flory model is that the localization of the crosslinks has been allowed (standard deviation in their position is (S&J,)""i) without fixing them to their affine deformation position. It is this localization as measured by w which decides whether the network is a rubber or not. Obviously if ta ~ V~% then the localization due to crosslinking is the same as that of confinement to the volume of the rubber and the process of gelling would have to be dealt with which is too complicated for a. 'mean field theory' such as this. It is easy to demonstrate how vital the short range repulsive forces are to the equilibrium of the system in the following way. So far the calculations done have been for Z(n) > e< V-V)_ j^Q attempt can be made to calculate {ep~F)<js over the X(® system. To do this first introduce collective variables in the X(e> system:

thus where

so that and

and

Doing the average over the JfW systems for /? ^ 0 as before (that is, (V— t/Xn^o), leaves

where On writing this out in collective variables

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The first term outside the functional integral is just the result previously calculated. The ~£i(pkP-ii)A(<j)) term corresponds to the closed loop correction (i.e. wasted crosslinks). The k

elasticity in the problem has in this theory come entirely from the k = 0 term! So the factor

should improve this calculation still further. The same is true for the Flory and James & Guth models. The term in the exponent is positive for small k which is where the physics of the problem lies. The exponent is

This is only an apparent singularity because on doing the pole integration one finds the functional integral becomes

This is in fact equal to Z(n) because the boundary conditions should be applied properly (i.e. there are no short range forces in the problem, thus a non uniform density distribution will arise so that
As long as ftA (&i) <^ Vk the fluctuation terms (i.e. k 5^ 0) are damped away and the cyclic boundary condition comes back (i.e. k = 0 term) giving the free energy F as before with the liquid like free energy just tacked on the end. (This corresponds to the A^A^ = 1 condition assumed before.) As fiA(u) — Fgoes through zero to become positive apparently some sort of transition occurs quite catastrophically. When /j,A(
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of polymer the formalism should provide a reasonable approximation to reality because the fluctuations in the system will be small, e.g. if/tA(w) — Fis negative, such as must be the case in swelling experiments. It is this case that will now be solved.

4. E X C L U D E D V O L U M E IN R U B B E R The excluded volume will now be put into the model rigorously and then solved in the spirit of the last section. The inclusion of the intra-chain forces through some potential V(R(S1) — R(S2)} will be modified by a solvent in the rubber and also by 'screening' of other parts of the polymer chains in the network. In short a theory of liquids is needed to solve this problem. However, polymer solutions at intermediate densities (that is not so low that there are large fluctuations in density and not so high that the second virial coefficient is no longer adequate) can become tractable when a simple form of 'pseudo-potential' is used. The basic idea is that the polymer density is comparable to that of a gas whereas the liquid it is dissolved in just serves to modify the interactions between chains. The excluded volume v of any system is defined by

which is just the second virial coefficient. If the system is at sufficiently low density then the free energy may be expressed in terms of this parameter quite adequately. The 'excluded volume' parameter can then be put into the partition function as an effective potential For a polymer solution this leads to a free energy dependence on v given by where m is the number of polymer chains, V the volume, mZ, the total chain length in solution and p the density of polymer (see Edwards 1966). The term which is linear in the polymer density isjust that which would be expected from a gas. The term which has the three halves power is specific to the chainlike nature of the polymer. It is due to interactions which may be drawn like this:

Figure 7

In a network this type of interaction will be modified by the presence of crosslinks, again the same sort of interaction will be present but this would be expected to be enhanced by interactions like:

Figure 8

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Not only then will there be excluded volume forces holding the networks apart which do not depend on crosslinking (as in the Flory-Huggins theory; see, for example, Yamakawa 1971; Flory 1942) but also terms depending on the crosslinking. In other words the free energy of the system is not obtained just by adding the free energy of the network to the free energy of a solution at the same concentration; there is also a cross term The way these 'mixing' interactions add to the free energy could modify both shear and bulk properties. With these thoughts in mind the general partition function will now be written down. It is the same as equation (3.2) modified by the excluded volume in each of the (n + 1) systems

where H^ is the Hamiltonian of the ath system. Whereas before ^xH(cf> was given by

now the ath system has excluded volume

Note, y'°* is the excluded volume of the system at fabrication, all the other t/"> are equal and are the excluded volume of the strained rubber. Therefore

Using the same transformation on the (J?w(^)} systems to go to the {XVn(s)} coordinates one finds that the

However, it is still possible to use the same procedure as in the previous section. The harmonic well potential is introduced as before and ( V — U) in this case is

which is the same as before with excluded volume added in. Averaging this over the {X(f(s)} systems except for the X(®(s) coordinate can be done for all the terms, except the excluded volume term, as before, giving

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where

The

average can be done by rewriting the delta function as

a Fourier transform

taking the lowest eigenfunction approximation for the GW Green function (3.7)

and completing the square. This approximation will be valid at large st — J2 (therefore small k).

Notice that in this approximation the average of the /? ^ 0 systems is independent of s^ — S2, thus allowing the introduction of collective variables for the X(0> system. It is a property of the transformation that

where Introducing collective variables for the centre of mass system

and transforming the excluded volume comes in its final form to be

Equation (4.22) will be denoted by:

The calculation as outlined by (3.84) can now be attempted:

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where 3? is the functional integral given by

The contribution to Z(«) outside the functional integral is given by the same answer as before plus excluded volume terms:

where and lc = minimum loop size. By using (3.22) for V(a)(K), the k = 0 term gives a contribution

(which provides the necessary stability condition). The sum over the k -•£ 0 modes is the self energy of the chain (me for the single system formula, which is the excluded volume equivalent of closed loops on the same chain). This has to be cut off in a similar way to the closed loop calculation due to chain flexibility, the cut offlength being smaller than (to)"1, the full harmonic Green function should be used to average the excluded volume term over the /? ^ 0 systems. Therefore, by using (3.59)

7 say, the self energy is really

On doing the k integration one obtains

The integral is dominated by the behaviour at small \st — s%\ if the cut offlength is smaller than the localization (which is the case for a rubber). Therefore the self energy becomes

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where

So finally This is just the form for the self energy that would be expected (i.e. independent of V and AJ. So for the non-functional part of Z(n) in (4.2) we obtain

where S- is given by (4.26). The evaluation of &, the functional integral, will now be made. The existence of S' will determine under what conditions the model is valid. If S' does not exist then syneresis has occurred, as discussed in the previous chapter. Combining equations (4.22) and (4.26) we have

and on doing the pt. integrals,

where

as in (4.22). The condition for the existence of y being that V(k) — A/i > 0 for all k. Thus

and if the fi integration is done by steepest descents then it is found that where /ia is the saddle point:

Figure 9

Therefore if the /* integration can be done with the contour going through the saddle point such that V—Ps-H > 0, then the even density boundary conditions hold. If the contour has to be closed so as not to include /«„, in order to keep v — A/J, positive, then the non-even density boundary conditions prevail.

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Thus for this model, therefore i.e. and

i.e.

to order n. The k integral in (4.5) is only valid up to k ~
provided w/a > 1. Thus doing the k integral in (4.5) we find

provided i.e.

This term gives the desired mixing of excluded volume and crosslinking. Putting all this back into the formula for Z(ri), (4.35), we find

and since the model is only valid for

then we may expand the exponent in powers of

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giving

Dropping powers of/t greater than the first and then doing the contour integration we find

where comes from Here again a similar argument as for the self-energy calculation should apply. The term is really In {e") and as such would be expected to give

i.e. just the same answer as from n +1 uncrosslinked systems. Thus

Therefore on looking for terms of order n as in formula (2.19) we find

The localization has been unaffected by the excluded volume, t = 6NJIL as before. The closed loop correction to the crosslinks has been modified by the excluded volume at fabrication so that less closed loops form as would be expected. The other JV-dependent term is the 'mixing' term expected and may be thought of as the modification to the heat of mixing by the crosslinks. The other terms are just the contribution from an uncrosslinked solution. The formalism used (collective variables) with the inherent approximations means that the formula derived is valid only when the density of polymer is low enough so that a second virial coefficient (or excluded volume parameter) provides a good description of the system. The density also must be high enough for gel formation above the syneresis point (i.e. there must be no separation of the gel from the solvent during formation of the network) (!/0)Z/2 V > NjL). 30

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(The system is homogeneous.) If the excluded volume effect is very large then presumably the localization of chains is effected, the chains becoming locked into position to form a glass. Restriction to the excluded volume as given by (4.48) sees that the localization is not affected by the excluded volume. The correction to the 'heat of mixing' is linear in N (the number of crosslinks) because the expansion in //, was only taken to first order in (4.52). Near the syneresis point higher ordered terms in fi will be more important thus terms of higher order in N could be expected near syneresis. Thus the excluded volume has contributed both to the shear properties by cutting down the wasted loops at the crosslinking process and the bulk properties by modifying the 'heat of mixing'. The free energy of the network is then

to which the usual solution term is added. The above formula being valid in the lightly crosslinked regime above the syneresis point and at low densities. 5. ENTANGLEMENTS IN RUBBER (Edwards 1967, 1968) In the models considered so far the phantom nature of the chains has been a major defect. Polymer chains cannot pass through one another. At low densities this will be a small perturbation on the phantom chain system, thus for swollen rubbers or rubbers gelled in a solvent at fairly low polymer densities excluded volume effects could well be more important. At higher and higher densities the chains must become more and more entangled and in crepe rubber it is these entanglements that give rise to the elasticity. At sufficiently high densities it could well be that a regime exists where the topological nature of the chains (i.e. their length and the fact that they are not phantom) is far more important than the details of the molecular forces. It is in this spirit that this section will be developed. In setting up the problem a specification for the topology of the network must now include the entanglements of the chain as well as the crosslinks. The entanglements will be specified by invariants. The basic idea is that a knot is a topological classification. A knot in one class cannot be topologically deformed into a knot of another. However, 'invariants' can be written down that also classify topologies, and although this classification is different from the knot classification (it appears an infinite number of invariants are needed to classify one class of knot) the first class of invariant seems an adequate way to specify the entanglement topology of the network. Consider the two diagrams

Figure 10

Figure11

An invariant /12 can be defined that distinguishes between (a) and (b). Figure 11 is the familiar integral of Gauss

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/12 is like a solid angle and is analogous to the scalar magnetic potential. (Note, in order that these Invariants are truly conserved, the chains have to be either infinite in length or closed.) The important point is that when /12 = 4it the loops cannot be pulled apart. A more complicated invariant /123 exists for three loops that can distinguish Borromean Rings (i.e. J12 = /23 = /13 = 0 but /I23 •£ 0). Borromean Rings can be pulled apart if any one ring is removed but not if all three are present.

Similarly there exist self invariants.

and higher invariants 71234 for four loops, etc. Thus a simple knot for one chain has to be specified by (Ai> Au> ^mi • • • ) • The most important restriction comes in the first invariant /12. If this is conserved in the network then hopefully the conservation of the other invariants will have a small effect. /12 is basically a two-body effect, the rest of the invariants three-body and higher. It should also be noted that in the continuum chain model used knots may exist on any length scale with equal 'measure' or weight (i.e. in lengths very much smaller than I the random flight step length). Thus putting in chain stiffness or cutting of integrals at S < I will exclude the existence of these knots which often lead to divergences in the theory. The specification of entanglements being settled on, now the general partition function for the n + 1 systems can be set up. The theory will be for dense rubbers (melts not gels), thus excluded volume effects will be left out since a full theory of liquids would be needed to put molecular forces in as discussed previously, also chain statistics are very little altered in the melt. The starting point will be (3.13)

this specifies the crosslinking. The conservation of the invariant /12 can be put in by the constraints Thus

Since the invariant 7 changes by multiples of 4;i the delta functions in

are Kronecker delta functions and not Dirac delta functions.

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We shall now use exactly the same approach to solve the functional integral for Z(n) as set out previously. The same transformation of variables TJ and the same Green functions will be used. The U— Fin formula (3.4) now becomes

where the ing collective variables

has been rewritten exp ds as before and averaging

Introducover all systems except

The first two terms are unchanged. In fact with excluded volume forces

looks like

So the problem is in the same form as before with the effect of entanglements added into the / n \ exponent as ( S hi#(/[.R(0)] -/[Bw]) ) which now will be expressed in terms of the collective \«-i / variables. Given that / changes by multiples of 471, the Kronecker function can be written as (A/ = 4™, n = 0, + 1, + 2,...). When many entanglements are present each contributing a small effect one can use an expansion for the sine:

Expansion of the Kronecker delta function to this order should be a good approximation provided that the system is well gelled, the justification becoming clearer as the calculation develops. Thus consider It is a property of the invariants that they are conserved under any topological deformation, e.g. /[J?(0>] = /[AH'0'] since the affine deformation does not alter the topology. Therefore if if&iS) is small, which is true when the system is well gelled, then Thus /fR te) ] —/[/?(0)] is a good expansion parameter when the localization is good (i.e. when 01 is large). Now we may use the formula for the invariants to write down the values of (A/)2, where Thus

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It is shown in appendix 3 that, provided the correlation term neglected, and

We may average over all directions of the chain tangents at.% and sz giving

Thus writing (A/)8 in terms of collective coordinates

where *f(K) is infinite but it must be recalled that the entanglements only have meaning on length scales greater than one step length /. Thus -f(K) may be cut off by taking R~* to be a phenomenological parameter C times the function

C being the order l~l. Thus

Averaging over the X(?> systems and introducing

by using This gives Now the invariant conserving constraint comes into the expression for the generalized partition function as In collective variables with the Kronecker delta function expanded to first order this looks like

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This is a finite product expression for the n systems and excludes terms

and those of higher order i.e.

A better approximation for 3(1) could be thought to be S(I) — e~(13), but this is no good because this includes

as may be seen on expansion,

which is correct to first order in 72 but includes the terms

It is important to exclude these terms because they come into the final answer at order n which means they contribute to F. Therefore an improved approximation to 5(/[/?
which in the limit A£-s»0 tends to exp( — '^lI2pkp_!l), but writing the full expression for the constraints, which explicitly excludes terms

because of the finite product nature of

the form. Thus we may write

This approximation has the correct properties to order n in the exponent. Now we may write down the full expression for Z(n) including entanglements using (5.8) and

In order to do this functional integral the even density assumption must be employed, even without excluded volume (the polymerization takes place at the even density of the monomer, thus locking the entanglements in at uniform density). The difference between this and the phantom

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chain calculation being that once the entanglements are locked in stability is given to the rubber. The rubber cannot 'clump up' into the centre of the box because the entanglements do not allow it. The short range forces S VkPtcP-k damp out large fluctuations in the density as argued in chapter two, thus

(even without crosslinks and short range forces this gives

for small n). Thus the crosslinks give the same result as before, (3.69),

with where the main contribution to the elasticity for the crosslinks will be from the k = 0 mode as before. The entanglements yield Using (5.21) for/2(£, w, n) and putting n = Oin the / 2 (k,
Expanding the logarithm out in a series about the X^ contribution since the fluctuations (i.e. i/'s) will be small, gives

Where the first term is entirely due to the JT(0) or 'centre of mass' coordinate and the other terms are due to localization around the X((fl coordinate. Only terms of order iju>i(ijii)i =
Thus rewriting (5.38) we have

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where The integrals have to be cut off at some k = ljle, where the cut off represents some property of chain stiffness. Thus (5.11) becomes on doing the integrals

(using (pkp-,c} = 6jO/tt2 for n = 0-see (3.8)). p = LjV, density at fabrication. The Z'°' contribution to the entanglements has been to give the well known Mooney-Rivlin (Mooney 1940; Rivlin 1948, 1949) term at the k — 0 mode, plus a term of the opposite sign due to the fluctuations in the A'(0) system. The fluctuation in the XW systems may now be calculated and the calculation completed. The JfW systems contribute approximately

where the sum is to be cut off at large k = l//e.

Sum

le is like a self-entanglement length. Putting all this in the formula for Z(ri), (5.32), we have

where Thus applying (2.19) to find the first order in n

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349

As before,
Therefore

When crosslinks dominate the localization: (independent of A4 as before) and when entanglements dominate:

self-entanglements not dominant: which for small strains eit (A; = 1 + e,-) and taking terms linear in e's only gives

Therefore the free energy is given by (5.46) with
where_f(«) is given by the w dependent terms in (5.46) but has two limiting cases: (1) Highly crosslinked rubber

(2) Lightly crosslinked rubber

for small strains. Summarizing equation (5.52) we have the normal crosslink contribution plus a term due to the affine deformation on the entanglements plus another term due to the non-affine part of the 31

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deformation on the entanglements. The/(ai) is due to the fluctuations about the mean deformed position of the chain. As can be seen from (5.52) an equilibrium is assured by the entanglements and excluded volume is no longer needed to provide the equilibrium. It should be noted that the density, p, in all these equations is the density at fabrication (i.e. independent of the A's). Various special problems could now be studied, for example the problem of polymer ring molecules formed at high density so that they are entangled. The only change in the mathematics being that (pkp_k') is now given where L now is the length of circumference of the ring. This solid would exhibit the MooneyRivlin term but the non-affine term and the fluctuation term would be different.

6. CONCLUSIONS The defects of the phantom chain model of a rubber have the consequence of not allowing an even density state for the rubber, unless the even density state is artificially imposed on it through cyclic boundary conditions. If this is done the rubber is not in equilibrium. Modification of the model to include entanglements and excluded volume forces give rise to the equilibrium even density state. At low polymer densities excluded volume effects dominate in providing this even density, at high densities entanglements alone are sufficient. The localization of the polymer chains which is so necessary in gel formation, for if the chains did not localize then the network would remain a liquid, has been found by a self-consistent field approach. The simple physical argument used by Flory to justify the — Nln A,,, AK As term in the free energy can only be used for solutions not gels. The argument is that each crosslink takes away one degree of freedom and the chains are in a volume VXX Ay Xs. Therefore the free energy must have a term like — A^mA^A^A;, in it. This is right provided the system has not gelled (and therefore is not a solid) having the whole volume FA^A^A^ available to it. In a gel, however, each monomer of the chain is localized in a volume of («.,.&>„«a)-1, thus jVln (
APPENDIX 1 It is required to calculate Now

therefore (all coordinates except tffi),

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351

where the integration is over all coordinates except rff. Now the average may be calculated

where ./f~ is the normalization.

which averaging over all ^ and st given 0 ^ s} ^ ss =S L yields

APPENDIX 2 An example of the transformation T£(t where Xf (s) =

APPENDIX 3 To calculate (jKpfa) Rf^s^y. Introduce the variables

T&Kf^s)

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So that

Thus

Also since Then

But

therefore neglecting the e~""'3' '"i-8"1 terms we have the desired form. REFERENCES Dolan, A. K. & Edwards, S. F. 1974 Proc. R. Soc. Land. A 337, 509. Dolan, A. K. & Edwards, S. F. 1975 Proc. R. Soc. Land. A 343, 427. Edwards, S. F. 1966 Proc. Phys. Soc. 88, 265. Edwards, S. F. 1967 Proc. Phys. Soc. 91, 573. Edwards, S. F. 1968 J. Phys. A 1, 15. Edwards, S. F. 1970 Statistical mechanics of polymerized materials. In 4//i International Conference on amorphous materials (ed. R. W. Douglas & B. Ellis). New York: Wiley. Edwards, S. F. 1971 Statistical mechanics of rubber. In Polymer networks: structural and mechanical properties (ed. A. J. Ghompff & S. Newman). New York: Plenum Press. Edwards, S. F. & Freed, K. F. 1969 J. Phys. A 2, 145. Edwards, S. F. & Freed, K. F. 1970 J. Phys. G 3, 739, 750, 760. Feynman, R. P. & Hibbs, A. R. 1965 Quantum mechanics and path integrals. New York: McGraw-Hill. Flory, P. J. 1969 Statistical mechanics of chain molecules. New York: Interscience. Flory, P. J. 1953 Principles of polymer chemistry. Ithaca: Cornell University Press.

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THE THEORY OF R U B B E R E L A S T I C I T Y Flory, P. J. 1942 J. chem. Phys. 46, 132. Flory, P. J. & Wall, F. T. 1951 J. chem. Phys. 19, 1435. James, H. M. & Guth, E. 1943 J. chem. Phys. 11, 455. Mooney, M. 1940 J. appl. Phys. 11, 582. Rivlin, R. S. 1948 Phil. Trans. R. Soc. Land. A 241, 379. Rivlin, R. S. 1949 Phil. Trans. R. Soc. Land. A 242, 173. Yamakawa, H. 1971 Modern theory of polymer solutions. New York: Harper & Row.

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17 SAM EDWARDS AND THE STATISTICAL MECHANICS OF RUBBER Paul M. Goldbart and Nigel Goldenfeld Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois 61801-3080, U.S.A. Abstract

The statistical mechanics of cross-linked macromolecules requires simultaneous treatment of random polymer configurations, excludedvolume interactions, and the quenched disorder of the cross-links, as well as the topological constraints imposed by impenetrable chains. Such a description was pioneered by Beam and Edwards, who attempted to derive the stress-strain relationships of rubber from first principles. We review this work and describe subsequent efforts to understand the unique elastic properties of networks as well as the critical phenomena of the vulcanization transition. 17.1

Introduction

In 1967, with self-avoiding walks and polymer solutions under his belt, Sam Edwards turned his attention to the elastic properties of vulcanized macromolecules. His aim was to explain the remarkable nonlinear elastic response of rubber from considerations of fundamental statistical mechanics. More precisely, Edwards wanted to develop a semi-microscopic theory which, in modern parlance, described the universal properties of rubber that emerged from any polymer system described by the following minimal model: (i) The atomic constitution of the macromolecules is encoded into a pair of phenomenological parameters, the length of each macromolecule and its orientation persistence length; (ii) the interactions between the molecules are idealized via the Edwards excluded-volume interaction; (iii) the random covalent bonds forming the crosslinks are idealized as constraints co-locating the connected parties; and (iv) the polymer chains may not pass through one another, ensuring that any non-trivial topology of a polymer network is conserved. Edwards found the rubber problem significantly more complicated than his earlier forays into polymer science, and with characteristic ingenuity devised a variety of novel techniques and physical pictures to meet the challenge, all of which have survived as essential components of our current understanding, as well as being of general importance in statistical mechanics (Edwards 1971, 1972). Unquestionably, the massive paper of Beam

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and Edwards (1976) represents a watershed in the theory, which for the first time incorporated all of these developments, and accordingly has been reprinted for this volume. In addressing this topic of vulcanized matter, Edwards exhibited three skills that he has drawn on over the entire course of his career to date. The first one might call Newtonization—the ability to identify new areas of science ripe for upgrading from a qualitative to quantitative description. The second is mathematization—the art of developing formulations and minimal models that capture the essence of the phenomenon at hand, and are sufficiently amenable to mathematical analysis that the topic at hand is illuminated and stimulated by results that come forth. The third one might call, whimsically, bulldozerization— the art of making brutal and unjustifiable approximations that still preserve the characteristics of the model being treated and enable a mathematical caricature of the solution to be developed. The problem of rubber is especially interesting because, in our opinion, this program has not been carried through to completion. Why this is the case can be appreciated from the degree of complexity exhibited by the system at hand. Consider, then, the following necessary ingredients of even a minimal model: • A strongly interacting system of macromolecules (a solution or a melt) undergoing thermal motion provides the first level of randomness: Brownian motion. In addition, one must incorporate the interactions between the chains that give rise to the solution or melt properties, as described in the present volume by Robin Ball and by Yoshi Oono and Takao Ohta. • Cross-links connecting the macromolecules, characterized by quenched random variables that specify which segments are constrained to lie near one another, provide a second level of randomness. • This holonomic quenched random information should, in principle, be complemented by anholonomic, or topological, information specifying how, in the presence of a given set of cross-links, the macromolecules are interwoven. That is, the topology of the network is invariant after formation, and somehow needs to be specified and averaged over in calculating the statistical mechanics. Despite serious efforts, this aspect remains an open challenge, but the related topic of topological field theory has made progress in recent years, as described in the present volume by Edward Witten. • The third level of randomness is associated with the state of matter that emerges, when a sufficient density of cross-links is introduced: the amorphous (or random) solid state. We shall describe below how this state can be described and its properties computed, both analytically and computationally. Because rubber is inherently maintained in a nonequilibrium state—more precisely a frozen and constrained equilibrium state—Edwards recognized very early that it was necessary to consider explicitly the process of formation. In the vulcanization process of Hayward and Goodyear, very long lived covalent chemical

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bonds (i.e. cross-links) are introduced between randomly selected atoms on randomly selected pairs of macromolecules so as to build up essentially permanent networks (i.e. clusters) of macromolecules having random architectures. If present in sufficient numbers, these cross-links induce the formation of a macroscopically large network. Such a network is, of course, distinguished by its connectivity; but what is remarkable and unprecedented is that this connectivity is associated with a thermodynamic phase transition from a polymer liquid to a new state of matter that has a non-zero modulus for small static shear deformations, i.e., is an equilibrium amorphous solid state. Rubber is an equilibrium state, in the sense that it inhabits a window of time-scales, ranging from the relaxation time for Brownian fluctuations of the macromolecules to the long time-scale describing the ultimate rupture of the covalent bonds enforcing the cross-links and even the integrity of the original macromolecules. This latter time-scale is potentially very long, compared with that of typical laboratory experiments, and for most practical purposes can therefore be regarded as practically infinite. The state is amorphous, in the sense that no two neighborhoods in the sample are identical, in contrast with, say, a crystalline sample. This is reflected in the absence of long-range periodicity in the mean locations of the atoms, as well as in the continuous distribution of the rootmean-square thermal displacements of the macromolecular segments. Rubber is solid meaning that it has a non-zero static shear modulus at the level of linear response. In a sense, vulcanized matter provides the only example of a real solid beyond the linear response regime, because as long as the cross-links are permanent, the impenetrability and topological entanglement of the polymer chains ensures that a static shear deformation can never vanish due to creep or other diffusive effects associated with vacancies or grain boundaries, in contrast with the case of a crystalline solid. Edwards recognized that vulcanized matter also provides an example of a glassy material, perhaps even 'the theorist's ideal glass,' to use his memorable turn of phrase, because it was possible to pinpoint precisely the origins of the equilibrium amorphous state; moreover, because there was no question that this system was in equilibrium in the sense described above, dynamical questions that bedevil the theory of conventional glasses could be finessed. However, Edwards' goal was more ambitious than providing a theoretical description of an esoteric emergent state with broken symmetry: Edwards was attracted to the problem because of the paradoxical nature of early attempts to explain the entropic linear elasticity theory of rubber, as well as the nonlinear elastic response of networks. A review of the field prior to Beam and Edwards' work has been given by Treloar (1973). These early theories, of which the most influential were those of Flory and Wall (1951) and James and Guth (1943), used 'phantom chain' models of polymer networks. As Beam and Edwards dryly noted in their introduction: '... phantom chains can pass freely through one another (and themselves); secondly, there are no molecular inter or intra chain forces. These effects have now been put into the model and the theory developed.'

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In other words, both entanglements and excluded-volume interactions were missing. The interactions are similar in spirit to those Edwards had already encountered and successfully dealt with in polymer solution theory, but the entanglements represented a fundamental new difficulty, in terms of their representation as well as their actual physical consequences. Edwards made several attacks on the entanglement problem, starting in f 967 with two very different approaches. In the first approach, Edwards (1967a) attempted to represent an entanglement as an explicitly conserved quantity that enters into the partition function as a delta-function constraint. For a real rubber, such a constraint would be a fantastically complicated expression depending upon the conformations of all the chains. Or in Edwards' (1967a) own words: 'There exist several substances such as rubber, glass and polymerized materials, whose molecules possess topological relationships to one another, and these relationships are either permanent, or survive long enough to be considered permanent in the calculation of thermodynamic properties.... The full problem of calculating, say, the equation of state of a glass, which will contain elastic constants depending on the way the various silica chains are linked up, is a very difficult problem or rather series of problems for different conditions. In this paper, therefore, the very simplest problems having the topological specification as their key ingredient will be studied.'

Accordingly, Edwards actually treated the statistical mechanics of a polymer in two dimensions with a point obstacle, discovering that the constraint could be precisely formulated in this case and entered the formalism in a manner akin to a gauge field. In the following year, Edwards investigated whether or not topological constraints could be represented by invariants (Edwards 1968, reprinted in this volume), functionals of the configurations having the desired property that each distinct topology corresponded to a different value of the functional. Whether or not such invariants exist is an open problem, but it has been known since Gauss (1877) that one can construct invariants in the sense that deformations of the linked curves do not alter the value of the invariant. The value of such invariants does not fully distinguish topologically distinct configurations, however; e.g., the Gauss invariant has value 0 for two unlinked closed curves, but this value is shared by 29 of the first 91 simple links (Michels and Wiegel 1986). Thus, the value 0 does not imply the absence of linkage, although the absence of linkage implies the value 0. Edwards (1968) attempted to calculate the probabilities of polymer configurations having a specific topology, introducing ideas that have close connection with recent developments in topological field theory, and which are reviewed in this volume by Edward Witten and in the book by Nechaev (1996). As far as networks are concerned, Edwards (1968) was surely discouraged by the complexity of the analysis and concluded that 'cruder methods will probably be needed to handle them (e.g. Edwards 1967b)' (citation in original). However, he was not discouraged enough it seems, because Beam and Edwards (1976) attempted to work with the topological invariants, acknowledging that in spite

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of their deficiencies, they seemed an 'adequate way to specify the entanglement topology of the network.' The second approach that Edwards introduced in 1967 is the idea that the topological constraints around a given polymer should be thought of as being represented in a mean-field sense as a confining tube (Edwards 1967b). Mathematically much simpler than the topological-invariant approach, the tube model has had enormous impact not only on problems of rubber elasticity, but most famously in the rheology of polymer melts. A review of this body of work can be found in the chapter by Graessley and McLeish in the present volume. The impact of the tube model in rubber elasticity has grown in recent years (Marrucci 1981; Gaylord 1982; Gaylord and Douglas 1987, 1990; Edwards and Vilgis 1988; Heinrich et al. 1995), as it has been augmented by a variety of phenomenological refinements, accounting for non-affine deformation (Rubinstein and Panyukov 1997), sliding slip-links representing the redistribution of polymer segments along the confining tube (Ball et al. 1981; Rubinstein and Panyukov 2002), and even double-tube models where cross-links and entanglements are treated on the same footing (Mergell and Everaers 2001). The success of these semi-empirical approaches is difficult to assess, regardless of their conceptual foundations. At the time that Beam and Edwards did their work, deviations from the classic form of the stress-strain curves predicted by phantom network theory were frequently represented in the Mooney-Rivlin plot (Mooney 1940; Rivlin 1948, 1949). Here, one considers a pure strain, with deformation ratio in the three principal directions \i(i = 1, 2, 3) produced by a specified force. For a simple uniaxial extension A in the x-direction of an incompressible unit cube in its unstrained state with a force F per unit unstrained area, the form

where C\ and Ci are fitting parameters, is frequently found to be quite a good fit to experiment. Beam and Edwards were presumably pleased to report that the results of their heroic analysis, including entanglement effects [eqns. (5.46) and (5.48)] reproduced the Mooney-Rivlin form with the addition of an extra term arising from non-affine deformations of the polymer chains. Buring the last twenty years or so, with the proliferation of fundamental theories of rubber elasticity as well as more phenomenological models, more exacting experimental data are available on better characterized networks, such as endlinked poly(dimethylsiloxane) networks, and it is clear that the Mooney-Rivlin form is not a particularly accurate representation of uniaxial strain behaviour (Kawamura et al. 2001). Moreover, it it now known that simple uniaxial extensions are not a strong test of the predictive power of rubber elasticity theories. As pointed out by Gottlieb and Gaylord (1983), all possible deformations of an incompressible network can be represented in terms of biaxial strains. Comparison of theory with experimental data for such strains provides the greatest possible resolution (Gottlieb and Gaylord 1987; Urayama et al. 2001), and indeed

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it seems that the Edwards-Vilgis slip-link tube model (Edwards and Vilgis 1986) has the best reproducibility over the range of deformations studied, although even with 3 adjustable parameters there are qualitative disagreements with the experimental data in some cases. As should be clear from the preceding discussion, the Beam and Edwards goal of deriving from first principles the stress-strain characteristics of rubber has not been achieved, without recourse to phenomenological modelling or assumptions. We have highlighted the difficulty of representing and incorporating topological constraints, but there is another even more fundamental reason for the failure of the early theories of rubber elasticity (e.g. Kuhn 1936; Wall 1942; James and Guth 1943; for a review see Treloar 1973): they do not have a proper description of the undeformed solid state. Rather, the early workers assumed that, as the cross-link density increased above a critical value, the network would be a bona fide solid, and thus exhibit elastic response. Yet, they did not demonstrate that or how this happens; and this turns out to be crucial, because such theories regard the polymer molecules, and certainly their cross-links and entanglements, as somehow embedded in a medium that deforms in an ad hoc way under strain—e.g., affine or non-affine. Of course this medium is fictitious but, at least in a rubber, there is only the polymer itself. The elasticity and the way in which elements of the network deform under strain is an emergent feature of the solid state, and a comprehensive theory would not require extra assumptions. This point was clear to Beam and Edwards, who calculated the localization of chains and cross-links in the solid state, and thus provided the first sound description of the rubbery state. Of less interest to Edwards at the time was the issue of the transition from the liquid to the solid state, which can be addressed in the Beam and Edwards model from an order-parameter point of view (Goldbart and Goldenfeld 1987, 1989a, 1989b; for a recent review of this and subsequent work, see Goldbart 2000). Whilst many of the features of the transition to the solid state are now understood, particularly the static properties (but less so the dynamics), a complete theory capable of addressing, e.g., time-dependent and nonlinear response to deformations in the well-cross-linked regime remains a task for the future. Now that we have introduced the Beam and Edwards work, let us proceed to a more detailed study of its innovations and formalism.

17.2 17.2.1

Edwards' formulation of the statistical mechanics of vulcanized macromolecular systems Idealized model

Edwards formulated the statistical mechanics of vulcanized macromolecules via the following semi-microscopic approach: • Begin with a single realization of the physical system: a macroscopic collection of J identical macromolecules in equilibrium in a D-dimensional hypercube of volume V subject to a macroscopic number E of cross-links.

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Model the macromolecular configurations as a collection of continuous random paths Rj-(s) (with f < j < J and 0 < s < I) in D-dimensional space, thus dispensing with the detailed chemistry of the macromolecules. Furnish the collection with a Boltzmann weight, a Wiener measure,

parametrized via the chain length L and the persistence length I, to account for the permanent structure of the individual molecules. Augment the statistical weight via an effective Edwards delta-function interaction,

parametrized via a single excluded-volume parameter aD to account for the steric repulsion between all pairs of chain segments. Model the cross-links via a random number E of constraints

that force spatial co-location of randomly-selected pairs of arc-length locations {se,s'e} on randomly-selected pairs of chains { j e , j ' e } (with e = 1 , . . . , E). The effect of these is to set to zero the weight associated with configurations that fail to satisfy all the quenched random constraints, whilst leaving untouched the relative weights of configurations that do satisfy them. Thus, Edwards dispenses with the detailed chemistry of the bonds constituting the cross-links. Phis approach leads quite straightforwardly to the partition function

for the system of interacting macromolecules, where / _DR denotes functional integration over the chain configurations, suitably normalized, if necessary. This partition function depends, inter alia, on the collection of variables {je, s e ;j' e , s'e} specifying the quenched random constraints. In fact, the account we have just given does not quite describe the formulation of the statistical mechanics of vulcanized matter given by

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Beam and Edwards (1976). A minor distinction is their restriction to three spatial dimensions, made naturally—given the aims of the day—but relaxed here. This provides access to the properties of vulcanized matter over a range of dimensions, including the particularly interesting cases of two (the lower critical dimension for the vulcanization transition) and six (its upper critical dimension), both of which are discussed further, below. More substantial is the DeamEdwards choice of considering one super-long macromolecule that spans the container, rather than a system comprising a thermodynamically large number of microscopic macromolecules. Again natural, given the aims of the day, this choice amounts to the introduction of a large number of non-random constraints that join the constituent molecules head-to-tail, so as to contruct the super-long macromolecule. Whilst presumably of little effect for the well-crosslinked regime, this device obscures the physics of the transition, triggered by the density of cross-links, from the liquid state to the amorphous solid state, exhibited by rubbery systems. 17.2.2 Quenched disorder: Treating the cross-links statistically Because of this dependence on a macroscopic number of quenched random variables this partition function is not, in practice, a very attractive quantity to work with, and neither is its logarithm, the free energy (up to a factor of minus the temperature T). Just as Edwards and Anderson did in the setting of spin glass, in the setting of vulcanized macromolecular matter Edwards recognized that a more useful quantity would be the free energy, averaged—with respect to a suitable distribution—over the quenched randomness. Why the free energy and not the partition function? Put briefly, the reason is that the free energy is a thermodynamically extensive quantity. Its consequent additive character allows one to interpret the averaging of the free energy over the disorder as the result of dividing the macroscopic system into regions each of size larger than an appropriate fluctuation correlation length. Such regions are then statistically independent from one another and each has its own realization of the quenched randomness, so that averaging over the sample is equivalent to making an average over the probability distribution of the disorder. The alternative, i.e., an average of the the partition function over the quenched randomness, would have the effect of treating the cross-link specifications as equilibrating variables, variables on the same footing as the macromolecular coordinates, which explore their configuration space during the course of measurements performed on macroscopic time-scales. This is not, of course, what Edwards was envisaging to be happening physically. Rather, he was envisaging settings in which the quenched randomness is unchanging over the course of an experiment. 17.2.3 Handling the quenched disorder via replicas In accordance with the original Deam-Edwards approach, let us follow the earliest view of replicas, viz., as a device for constructing the free energy averaged over

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the distribution of quenched randomness. This hinges on a well-known property of the power function,

which enables one to write

where the brackets [• • • ] denote an average over the quenched disorder, and shows that one would be well positioned if one could make progress with the computation and interpretation of the disorder-averaged replicated partition function (Zn\. The replica method entails writing Zn as a product of a positive integral number n of copies of the original partition function,

where the product over replicas extends over the last two terms, i.e., in terms of n independent copies of the original system, each with the identical realization of the quenched disorder, and then passing to the n —> 0 limit. With it, one is able to perform the disorder average before the Gibbs sum over configurations, and doing so yields

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Typical of replica theories, this generates an effective partition function associated with an effective system in which there is no quenched disorder but there are replicas and couplings between them. f 7.2.4 Modelling the statistics of the cross-links Now, in choosing to consider the disorder-averaged free energy, Edwards was putting himself in the position of having to adopt a model for the distribution of the quenched random constraints, viz., the cross-links. Recall that in other settings involving quenched randomness, such as spin glasses or electron motion in disordered conductors, one often chooses a distribution for the quenched randomness on the basis of the tractability that it confers on the subsequent analysis, provided, of course, that the distribution is physically reasonable. Gaussian distributions of quenched random exchange interactions are commonly adopted for spin glasses (see the contributions to this volume by Anderson, Sherrington, Parisi, and Mezard), although significant consequences can arise when alternative distributions are chosen, such as the dilute distribution adopted by Viana and Bray (1985); this particular choice happens to share certain features with vulcanized matter. Quenched random potential fluctuations are often modeled as Gaussian with short-ranged correlations when analyzing electron motion in disordered conductors. At least when considering critical phenomena, the mechanism of universality often supports the idea that the details of the quenched randomness are not important, and that one should adopt the simplest model in the appropriate universality class, presumably characterized by a very small number of independent parameters. Moreover, one normally assumes that the sample-preparation conditions permit one to understand the statistical features of the quenched randomness separately from the statistical mechanics of the freedoms that are to be subjected to forces or constraints described by quenched random variables. How do these issues play out in the setting of vulcanized matter? Imagine a solution or melt of macromolecules in equilibrium, and begin some process, such as radiation bombardment or chemical cross-linking, by which nearby chain segments become bonded together covalently (and hence essentially permanently), i.e., cross-linked. Terminate the process after some time interval, after which some density of cross-links has been introduced. Evidently, the probability distribution for the quenched disorder, i.e., the distribution governing the likelihood of finding a specific number of cross-links forcing co-location of specific pairs of chain segments, is rather complicated, depending on the non-equilibrium, non-stationary properties of the irreversibly evolving macromolecular system. Beam and Edwards found an elegant resolution of this issue of identifying a physically reasonable and yet analytically tractable model for the statistics of the cross-linking process. They considered a solution or melt of macromolecules in equilibrium, but imagined an essentially instantaneous process in

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which cross-links are introduced, with some independent probability, between all pairs of chain segments that happen to be close at some particular instant. The larger the probability, the larger the density of constraints introduced. The resulting probability distribution governing the likelihood of finding a specific number of cross-links that co-locate specific pairs of chain segments is not known at this stage; but a moment's reflection reveals that all the information needed to construct the cross-link distribution is already contained in the high-order correlator of the uncross-linked macromolecular system that determines the likelihood of finding the to-be-cross-linked chain segments nearby one another, augmented by a simple factor governing the likelihood that they are then cross-linked. The former factor is, up to normalization, none other than the very partition function that we are seeking; the latter is a simple Poissonian factor. So the probability distribution for the quenched randomness is proportional to

As we shall see shortly, this tight connection between the distribution governing the quenched disorder and the partition function we are seeking conveniently extends the replica permutation symmetry from Sn to S*n+i. Note that one can easily envisage experimental situations for which the Beam and Edwards distribution would need to be generalized, thus reducing this symmetry, e.g. if the conditions at cross-linking, such as the temperature, differ from those at which measurements on the sample are made, or if the sample is deformed after cross-linking, as it would be in elasticity experiments. Some other generalizations of the Beam and Edwards cross-link distribution have been explored by Broderix et al. (2002), and these have been useful in accessing the well-cross-linked regime. We now explore the effect of the Beam and Edwards distribution on the disorder-averaged free energy. Focusing on the term [• • •] in eqn. (17.9), which couples the replicas, and making use of the distribution for the quenched randomness, eqn. (17.10), one finds

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This straightforward exercise shows the form of the effective inter-replica coupling to be a short-ranged attraction, operative only when all Cartesian components of all replicas of a pair of chain segments are co-located. 17.2.5 Effective pure theory of coupled replicas What does the effective pure theory of coupled replicas look like? By using the result (17.12) for the disorder average of the term that couples the replicas, and recalling the basic replica limit (17.7), one sees that the replica approach gives the disorder-averaged free energy in terms of the partition function Zi+n for a system of n + 1 interacting replicas:

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Let us pause to examine the form of this effective theory. There is one more replica than usual—the zeroth replica—present to construct the distribution of quenched disorder. Notice that it appears symmetrically, relative to the standard replicas, a consequence of the cross-linking process envisioned by Beam and Edwards. The replicated Wiener measure ensures the integrity of the macromolecules in each replica. The replicated excluded-volume interaction tends to maintain homogeneity of the density separately within each replica. The term coupling the replicas tries to align their configurations so that the disorder contribution to the free energy is minimized. Thus, the latter two terms compete with—or frustrate—one another. The form and structure of the amorphous solid state—the state of matter that emerges when a sufficient density of cross-links is introduced—is determined by the resolution of the tension between these two terms. Let us also pause to examine the symmetry content of the replica theory, as the pattern in which symmetry breaks determines the nature of the amorphous solid state. By examining eqn. (17.14) one sees that the theory is invariant under independent translations of the replicas, independent rotations of them, and (discrete) permutations of them. It would also be invariant under (continuous) rotations that mix the replicas, but this symmetry is explicitly broken by the excluded-volume interaction. 17.3

Predictions of the Dearn Edwards theory

Beam and Edwards evaluated the replicated partition function using a variational method based on the equality (exp(x)} > exp((x}), which is widely used in the evaluation of partition functions, generating an upper bound on the free energy. The choice of propagator used in the evaluation of the average is significant: a propagator describing chains anchored at the walls of the system has a density profile that falls to zero near the walls and rises to a maximum value within the centre of the system. On the other hand, a propagator that is periodic has uniform density within the system. The former corresponds to the formalism of James and Guth (1943), while the latter corresponds to the Flory and Wall (1951) model. These models made additional assumptions: in Flory and Wall (1951), the cross-links are assumed fixed in the elastic medium of the rubber, whereas in the James and Guth (1943) model, cross-links at the surface experience the deformation of the large-scale system, and the cross-links in the bulk fluctuate about mean affinely-deformed positions. Beam and Edwards' model went beyond these assumptions by explicitly including the excludedvolume forces that hold the network out, and participate in stress transmission. They were the first to understand that the excluded-volume interactions and the cross-links effectively compete in the one replica sector (see Section 17.4), giving rise to a phase diagram for a network that includes the possibility of collapse (i.e. syneresis). Beam and Edwards also attempted to include the effects of entanglements, in Section 5 of their paper, giving rise to a Mooney-Rivlin form for the free energy of a deformed network. The Beam and Edwards model

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was subsequently approached without using replicas by Solf and Vilgis (1995), who softened the delta-function constraints in the Beam and Edwards partition function using a Gaussian regularization. Subsequent work (Ball and Edwards 1980) used a somewhat different approach to the Beam and Edwards model, invoking collective coordinates throughout to treat semi-dilute gels and dense rubbers where screening effects are important. The interesting question of whether or not there is a network analogue of the universal equation of state found in semi-dilute solutions (see Ch. 8 by Oono and Ohta) has not been explored to our knowledge, and would be an important way to test the starting point of theories of rubber elasticity at infinitesimal deformations, which in our view, is where the most basic level of confrontation between theory and experiment should be carried out. Further significant work on the problem of a super-long randomly cross-linked macromolecule, formulated in Beam and Edwards (1976), has been done by Panyukov and Rabin and their co-workers. For a detailed account of this work, readers are referred to their review (Panyukov and Rabin 1996).

17.4

Nature of the vulcanization transition

Although not amongst Edwards' original aims, his approach to the statistical mechanics of vulcanized matter can be invoked to develop a picture of the phase transition from the liquid to the amorphous solid state that occurs upon sufficient cross-linking. We shall see that the replica theory not only predicts the order of the transition, but predicts (Peng and Goldbart 2000; Janssen and Stenull 2001; Peng et al. 2001) that the universality class is that argued for by de Gennes previously, namely bond percolation (de Gennes 1977). This latter development is completely non-trivial, because the description based upon Beam and Edwards describes thermodynamics and architecture, whereas percolation focuses exclusively on architecture. That rigidity can emerge from a random collection of bonds that are themselves rigid is not especially surprising. However, that rigidity can arise from the assembly of a bunch of floppy objects in thermal equilibrium is rather more spectacular, and requires explanation. As we shall discuss more in Section 17.5.1, the description of the critical phenomena also predicts the structure and heterogeneity of the solid state, its response to shear deformations, and the effects of Goldstone-type fluctuations of the solid state. In order to obtain a field theory description, we need to use collective coordinates defined on replicated real space, similar in spirit to those used in semi-dilute solutions (see Ch. 8 by Oono and Ohta):

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or equivalently replicated wave-vector space,

These collective coordinates were first introduced in the present context by Ball and Edwards (1980) in their work on screening phenomena in vulcanized matter. The physical interpretation of their expectation values, and especially their ability to detect and diagnose the random solid state, will be described below, in Section 17.5. There are several schemes for moving from the semi-microscopic description in terms of replicated macromolecular chain coordinates, eqn. (17.14), to a description in terms of the collective coordinates f2. One way is via the insertion of a delta functional, which serves to relate f2 to the chain degrees of freedom via eqn. (17.15) or eqn. (17.16). A second way is to employ Hubbard-Stratonovich transformations to decouple the replicas of each chain from the replicas of each other chain. In this scheme, the decoupling is effected by writing the delta functions of the excluded-volume and cross-link terms in their Fourier representations. Hence one sees that these terms have the form of sums of squares, which are amenable to Gaussian decouping. Regardless of the scheme, one is left with a theory describing the replicas of a single macromolecular chain, interacting with the fluctuating collective coordinates f2. It is helpful to introduce the convenient language of sectors of the field f2(k°, k 1 , . . . , k n ). If all vectors in the argument (k°, k 1 , . . . , k n ) are zero, we say that the field lies in the zero-replica sector (ORS). If exactly one vector in the argument is non-zero, we say that it lies in the one-replica sector (IRS). Otherwise, we say that it lies in the higher-replica sector (HRS). The convenience of this decomposition is as follows. First, it shows that the excluded-volume interaction is decoupled via solely IRS fields. Second, it shows that the IRS contribution to the cross-link term renormalizes precisely the excluded-volume term. A phase transition to the amorphous solid state, if it should occur, would show up as a condensation of the order parameter field fj, i.e., by this field acquiring a non-zero expectation value (which we shall call the order parameter) . More precisely, if the emerging state is to be the amorphous solid state and not, say, a crystal, what should become non-zero is the expectation value of f2 only at values of k 0 ,kV ,k™ that sum to zero: ^™=o k " = °- WnJ" and how this field detects and diagnoses the amorphous solid state is discussed below, in Section 17.5. Let us assume that there is a transition to the amorphous solid state, which is triggered by increasing the cross-link density control parameter jj, through a certain critical value, and let us further assume that the transition is continuous, so that the order parameter grows continuously from zero at the transition. We shall see that this assumption is self-consistent, and that even though there are symmetry-allowed cubic invariants, they do not prevent us from developing a Landau theory description. An analogous

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situation holds for spin glasses and percolation. Cubic invariants are not forbidden, but first-order transitions would yield unphysical states—here a state with a negative fraction of localized particles. Then we can begin building a theory of the transition by expanding, perturbatively in powers of the fj, the usual log-trace term that arises after the introduction of the collective coordinates. In the present setting, this term amounts to the free energy of an effective system comprising the replicas of a single polymer coupled to the fluctuating collective coordinate. It is adequate to retain terms simply to cubic order. It is also adequate to make a gradient expansion, i.e., to expand perturbatively in powers of the wave-vectors, retaining terms to quadratic order in the f2 2 term and only to zeroth order in the f23 term. Thus one arrives at the LandauWilson effective free energy for the amorphous solidification transition (Peng etal. 1998):

where c (=N/V) is the number of entities being constrained per unit volume, and {ar, £,o,g} are, respectively, the control parameter for the density of constraints, the linear size of the underlying objects being linked, and the nonlinear coupling constant controlling the strength with which the fj fluctuations interact. HRS indicates that only wave vectors in the higher-replica sector are to be included in the summations; other degrees of freedom are not critical. The Deam-Edwards (1976) semi-microscopic model of vulcanized macromolecular matter yields a = 1/2, r = (jj? — Mc)/Vc)£o = Li/ID and g = 1/6, where jj, controls the mean number of constraints (and has critical value /xc = 1), i is the persistence length of the macromolecules, and L/t is the number of segments per macromolecule. Before delving into the nature of the amorphous solid state itself, let us pause to glance at a relatively recent result concerning the nature of the solidification transition as it is approached from the liquid side. Larger and larger random macromolecular structures are built up, as the transition is approached, and these induce order-parameter correlations of increasingly long range. Quite naturally, the physical question addressed by these correlations is mutual localization: If we localize a particle (to a particular degree) at some point, over what region of space are we likely to find other particles localized to some extent? Via a renormalization-group analysis of the Landau-Wilson free energy (17.17), Peng and Goldbart (2000) found evidence that the critical phenomenology of the vulcanization transition— or, more specifically, the aspects of it pertinent to the liquid and critical states — were correctly captured by percolation theory, as had been argued long ago by de Gennes (1977). The full demonstration of this point, to all orders rather than just first order in an expansion around

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six dimensions, came shortly thereafter, from Stenull and Janssen (2001) and Peng et al. (2001). Of course, percolation theory, with its single ensemble, does not capture the physical properties of the thermally fluctuating, architecturally disordered, amorphous solid state, and it is to such properties that we now turn. 17.5

The emergent amorphous solid state

17.5.1 Microscopic character The form (17.14) of the inter-replica coupling that emerges from the disorder average suggests that the following entity should play a central role in the theory of the amorphous solid state:

Here, {• • •} denotes an expectation value taken in the presence of a specific realization of the quenched randomness, and [• • •] denotes averaging over the number and specification of the quenched random constraints. The number A and values {k1, k 2 , . . . , k A } of the wave-vectors (which are, by fiat, all non-zero) are free to be chosen and, as we shall see, allow the diagnosis of statistical features of the system. This connection between a coupling and an order parameter is well known from many settings, including magnetism and Gorkov's approach to superconductivity. In these settings, as well as in the present one, the introduction of appropriate collective coordinates provides a direct and informative route via which the connection is made. How does this quantity (17.18) detect and diagnose the amorphous solid state? To see this, imagine a situation in which some fraction Q of the chains are localized, let us say harmonically, with a random R.M.S. displacement £,j(s) from a random mean position (Rj(s)}, the remaining fraction being delocalized. Assembling the contributions to (17.18), one finds

where p, the distribution of localization lengths of the chain segments that constitute the localized fraction, is given by

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Thus one sees that the order parameter (17.18) captures amorphous solid order in much the same way that the Edwards-Anderson order parameter captures spin-glass order: the inclusion of multiple equilibrium expectation values evades the random phases (or, in the spin glass case, random magnetic moment orientations) that would cause the order parameter to vanish. At least for states of the replica theory [eqns. (17.13) and (17.14)] that retain the full symmetry of replica permutations (and no other states have yet been found), one can infer the value of the physical construct (17.18) from the computation of the expectation value of the collective coordinate (17.16). So, what is actually found for (17.19)? At the mean-field level, in which one seeks values of fj(fc) that make stationary the Landau-Wilson effective free energy (17.17), one finds that the form (17.19) precisely accompishes this, provided A is taken to be 1 + n and Q and p(£) obey the (replica-independent) conditions given in Castillo et al. (1994); see also Goldbart et al. (1996) and Goldbart (2000). The physical consequences are as follows. For a subcritical density of cross-links the state of the system is fluid, with all macromolecules delocalized. When the critical cross-link density is exceeded, there is a continuous transtition to the amorphous solid state. This state is characterized by a non-zero fraction Q, growing with the cross-link density, of localized macromolecules, and a specific form for p(£) that describes the heterogeneity of the position-fluctuations of the localized macromolecules. In particular, for all nearcritical cross-link densities, £>(£), when scaled by the most probable localization length, is completely determined by a parameter-free universal scaling function. The universality of this distribution is akin to that of, say, the exponent (3 governing the growth of the magnetization with temperature, at zero magnetic field, as the temperature is reduced below its critical value. Its precise form does not depend on the values of the coefficients in the Landau-Wilson free energy but only on the structure of the theory. These predictions have been found to be in satisfactory agreement with computer simulations (Barsky and Plischke 1996). They are currently being tested in experiments performed on gels and colloidal systems in A. D. Dinsmore's laboratory at the University of Massachusetts at Amherst. Preliminary data (Dinsmore et al. 2002) show reasonable agreement with the theoretical prediction that the distribution of localization lengths should be collapsible on to a universal distribution, for various cross-link densities. They also show fair agreement with the predicted form of the universal distribution, although the data are from regimes insufficiently close to the critical point for good agreement to be expected. Let us pause to emphasize the symmetry content of the form (17.19) for the amorphous solid order parameter. In the fluid (i.e. Q = 0) state, this order parameter takes the value ria=i^k a ,o, which is invariant under independent translations and permutations of the replicas, i.e., is as symmetrical as the replica theory itself. In the amorphous solid state this symmetry is reduced but not completely eliminated. Amongst the independent translational invariances, all are lost except invariance under common translations of the replicas, conferred by the factor This embodies the fact that, whilst being

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microscopically symmetry-free, the amorphous solid state is statistically (and hence macroscopically) homogeneous. Even if the state is not quantitatively of the form given in (17.19), it will still have this symmetry content. What does the order parameter look like in replicated real (rather than wavevector) space? Fourier transformation gives

This form can be interpreted as describing the joint probability density for the positions of an A-fold replicated collection of J macromolecules in a state in which the replicas of a fraction Q of the segments are bound into molecules of a distribution p of sizes £, the centres of mass of which are distributed homogeneously over space. The symmetry of the state under common translations is particularly evident in this formulation: the integration over x induces it. This view shows the formation of the amorphous solid state to be associated with a condensation or bound-state formation amongst replicas, this condensation being central to the Deam-Edwards approach. The replicated real-space picture of the order parameter is especially useful when one comes to look for the Goldstone excitations of the amorphous solid state and their implications, an issue that we shall touch on in Section 17.5.3. 17.5.2 Macroscopic character In Section 17.5.1 some light was shed on the microscopic character of the amorphous solid state via the identification, interpretation and (mean-field-level) computation of the order parameter. This revealed the essential microscopic attribute of the amorphous solid state, at least in sufficiently large numbers of dimensions, viz., the fundamentally random localization of particles. What about the macroscopic character? To address this, at least two strategies can be followed. The first is direct, asking the question: by how much does the free energy of the amorphous solid state increase when the boundaries of the sample are distorted in such a manner as to change the shape of the sample without changing its volume (i.e. a shear deformation)? In other words, is the amorphous solid state rigid and, if so, what is the value of the parameter(s) that characterize its elasticity? Evidently, one must analyze the situation in which the distortion is made after cross-linking (cf. distorting a container of water before and after freezing), a scheme that is readily—at least in principle—accomplished within the replica formalism. In the context of the Landau approach of Section 17.4, this amounts to determing the change in the Landau free energy owing to the change in shape of replicas 1 , . . . , n, computing the resulting change in the classical value of the order parameter, and using this to ascertain the increase in the classical value of the free energy.

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Owing to the deformation of the grid of allowed wave vectors that results from the shape change, and the concommitant reduction in the symmetry of permutations of the replicas (from n+1 to n), this is a surprisingly awkward task. Nevertheless, it can be seen through, both from the starting point of the Landau theory and from the semi-microscopic starting point (Castillo and Goldbart 1998, 2000), and yields the following results. The amorphous solid state is indeed solid, in the sense that at the classical level it possesses a non-zero static shear modulus. At the classical level, this shear modulus is given, up to numerical factors, by k^Tcr3, where r is the fractional amount by which the cross-link density exceeds its critical value. Moreover, the statistics of localization are found to have the following properties. Although the sample is deformed, and with it the mean positions, the probability clouds remain essentially spherical and the distribution of localization lengths is unaltered. This entire approach to the elastic properties of the amorphous solid state constitutes an implementation of that considered from a variational perspective by Beam and Edwards (1976). 17.5.3 Goldstone fluctuations; low dimensions We have seen that there are two alternative descriptions of the liquid to solid transition, one based on an analogy with percolation, the other based on the standard Landau theory of phase transitions, and shown how they lead to a common description of the onset of the solid phase, as far as percolative questions are concerned. Are there any differences between these two approaches? Clearly we would expect there to be differences in the dynamics, which are not intrinsic to the percolation picture. However, there are also differences that arise in the statics. The basic reason is that the transition corresponds to the spontaneous breaking of translational invariance—a continuous symmetry. Thus, the lower critical dimension and the nature of the low-lying fluctuation spectrum are different from that predicted by a percolation model, which does not possess a continuous symmetry at all. In dimensions larger than two, the main prediction of the thermodynamic approach (Mukhopadhyay et al. 2004) is a simple shift of the distribution of (squared) localization lengths. More strikingly, this approach predicts that the lower critical dimension should be 2, while the percolation picture predicts a lower critical dimension of 1 (Mukhopadhyay et al. 2004). At the lower critical dimension, following the standard phenomenology of such phase transitions (see, e.g., Goldenfeld 1992), the low-lying Goldstone-type fluctuations are particularly potent: they lead to the loss of true localization, the suppression to zero of the order parameter, and to unusual power-law order-parameter correlations, similar to those associated with two-dimensional melting systems (see, e.g., Nelson 2001). Thus, one has instead a quasi-amorphous solid state characterized by having a non-zero fraction of quasi-localized particles whose localization lengths diverge only logarithmically with the system size. Such a state possesses rigidity with respect to shear deformations but no true long-range order.

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Between two and one dimensions, the thermodynamic approach suggests the existence of the percolation transition associated with macroscopic cluster formation, but the absence of rigidity. On the other hand, the architectural/percolative approach does not even hint at any new physics until the dimension is reduced all the way to its lower critical dimension, 1. 17.6

Ongoing directions: Dynamics at the liquid to solid transition

We conclude this chapter with a brief snapshot of ongoing activity and further developments. Although there have been significant improvements both in our understanding of the solid state of cross-linked macromolecules and in our ability to model the processes that contribute to elasticity, many fundamental questions remain unresolved. Arguably the most significant development in the subject, from the point of view of fundamental rheology, has been the discovery of power-law behaviour in the frequency-dependent shear-stress relaxation function at the vulcanization transition (Chambon and Winter 1985; Winter and Chambon 1986; Chambon and Winter 1987). This has a natural interpretation in terms of dynamic critical phenomena (Goldenfeld and Goldbart 1992); for a detailed review, see Winter and Mours (1997). The Deam-Edwards model predicts that the transition is a continuous transition (Goldbart and Goldenfeld 1987) through a non-trivial analysis of the replica theory, and this has unusual (for a solid) and at the same time generic (for a continuous transition) implications (Goldenfeld 1992) for the behaviour of the deformation response as the critical density of cross-links nc is approached (Goldenfeld and Goldbart 1992). On the liquid side of the transition, the shear viscosity diverges as r\ ~ e~ 7 , where c = nc — n, while on the solid side of the transition, the zero frequency shear modulus grows from zero as GO ~ e'3. Moreover, at the transition point itself, a variety of universal predictions can be made, including power-law behaviour of the frequency-dependent rheological properties, universal phase lags between the real and imaginary parts of the complex viscosity—a generalization of viscosity that captures both elastic and rheological response in the frequency domain— and scaling relations between exponents describing the critical and near-critical behaviour. There are, however, complications to this superficially rosy picture: the observed exponents do not seem to be universal. For example, the timedependent response G(t) to a small step-function shear-strain decays as a power law in time t with an exponent m whose value is reported (Winter and Mours 1997) to be in the range 0.19 < m < 0.92. Similarly, the divergence of the viscosity near the transition yields in different experiments values for the exponent 7 in the range 0.6 < 7 < 1.3 (Adam et al. 1985; Durand et al. 1987; Lusignan et al. 1995). It is possible that these variations reflect a dependence of the universality class on the concentration. However, there are other sources of variation, even error. For example, it is notoriously difficult to define precisely where the critical point occurs and multi-parameter fits to the data often yield

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a strong sensitivity to the critical-point position (Goldenfeld 1992). In the rheological literature, following Chambon and Winter's (1985) suggestion, the critical point is identified with the point at which is observed power-law scaling with frequency of both real and imaginary parts of the shear modulus. However, small deviations from this point may be hard to detect when there is limited frequencyresolution. Moreover, cross-over effects due to the system not being at criticality can cause complications in the interpretation of data (Goldenfeld 1992). Computer simulations of the static and dynamic critical behaviour of crosslinked networks tend to support the conclusion of this body of experimental work. In a series of papers by Plischke and collaborators (Vernon et al. 2001; Jespersen and Plischke 2003) the critical dynamics has been explored, and recent work has included solvent effects (Plischke et al. 2003). In the latter case, which is relevant to experiments on gels, it is found that the liquid-solid transition is characterized by a divergent viscosity with a non-universal concentration-dependent exponent. The dynamic behaviour near the transition can be understood as reflecting the transport properties of incipient clusters close to the percolation threshold, and recent theoretical work has attempted to treat this using Rouse (Broderix et al. 1999; Miiller 2003) and Zimm (Kiintzel et al. 2003) dynamics. 17.7 Concluding remarks In this chapter, we have followed the development of the theory of networks, starting from the Beam and Edwards article. Bespite impressive technical advances in theory, ingenious simulation schemes, and refined experimental probes, it is fair to say that a complete and fundamental understanding is lacking. We have focused on a small part of the existing literature on the subject; beyond the Beam and Edwards sphere of influence, phenomenological approaches to the elasticity of not only simple polymer networks, but more complex materials have been pursued (for a review, see, e.g., Warner and Terentjev 2003). The subject will continue to expand, and it is clear that the Beam and Edwards paper will continue to be cited and influential for many years to come.

Acknowledgments PMG would like to express his gratitude to the many people with whom he has had the pleasure of collaborating on vulcanized matter (in addition to NG!), including Horacio Castillo, Tony Binsmore, Xiaoming Mao, Alan McKane, Swagatam Mukhopadhyay, Weiqun Peng, Konstantin Shakhnovich, Xiangjun Xing, and especially Annette Zippelius. He would also like to thank Sam Edwards for creating a research field that 'has proved of abiding interest! NG would like to take this opportunity to thank Sam Edwards for his inspiring introduction to the subject-matter of this article, and to express his gratitude for the life-changing experience of being his student. He thanks Richard Gaylord for helpful discussions on the tube model of rubber elasticity. He also thanks the Bepartment of Applied Mathematics and Theoretical Physics at the University of Cambridge, U.K., for its hospitality during the course of this work.

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PMG would like to thank for its hospitality the Kavli Institute for Theoretical Physics at the University of California at Santa Barbara, where part of his work on this article was done. Support from the U.S. National Science Foundation through grants DMR02-05858 and PH99-07949 is gratefully acknowledged. References Adam, M., Delsanti, M. and Durand, D. (1985). Macro-molecules, 18, 2285. Ball, R. C., Doi, M., Edwards, S. F., Warner, M. (1981). Polymer, 27, 483. Ball, R. C. and Edwards, S. F. (1980). Macromol., 13, 748. Barsky, S. J. and Plischke, M. (1996) Phys. Rev. E, 53, 871. Broderix, K., Lowe, H., Miiller, P. and Zippelius, A. (1999). Europhys. Lett, 48, 421. Broderix, K., Weigt, M. and Zippelius, A. (2002). Eur. Phys. J. B, 29, 441. Castillo, H. E. and Goldbart (1998). Phys. Rev. E, 58, R24-27. Castillo, H. E. and Goldbart (2000). Phys. Rev. E, 62, 8159. Castillo, H. E., Goldbart, P. M. and Zippelius, A. (1994). Europhys. Lett., 28, 519. Chambon, F. and Winter, H. H. (1985). Polym. Bull., 13, 499. Chambon, F. and Winter, H. H. (1987). J. Rheol., 31, 683. Dinsmore, A. D., Goldbart, P. M. and Weitz, D. A. (2002). Bull. Am. Phys. Soc. D30.009. Beam, R. T. and Edwards, S. F. (1976). Phil. Trans. R. Soc., 280A, 317. Durand, D., Delsanti, M. and Luck, J. M. (1987). Europhys. Lett., 3, 297. Edwards, S.F. (1967a), Proc. Phys. Soc., 91, 513. Edwards, S. F. (1967b). Proc. Phys. Soc., 92, 9. Edwards, S. F. (1968). J. Phys. A, 1, 15. Reprint in this volume. Edwards, S. F. (1971). In: Polymer Networks: Structure and Mechanical Properties, pp. 83-110. Proceedings of the ACS Symposium on Highly CrossLinked Polymer Networks, held in Chicago, Illinois, September 14-15, 1970. Chompff, A. J. and Newman, S. (eds.). Plenum, London. Edwards, S. F. (1972). In: Amorphous Materials: Papers Presented to the Third International Conference on the Physics of Non-crystalline Solids held at Sheffield University, September 1970, pp. 279-300. Douglas, R. W. and Ellis, B. (eds.). Wiley-Interscience, London. Edwards, S. F. and Vilgis, T. A. (1986). Polymer, 27, 483. Edwards, S. F. and Vilgis, T. A. (1988). Rep. Prog. Phys., 51, 243. Flory, P. J. and Wall, F. T. (1951). J. Chem. Phys., 19, 1435. Gauss, C. F. (1877). Koenig. Ges. Wiss. Goettingen, 5, 602. Gaylord, R. J. (1982). Polym. Bull, 8, 325. Gaylord, R. J. and Douglas, J. F. (1987). Polym. Bull, 18, 347. Gaylord, R. J. and Douglas, J. F. (1990). Polym. Bull, 23, 529. de Gennes, P.-G. (1977). J. Physique Lett., 38, L355. Goldbart, P. M. (2000). J. Phys.: Condens. Matter, 12, 6585. Goldbart, P. M., Castillo, H. E. and Zippelius, A. (1996). Adv. Phys., 45, 393.

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Goldbart, P. M. and Goldenfeld, N. D. (1987). Phys. Rev. Lett., 58, 2676. Goldbart, P. M. and Goldenfeld, N. D. (1989a). Phys. Rev. A, 38, 1402; ibid. 1412. Goldbart, P. M. and Goldenfeld, N. D. (1989b). Macro-molecules, 22, 948. Goldenfeld, N. (1992). Lectures on Phase Transitions and the Renormalization Group, Addison-Wesley, Reading, Mass. Goldenfeld, N. D. and Goldbart, P. M. (1992). Phys. Rev. A, 45, R5343. Gottlieb, M. and Gaylord, R. J. (1983). Polymer, 24, 1644. Gottlieb, M. and Gaylord, R. J. (1987). Macro-molecules, 20, 130. Heinrich, G., Helmis, G. and Vilgis, T. (1995). Kautschuk, Gummi, Kunststoffe, 10, 689. James, H. M. and Guth, E. (1943). J. Che-m. Phys., 11, 455. Jespersen, S. N. and Plischke, M. (2003). Phys. Rev. E, 68, 021403. Kawamura, T., Urayama, K. and Kohjiya, S. (2001). Macromolecules, 34, 8252. Kuhn, W. (1936). Roll. Z., 76, 258. Kiintzel, M., Lowe, H., Miiller, P. and Zippelius, A. (2003). Preprint cond-mat/0303578. Lusignan, C. P., Mourey. T. H., Wilson, J. C. and Colby, R. H. (1995). Phys. Rev. E, 52, 6271. Marrucci, G. (1981). Macromol, 14, 434. Mergell, B. and Everaers, R. (2001). Macromolecules, 34, 5675. Michels, J. P. J. and Wiegel, F. W. (1986). Proc. R. Soc. Land. A, 403, 269. Mooney, M. (1940). J. Appl. Phys., 11, 582. Mukhopadhyay, S., Goldbart, P. M. and Zippelius, A. (2004). Preprint cond-mat/0310664. Goldstone fluctuations in the amorphous solid state, Europhys. Lett, (in press). Miiller, P. (2003). J. Phys. A: Math. Gen., 36, 10443. Nechaev, S. K. (1996). Statistics of knots and entangled random walks. World Scientific. Nelson, D. R. (2001). Defects and Geometry in Condensed Matter Physics. Cambridge University Press, Cambridge, U.K. Panyukov, S. and Rabin, Y. (1996). Phys. Rep. 269, 1. Peng, W., Castillo, H. E., Goldbart, P. M. and Zippelius, A. (1998). Phys. Rev. B, 57, 839. Peng, W. and Goldbart, P. M. (2000). Phys. Rev. E, 61, 3339. Peng, W., Goldbart, P. M. and McKane, A. J. (2001). Phys. Rev. E, 64, 031105. Plischke, M., Vernon, D. C. and Joos, B. (2003). Phys. Rev. E, 67, 011401. Rivlin, R. S. (1948). Phil. Trans. R. Soc. Lond., 241A, 379. Rivlin, R. S. (1949). Phil. Trans. R. Soc. Lond., 242A, 173. Rubinstein, M. and Panyukov, S. (2002). Macromolecules, 35, 6670. Solf, M. P. and Vilgis, T. A. (1995). J. Phys. A: Math. Gen., 28, 6655. Stenull, O. and Janssen, H.-K. (2001). Phys. Rev. E, 64, 026119. Treloar, L. R. G. (1973). Rep. Prog. Phys., 36, 755.

References

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Urayama, K., Kawamura, T. and Kohjiya, S. (2001). Macromolecules, 34, 8261, Vernon, D., Plschke, M. and Joos, B. (2001). Phys. Rev. E, 64, 031505. Viana, L. and Bray, A. J. (1985). J. Phys. C: Solid State Phys., 18, 3037. Wall, F. T. (1942). J. Chem. Phys., 10, 485. Warner, M. and Terentjev, E. M. (2003). Liquid Crystal Elastomers, Oxford University Press. Winter, H. H. and Chambon, F. (1986). J. Rheol., 30, 367. Winter, H. H. and Mours, M. (1997). Adv. Polym. Set., 134, 165.

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REPRINT DYNAMICS OF CONCENTRATED POLYMER SYSTEMS PART 2.—MOLECULAR MOTION UNDER FLOW by M. Doi and S. F. Edwards Journal of the Chemical Society, Faraday Transactions //, 74, 1802—1817 (1978).

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Offprinted from the Journal of The Chemical Society, Faraday Transactions II, 1978, Vol. 74.

Dynamics of Concentrated Polymer Systems Part 2.—Molecular Motion under Flow BY MASAO Doi f AND S. F. EDWARDS* Cavendish Laboratory, Madingley Road, Cambridge CBS OHE Received 22nd March, 1978 The primitive chain model presented in Part 1 is extended to the case in which the system is macroscopically deformed. The molecular expression of the stress due to the primitive chain is given, and the stress relaxation after a sudden deformation is calculated as an example. 1. INTRODUCTION In the previous paper * (Part 1), we described the Brownian motion of the primitive chain in equilibrium. Now we discuss its dynamical behaviour when the system is macroscopically deformed. In the equilibrium state, the primitive chain model is constructed based on the following three assumptions: (A) the real polymer chains are moving in a mean field, called the cage. In this cage field each chain is confined in a tube-like region, the central line of which is the primitive chain, defined in Part 1. (B) The equilibrium conformation of the primitive chain is a random walk with step length a and arc length L. (C) The primitive chain moves as one Brownian particle along itself with a curvilinear diffusion constant D. The purpose of this paper is to add a further assumption so that we can discuss dynamical properties under flow, particularly the rheological properties. The argument of this paper is the basis of the next paper 2 in which the rheological constitutive equation is constructed. The constitutive equation is obtained if we know (i) how the stress can be calculated from the primitive chain model and (ii) how the primitive chain deforms under the macroscopic flow. To answer these problems we follow the classical theory of rubber elasticity,3' 4 which gives an important clue to the answer to the first problem. The success of the simple phantom chain model suggests that even in a condensed system the stress is mainly due to the intramolecular entropic force; the intermolecular force acts primarily to keep the volume of the system constant and is not important for the anisotropic part of the stress. To pursue this idea we shall present an alternative to the cage model, a slip-link network, whose junctions are not permanent crosslinks but small rings through which the chain can pass freely, and use the stress formula in rubber elasticity;

Here rt is the end-to-end vector of the chain segments between two junction points, n, the number of monomers in this chain segment, kT the temperature, b the bond t Permanent address: Department of Physics, Faculty of Science, Tokyo Metropolitan University, Setagaya, Tokyo, Japan. 1802 1

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length of the real chain, and P is the pressure, which is determined from the condition that the system is incompressible. The idea of applying the theory of rubber elasticity to polymer melts and concentrated solution is not new. One of the earliest Theological models of polymeric liquids is a rubber model whose cross-link points are not permanent but have certain creation and breakage probability.5"8 This model has been widely studied and modified in various ways. However, in such models the mechanism of the breakage and creation of the chain segment was introduced phenomenologically and was not clarified on molecular level. The slip-link network model gives a simple molecular mechanism of the breakage and creation process, i.e., it occurs by the sliding motion of the chain through the slip-links. The sliding motion of the chain has another important effect. In rubber the number of monomers nt in a chain segment remains constant, but in a slip-link network, nt is changeable. As we shall show in section 5, this can explain the nonGaussian elastic response of the polymeric liquid. The aim of this paper is to discuss these physical aspects of the slip-link network model. For that purpose we limit ourselves to a simplest problem, i.e., the relaxation of the chain after a sudden application of deformation. The relaxational behaviour is discussed not only in stress but also in the conformation of the individual chain which would be measured by some scattering experiment for a labelled chain. (The conformational relaxation of a chain was first discussed by Daudi,9 but his basic equation is different from the present one). 2. SLIP-LINK NETWORK MODEL Fig. 1 shows three equivalent models which we shall use interchangeably in our argument; (a) the cage model, (b) the tube model and (c) the slip-link network model. The first two models have already been explained in Part 1.

FIG. 1.—Three equivalent models: (a) the cage model, (6) the tube model and (c) the slip-link model.

The major constituents of the slip-link network are slip-links which are small rings through which the chain can pass freely. In equilibrium state, the slip-links are separated by a distance a, the mesh size of the cage (or the radius of the tube). In this model the primitive chain is defined as a line joining these slip-links and the primitive chain segment is defined as a line segment between two slip-links. In the slip-link model the interactions between diiferent polymer chains are neglected except for slip-links, and the polymer is assumed to move freely between the slip-links. The slip-links are an alternative to the effect of the cage and need not be considered as real junctions at which two polymers are tightly entangled. It is, however, important to note that the effect of the cage cannot be expressed only by slip-links. Actually if the only constraint for the polymer is the slip-links, the arc length of the

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polymer will soon shrink as is shown in fig. 2. In the cage model (or tube model) such shrinkage does not occur because the lateral expansion of the chain accompanying the longitudinal shrinkage is forbidden by the cage (or tube). To avoid this difficulty in the slip-link model, we assume two Maxwell demons which are pulling out the chain at the chain ends with constant tensile force F^ [fig. 2(aj\. This force is estimated as follows. According to the elementary theory of statistical mechanics of chains, a force of magnitude 3kTl/nb2 is needed to keep the chain segment consisting of » monomers at relative distance /. Since the arc length is L - N0b2la (N0 being the degree of polymerization), F,9 is given by

This hypothetical force is of course an artifact. If we start from the tube model, we can construct a theory without introducing such a hypothetical force. In fact, the tensile force appears in our theory in order to determine the equilibrium number of monomers between two slip-links [see eqn (4.2)], which is also derived from the tube model (see Appendix A).

Fie. 2.—In the slip-link model a hypothetical tensile force Feq is necessary to keep the polymer being trapped by the slip-links (a). Otherwise the polymer will soon shrink in the longitudinal direction and disengage from the slip-links (6). In the tube model such longitudinal shrinkage does not occur because of the tube potential shown by the broken lines.

Before ending this section a remark is made about the meaning of the parameter a. Our slip-link model is based on the following physical picture of the dense polymer system: even in a dense system of polymer chains, entanglement would not be important on a small length scale because, for a small movement of the chain, other chains would respond by a local rearrangement of their position and would behave as a normal viscous liquid. On the other hand for a large distance, the entanglement effect would be quite important and its characteristic effect would be as follows; for a chain to move in its lateral direction (the direction perpendicular to the chain axis), the chain has to drag many other chains and feels a very large resistance, whereas in the longitudinal movement, it feels much less resistance; i.e., the drag force due to other chains is highly anisotropic. The slip-link model is a simple realization of such situation. The length a, therefore, has to be understood as a critical length at which the response of other chains begins to show the above qualitatively different behaviour.

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The existence of such critical length is suggested by various observations: first, in rubber when the distance between the cross-link points is short, the conventional ideal gaussian network is known to work well as a first approximation.4' 10 Second, in a polymer melt, if the molecular weight is small, the Theological data are well described by the Rouse theory, which does not take into account the entanglement effect.10 These observations suggest that when the chain is short enough, the free chain picture is approximately correct even if the chains are closely packed. Determination of such critical length is an important problem to be solved theoretically, but here we proceed regarding it as an adjustable parameter. 3. RESPONSE TO AN INSTANTANEOUS DEFORMATION Let us consider the following experiment: at time t = 0, a polymer melt (or concentrated solution) is suddenly deformed homogeneously and the deformation is kept constant after that (fig. 3). Since the system is basically a liquid, it eventually returns to the equilibrium state which is identical to the one before deformation. Then the problem is how this relaxation occurs.

FIG. 3.—Schematic illustration of the relaxation process after the sudden deformation, (a) Initial equilibrium state, (A) immediately after the deformation: each part of the chain is stretched or compressed, (c) after the first relaxation process: the primitive chain recovers its equilibrium arc length, but the conformation of the primitive chain is still in a non-equilibrium state, (d) the second relaxation process: the chain disengages from the deformed slip-links and returns to the final equilibrium state.

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To monitor this relaxation process we consider two quantities. is the tensor average of the end-to-end vector V(t) of the chain.

5

The first quantity

where rt(t) is the end-to-end vector of the fth primitive chain segment of a chainThe second quantity is the stress tensor eqn (1.1), which is now written as

where c is the number of polymers in unit volume, and the summation is taken over the primitive chain segment of a single chain. First we calculate the initial values of these quantities. Let E be the tensor describing the deformation such that the material point located at the Cartesian coordinate xx(a = x, y, z) changes its position to x't = "LfE^Xf. As in the conventional theory of rubber elasticity we make the assumption of affine deformation. Assumption (Dl): by deformation each slip-link point changes its position affinely. Thus the end-to-end vector r, of the fth slip-link segment is changed to [fig. 3(6)] r£=E.r,. (3-3) (In this section primes are used to denote the quantity after deformation). The initial value of is obtained from eqn (3.1) and (3.3)

where N = N0 b2la2 is the number of primitive chain segments in the equilibrium state and < . . . > 0 means an average over the equilibrium distribution. Since

we get /

In the frame that Eaf is diagonal with principal values A»,

Because of the inequality the mean square end-to-end distance < F'2> is always greater than the equilibrium value 0 for any deformation which does not change the volume of the system. Similarly,

Since

We get

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which, of course, is a result of the conventional theory of rubber elasticity. (cN is the number of chain segments in unit volume). In the permanently cross-linked network, these initial values are kept constant. In the slip-link network two relaxation processes occur. First the stretched primitive chain will shrink to some finite length by sliding through the slip-links [fig. 3(c)]. This first relaxation process finishes in a relatively short time, i.e., in 7^, the characteristic relaxation time of the wriggling motion. Second the primitive chain disengages from the deformed slip-link [fig. 3(rf)]. This process occurs slowly with the relaxation time TA. We shall now discuss these two processes in more detail. 4. FIRST RELAXATION PROCESS Immediately after the deformation, the lengths of the primitive chain segments \r't\ — |Erj| are not equal, and the tensile forces acting on the primitive chain segment

are not balanced (note «,; = «( = «2/62). In the first relaxation process, the monomers are redistributed so as to balance these forces. After this relaxation process is completed, the tensile forces have to be equal to the equilibrium value Feq = 3 kT/a, because there is no special constraint at the chain ends. Thus after this process the number of monomers contained in the ith primitive chain segment is (The double prime indicates a quantity after the first relaxation process).

FIG. 4.—Polymer chain in (a) an undeformed cage and (6) deformed cage.

Strictly speaking, eqn (4.2) is an assumption. We may state this assumption in the following form; Assumption (D2): When the wriggling motion reaches equilibrium, the monomer density per unit arc length is equal to the equilibrium value ajb2. Since the deformation generally changes the local environment of the individual chain, the validity of this assumption is not obvious, but a plausible argument may be given to justify this assumption. Suppose a regular cage field is stretched in the ^-direction (fig. 4). Because of the incompressibility condition, this causes a contraction in the j-direction. Now consider the squeezing force of the cage for the chain lying in the .^-direction. The squeezing force may increase due to the decrease in af, the separation of obstacles in the ^-direction, but at the same time it may decrease due to the decrease in the

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obstacle density per arc length (1M,). Therefore it is not implausible to assume that the monomer density per unit arc length remains constant as a result of these two balancing effects. After this first relaxation process, some of the primitive chain segments are destroyed, and the number of the primitive chain segments decreases. Let N'{ and N'z be the segments in which the first and the last monomers come after the first relaxation process, then the arc length of the primitive chain is given by

Using eqn (4.2) and the fact that the total number of monomers is equal to N0, we get

i.e., the arc length after the first relaxation process is equal to the equilibrium value, which is a natural consequence of the assumption (D2). The average number of primitive chain segments N" is determined from eqn (4.2) as

Since \rt\ = a, the average in eqn (4.5) is an average over the orientation of rt. Letting u be a unit vector of isotropic distribution, we write eqn (4.5) as

We now calculate the tensor after the first relaxation process. that the number of the primitive chain segment is now N", we get

Noting

i.e., < V* K^'> is smaller than the initial value < V'x K£> by the factor 1 /<|E . «|>0. [The inequality <|E . «|>0 > 1 can be proved for any incompressible deformation (see Appendix B)]. The stress a'^ after the first relaxation process is calculated from eqn (3.2) using eqn (4.2) and (4.6)

5. SECOND RELAXATION PROCESS The second relaxation process is a disengagement process of the primitive chain from the deformed tube. This process can be analysed by the kinematics of the primitive chain discussed in Part 1. Let us first calculate the relaxation of the stress
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When the wriggling motion reaches equilibrium, n^i) = rt(t)alb and the stress tensor is written as

where «,(*) is a unit vector parallel to the /th primitive chain segment. Eqn (5.1) is rewritten by use of the arc length coordinate s (0 < s ^ L) as (note 5>( is replaced by

where u(s, t) = dR(s, t)/8s is now a unit vector tangent to the primitive chain at the arc length coordinate s, or by absorbing the isotropic tensor 6,f to the pressure term

where The time evolution of Sxp(s, t) is calculated in the same way as in Part 1; there a discrete hopping model was used in order to resolve the subtle problem about the boundary condition, but here, since there is no fear of confusion, we start from a continuous model. Suppose the primitive chain moves by a curvilinear distance A* in a small time interval Af, then This is a continuous version of the Langevin equation of Part 1 [eqn (3.2) of Part 1]. In eqn (5.5) As is a random variable whose distribution is characterized by the moments Now according to eqn (5.5) Expanding the right hand side and using eqn (5.6)

i.e.,

The boundary condition for Sap(s, f) is that at the chain ends the distribution of u(s) is random, i.e.,

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[Strictly speaking, eqn (5.10) is for the hypothetical element for s = —a and s = L+a, but since L $> a we may replace this condition as eqn (5.10)]. The initial condition of S^ is given by eqn (4.8). Since the first relaxation occurs very quickly in the time scale of the primitive chain, we may write this as

The differential eqn (5.9)-(5.11) is easily solved

where Td = L2/Dn* is the disengagement time already introduced in Part 1. the stress is calculated as

Hence

where

This equation is compared with experiment in the next section. The relaxation of the tensor is calculated in almost the same way. In the arc length coordinate, the tensor is written as

If s and s' belong to different primitive chain segments <««(«, t) up(s, f)} vanishes [see eqn (3.5)], i.e., K being the largest of 4,. Hence if the deformation is not very large, i.e., if A < L/a, the integral converges rapidly, and eqn (5.16) can be written as

The relaxation of Vxf(s, t) can be calculated in exactly the same manner as S^(s, t). It satisfies the differential equation

and its boundary conditions are now

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with the initial condition Hence

where Therefore Experimentally, the tensor is not easy to measure. A more easily measurable quantity is the tensor

where RG is the position of the centre of mass

We call Ixf the radius of gyration tensor because the trace of Ixf gives the mean square radius of gyration <52>. The well known identity

relates 1^ to the (coherent) structure factor -*> in the small k region.

Thus the tensor Ixf can be measured by some scattering experiment (neutron or X-ray) for a labelled polymer. By eqn (5.28) /.. is expressed as

Using the same approximation as in < Vx(i) Vf(t)) we get

Substituting eqn (5.27), we finally get

i.e., Itf(t) decays in almost a single exponential manner with relaxation time TA.

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6. COMPARISON WITH EXPERIMENT

Here we briefly compare the result (5.13) with the stress relaxation experiment. This was done by Osaki and his coworkers11'13 for shear deformation and by Tschoegl's group for uniaxial stretching.14 The deformation tensors of these deformations are shear: uniaxial stretching:

Both experiments showed that the relaxation function att(t; A) is factorized into two functions, one of the time and the other of strain (A), which agrees with our result (5.13).

FIQ. 5.—Strain dependent part of the stress relaxation function for uniaxial stretching [eqn (6.3)]. Circles, observed values [after ref. (14): sample, uncross-linked SBR at 23°C]. Solid curve, eqn (6.5). Broken curve, eqn (7.4). Dashed curve, the result of ideal gaussian rubber. The parameter G0 is determined from the initial slope for the respective cases.

Tschoeel et al. proposed the following experimental equation where G0 is a constant and n(f) and /st(A) are the functions which depend only on time t and A respectively. The function/st(A) corresponds to our Q,i— Qx*, i-e.,

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The result of the integral is

with In fig. 5 this theoretical curve is compared with the observed one. The agreement is fairly good.

FIG. 6.—Strain dependent part of the stress relaxation function for simple shear [eqn (6.7)]. Circles, observed values [after ref. (11): sample, polystyrene solution in diethyl phthalate; molecular weight, 3 x 10s ; concentration, d 0.166g cm-3, O 0.221 g cm-3, Q 0.275 g cm-3]. Solid curve, eqn (6.8). Broken curve, eqn (7.4). In the ideal gaussian rubber fs^/X is constant.

Similar experimental equation was shown to hold for a polystyrene solution with molecular weight M = 3 x 10s and concentration 15 ^ p ^ 30 % by Osaki et al., The theoretical form of/8h(A) from the present work is

which was evaluated numerically and is compared with the experimental function in fig. 6. The agreement may be said to be good considering the wide range of A and that there is no adjustable parameter in the theoretical curve. [Note that for the gaussian rubber/5h(A)/l is constant.] Good agreement was not obtained for the time dependent part n(t) in both cases, but this perhaps should not be considered as a serious difficulty because the time dependent part is very sensitive to the molecular weight distribution. The molecular weight distribution is discussed in Part 3.

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7. INDEPENDENT ALIGNMENT APPROXIMATION The argument given in sections 4 and 5 is summarized in the following assumption. The assumption is most conveniently stated referring to the tube model. Assumption (D) : when the system is maoroscopieally deformed the tube .deforms affinely and the primitive chain occupies a region of arc length L inside this tube. (If the arc length of the deformed tube is shorter than the original one, the conformation of the newly created tube is assumed to be a random walk with step length a). When the system is continuously deformed the above transformation rule applies to each small time interval At. In this case the Brownian motion of the primitive chain occurs concurrently with the deformation. Note that in the assumption (D) even though the tube is assumed to deform affinely, the individual parts of the primitive chain do not deform affinely. For example let us consider a change of 'the tangent vector u(s) under deformation. If « is a unit vector embedded in the material, it changes as

However the deformed tangent vector u'(s) is not equal to the affinely deformed one of u(s) itself, but the deformed one of some other part E«(/)/|Eii(,j')| because the chain has moved inside the tube so as to keep its arc length constant. As a consequence, u'(s) is a function of the entire conformation of the chains, and the transformation rule of a(y) is in general not simple. Nevertheless it may be still a sensible approximation to assume that individual tangent vector u(s) deforms according to eqn (7.1) independently of other parts of the chain. We call this model the independent alignment model. The independent alignment model is the simplest model which satisfies the condition that the arc length of the primitive chain remains constant when the wriggling motion reaches equilibrium. It is simple because u'(s) is now a function of u(s) only, and this simplicity enables a rigorous mathematical treatment to be completed. A cautionary remark is needed for the independent alignment model. This model is an approximation, and .should not be regarded. as a general dynamical model of the primitive chain. For example it would not be appropriate to use this model for the calculation of a global quantity of the chain such as < Vt(t)Vt(t)y. Actually according to this model, (Vx(t)Vft(t)y at t = 2"eq is given by

which is significantly different from eqn (4.7), particularly for large deformation. On the other hand this model is expected not to give a serious difference for the stress tensor because the stress is proportional to the averaged orientational order parameter

[see eqn (5.2)], which is rather insensitive to the nature of the approximation included in the independent alignment model. In fact the independent alignment model leads to only a slight modification of the stress relaxation formula eqn (5.13): the modification is that 2n«(E) is now given by

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instead of eqn (5.11). The difference between eqn (5.11) and (7.4) is not serious. In fig. 5 and 6, the results calculated by eqn (7.4) are shown by broken lines. The difference is as much as 30 % for these two types of deformation. In Part 3 we shall use the independent alignment model and show it actually yields a simple but non-trivial constitutive equation. 8. CONCLUSION In this paper we have described a plausible model which explains how the primitive chain changes it conformation under a macroscopic deformation. The model is shown to agree well with experiments of stress relaxation. A more direct check of the model would be given by a measurement of the relaxation of the radius of gyration tensor, for which our model predicts; (i) Immediately after the deformation, Ixf is equal to the affinely deformed one

when <S2>0 is the equilibrium value of the mean square radius of gyration, (ii) In a relatively short time 7"eq, all the components of Ixf get smaller; (Note that the smaller component of Ixe also becomes smaller in this first relaxation process). (iii) After that Ixf(f) returns to the equilibrium value in almost a single exponential curve. The experimental verification of these points would be quite interesting. The authors thank Mr. R. Ball for helpful discussions. M. D. is grateful to the S.R.C. for financial support. APPENDIX A In this Appendix we show that when the polymer is trapped in a tube of radius a, the equilibrium monomer density per tube length is of the order of ajb2. Let us consider the system shown in fig. 7 : part of a very long chain is confined in a straight tube of length /. The chain outside the tube is assumed to be free. Our problem is to show that the mean number of monomers inside the tube is of the order of aljb2.

FIG. 7.—System considered in Appendix A.

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We label the monomers such that the monomer at the left-end of the tube is 0. Then the probability that there are n monomers inside the tube is given by

where R2 and R2 are the positions of the tube ends [IZj = (0, 0, 0) and R2 - (0, 0, /) in the coordinate system shown in fig. 7] and V(r) is the tube potential, which is assumed to be harmonic The integration 5R(m) should be carried out for all monomers (— oo < m < oo), but since the integral for m > n and m < 0 cancels between the numerator and denominator, we may write eqn {A. 1) as

The functional integral is evaluated by the Green function.

i.e.,

as

The parameter w is determined by the condition that the fluctuation <x 2 > is equal to a1

i.e. The most probable value of n is given by «* which maximizes eqn (A.6); hence which is the required result. This result is also obtained by the condition that the entropic force F = 3kTI/nb2 is balanced by the hypothetical force FC9 = 3kTja :

One should also note that the relative fluctuation of n is small, i.e.,

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which means that the line density of monomers along the tube is a well defined thermodynamic quantity. APPENDIX B PROOF OF THE INEQUALITY <|E . »|>0 ^ 1

The inequality to be proved is for any tensor E provided det|E| = 1. This inequality is related to the theorem, originally suggested by Edwards,15 that the mean arc length of a random walk embedded in a material always becomes longer when the material is deformed incompressibly. First we show the identity where E"1 is the inverse of E. To prove this we start from the definition

Changing the variable r to r' = E . r, we get

Now using this identity and the two general inequalities <4 2 > 0 ^ <^)o and <^4>o
which leads to eqn (B.I). 1

M. Doi and S. F. Edwards, J.C.S. Faraday II, 1978, 74, 1789. M. Doi and S. F. Edwards, J.C.S. Faraday II, 1978,74, 1818. L. R. G. Treloar, The Physics of Rubber Elasticity (Oxford University Press, London, 3rd edn, 1975). 4 P. J. Flory, Principles of Polymer Chemistry (Cornell University Press, Ithaca, New York, 1953). 5 A. S. Lodge, Elastic Liquids (Academic Press, New York, 1964). 6 R. B. Bird, R. C. Armstrong, O. Hassager and C. F. Curtiss, Dynamics of Polymeric Liquids (John Wiley and Sons, New York, 1977), vol. 1 and 2. 7 M. S. Green and A. V. Tobolsky, /. Chem. Phys., 1946,14, 80. 8 M. Yamamoto, /. Phys. Soc. Japan, 1956,11, 413 ; 1957,12, 1148; 1958,13,1200. 9 S. Daudi, /. Physique, 1977, 38, 731. 10 J. Ferry, Viscoelastic Properties of Polymers (John Wiley and Sons, New York, 1970). 11 M. Fukuda, K. Osaki and M. Kurata, J. Polymer Sci. A, 1975,13, 1563. 12 K. Osaki, M. Fukuda, S. Ohta, B. S. Kim and M. Kurata, J. Polymer Sci. A, 1975, 13, 1577. 13 K. Osaki, S. Ohta, M. Fukuda and M. Kurata, /. Polymer Sci. A, 1976, 14, 1701. 14 W. V. Chang, R. Bloch and N. W. Tschoegl, J. Polymer Sci. A, 1977, IS, 923. 1 1 S. F. Edwards, Polymer, 1977, 6, 143. 1

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Reprinted from PHILOSOPHICAL MAGAZINE, Vol. 3, No. 33, p. 1020,

September 19S8

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THE DOI-EDWARDS THEORY W. W. Graessley1 and T. C. B. McLeish2 1

Princeton University; 7496 Old Channel Trail, Montague, Michigan 49437, U.S.A. 2 IRC in Polymer Science and Technology, Polymers and Complex Fluids, Department of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom Polymeric substances have always played an important role in rheology, a field of study rooted in classical mechanics and condensed matter physics that deals with stress-strain relationships in material bodies. Linear relationships such as Newton's law and Hooke's law suffice for many substances and purposes, but there are numerous exceptions that defy, sometimes spectacularly, such narrow classifications. Departures in the form of non-linear response and timedependent phenomena are especially prominent when the substances contain polymer molecules. The continuum aspects of rheology grew from beginnings in the early 19th century. Many prominent scientists and mathematicians—Cauchy, Green, Maxwell, Boltzmann, to name only a few—laid the foundations that finally led to such modern formulations as linear viscoelasticity and finite strain elasticity. Molecular aspects of the subject have a more recent origin. Dating from the 1930s, about when the macromolecular nature of polymeric materials had become a settled matter, molecular rheology centered on the flexible-chain polymers, the basis of the then-emerging industry of plastic materials. Its development as an experimental science was well established by the mid-1970s when the pieces leading finally to the Doi-Edwards theory of polymer dynamics were being assembled. The impact of the theory is best appreciated from the perspective of that period. Dilute solution theory of polymers, epitomised by the 1971 Yamakawa treatise (Yamakawa 1971), had already reached a high level of refinement. The random coil picture, with sizes and dynamics modified by intramolecular volume exclusion and hydrodynamic interaction, was well established. Detailed studies of isolated chain dynamics, supported by the Rouse and Zimm molecular models, were well underway by the mid-1970's (Ferry 1980). The synthesis of macromolecules had meanwhile grown increasingly sophisticated. Fundamental work had become much less dependent on elaborate fractionation procedures to obtain structurally uniform samples. 'Living' polymerisation methods (Morton

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and Fetters 1975), capable of furnishing nearly monodisperse linear, star-shaped (Hadjichristidis and Roovers 1974) and even comb-shaped polymers (Roovers 1975) of several species, permitted systematic studies of the relationships between large-scale molecular architecture and polymeric properties. The increasing availability of structurally uniform materials greatly benefited studies of the polymeric liquid state at all concentrations. Small-angle neutron scattering, introduced in the early 1970's, permitted for the first time the direct determination of coil dimensions at elevated concentrations. Flexible chain polymers were shown to have random walk configurations in the undiluted liquid state (Kirste et al. 1972; Cotton et al. 1974), removing nagging uncertainties about the equilibrium structure and thereby clearing the way for theoretical considerations of both melt state dynamics and the elasticity of polymeric networks. Experiments on the dynamics of polymer melts and solutions were aided by advances in continuum rheology, which has always provided an important conceptual framework. The earlier discovery of time-temperature superposition (Markovitz 1975) greatly expanded the effective dynamic range of experiments and helped to uncover and characterise an interaction between chains—the entanglement interaction—that arises from mutual backbone uncrossability and produces a temporary network-like mechanical response (Ferry 1970). Indeed, the association between network elasticity and liquid viscoelasticity evolved into the notion that polymeric liquids can be treated as relaxing networks. The Lodge elastic liquid (Lodge 1964), the Coleman-Noll simple fluid (Coleman and Noll 1961), and the relaxing Rivlin rubber of Bernstein, Kearsley and Zapas (1965) epitomise continuum formulations from the 1955-70 era. The identification of linear viscoelasticity (Ferry 1970) and steady-state simple shear flow (Coleman et al. 1966) as fully analyzable deformation-response classes and the development of instruments capable of determining the relevant properties date from that period as well. This constellation of fortunate circumstances produced by the mid-70s a set of species-independent experimental laws for monodisperse linear polymers, relating chain length and viscoelastic properties in the long-time response region, the terminal dispersion that determines flow behaviour. Thus, the plateau modulus G°, a temperature and chain-length insensitive property of the species, defines the rubber-like elastic response at intermediate frequencies and sets the stress magnitude at the onset of the terminal region (Ferry 1970). The zero-frequency viscosity 770, a property of the terminal region, is linear in chain length for short chains, as predicted by the Rouse model, but crosses over to a stronger power-law beyond some characteristic chain length (Berry and Fox 1968):

in which M is the polymer molecular weight, a chain-length measure, and both Mc and K(T) are species-dependent properties. The zero-frequency recoverable compliance Js°, a parameter that quantifies elastic behaviour in steady flows,

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crosses over from the Rouse prediction, Js° oc M, to chain-length independence for long chains (Plazek and O'Rourke 1971). In the long-chain region, the product of plateau modulus and recoverable compliance, a measure of terminal relaxation time distribution, is a species independent parameter (Graessley 1974):

The fundamental aspects of the flow curve, the non-linear viscoelastic relationship between steady-state viscosity and shear rate, were also becoming clear. Thus, for monodisperse long chains at high concentrations (Graessley 1974),

where F is a universal, monotonically decreasing function that starts at unity for small dimensionless shear rates, 7770 J$ ^ 1) and g°es over to a power law, t] oc \j\-P,p ~ 0.9 for 7770 Js° > 1. Other generalizations were beginning to emerge as well. Evidence was accumulating, e.g., that strain and time dependence of stress following a large step strain could be expressed as a product of separate functions (Zapas and Phillips 1971; Fukuda et al. 1975):

A body of information also suggested that universal laws governed the concentration dependence of properties such as Js° and G°, and that other non-linear viscoelastic properties had similar, species-independent properties (Graessley 1974). After some false starts, it soon became clear that the rules of behaviour for linear chains, such as eqns. (19.1) and (19.2), are quite different for chains with long branches and that sometimes the differences are enormous (Long et al. 1964; Kraus and Gruver 1965). Observations of strange flow behaviour for samples of some species were attributed to branching (Hogen et al. 1967), rightly so it turned out, even though the spectroscopic evidence of branches was marginal at best. The remarkable phenomenon of shear refining was also attributed to long branches (Pritchard and Wissbrun 1969; Fujiki 1971). Understandably, long branching became for a time an explanation of last resort for unanticipated flow behaviour. Attempts were made to organise the various observations for linear chain systems through simple modifications of the Rouse model (Ferry 1970), multiplying its frictional contribution by a chain-length-dependent term beyond Mc to generate a terminal region shifted to longer times, as observed, and thereby to force agreement with eqn. (19.1). However, that modification alone cannot explain the observed long-chain behaviour of Js°. A rather good account of the flow curve—the power law exponent, and the effect of molecular weight distribution— was obtained from the idea of flow-induced disentanglement with concomitant relaxation-time reduction (Graessley 1967). However, attempts to generalize that

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approach to transient flows or other flow classes were unsuccessful. The field was left with an intriguing but seemingly disconnected collection of what appeared to be universal structure-property relationships for highly entangled flexiblechain liquids. There was simply no theoretical framework to hang them on or build from. Edwards introduced the tube idea in 1967 (Edwards 1967), when considering ways to understand the contribution of uncrossability effects to the modulus of permanent networks. De Gennes introduced the idea of reptation in 1971 (de Gennes 1971), also in the context of networks, when considering how unattached chains would diffuse and rearrange their conformations in an environment of uncrossable network strands. In 1974, Doi introduced the crucial connection with mechanical behaviour (Doi 1974), associating the fraction of stress remaining at time t after a step strain to the average fraction of initial tube still occupied at that time. In the fullness of time, these pieces were brought together to form a molecular theory of dynamics and rheology for entangled flexible-chain liquids, presented in four papers during 1978-9 (Doi and Edwards 1978a, 1978b, 1978c, 1979). The second of these papers, subtitled 'Molecular motion under flow,' was selected for reprinting in this volume because it contains the explanations of many puzzles and came as a revelation to many working in the field. One of us was swept away by finding explanations for things he had worried about for many years. (The other author was otherwise occupied in 1978 and recalls experiencing no such epiphany.) The unforced triumphs of the Doi-Edwards theory—Js° independent of chain length, r/o a power law with almost the right exponent, factorability of (7(7, t) with the right strain dependence, the connection between rheology and non-mechanical dynamics such as diffusion—made its limitations immediately forgivable. The pleasure was intense enough to provoke a paper pointing out some flaws and offering some not-very-successful quick fixes (Graessley 1980). Despite its successes, the real power of the Doi-Edwards theory was not its detailed predictions about the dynamics of entangled polymeric liquids, but rather its establishing of a unified molecular framework to build upon, where none had existed previously. Additional physics such as contour length fluctuations and constraint release, and fruitful approximations such as dynamic dilution and double reptation, became parts of the structure over time. The same molecular framework also proved the key to understanding the strong effects of long branches. We now discuss some of these subsequent developments. One astonishing and unusual aspect of the Doi-Edwards series of papers was the result that, once the tube concept had been granted validity in melts and entangled solutions, there then followed not only an account for these systems in linear response, but also a fully non-linear theory at the same stroke. This was possible because of the dual role played by the tube diameter, a. This lengthscale interacts with the chains by constraining their lateral motions to amplitudes smaller than a, but also acts as the scale above which the local deformation of the melt follows the bulk flow in an affine manner. Futhermore, the size of the tube

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diameter, even in polymer melts, is large enough for the sub-chains contained within it to have reached the limiting statistical behaviour of Gaussian chains. Prom the first property of the tube, the non-linear problem of molecular rheology then becomes, in mean field, the theory of statistical chains in deforming tubes. But the second property implies that even large strains at the macroscopic level represent small deformations at the level of bonds. It is the small level of local perturbation to the melt structure that allows the development of the theory deep into the non-linear regime. Taken together, the simple processes of reptation, convection and retraction of chains in their deforming tubes allow for mathematical formulation. Universal predictions for both functions ^(7) and g(t) [as defined in eqn. (19.4) above] emerge naturally. Especially striking was the parameter-free prediction of the 'damping function' h(i). Careful experiments on highly entangled monodisperse solutions by Osaki (Osaki et al. 1982) were in more or less exact agreement with the predicted form in shear. Another contribution to the impact of the theory was the derivation, in Paper III of the series, of a complete constitutive equation from the tube picture. Although this required a rather heavy further mathematical assumption (known as the 'independent alignment approximation'), it did allow the encapsulation of the theory in the form of a 'K-BKZ' integral equation (Bernstein et al. 1965), already familiar to those in the chemical engineering community working in the field. The derivation of such a form from molecular considerations of melt physics was unprecedented. These milestone papers gave the community conceptual and theoretical tools to work with, but they also left the challenges that motivated the research of the subsequent two decades. Pike all successful 'first theories,' the Doi-Edwards treatment left plenty of known physics out of the formulation. This had the advantage of maintaining clean lines in the original presentation of the tube theory, but naturally created areas of interest where agreement with experiment was less spectacular than in the case of the damping function, but suggestive of further exploration. In rheological experiments, two issues have been prominent in subsequent research. One is the form of the linear relaxation spectrum G(t) itself, much broader than the predicted simple reptation spectrum, which is almost single-exponential. Closely associated with this is the mismatch between the theoretical scaling of melt viscosity with molecular weight r/ ~ M3 and the experimentally observed r/ ~ M 3 - 4 . The '3.4' issue has spawned a remarkable literature and a long list of possible explanations, but the consensus is now simply that the proper inclusion of all internal Rouse modes of the chain along the curvilinear tube contour, so not constrained by it, is the predominant cause (Milner and McPeish 1998). The apparent power-law is a very slow cross-over to the long-chain limit of a 'reptation only' theory, from a viscosity at low molecular weights reduced by such 'contour length fluctuations' (CPF). Although identified as the leading candidate early on (Doi 1981), and implicit in the original Doi-Edwards treatment, it has taken considerable effort in theory and simulation to make properly constrained predictions that bear detailed comparison with experiment

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(Likhtman and McLeish 2002). Of special significance, here and elsewhere, has been the application of more directly molecular experimental methods than rheology. For example, the internal Rouse motion along the tube responsible for the '3.4' behaviour also leads to a remarkable series of non-Fickian regimes of diffusion for single monomers, as well as for the dynamic structure factor of labelled chains. Laid out as a detailed prediction in the original series of papers, experimental confirmation of tube-confined Rouse motion has had to wait for developments in the techniques of Neutron Spin Echo (NSE) (Wischnewski et al. 2002) and Field-Gradient Nuclear Magnetic Resonance (FGNMR) (Komlosh and Callaghan 1998) of the last decade. The considerable task of simulating a truly entangled melt numerically also contributed to our understanding of the interplay of chain reptation and internal chain dynamics. By 1990, the clear signal of tubelike motion had been identified in a large simulation (Kremer and Grest 1990). Statistics on individual monomer motion sufficient to compare with NSE and FGNMR data had to wait another ten years (Piitz et al. 2000). A further illustration of the way that different experimental methods work together is served by the problem of the self-diffusion of entangled chains. Reptation theory predicts that viscosity depends on molecular weight to the power 3 and the chain self-diffusion constant on molecular weight to the power —2. We now know that these are indeed the correct power laws for asymptotically large molecular weight. However, in the 1980s, measurements had not accessed sufficiently large molecular weight, and crossover or effective power laws were measured. Thus, it was problematic when the experimental picture seemed to indicate that the viscosity power law was actually 3.4, while the self-diffusion exponent was widely reported to be —2. Frischknecht and Milner (2000) have shown theoretically that for an effective viscosity power law of 3.4, the selfdiffusion exponent should be —2.4, not —2. Some confusion reigned over the experimental picture for a time, arising from disparate and limited datasets for various species in entangled melts and solutions, but a recent comprehensive examination of available data (Lodge 1999) has shown that the self-diffusion exponent is in fact consistent with —2.4. Accordingly, even in the crossover regime that precedes the asymptotic reptation regime, the effective exponents for viscosity and self-diffusion are consistent with one another. A second development in the tube model shares with the story of CLF an early anticipation, the advent of complementary experimental techniques, and steady theoretical progress. Very early, it was recognised that the effective tube constraints could not be permanent in a melt, as they were in a network, for they themselves arose from only temporary topological interactions. When one of the chains in the neighbourhood of a test-chain diffuses away, a constraint is temporarily released. It was pointed out (Graessley 1982) that a linear chain of such 'constraint release' (CR) events might be modelled as a Rouse-like object, and indeed a self-consistent picture of reptating chains inside Rouse-like tubes is consistent with an increasingly rich field of data on bimodal entangled blends. Here, the two molecular weights and the relative weight fraction of the components create a wide space of phenomena when the rates of Rouse-tube and

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chain-reptation are independently varied (Viovy et al. 1991). Experimentally, the complementary use of linear rheology and dielectric relaxation of main-chain dipolar melts has proved a very powerful technique (Watanabe 2001) for separately identifying the contribution of CR to relaxation dynamics. The cost to a universal theory is the determination of one more dimensionless number. It is not known fundamentally how many neighbouring CR events are required for one 'Rouse hop' of a tube segment, although this number must be of order unity, or slightly smaller if several chains cooperate in defining a single tube segment. But this is arguably a small price to pay for a melt theory that is able quantitatively to deal with the large polydispersity that arises in all practical situations. An approximation to the 'Rouse-tube' approach (though originally arising from a different physical picture) known as 'double reptation' is now a common tool in industry, used in the design of new products (Tuminello 1986). In linear response of monodisperse melts, CR has only a moderate effect on the rheology and dynamics, but quite the opposite has proved to be the case in non-linear deformation, especially in strong shear. When chains are confined to permanent tubes that are shearing much faster than the chains can escape by reptation, they are rapidly rendered closely parallel to the flow direction, with fluctuations confined to the tube diameter. This extreme prediction has consequences for rheology, and also for scattering via the single chain structure factor. Since such highly aligned chains are not able to contribute significantly to the shear stress, a prediction of permanent tubes is that, as the shear rate is increased, the resulting stress passes through a maximum at approximately the inverse reptation time. Although an early hope was expressed (in the fourth paper of the Doi-Edwards series, and in subsequent publications) that this might be connected with melt-flow instabilities, no direct evidence of such a maximum emerged in controlled experiments on monodisperse materials. Furthermore, when deuterium-labelled experiments on monodisperse polystyrene in strong shear became available (Muller et al. 1993), it became clear that the resulting chain anisotropy was also much less than permanent tubes would predict. Fortunately, the seeds of an explanation were all embedded in the original theory, for in flow, the Rouse like constraint-release process will be speeded up by the retraction of neighbouring chains in their deforming tubes in addition to pure reptation. The correct accounting of retraction events was furthermore implicit in the calculation of the damping function ^(7) in the Doi-Edwards theory. Theories of such 'convective constraint release' (Marrucci 1996) (CCR) have finally begun to address long-standing data-sets on carefully-prepared entangled solutions (Menezes and Graessley 1982) in strong shear. Significantly, assuming a physical value of the dimensionless constraint release rate that removes the shearstress maximum also accounts for the chain anisotropy seen in the Strasbourg neutron experiments (Likhtman et al. 2000). As an illustration of the fruitful way that the original formulation of Doi and Edwards has accommodated the additional physics implicit in the tube model, we give here a recent version of the stochastic equation for the change in conformation over a time interval At

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of a chain R(s,t) in terms of its contour co-ordinates (Graham et al. 2003):

where A£ is a Brownian noise describing displacements along the tube (reptation), K is the velocity gradient tensor, g describes the motion due to random constraint release events, Z = M/Me, and A is a time-dependent parameter that maintains the chain at its equilibrium length. Thus, the terms on the right represent, in order, reptation, deformation by flow (these are both inherited from the original Doi-Edwards formulation), constraint release (it appears as a diffusive Rouse-like operator), the stochastic noise that arises in partnership with the stochastic CR relaxation, and finally convective retraction along the tube contour. Although the Doi-Edwards papers focussed exclusively on linear chains, there were already predictions from the tube model that indicated natural explanations of the puzzling rheology of star-polymer melts, and of more complex branched polymers generally (de Gennes 1975). Since the model is topological at its heart, it has been very appealing to use it to address physical effects that depend on such topological changes in the molecular structure. The observed exponential dependence of the melt terminal relaxation time on the molecular weight of each single arm of a symmetric star molecule (with no dependence otherwise on the overall molecular weight) was a simple and direct consequence of the tube theory with reptation suppressed by the branch point. However, in early attempts to calculate the rheology of well-controlled star polymer melts quantitatively, rather large ad hoc changes were required to the model's dimensionless constants (Pearson and Helfand 1984). Then came the realisation that CR effects were much more significant in star polymers than in linears, since additional relaxations modified terms inside exponentials in the degree of entanglement, rather than power laws (Ball and McLeish 1989; Milner and McLeish 1997). A quantitative account of a range of experiments on carefully-synthesised architectures, including H-shaped polymers and combs (McLeish and Milner 1998) has resulted. A current active issue is the attempt to understand how the approximate calculation of CR, known as 'tube dilation,' used in these models of branched polymers, is related to the fundamental tube-Rouse dynamics seen in linear blends. New experiments are again of vital importance here-in this case another application of dielectric spectroscopy (Watanabe et al. 2002). Current theory (McLeish 2003) and simulation (Shanbhag et al. 2001) are trying to understand afresh the complexities of an entangled melt whose constraints possess a very broad range of relaxation times. The prospects for the tube approach to entangled polymers are clearly as active and potentially rich as the last two decades have proved. It is a remarkable piece of science that is able to touch deep fundamental issues of physics, such as

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topologically constrained dynamics, on the one hand, and yet address common industrial engineering issues, such as polymer melt processing, on the other. Neither would be possible to address without the imaginative imposition of that emergent dynamic object: the entanglement tube. References Ball, R. C. and McLeish, T. C. B. (1989). Macro-molecules 22, 1911. Bernstein, B., Kearsley, E. and Zapas, L. J. (1965). Trans. Soc. Rheol. 9, 27. Berry, G. C. and Fox, T. G. (1968). Adv. Polym. Sci. 5, 262. Coleman, B. D., Markovitz, H. and Noll, W. (1966). Viscometric Flows of NonNewtonian Fluids. Springer Verlag, Berlin. Coleman, B. D. and Noll, W. (1961). Arm. N. Y. Acad. Sci. 89, 672. Cotton, J. P., Decker, D., Benoit, H., Farnoux, B., Higgins, J., Jannink, G., Ober, R., Picot, C. and Des Cloizeaux, J. (1974). Macro-molecules 7, 863. Doi, M. (1974). Chem. Phys. Lett. 26, 269. Doi, M. (1981). J. Polym. Sci. Polym. Lett. Edn. 19, 265. Doi, M. and Edwards, S. F. (1978a). J. Chem. Soc., Faraday Trans. 2 74, 1789. Doi, M. and Edwards, S. F. (1978b). J. Chem. Soc., Faraday Trans. 2 74, 1802. Reprinted in this volume. Doi, M. and Edwards, S. F. (1978c). J. Chem. Soc., Faraday Trans. 2 74, 1818. Doi, M. and Edwards, S. F. (1979). J. Chem. Soc., Faraday Trans. 2 75, 38. Edwards, S. F. (1967). Proc. Phys. Soc. Lond. 92, 9. Ferry, J. D. (1970). Viscoelastic Properties of Polymers. 2nd ed., J. Wiley and Sons, New York. Ferry, J. D. (1980). Viscoelastic Properties of Polymers. 3rd ed., J. Wiley and Sons, New York. Frischknecht, A. E. and Milner, S. T. (2000). Macromolecules 33, 5273. Fujiki, T. (1971). J. Appl. Polym. Sci. 15, 47. Fukuda, M., Osaki, K. and Kurata, M. (1975). J. Polym. Sci., Polym. Phys. Ed. 13, 1563. de Gennes, P.-G. (1971). J. Chem. Phys. 55, 572. de Gennes, P.-G. (1975). J. Phys. (Parts) 36, 1199. Graessley, W. W. (1967). J. Chem. Phys. 47, 1942. Graessley, W. W. (1974). Adv. Polym. Set. 16, 1. Graessley, W. W. (1980). J. Polym. Sci., Polym. Phys. Ed. 18, 27. Graessley, W. W. (1982). Adv. Polym. Sci. 47, 67. Graham, R. S., Likhtman, A. E., McLeish, T. C. B. and Milner, S. T. (2003). J. Rheol. 47, 1171. Hadjichristidis, N. and Roovers, J. E. L. (1974). J. Polym. Sci., Polym. Phys. Ed. 12, 2521. Hogen, J. P., Levett, C. T. and Werkman, R. T. (1967). SPE J., November 1967, 87.

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20

REPRINT THE SURFACE STATISTICS OF A GRANULAR AGGREGATE by S. F. Edwards and D. R. Wilkinson Proceedings of The Royal Society of London. Series A, Mathematical and Physical Sciences, 381, 17-31 (1982).

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Proc. X. Soc. Land. A 381, 17-31 (1982) Printed in Great Britain

The surface statistics of a granular aggregate BY S. F. EDWARDS, F.R.S., AND D. R. WILKINSON Theory of Condensed Matter, Cavendish Laboratory, Madingley Road, Cambridge, CBS QHE, U.K. (Received 5 August 1981) The problem of the surface fluctuations in a settled granular material is posed. A simple model is given which describes the process by which a particle settles and comes to rest on the existing surface of the packing and from this a set of Langevin equations for the Fourier modes of the surface are derived. These equations imply that the Fourier amplitudes behave like the velocities of a set of independent Brownian particles. We show that this results in logarithmically divergent surface fluctuations if the flux of particles onto the surface is random, the divergence being removed by a more accurate description of the settling material, for example by having the granules fall through a sieve.

1. I N T R O D U C T I O N Suppose that we take a bin and gently and uniformly pour in a granular material. As the material in the bin builds up we can identify a surface and ask the question. ' What is the magnitude of the fluctuation in the height of surface (measured from the base of the bin) ?' Also of interest is the length scale of the surface fluctuations and how they behave dynamically as more material is added. The statement that the material is added gently and uniformly is a statement about the flux of material into the bin. This statement will have to be mathematized eventually but we make the following assumptions which lead to the simplest problem, but which are physically realizable: (i) The flux is sufficiently weak so that one can ignore any correlation between incoming particles. (ii) The particles settle gently (as if sedimenting from a viscous liquid) in such a way that once each particle has settled under gravity it does not then move when other particles settle above it. Clearly a packing made according to this prescription will be more dilute than one which is shaken where a cooperative reorganization of the particles is allowed to take place. These assumptions can be considered as part of the definition of the problem. In § 2 we go on to develop a model of the surface behaviour but first we must define the surface and the probability of finding it in some particular shape. It is possible to devise an operational definition for a surface e.g. for each triplet of particles in the ensemble construct the plane joining the three centres of mass (the plane is not extended to infinity but is a finite triangle whose vertices are the centres 17

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of mass). The surface is formed by those portions of each plane which do not lie vertically below part of another plane. Having defined the surface we describe it using a single valued function z(p, N ) , where 2 is the height of the surface vertically above a point on the base of the bin, defined by the two dimensional vector p = x, y and N is the number of spheres in the bin at the time the surface is described. After N spheres have been added to the bin there will be some probability of finding the surface described by a function z(p). This probability is a function of the value of 2 at every point p, i.e. it depends on the whole function z(p). The probability is a functional of 2 and is denoted In addition we can define a Green functional by asking for the probability that the surface is z(p) after AT spheres have been added given that after A" spheres it was z'(p). This is denoted by In the usual way we can write down a functional integral equation which expresses the law of compound probabilities

where the integral is over all possible functions z'(p). In §2 we construct a microscopic model of the Green functional for N = N' + i. From this we derive a Langevin equation and hence derive the statistics of the surface following the theory of Brownian motion. 2. THE B L O B M O D E L When a particle lands and settles on the existing ensemble of particles, the surface changes. Consider the addition to the surface as a blob. Some of this blob will be made up of particle and some of the blob will consist of space. Ignore the distinction between particle and space and think only of blobs being added to the surface. The shape of blobs added to the surface will depend on the shape of the surface, just before the moment the blob lands, as well as the shape of the particle landing. Also the position of each blob added will depend on the shape of the surface locally because newly added particles will tend to seek favourable locations on the surface. All the information relating the probable shape and position of a blob added to the surface to the shape of the surface just before the blob is added is contained in the Green functional introduced in § 1. Now the Green functional cannot depend on N explicitly as the laws which determine the behaviour of the system do not change with N. (The Green functional

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may depend on the number of particles in the bin but only in as much as z'(p) is a function of that number.) We also define a variable v(p) as z(p) — z'(p), which represents the shape of a blob. Thus the Green functional becomes

where W[v(p); z'(p)] is the probability functional that a blob added to a surface described by z'(p) will have shape v(p). The functional equation (1.1) for A.ZV equal to 1 becomes

At this point we must input the microscopic physics of the process of building up the surface one particle at a time and hence find a model for the functional W. The crudest possible form of W is obtained by completely ignoring the dependence of the function v(p), which describes the blob, on the function z(p) which describes the surface. In these circumstances v(p) becomes a random variable. A suitable model for W[»(p)] would then be

The interpretation of this equation is as follows. Each blob is a shape/(p) centred at a point p0, where p0 is a random point in the bin, whose area is LP. The function/(p — p0) might be some given function or it may itself be distributed with some probability functional. The only important feature of/(p — p0) is that it must have a sharp cut off at p—p01 = a, where a is the width of the blob. For example

where h is the total volume of each blob, would be a suitable model for the blob shape. In §3, we shall show that the crude model of W given by (2.2) gives rise to an indefinite increase in the magnitude of the surface fluctuations as the packing builds up. The root mean square fluctuations increase like the square root of the depth of the packing, in fact. This behaviour stems purely from the fact that we have put v(p) equal to a random variable, uncorrelated with z(p). The crucial difference between the real system and the one represented by (2.2) is the fact that particles will tend to settle in local minima of the existing surface. If a particle descends centred at p0 its final resting place on the surface will be near p0 but will be in a local minimum of z(p). We might, at first sight, expect that particles will tend to find resting positions of low potential energy in a global sense, i.e. will tend to sit in regions of the surface where z(p) is smaller than average. This, however, is not the case. The phenomenon of settling is entirely local as we have assumed that the particles settle without kinetic energy and they move, on the

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surface, only so far as to find a resting position in which they are stable under gravity; i.e. a local minimum of the surface. Thus v(p) which describes the blob added to the surface is no longer a random variable but depends on z(p). A very simple model for W which describes this situation can be obtained by dividing the bin into a square lattice. Thus z(p) becomes a discrete set of values za and similarly v(p). In this representation a model for W in which v(p) is uncorrelated with z(p) would then be

where M2 is the number of lattice squares in the bin. For the function (2.4), the value of z in the lattice square labelled by « increases by I and all the other values of 2 are unchanged. The selected lattice square a, is chosen randomly. Suppose we allow a change in the values of za_lx, za+lx, z^_iy and za+iy (the four nearest neighbours of the lattice square a), as well as za, then we effectively allow some local movement of the particle. If in addition we increase say zn_lx if ztt+lx is less than za then this represents a net movement of the blob towards a local minimum. For instance consider

Notice that for (2.5) the total volume increase of the system is I multiplied by the lattice spacing squared. Thus the form of W represented by (2.5) allows the movement and spreading out of a fixed volume blob. Suppose that za were large in comparison with its four neighbours, then (2.5) allows the blob to spread out equally over the four neighbouring lattice squares, whereas a more physical model would allow the blob to move at random to one of the four neighbours. The model is therefore only crude but improving the details will not change the results derived below except perhaps for numerical constants. Given the model function (2.5) we need to relate the imaginary lattice spacing to a physical length. The lattice spacing is the typical distance moved by the particle between landing on the surface and finding a suitable resting place. For a packing of identical hard spheres this will be approximately equal to the sphere radius. At this point we could solve (2.1) with W given by (2.5), however it is more useful to extract a Langevin equation from the model and then evoke the central limit theorem to calculate the probability functionals which describe the surface. In this way it is clear that the exact specifications of the model functional W[v(p); z(p)] are unimportant. We proceed by considering the finite set of complex fourier coefficients

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where a is the lattice spacing. Hence the number of lattice squares is given by Mz = L2/az. The inverse of (2.6) is

(There is never any ambiguity in using the same symbol z for these two functions.) We now calculate TF[{wA}; {zk}], the probability function for the set of vk given the set of zk. This is given by

where clearly the set of zk is given if the set of za is. The average of vk, for some particular fourier mode klt is given by

For the function W[{»„}; {za}] given by (2.5) this average is

plus three similar terms representing contributions from va_lx, va_ly and vt+ly. Performing the sum over a gives

Now for |fe| much smaller than I/a we can expand the cosines to second order in k to obtain where /i is the total volume increase of the system (i.e. h = la?). For values of | fej| close to I/a the lattice model breaks down as it is not a precise description, of the process of settling, over very short length scales. In the lattice model wavelengths less than the lattice spacing are not included in the sum over k in (2.7). In a blob model without a lattice the shortest wavelengths allowed correspond to the width of the blob. Thus we conclude that (2.8) is a good approximation for most of the k modes of interest. The first term on the right hand side of (2.8) represents the volume increase of the system per particle added. The second term shows that for any zk which becomes large there is a tendency for the arrival of additional spheres to reduce this value of zk. This effect is most pronounced for large values of k (corresponding to short wavelength fluctuations of the surface).

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We now rewrite (2.8) in real space. Multiplying by e~1*1''"/i2 and summing over fe, gives

or

At this point we introduce a time variable, t, and an average rate of landings per unit time per unit area, r. Because of the assumptions about the flux of particles being weak and particles not moving once settled we do not expect r to appear in the final formula for the fluctuations. The expected change in z(p) per unit time is thus

This equation states that two effects contribute to the expected rate of change of z(p). First there is the average steady increase in z(p) equal to the volume of blobs added to the system per unit area per unit time (i.e. rh). Secondly for regions of p where V2z is large we expect z(p) to increase more than average. This is because where Vaz is large there is a minimum in the function z(p) and particles are expected to settle in minima and hence z(p) is expected to increase. In regions where V2z is large and negative, z(p) has a maximum and is expected to increase more slowly than average. We could have written down (2.10), for the rate of change of z(p), from the original discussion of what was required of the model functional TP[*>(p)]. The particular model (2.5) merely shows that the coefficient of ra*V2z is one. A slightly different model would give a different coefficient of order one. Returning to ^[{v*}; {zk}], we can calculate the second moment given by

Summing over « gives As above, values of k^ greater than I/a are not allowed so we write The cut off is exact for the functional W given in (2.2) by using (2.3) for the blob shape f(p-p 0). We are now in a position to write down a set of Langevin equations for the fourier modes of the surface; where £k is a random variable, uncorrelated with zk and with zero mean. The variable £,k for k = 0 is zero for identical blobs. (It will be non-zero for blobs with a

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volume distribution. The contribution to the surface fluctuations from £ft=0 is discussed in §3, vi.) Thus The equivalent real space Langevin equation is

The terms rh + £(p, t) represent the flux of particles onto the surface, where rh is the steady flux and £ is the zero mean, random fluctuation in the flux. The term in V2 z(p, t) represents the fact that the change in the surface depends on the existing surface as well as the flux, but note that this term is proportional to r and thus is connected with the behaviour of incoming particles. The discussion above gives some justification for what is essentially a phenomenological equation which describes the behaviour of the surface. There is some further discussion in § 5 but we note here an important difference between the blob model and the physical system of a granular material. The blob model allows local movement of the blob on settling and allows a change of shape in the blob to accommodate the existing surface. The volume of each blob is however independent of its surface landing position whereas the increase in volume of the real system will depend on where that particle settles.

3. THE CALCULATION OF THE SURFACE FLUCTUATIONS FEOM THE LANGEVIN EQUATION The Langevin equations having been constructed for each of the fourier modes of the surface the problem becomes mathematically equivalent to a set of independent Brownian particles in which the fourier amplitude zk behaves like the velocity of a Brownian particle with a viscous damping proportional to k2. Consequently many results can be written down at once from the theory of Brownian motion (see for instance Chandrasekhar (1943), Reif (1965) and Resibois & De Leener (1978^ For example the steady state probability distribution for the set of fourier amplitudes, {«*}> is given by

and hence changing variables (see for instance Edwards (1973))

where Here we have ignored the distribution for the total volume of the system (ZA=O).

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Where this problem differs from that of Brownian motion is in the physical interpretation of the moments of the probability distributions. We therefore derive these now directly from the Langevin equation

This can be simplified by redefining z so that it is measured relative to its average value rather than the base of the bin. Thus the transformation reduces the equation to If we define the complex fourier coefficients zk^ by

where T is some very long time, then by the Wiener-Khintchine theorem (see for instance Reif (1965)), in the limit of large L and T, the correlation function
Fourier transforming the Langevin equation (3.1) gives

and hence the ensemble average required for (3.2) is given by

Thus to calculate the correlation function (z(p,t)z(p + p', t + t')} we require the ensemble average <£*>U£_A><>Consider a function £,(p, t) given by

which represents a series of incoming blobs, of shape/j, arriving centred at position pt and at time tt. Initially we assume that each event (particle landing) can occur at a time ti randomly distributed in 0 < ti < T. Similarly we assume that pi = xt, yi is found at random in 0 < xi < L and 0 < yi ^ L. Then the average value of £,k w is given by

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where {/A} is the fe fourier component of the shape of a blob, averaged over all possible blob shapes. To remove the average flux we put {/A=0) equal to zero and hence The average of

is

The integrals are only non zero in the case where i — j and hence

where rL2T is the total number of events under consideration. The function (fk /_ft.> is equal to the average square volume of the blobs multiplied by a sharp cut off at |fe| equal to the reciprocal of a typical blob width. For identical circularly symmetric Gaussian blobs for which/(p — p0) is given by

and
is

Combining (3.2), (3.3) and (3.5) yields

Performing the w integral gives

The integral as it stands is divergent. We must remember that (£*£_*) is zero for k = 0 reflecting the fact that the average of the flux is treated separately (i.e. it is not included in f). The sum over k is from |fe| = n/L only. (The factor is m not 2it because we impose the boundary conditions that Vz is zero at the edge of the box. Thus the first allowed fourier components are oos(nx/L) and oos(ny/L). However this does not mean that we have included only one quarter of the fourier components in the sum over k as we have allowed negative k as well as positive. The only effect is to change the bottom limit of the integral. Note also that one of the components of fe is allowed to be zero provided the other is not.) Thus we replace, what is correctly, the sum over fe by an integral over k with |fe| > n/L.

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We proceed to evaluate the integral (3.6) for certain special cases: (i) The mean square value of z(p) measured relative to its average value Putting p' = 0 and T' = 0 in (3.6) gives

which evaluated gives an exponential integral. For a/L small we can replace the Gaussian cut off by a top-hat cut-off to obtain

The term hP/^na* is a quantity proportional to the square of the height of each blob, as we might expect. The term In (L/na) is at first sight somewhat surprising as the bin size appears in the formula for the surface fluctuations. The result is due to the assumption that each particle in the flux lands at random anywhere in the bin. Suppose that the particles were dropping from a sieve above or sedimenting after precipitation in a chemical reactor. In the sieve case the flux is uniform to the extent that over a long period of time the flux of particles into a region of the bin (small compared to the total area of the bin) is constrained to be equal to its expectation value. It is possible to define a length 6 such that modes of wavelength greater than 6 are constrained is such a way that the quantity <j^d£'£t(i') JJd£"£_A(r)> is not allowed to increase indefinitely as the time, t, increases. Consequently the quantity (£*, w£-/t" -«•) ig zero for w = 0 for these constrained k modes, and the divergence of the Wiener-Khintchine, integral over w and k, at ui = 0, fe = 0 is removed. The effect is to replace the lower limit of the integral (3.6) by 2n/b instead ofn/L. Thus jthe quantity 2L (representing the longest allowed wavelength present in the flux) in (3.7) should be replaced by the length b. Returning to (3.7), it is interesting to insert some typical values to see just how weak the logarithmic divergence is. Consider particles of size 1 mm in a bin of dimension 3 m. Then the logarithmic term in (3.7) gives rise to a factor of (2.6) in the root mean square fluctuation of the surface. Now suppose we increase the bin size to 3 km, then the factor increases to (3.7). Thus for a 1 million fold increase in the surface area of the bin the magnitude of the fluctuations increases by only 50 %. Notice that the result (3.7) does not imply a surface 'roughness' dependent on the bin size. The result comes from very long wavelength fluctuations, in the surface, which have a small slope but a large amplitude. (ii) The static behaviour of the fluctuations Consider the function

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This is the expectation value of the square of the height change observed when moving on the surface, hi a straight line, distance p . The origin, p = 0, is chosen arbitrarily. Prom (3.6) the function is given by

In this expression the divergence at the origin of the integral is removed because (1 — J0(k-p)) is of order &2 for small A;. Hence the bottom limit of the integral can again be replaced by zero provided \p\ < L. The integral is given approximately by

This is also a strange result. The value of ((z(p, t) — 2(0, J))2) increases indefinitely as p increases. In practice the function will reach a steady value when \p\ is a length of order L. Thus the fluctuations have no microscopic length scale. Some physical justification for this result can be found as follows: Consider a one dimensional surface in the lattice model. If the value of zx at each lattice point were constrained to be z^-v +1, then the function z(x) would make a random walk and for large x we obtain which is Einstein's famous result. In two dimensions the equivalent constraint becomes that for each lattice square the value of z differs by I from the value of z in each of the four nearest neighbour lattice squares. The lattice problem as posed is not easily soluble because the lattice square at the origin effects the lattice square at p by every possible path of nearest neighbours joining 0 to p. The effect of this is that the fluctuations are strongly constrained as a result of the' frustration' caused by all the paths. Thus the function x/a in one dimension is replaced by the function In (\p\/a) in two dimensions. Note that the equivalent problem in three dimensions which may represent temperature fluctuations in a block of metal, for instance, shows none of the divergent behaviour. The results (3.7) and (3.9) are directly related to the divergence at k equals 0 of the integral (3.6). The long wavelength modes are not strongly damped because the slope of the surface associated with such a mode is very small for a given amplitude. The Langevin equation (4.1) implies that the system has a resistance to high slopes, i.e. high values of |Vz|. A Langevin equation of the form

would correspond to a resistance of the system to high values of 22. (Remember that z here is defined relative to its average value.) The divergence of the integral (3.6) would be removed and a new scale of length, given by a2/y^, would characterize the fluctuations.

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The equation (3.10) would correspond to a tendency for incoming particles not to settle in areas where z(p) is higher than average. Although this sounds reasonable it is in fact unphysical because a particle settling can only tell whether it is on a steep slope or at a minimum, etc. it has no' knowledge' of its value of z(p) in a global sense. The corresponding statement about the steady state probability functional p[z(p)], is that it does not depend on z(p) explicitly, only on the derivatives of z(p). The resistance of the system to high slopes is clearly related to the angle of repose for a granular material. The tangent of the angle of repose is the steepest slope the material can sustain under gravity. (iii) The dynamical behaviour of the fluctuations Consider the function where z is defined relative to its average value at time t. From (3.6) the function is given by The lower limit of the integral can be taken as zero provided ra?t' <^ L?/a2. The integral can be evaluated to give

For t' much less than the time to add just enough particles to cover one layer on the packing (t' <^ 1/ra2) (3.11) reduces to In other words for short times the function z(p, t) describes a random walk of the usual kind, with the fluctuation increasing like $. For longer times this walk is constrained as the value of z at some point p 'realizes' that it is effected by neighbouring values of z. The fluctuations then increase very slowly with time until raH' is of order JD2/a2. (That is the number of monolayers added to the packing is of order the number of particles in a monolayer.) After this very long time the fluctuation reaches its steady value given by (3.7). (iv) The fluctuations in the gradient of the surface By analogy with the derivation of (3.2) the correlation function in the slope or gradient of the surface is given by

combining this equation with (3.3) and (3.5) and integrating over w and k gives

where I is the unit tensor in two dimensions.

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Thus the fluctuations in Vz show none of the bad behaviour exhibited by the fluctuations in z. The fluctuations decay with a length scale and a timescale which are well defined in terms of microscopic properties. (v) The solution to the problem in which the blobs settle completely at random The process by which a surface builds up by blobs landing at random and simply resting where they land, is also of some interest. For instance if lumps of a substance, such as clay, are deposited at random and just stick where they land, remaining rigid, then the problem will be described by a blob model in which the settling position of the blobs is uncorrelated with the existing surface. In this case we can derive a Langevin equation without a drag term. The fourier modes zk behave like the positions of Brownian particles and consequently the surface fluctuations increase with time and are given by where (z) = rht. Other problems of interest are those where the blobs really do exist, for example blobs of a very viscous liquid. When these land their shape immediately changes to accommodate the existing surface but then the surface slowly relaxes. In addition a granular material may be added to a shaken container. In these problems a Langevin equation of the form (2.13) may still l& valid except that the coefficient of V2z will no longer be proportional to the rate of addition of particles, r. Consequently as soon as the flux of particles onto the surface is turned off the steady state fluctuations will die away exponentially leaving a flat surface. (vi) The contribution of the mode k = 0 to the surface fluctuations Within the blob model the volume of each blob added is a random variable uncorrelated with the existing volume of the system. Consequently if the blobs have a volume distribution we should expect the fluctuation in the fourier amplitude Zfc_ 0 to grow indefinitely with time and to be given by The k = 0 term should then be included in the Wiener-Khintchine sum and consequently contributes a-term to the mean square surface fluctuation. However it is interesting to note the following points: (a) For a packing whose average height is equal to its width (i.e. rt (/0) = L) this term is of order {/0}/JD and is consequently negligible in comparison with the expression given in (3.7). (6) If we are studying a single system at a single time rather than an ensemble of systems we can measure the fluctuations in the surface relative to the average height of the surface for this particular system (i.e. Z0/LZ). Fluctuations measured thus do not include a contribution of the form (3.14).

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(c) For a real system we might expect there to be a local correlation in 'blob' volumes. Thus particles will settle in such a way that the increase in volume of the system is large or small depending on whether the recent volume increases occurring in the same neighbourhood were small or large respectively. Hence for a real system we might expect the volume fluctuations to be smaller than predicted by the blob theory.

4. CONCLUSIONS The problem posed and the assumptions made in § 1 define a precisely specified physical problem. We have solved this by deriving a Langevin equation in § 2. This equation is of sufficiently simple structure to be soluble. Any criticism of the theory will therefore be made in §§ 1 or 2. We should ask whether the precisely defined physical problem, formulated in § 1, is the problem which is really of interest in practical applications. The following assumptions have been made in deriving the Langevin equation: (i) That the local movement of particles on the surface is a sufficiently accurate description of the settling process. So long as this process is truly local the Langevin equations for the fourier amplitudes zk will remain an uncoupled set. We might imagine the situation arising where a particle rolls a long way down a steep slope to find a minimum in which to rest. If one argues that this behaviour is not well described by a local movement of the particle, then the theory can be defended by pointing out that the blob model predicts that the probability of finding a long steep slope on the surface is very small. This is because the correlation function in the gradient of z(p) is short ranged. Hence the blob theory is in this respect self consistent. (ii) That terms linear in z(p, t), and hence zk(t), are a sufficiently accurate description of the settling process. For instance one might argue that a term such as — Vz(p)-Vz(p) should appear in the real space Langevin equation, in addition to V2z(p). Such terms make the problem considerably more difficult because fourier transformation will not diagonalize the problem. (iii) That the term — Jc2zk or equivalently V2z(p) is a reasonable description of the settling process. In fact k2 is only the leading term in an even power series in k. The higher terms make no significant difference to the predictions of the blob theory because the divergence of the integral over k is unaffected. A term independent of k (i.e. — %) is ruled out for reasons explained in the text. Of the various criticisms of the theory it is the question of linearity which is most difficult to defend on physical grounds. If we accept the Langevin equation we can rigorously show that, in the limit of large bin area, the fourier modes behave as a set of independent Brownian quasiparticles, where the mode amplitude, zk, corresponds to the velocity of the quasiparticle. We can also show that the surface behaviour is dominated by the long wavelength modes of the system. In a system in which the flux of particles has fluctuation modes with wavelengths equal to the bin dimension, the effect is to give surface fluctuations whose steady state, mean square value is proportional to the

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logarithm of the bin dimension. Any particular fourier mode, however, has a well defined contribution to the fluctuations, which is independent of the bin dimensions. It is only when the sum over the fourier modes is performed that the bin dimension appears. The theory has a sort of self consistency in that it is most strictly valid for the long wavelength modes. If the flux of particles is uniform to the extent that long wavelength fluctuations are not allowed to increase with time, then the bin size dependence of the surface fluctuations is removed. However, the longarithmic behaviour, which is typical of two dimensional systems, remains but only over a short range. The relevance of this point to such systems as a solid precipitating from a chemical reaction and settling is not clear. For practical purposes it makes very little difference whether the logarithmic dependence on bin size is present or not because of the incredibly slow increase in the root mean square surface fluctuation with bin area, which the square root of the logarithm of the square root of the area represents. The steady state probability functional which describes the surface is predicted to be not explicitly dependent on the function z(p) but to depend on the slope of the surface Vz(p). (With the proviso that the total volume of the system is given.) This reflects the fact that a particle landing at a random position in the bin, p, is' unaware' of the value of z(p). Thus there is no tendency for the surface to increase more than expected in a region where z(p) is low, i.e. in a global minimum of the surface. It is only the value of V2z(p), or the curvature of the surface, which is important. The Langevin equation of § 2 is a precise description of the behaviour of the surface of a certain class of randomly deposited granular material and it is our contention that it incorporates most of the essential physics of the problem and, therefore, that its predictions are, at least qualitatively, correct for the more difficult case where nonlinear terms matter. Various approximate methods are available for handling this problem, but it is our belief that the precise definition of improvements in the description of the physics of the problem is more difficult than the resulting mathematical problem. D.B.W. gratefully acknowledges the financial support of the S.R.C. and I.C.I, in a CASE studentship, and Dr Peter Cardew for help and encouragement. REFERENCES Chandrasekhar, S. 1943 Rev. mod. Phys. 15, 1. Edwards, S. F. 1976 Molecular fluids (eds. R. Balian and G. Weill). New York: Gordon and Breach. Reif, F. 1965 Statistical and thermal physics, oh. 15. New York: McGraw-Hill. Resibois, P. & De Leener, M. 1978 Classical kinetic theory of fluids. Wiley.

21 THE SURFACE STATISTICS OF A GROWING AGGREGATE Mehran Kardar Physics Department, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A. Abstract

The paper on ' The surface statistics of a granular aggregate' by Edwards and Wilkinson (1982) established an enduring framework for characterizing dynamic scaling phenomena for growing surfaces. More generally, it provided a new way of analyzing fluctuation phenomena in a variety of nonequilibrium circumstances. After briefly reviewing some key concepts of the Edwards-Wilkinson equation, I describe a number of its generalizations in connection with different types of growing surfaces. Growth equations are related by mathematical links to sums over directed paths in random media. The latter reveals further connections to the contributions of Edwards in polymer physics (path integrals) and disordered systems (the replica formalism). 21.1

The Edwards Wilkinson equation

Rough surfaces abound in nature, from microscopic (cracks) to macroscopic (mountains) scales. A statistical self-similarity often underlies the fluctuations in topography, in which case they can be described as self-affine fractals (Mandelbrot 1982). How can local stochastic events that shape and modify these surfaces give rise to correlations that extend over long length scales? The paper by Edwards and Wilkinson (henceforth EW) sets up a methodology for answering this question in a specific context (sedimentation), but provides a general framework that has been employed for addressing a rich variety of problems in nonequilibrium systems (Edwards and Wilkinson 1982). 21.1.1 Derivation The question posed in Edwards and Wilkinson (1982) is: 'What is the magnitude of the fluctuations in the height of surface [when] we take a bin and gently and uniformly pour in a granular material?' Since this issue has little to do with the forces and interactions that control the positions of individual grains, the standard methods of physics do not provide

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much insight into this query. However, EW demonstrate that the methods of stochastic processes, in particular Langevin and Fokker-Planck equations, can be successfully adapted to address this problem. Along the way, they emphasize how the general concepts of symmetries and conservation principles constrain the equations that govern the statistics of surfaces. Given the stochastic nature of the problem, EW start by considering the probability functional p[z(p), N] of finding the surface with a shape described by its height z above a point p, after N particles have been added to the aggregate. Through the addition of grains the surface profile changes, and the corresponding probability is modified according to the general evolution eqn. (2.1) of this paper. To proceed further, EW need to make assumptions about the form of the evolution kernel, notably its locality (only neighboring sites are affected) and independence of the absolute height. In a linear approximation, they can characterize the statistics of the Fourier modes of the surface profile, and show that they behave as solutions of the Langevin equation (2.12). Finally returning to real space, they end up with

where r is the rate of deposition (per unit time and area) of grains of size a and volume h. This is eqn. (2.13) of Edwards and Wilkinson (1982), which is now known as the Edwards-Wilkinson (EW) equation in the surface growth community. The stochastic nature of the aggregation process is now captured by the random noise £(p, t) which encodes the non-uniformities of the deposition process. 21.1.2 Results In the linear approximation, the Fourier modes of the surface profile are related to those of the stochastic noise by

Random and uncorrelated deposition of grains of size a leads to a noise £k,w of zero mean, with two-point correlations [eqn. (3.5) of EW, with a factor L 2 T set to unity]

From eqns. (21.2) and (21.3) various two-point correlation functions of the surface can then be extracted. After the surface reaches a steady state at a sufficiently long time t, the static fluctuations in the profile between two points at a separation p behave as in two dimensions; in one dimension.

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EW also discuss the dynamic behaviour of the fluctuations, and in two dimensions find

While there are no correlations in the random process of deposition, the local but collective rearrangements of the grains upon settling result in correlations of surface fluctuations that span much longer length- scales. The above results are simple to derive from the linear diffusion equation. What is significant is the overall vision of how methods of stochastic processes can be adapted to an inherently nonequilibrium situation: First the object of interest is identified (the fluctuating surface) and represented by a continuous field. Equations describing the time evolution of this field (or the corresponding probability functional) are then constructed by appealing to general concepts of locality, symmetry, and (implicitly) conservation principles. The correlations (in space and time) of the field can then be calculated by standard methods. While the focus of Edwards and Wilkinson (1982) is the linearized equation, in the conclusions they state that nonlinear terms could well appear in this process and lead to quantitative changes. These comments anticipate the later work in this subject, as described further on. 21.1.3 Numerical simulations Numerical simulations have played an important role in the field of aggregation and surface growth. One of the influential papers characterizing the self-affine fluctuations and dynamic scaling of growing surfaces is that of Family and Vicsek (FV; 1985). In this work particles aggregate into a two-dimensional cluster (with a one-dimensional surface) through a process of ballistic deposition. (A new particle starts at a random point above the surface, drops along a straight line, and is incorporated into the cluster at the point of first contact.) The starting flat surface becomes rough, with fluctuations on large scales. In analogy to dynamic scaling phenomena close to a critical point, FV characterize the fluctuations of the surface by an ansatz that is now widely used in studies of self-affine roughness. Using the currently standard notation, the width of the surface w at a time t after deposition starts (t is proportional to the average height of the surface) on a substrate of linear dimension L satisfies the scaling form

The dynamic exponent z describes the evolution of correlated regions with time: initially, different parts of the surface are independent; but regions of correlated roughness form over time, their size growing as £(t) oc tl/z . In each correlated region, the width of surface grows as the observation scale raised to the roughness exponent a. From eqn. (21.4), we note that the EW equation implies a = 1/2 for a onedimensional surface, while z = 2 due to the diffusive character of fluctuations in

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this equation. The latter should be manifested in the short-time growth of the surface width, before saturation effects from the finite size set in. For the width w in eqn. (21.6) to be independent of L, we must have lim^^o f ( x ) ~ xa/z, leading to w(t)^t^, with (3 = a/z. The results of numerical simulations in Family and Vicsek (1985) could indeed be fitted to a « 0.5, while [3 K, 0.30 was somewhat different from the prediction of [3 = 1/4 from the EW equation. Later simulations have confirmed that this difference is significant, and is related to the importance of nonlinearities in the ballistic deposition process, as will be described in the next section. In a subsequent paper, Family (1986) performed a simulation closer in spirit to the sedimentation process, in which the added particles could diffuse on the surface. For this model, he indeed finds values of a K 0.5 and (3 « 0.25, as predicted by the EW equation. 21.2

The Kardar Par is i Zhang equation

21.2.1 Derivation Following the example of EW, in this section we take a general approach to describing the statistics of a growing surface. Changing notation from the previous section, the height of the surface will be denoted by /i(x, t), rather than z ( p , t ) . As long as the stochastic evolution of the surface is local, it can be decomposed as

• Particle deposition is described by the first term. Thus ry(x, t) is a random function whose mean value gives the average particle flux at x, and its fluctuations represent the shot noise in deposition. We shall assume that the noise is uncorrelated at different sites and different times, i.e., it is a Gaussian process with

[The noise considered by EW in eqn. (21.3) also takes into account the finite size of the particles.] • Surface relaxation subsequent to deposition is described by the functional $, and may depend upon various properties of the height, such as its slope V/i, or curvature V 2 /i. We shall assume that the relaxation is local, i.e., it can be adequately described by the first few terms of an expansion of $ in h and its gradients. Which terms can be included in such an expansion is then determined by the underlying symmetries and conservation laws appropriate to the dynamics. The basic idea is that any term that is not excluded for fundamental reasons of symmetry or conservation will be generically present. Let us now consider a number of cases.

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• Random deposition with no subsequent relaxation corresponds to $ = 0. This is also the first case considered in EW. Integrating dth = ry(x, t) yields

from which we can immediately obtain corresponding to /? = 1/2, and a correlation length of zero. • Settling, as discussed by EW is, at the lowest order, proportional to the local curvature, leading to This is of course the EW equation (21.1) in new notation; averaging its solution in Fourier space gives

Recasting the above result in real space then leads to the exponents

for the d-dimensional surface of a d+ 1 dimensional aggregate. In particular, in d = 2, a = 1/2 and /? = 1/4, while for d = 3, as in eqns. (21.4) and (21.5), the mean-square width grows logarithmically in both space and time. • Ballistic deposition is generically a nonlinear process, as anticipated by EW. There is no a priori reason why the relaxation function $ should not depend on the slope V/i. (A direct dependence on h itself is ruled out by the translational symmetry, h —> h + constant, of the underlying dynamics.) By symmetry, the relaxation should be the same for slopes ±V/i, and hence the first term in an expansion in powers of slope starts with (V/i) 2 , leading to

Higher-order terms can also be added, but are irrelevant in that they do not change the scaling properties. There are several excellent reviews of eqn. (21.14), known as the Kardar-Parisi-Zhang (KPZ) equation (Kardar et al. 1986). Since the origin and properties of the nonlinear term are discussed in detail in these reviews (Halpin-Healy and Zhang 1995; Barabasi and Stanley 1995), I shall not elaborate on them any further. As described in the next section, in d = 1 the nonlinear equation leads to the exact exponent values of a = 1/2 and /? = 1/3 (as compared to /? = 1/4 for the EW equation), in excellent agreement with the numerical simulations (Family and Vicsek 1991; for a review, see Meakin 1993). There are no exact solutions in d = 2, but based on simulations we can estimate a K 0.38 and (3 « 0.24. The KPZ equation appears to describe the asymptotic behaviour of most local, random growth processes.

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21.2.2 Scaling behaviour in one dimension It is interesting to note that the KPZ and EW equations have exactly the same static properties in steady state lor d = 1, as can be proved by examining the probability distribution P[h]. As the surface changes in time according to eqn. (21.14), the corresponding probability evolves according to the FokkerPlanck equation

The term in the square brackets is due to the deterministic probability current, the remainder comes from the stochastic noise. A steady state solution is one for which dtP = 0. In the absence of the nonlinear term, the steady state solution to eqn. (21.15) is a simple Gaussian,

where TV is a normalization constant. In general, this is not a steady state for A ^ 0. In one dimension, however, the contribution of the nonlinear term to the probability current can be simplified to

a surface integral safely set to zero in an infinite system. Thus, the steady-state spatial correlations are not influenced by the presence of the nonlinearity, and both EW and KPZ equations share the exponent a = 1/2 in d = I . To get the exponent /?, note that the slope v(x, t) = — AV/i(x, t) satisfies

which is known as the Burgers equation (Burgers 1974). It describes the velocity v in a vortex-free fluid (since V x v = — AV x V/i = 0) of viscosity v1 which is randomly stirred by a force AVry. Its scaling properties were first discussed through renormalization-group methods by Forster, Nelson, and Stephen (1977). Since the underlying Navier-Stokes equation is derivable from Newton's laws of motion for fluid particles, it has the Galilean invariance of changing to a uniformly moving coordinate frame. This symmetry is preserved under renormalization to larger scales, and requires that the ratio of the two terms on the left hand side of eqn. (21.18) (dtv and v- Vi>) stays at unity. In terms of eqn. (21.14), this implies the exponent identity

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This identity is indeed satisfied by all numerical simulations in the KPZ universality class. For d = 1, in conjunction with a = 1/2, it leads to (3 = 1/3 which corresponds to super-diffusive dispersion of fluctuations. 2f .2.3 Conservative growth We may ask why the KPZ nonlinear!ty is not present in the model of sedimentation simulated by Family (f 986), which exhibits the EW exponent of /? = f/4. The reason is that it is forbidden by a conservation law. If the growth process does not allow the formation of overhangs or voids, and there is also no desorption, then all the incoming flux is incorporated into the growing aggregate. This means that the net mass of the aggregate, proportional to H(t) = j ddx /i(x, t), can only change due to the random deposition; the relaxation processes should conserve H(t). This immediately implies that the relaxation function must be related to a surface current, i.e.,

The KPZ nonlinearity is thus ruled out, as it cannot be written as the divergence of another function. Following observations of Villain (1991), many studies have focused on such conservative models in the context of MBE growth. The basic idea is that aggregates formed by the MBE process are typically free from holes and defects, and that the desorption of the adsorbed particles is negligible. It is thus argued that, at least over some sizeable pre-asymptotic regime, the relaxation processes should be conservative. Some examples of such conservative models are mentioned next. • Surface diffusion currents in equilibrium can be related to variations of a chemical potential via j oc —Vy«. Since, in the equilibrium between two phases, the chemical potential is proportional to local curvature (n oc — V 2 /i), this leads to an equation of motion

This equation is again linear, and as in eqn. (21.12) can be solved by Fourier transformation to yield,

The corresponding exponents in real space are,

This leads to a = 3/2 and /? = 3/8 in d = 1, and a = I and /? = 1/4 in d = 2. Note that the surface remains self-affine, maintaining a well-defined orientation, only as long as a < 1. The above large values of the exponent a indicate a breakdown of validity of the above equation for d < 2.

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• Nonlinear 'MBE' models have been proposed partly to remedy the breakdown of the linear equation for a > I . One such model starts with a nonlinear chemical potential introduced by Sun, Guo and Grant (1989), resulting in

Despite its nonlinear form, eqn. (21.24) can in fact be analyzed to yield the exact exponents (Lai and Das Sarma 1991)

In particular, a = 2/3 and /? = 1/5 in d = 2. However, in nonequilibrium circumstances, there is no reason for the surface current to be derivable from a chemical potential. Removing this restriction allows the inclusion of other nonlinearities (Lai and Das Sarma 1991), such as V(V/i) 3 , which are in fact more relevant. More importantly, nonequilibrium currents should generically include a term proportional to V/i, which is the dominant gradient, as discussed next. • Diffusion bias refers to generic nonequilibrium currents that are proportional to the local surface slope (Krug et al. 1993). One possible origin of such currents is in the Schwoebel barriers (Schwoebel 1969): Atoms on a stepped surface are easily incorporated at the step to a higher ledge, but are reflected by a barrier on attempting to jump to a lower ledge (Burton et al. 1951). This sets up a net up-hill current (Villain 1991) j = v' V/i, leading to an equation of motion

Due to the different sign from the EW equation, this leads to an unstable growth of fluctuations, and therefore higher-order terms are necessary to ensure stability. For example, Johnson et al. (1994) have proposed the following nonlinear equation

The instabilities in this equation develop into a complex array of mounds dubbed SLUGs (Super Large Unstable Growths). Finally, it is also possible for the nonequilibrium currents to be oriented downhill (as in sedimentation), in which case the behaviour is the same as for the EW equation. The equations presented in this section are but a small sample of many that have been proposed, and which in some way can be traced back to the EW equation. 21.2 .4 Experiments An excellent review of the experimental research in this subject is provided by Krim and Palasantzas (1995). The observational methods include diffraction (specular or diffuse X-rays, RHEED, LEED, HR-LEED, and helium scattering),

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direct imaging (STM, AFM, SEM, TEM), and surface adsorption. A variety of metallic (Ag, Au, Cu, Fe) and other (Si, InP, NbN, polymer) surfaces grown under a host of different conditions have been examined by such probes. Some of these surfaces exhibit unstable growth, while others appear to satisfy selfsimilar scaling. However, there is usually no clear-cut identification of the exponents with the theoretical models. Some experiments on Au and Ag give roughness exponents consistent with the KPZ value, but larger values of /?. Other surfaces give larger values of a, consistent with those of the nonlinear 'MBE' equation (21.24). The reader is referred to this review article (Krim and Palasantzas 1995) for the details. Perhaps the following statements at the end of the review are most revealing of the experimental situation: 'Over 50% of the experimental work reported on here was published in the interval from January 1993 to August 1994. The pace of experimental work is clearly accelerating, and rapid advances in the field can be expected.' Given the discrepancies between experiment and theory, we can also ask if important elements have been left out of the analysis. The formalism presented so far deals solely with a single coarse-grained variable, the height /i(x, t ) . Other variables may play an important role in the evolution of h. For example, in many cases the roughness is intimately related to formation of micro-crystalline grains. Variations in crystallinity have so far been left out of the theoretical picture. In principle, one could introduce an additional 'order parameter' describing the local degree of crystallinity. Surface relaxation may then depend on this order parameter, as explored further in the final section. 21.3

Directed paths in random media

The Cole-Hopf transformation maps the KPZ equation to the problem of directed polymers in random media. While the latter is not related to the statistics of growing surfaces, it makes connections to other areas greatly influenced by the work of Edwards, notably the path-integral formulation for polymers, and the replica approach to disordered systems. It also allows me to recall how what I learned from Edwards (in a course at Cambridge University) helped at a crucial stage in my research. 21.3.1

The Cole-Hopf

transformation

The nonlinear KPZ equation (21.14) can in fact be changed to linear form (Burgers 1974) with the aid of a 'Cole-Hopf transformation,

The function W(x,t) evolves according to the diffusion equation, but unlike in the EW equation (21.1), the noise is multiplicative, as

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353

In the noiseless case, this mapping allows us to ask what happens in the case of a slow and uniform snowfall on to an initial profile which at t = 0 is described by ho(x). For ry(x, t) = 0, eqn. (21.29) can be solved subject to the initial condition W(x,t = 0) = exp[A/io(x)/2z/] using the diffusion kernel, leading to a profile at time t given by

It is instructive to examine the v —> 0 limit, which is indeed appropriate to snowfalls, since there is not much rearrangement after deposition. In this limit, the integral in eqn. (21.30) can be performed by the saddle-point method. For each x we have to identify a point x' that maximizes the exponent, leading to a collection of paraboloids described by

Such parabolic sequences are quite common in many growth processes in Nature, from biological to geological formations. The patterns for A = 1 are identical to those obtained by the geometrical construction of Huygens, familiar from optics. The growth profile (wave front) is constructed from the outer envelop of circles of radius t drawn from all points on the initial profile. The nonlinearity in eqn. (21.14) thus algebraically accounts for their origin. As growth proceeds, the surface smoothens by the coarsening of the parabolas. What is the typical size of the features at time t? In maximizing the exponent in eqn. (21.31), we have to balance a reduction |x — x' /2At, by a possible gain from /IQ(X') in selecting a point away from x. The final scaling is controlled by the roughness of the initial profile. Let us assume that the original pattern is a self-affine fractal of roughness a, i.e.,

[According to Mandelbrot (1982), a « 0.7 for mountains.] Balancing the two terms in eqn. (21.31) gives

For example, if the initial profile is like a random walk in d = 1, a = 1/2, and z = 3/2. This leads to the spreading of information along the profile by a process that is faster than diffusion, since Sx ~ t 2 / 3 .

354 21.3.2

Surface Statistics of a Growing Aggregate by Mehran Kardar Directed polymers

For nonzero noise, we can still formally solve eqn. (21.29), casting the solution as a path integral:

The functional integral is over all paths x(t') that start at the origin and end up at point x after time t. The weight for each path can be regarded as a Boltzmann weight with two contributions to the energy. The first is proportional to the 'kinetic energy' (ctx/cfe) 2 , which in the polymeric context can be regarded as an effective elasticity of entropic origin. This is of course a key component in the path-integral formulation of self-avoiding polymers developed by Edwards (see, e.g., Doi and Edwards 1986). The second effect is due to the multiplicative noise, which can be regarded as a frozen random energy landscape proportional to ry(x, t). An overall 'potential energy' is then obtained by adding all the random energies along the path. The different space-time trajectories can also be regarded as configurations of a polymer directed along the t direction, now regarded as an additional spatial direction. Equation (21.34) is then the partition function of all directed paths (with constrained end points) in a medium with quenched random energies. The problem of directed paths in random media (DPRM) occurs in a great many physical situations. One of the original motivations was to understand the domain wall of a two-dimensional Ising model in the presence of random bond impurities (Huse and Henley 1985). If all the random bonds are ferromagnetic, in the ground state all spins are up or down. Now consider a mixed state in which a domain wall is forced into the system by fixing spins at opposite edges of the lattice to + and —. Bonds are broken at the interface of the two domains, and the total energy of the defect is twice the sum of all the the random bond energies crossed by the interface. In the Solid-On-Solid approximation, configurations of the domain wall are restricted to directed paths. In the continuum limit, the partition function is similar in form to W(t) in eqn. (21.34) for d = 1. In the limit of zero temperature (or v —> 0), eqn. (21.34) reduces to finding the directed path of lowest energy connecting the points (0,0) and (x, t ) . Unlike the traveling salesman problem, to which it bears some resemblance, the directed version can be solved in polynomial time. For concreteness, consider the ensemble of all paths directed along the diagonal of a square lattice, with random energies {Jx,t} assigned to each bond. Denoting by E(x,t) the minimum in the energy of all paths connecting (0,0) to (x, t), it is possible to construct the recursion relation, To find the actual configuration of the path, it is also necessary to store in memory one bit of information at each point (x,t), indicating whether the

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minimum in eqn. (21.35) comes from the first or second term. This bit of information indicates the direction of arrival for the optimal path at ( x , t ) . After the recursion relations have been iterated forward to 'time' step t, the optimal location is obtained from the minimum of the array {E(x,t)}. From this location the optimal path is reconstructed by stepping backward along the stored directions (Kardar and Zhang 1987). These optimal paths have a beautiful ultrametric structure that resembles the deltas of river basins, and many other natural branching patterns. The statistics of the E(x, t) are identical to those of \nW(x,t): the optimal path wanders as t 2 / 3 , while the fluctuations in E(t) scale as t 1 / 3 . It has been suggested that optimal paths are also relevant to fracture and failure phenomena; see, e.g., Herrmann and de Arcangelis (1990). Imagine a two-dimensional elastic medium with impurities, e.g., a network of springs of different strengths and extensions. If the network is subjected to external shear, a complicated stress field is set up in the material. It is possible that nonlinear effects in combination with randomness enhance the stress field along particular paths in the medium. Such bands of enhanced stress are visible by polarized light in a sheet of plexiglass. In fact, the minimal-energy directed path was proposed in 1964 as a model for the tensile rupture of paper (Tydeman and Hiron 1964). The variations in brightness of a piece of paper held in front of a light source are indicative of nonuniformities in local thickness and density /o(x). Tydeman and Hiron suggested that rupture occurs along the weakest line for which the sum of /o(x) is minimum. This is clearly just a continuum version of the optimal energy path in a random medium. (Since the average of /o(x) is positive, the optimal path will be directed.) The three-dimensional DPRM was introduced (Kardar and Zhang 1987) as a model for a polyelectrolyte in a gel matrix. Probably a better realization is provided by defect lines, such as dislocations or vortices, in a medium with impurities. There has been a resurgence of interest in this problem since it was realized that flux lines in high-temperature ceramic superconductors are highly flexible, and easily pinned by the oxygen impurities that are usually present in these materials (Nelson 1988; Nelson and Seung 1989). Pinning by impurities is in fact crucial for any application, as otherwise the flux lines drift due to the Lorentz force, giving rise to flux flow resistivity (Blatter et al. 1994). 21.3.3 The replica approach Summing over directed polymers is a simple, yet non-trivial, example of statistical mechanics in the presence of quenched random impurities. The primary tool for dealing with disordered systems is the replica formalism, invented by Edwards in connection with rubber networks, and developed further with Anderson in the context of spin glasses (Edwards and Anderson 1975). This method also provides valuable information on the problem of DPRM. The n th moment of the partition function W in eqn. (21.34) is computed by introducing n replicas of the original path. Averaging over the random energies ry(x, t) then leads to a

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partition function for n interacting paths, according to

where u = A 2 _D/(4i/ 2 ). The price for removing the quenched randomness is the interaction introduced between the replicas; in this case an attraction of strength u between any pair of paths that come into contact. In the same way that eqn. (21.34) was obtained by integrating eqn. (21.29), the above path integral is the solution to the differential equation

We are mostly interested in the asymptotic (thermodynamic) limit of large t, in which case the behaviour of Wn is dominated by the ground-state energy of the 'quantum' Hamiltonian Hn. In one dimension, the ground-state wave-function can be obtained by a simple 'Bethe ansatz,' (Thacker 1981) as

The resulting ground state energy energy is n(r? — 1) x (w 2 /48i/). In Kardar (1987), the absence of the n2 and the presence of the n3 term in the energy lead to the deduction that the typical fluctuations in the quenched free energy scale as t^ with /? = 1/3 in agreement with the arguments of the previous section. Unfortunately, the exact dependence of the bound-state energy on n is not known in higher dimensions. Elementary results from quantum mechanics tell us that an arbitrarily small attraction leads to the formation of a bound state in d < 2, but that a finite strength of the potential is needed to form a bound state in d > 2. Thus, in the most interesting case of 2 dimensions we expect a non-trivial probability distribution, while the replica analysis provides no information on its behaviour. In higher dimensions, there is a phase transition between weak and strong randomness regimes. For weak randomness there is no bound state and asymptotically Wn(t) = W(t) ™, indicating a sharp probability distribution. At one time it was hoped that a renormalization-group study of the KPZ equation could provide information about the scaling behaviour in the strongcoupling regime. The above analogy to attracting particles in quantum mechanics negates this possibility. In particular, Wiese (1998) has shown that to all orders

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in perturbation theory the coupling constant g = A 2 _D/V 3 evolves under change of scale (by a factor b = ee) as

The above equation merely confirms the expectations based on the replica analysis: there is flow to strong coupling for d < 2 (formation of a bound state), while there is a transition between weak (unbound) and strong (bound) coupling behaviour in higher dimensions. While the renormalization group provides some exact information about the transition between the two regimes, eqn. (21.39) sheds no light on the nature of the nontrivial phase. This has lead to much speculation about the exponents in the rough phase of the KPZ, and the presence or absence of an upper critical dimension (see, e.g., Lassig 1998). One of the promising approaches to estimating the exponents in the strong-coupling phase is due to Schwartz and Edwards (1998, 2002), and relies on a self-consistent evaluation of the correlation functions via the Fokker-Planck equation for the KPZ equation. 21.3.4 Many directed polymers I would like to take advantage of the opportunity offered by this tribute to acknowledge a personal scientific debt to Sam Edwards. In the mid-eighties I collaborated with David Nelson on the nature of commensurate to incommensurate transitions in adsorbed layers with quenched random impurities (Kardar and Nelson 1985). This problem can be formulated as a collection of N directed paths in a landscape of random energies, generalizing eqn. (21.34) to

The potential U(x — x') prevents any two paths from coming into contact, and can be replaced with the condition that the paths should not intersect. In 1+1 dimensions, the latter constraint can be implemented by regarding the paths as world-lines of fermions. After replication and averaging over disorder, the analog of eqn. (21.36) is

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The self-avoiding condition is now implicit, with the quantum Hamiltonian HN^ describing n species of fermions, with N fermions per family. Fermions belonging to different species attract on contact with strength u. The replicated partition function is dominated by the ground-state energy of H^m which has the form

The first term is simply N times the bound-state energy obtained for a single replicated path in the previous section. Loosely interpreted, the second term expresses the fact that the N molecules cannot occupy the same ground state (because of their fermionic character), and must occupy states of higher kinetic energies. In the absence of the interaction u, the momenta {ki} are equally spaced with separation 2vT/L, where L is the system size. Summing over all momenta in a Fermi sea then leads to a ground-state energy that grows as r3, where r = N/L is the particle density. Because of the interactions between particles of different species, {k^} are no longer equally spaced. Instead, their density p(k) satisfies the integral equation (Kardar 1987; see also Emig and Kardar 2001)

To obtain the quenched average free energy, eqn. (21.42) has to be continued to n —s- 0, and in the interesting limit of low densities (r —> 0) the corresponding integral equation is

While struggling with this problem in 1984, I had managed to get as far as eqn. (21.44), but had no idea of how to proceed. The kernel of the above integral equation was very different from the ones encountered in the usual references on the Bethe ansatz (Thacker 1981), which I was pursuing at the time. I then recalled that in the summer of 1978, while a student at Cambridge University, I had taken a course on mathematical physics from Sam Edwards, which covered integral equations along with a multitude of other topics. I looked over my lecture notes and was overjoyed to find that this equation had indeed been discussed. (Apparently it arises in the context of the lift of aerofoils, and is attributed to Carleman.) The resulting density is a semicircle,

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359

and leads to a free energy that behaves at low densities as r2 (as opposed to r3 in the pure problem). This leads to a new universality class for commensurate to incommensurate transitions in adsorbed layers with disorder (Kardar and Nelson 1985) which, incidentally, bears the same relation to the corresponding transition in pure systems (Pokrovsky and Talapov 1979) that the KPZ exponents have to those of EW. 21.4

Perspectives

In the previous two sections I have touched upon a variety of topics that can be traced back to the paper by EW. However, this represents only a small sample of a much broader set of subjects ranging from mathematical optimization to granular media, from flux line to traffic flow. Indeed, if we could visualize the growing cluster of ideas aggregating to the original seed of the EW paper, the shape would be very different from the compact structures studied by EW, resembling more a highly branched polymer. Providing a perspective on such a dynamic object would be foolhardy; instead I shall briefly mention a couple of interesting extensions. 21.4.1 Sequence alignment The ability to rapidly sequence the DNA from different organisms has made a large body of data available, and created a host of challenges in the emerging field of bioinformatics. Let us suppose that the sequences of bases for a gene, or (equivalently) the sequence of amino acids for a protein, has been newly discovered. Can one obtain any information about the potential functions of this protein, given the existing data on the many sequenced proteins whose functions are (at least partially) known? A commonly used method is to try to match the new sequence to the existing ones, finding the best possible alignment (s) based on some method of scoring similarities. Biostatisticians have constructed efficient (so-called dynamic programming) algorithms whose output is the optimal alignment, and a corresponding score. How can one be sure that the resulting alignment is significant, and not due to pure chance? The common way of assessing this significance for a given scoring scheme is to numerically construct a probability distribution for the score by simulations of matchings between random sequences. This is time-consuming (especially in the relevant tails of the distribution), and any analytical information is a valuable guide. It was noted by Hwa and Lassig (1996) that finding the optimal alignment of two sequences {sj} (i = 1, 2 , . . . , / ) and { s j } (j = 1, 2 , . . . , J) is similar to finding the lowest-energy directed path on an / x J lattice. Each diagonal bond [from (i, j) to (i + 1, j' + l)] is assigned an 'energy' equal to the score of the local pairing [si and Sj], and there are additional costs associated with segments along the axes (corresponding to insertions and/or deletions). The 'dynamic programming' algorithm is an appropriate variant of eqn. (21.35) used to recursively obtain the directed path (alignment) of optimal energy (score). Some aspects of the probability distribution for the score can then be gleaned from the knowledge of

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the distribution for the energy of the directed paths in random media. Indeed, some of these results are going to be implemented in the widely-used alignment algorithms (PSI-BLAST) disseminated by the National Center for Biotechnology Information. 21 .4.2 Textural growth I would like to conclude by returning to the problem of the roughness of a growing surface. While the experimental situation is not clear-cut (Krim and Palasantzas 1995), most observations report roughness exponents that are usually different from those expected on the basis of the KPZ equation (Halpin-Healy and Zhang 1995; Barabasi and Stanley 1995; Family and Vicsek 1991; for a review, see Meakin 1993). One possibility is the importance of a conservation law, as emphasized in the models of MBE discussed earlier (Sun et al. 1989; Lai and Das Sarma 1991; Siegert and Plischke 1994; Krug et al. 1993). Another possibility is that some relevant fields, necessary to the description of the problem, have been left out. The growth rate of a material characterized by an order parameter, such as in a magnet, may well be influenced by the local values of this parameter. Conversely, the evolution of the order parameters on the surface may depend on its profile. In Kardar (1999) the pair of coupled equations

where employed to describe the coupling of a KPZ surface to an angular order parameter 9. (The latter field is only allowed to change on the exposed surface, and is assumed to be frozen once incorporated into the bulk.) It was demonstrated that such growth (in the noiseless limit) leads to domains separated by sharp domain walls, a texture very different from the soft modes and vortices that characterize diffusive relaxation of spins. With such a perspective, we may regard the KPZ equation as applicable to the growth of an amorphous (glassy) material with no internal order. The growth of a crystal, however, inherently involves long range order, and the breaking of symmetry results in surface phonons which may interact with the height fluctuations. In the case of an isotropic surface, the appropriate generalization of eqns. (21.46) is (da Silveira and Kardar 2003)

where u(x, y, t) is the surface displacement vector field, and /zj,, z/j, are the usual Lame coefficients. It would be interesting to know if the above coupled equations

References

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involve scaling behaviours distinct from the usual KPZ equation. Standard fieldtheoretic renormalization-group methods will probably result in strong-coupling flows that shed no light on the exponents, and simulations are likely to be difficult, due to the complexity of the equations. One potential avenue of exploration is the appropriate generalization of self-consistent schemes, such as that of Schwartz and Edwards (1998, 2002) for the KPZ equation. Acknowledgments

This work was supported by the National Science Foundation through grant No. DMR-01-18213. I am grateful to Rava da Silveira for critical reading of the text. References

Barabasi, A.-L. and Stanley, H. E. (1995). Fractal Concepts in Surface Growth. Cambridge University Press. Blatter, G., Feigelman, M. V., Geshkenbein, V. B., Larkin, A. I. and Vinokur, V. M. (1994). Rev. Mod. Phys. 66, 1125. Burgers, J. M. (1974). The Nonlinear Diffusion Equation. Riedel, Boston. Burton, W. K., Cabrera, N. and Frank, F. C. (1951). Phil. Trans. R. Soc. Land. A 243, 299. Charmet, J. C., Roux, S. and Guyon, E. (eds.) (1990). Disorder and Fracture. NATO ASI Series B, Physics 235. Plenum Press, New York, da Silveira, R. and Kardar, M. (2003). Phys. Rev. E. 68, 046108. Doi, M. and Edwards, S. F. (1986). The Theory of Polymer Dynamics. Oxford University Press, New York. Edwards, S. F. and Anderson, P. W. (1975). J. Phys. F5, 965. Reprinted in this Volume. Edwards, S. F. and Wilkinson, D. R. (1982). Proc. R. Soc. Lond. A 381, 17. Reprinted in this Volume. Emig, T. and Kardar, M. (2001). Nucl. Phys. B604, 479. Family, F. (1986). J. Phys. A19, L441. Family, F. and Vicsek, T. (1985). J. Phys. A18, L75. Family, F. and Vicsek, T. (eds.) (1991). Dynamics of Fractal Surfaces. World Scientific, Singapore. Forster, D., Nelson, D. R. and Stephen, M. J. (1977). Phys. Rev. A 16, 732. Halpin-Healy, T. and Zhang, Y.-C. (1995). Phys. Rep. 254, 215. Herrmann, H. J. and de Arcangelis, L. (1990). In Charmet et al. 1990. Huse, D. A. and Henley, C. L. (1985). Phys. Rev. Lett. 54, 2708; see also Hwa (1999). Hwa, T. (1999). Nature 399, 43. Hwa, T. and Lassig, M. (1996). Phys. Rev. Lett. 76, 2591. Johnson, M. D., Orme, C., Hunt, A. W., Graff, D., Sudijono, J., Sander, L. M. and Orr, B. G. (1994). Phys. Rev. Lett. 72, 116. Kardar, M. (1987). Nucl. Phys. B290, 582.

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Kardar, M. (1999). Physica A 263, 345. Kardar, M. and Nelson, D. R. (1985). Phys. Rev. Lett. 55, 1157. Kardar, M., Parisi, G. and Zhang, Y.-C. (1986). Phys. Rev. Lett. 56, 889. Kardar, M. and Zhang, Y.-C. (1987). Phys. Rev. Lett. 58, 2087. Krim, J. and Palasantzas, G. (1995). Int. J. Mod. Phys. B 9, 599. Krug, J., Plischke, M. and Siegert, M. (1993). Phys. Rev. Lett. 70, 3271. Lai, Z. W. and Das Sarma, S. (1991). Phys. Rev. Lett. 66, 2348. Lassig, M. (1998). Phys. Rev. Lett. 80, 2366. Mandelbrot, B. B. (1982). The Fractal Geometry of Nature. Freeman, San Francisco. Meakin, P. (1993). Phys. Rep. 235, 191. Nelson, D. R. (1988). Phys. Rev. Lett. 60, 1973. Nelson, D. R. and Seung, H. S. (1989). Phys. Rev. B 39, 9153. Pokrovsky, V. L. and Talapov, A. L. (1979). Phys. Rev. Lett. 42, 65. Schwartz, M. and Edwards, S. F. (1998). Phys. Rev. E 57, 5730. Schwartz, M. and Edwards, S. F. (2002). Physica A 312, 363. Schwoebel, R. L. (1969). J. Appl. Phys. 40, 614. Siegert, M. and Plischke, M. (1994). Phys. Rev. E 50, 917. Sun, T., Guo, H. and Grant, M. (1989). Phys. Rev. A 40, 6763. Thacker, H. B. (1981). Rev. Mod. Phys. 53, 253 (1981); and references therein. Tydeman, P. A. and Hiron, A. M. (1964). B. P. & B.I.R.A. Bulletin 35, 9. Villain, J. (1991). J. Phys. I 1, 19. Wiese, K. J. (1998). J. Stat. Phys. 93, 143.

22 REPRINT THEORY OF POWDERS by S. F. Edwards and R. B. S. Oakeshott Physica A: Statistical and Theoretical Physics, 157, 1080-1090 (1989).

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Physica A 157 (1989) 1080-1090 North-Holland, Amsterdam

THEORY OF POWDERS

S.F. EDWARDS and R.B.S. OAKESHOTT Cavendish Laboratory, Madingley Road, Cambridge CBS OHE, UK Received 20 February 1989

Starting from the observations that powders have a large number of particles, and reproducible properties, we show how statistical mechanics applies to powders. The volume of the powder plays the role of the energy in conventional statistical mechanics with the hypothesis that all states of a specified volume are equally probable. We introduce a variable X - the compactivity - which is analogous to the temperature in thermodynamics. Some simple models are considered which demonstrate how the problems involved can be tackled using the concept of compactivity.

1. Introduction

There is an increasing interest in applying the methods of statistical mechanics and of transport theory to systems which are neither atomistic, nor in equilibrium, but which still fulfil a remaining tenet of statistical physics which is that systems can be completely defined by a very small number of parameters and can be constructed in a reproducible way. Powders fall into this category. If a powder consists for example of uniform cubes of salt, and is poured into a container, falling at low density uniformly from a great height, one expects a salt powder of a certain density. Repeating the preparation reproduces the same density. A treatment such as shaking the powder by a definite routine produces a new density and the identical routine applied to another sample of the initial powder will result in the same final density. Clearly a Maxwell demon could arrange the little cubes of NaCI to make a material of different properties to that of our experiment, but if such demonics are ignored, and we restrict ourselves to extensive operations such as stirring, shaking, compressing - all actions which do not act on grains individually - then well defined states of the powder result. In this paper we will set up a framework for describing the state of the powder, basing our development on anologies with statistical mechanics. Some attempts have been made to apply information theory ideas directly to powders 0378-4371/89/S03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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[1-3], but we want to introduce a formulation closer to conventional statistical mechanics in order to use the powerful ideas developed there. Although much information is available on powders and a notable literature exists [4-6], one is struck by the rather advanced situations discussed. It seems to us that the most elementary problems have not been fully addressed, so that this paper will study some powder problems at a very elementary level of statistical mechanical technique. We aim at new physical ideas and offer no new mathematical ideas. The number density of a powder lies between 10° mF1 and lO^ml"1. Although much less than the 1022 of atoms, this range takes one into the region where Van der Waals forces and plain electric charges become dominant. Although we do not need to refer to a density, we have in mind the range 103 to 106, i.e. friction is important but attractive forces are not dominant. For bulk samples, the numbers are still very large so that statistical arguments are entirely appropriate. We will argue that powders have an entropy, but it will be the volume which plays the role of the energy in normal statistical mechanics: the energy corresponding to the powder's thermal temperature 300 K is negligible. In statistical mechanics the most fundamental entry is via the microcanonical ensemble. This is a closed system, contained in a volume V, which is assumed to take up all configurations subject to the Hamiltonian function taking the value E with equal probability. Thus the distribution function is

where the entropy 5 is defined in terms of the total number of configurations

kB being Boltzmann's constant which harmonizes units of measurement. Although the microcanonical ensemble is the most fundamental, E is not easy to measure and the S function is difficult to handle, so it is easier to use the canonical ensemble and define the free energy F by the distribution function

where

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and

The energy is now given by

The most important variable describing the state of the powder is its density, or equivalently, but more usefully its volume V. The volume here is the actual volume taken up by the powder - i.e. as would be measured by letting a piston rest on the top of the powder. The volume therefore depends on the configuration of the particles - unlike the conventional case where the volume is set externally, and only the energy depends on the configuration of the particles. In principle other variables may be necessary to describe the state of a powder, i.e. if a powder is sheared, then it may develop an anisotropic texture, but we will not consider this possibility here. We now introduce a function W of the coordinates of the grains which specifies the volume of the system in terms of the positions and orientations of the grains. The form of W will depend on how the configuration of the grains is specified, and on the shape and size of the grains. We also introduce a function Q which picks out valid configurations of the grains - that is stable arrangements, where the particles can remain at rest under the influence of the confining forces, and with no overlap between particles. On the assumption that for a given volume all these configurations are equally probable, a table can be drawn up developing the analogy. (See table I.) There are model systems with very simple Hamiltonians available, e.g. the perfect gas or the Ising model, but in general H is not simple, for example a true ferromagnet has complicated many ion potentials. For powders, W and Q are never simple, e.g. hard spheres already offer a major problem from this point of view (though they can be written simply as is well known and discussed below). In table I we have introduced two new variables X and Y. X is the analogue of temperature and measures the "frothiness" or "fluffiness" of the powder. Since X = 0 corresponds to the most compact powder and X = °° to the least, we may name it the compactivity of the powder; it is the inverse of the compaction. A powder prepared with a larger volume than that corresponding to X = 00, and so with X<0, would be unstable: given any vibration the powder would tend to compact, increasing its entropy, and reducing its

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Table I Statistical mechanics

Powders

E H S kB (which gives 5 the dimensions of energy)

V W S A (which gives S the dimension of volume)

potential energy. Uur assumption tnat all configurations 01 a given volume are equally probable implies that the same X should characterize arrangements around particles in the powder with different sizes densities, coefficients of friction, etc. An obvious name for Y would be the free volume, but that phrase is already appropriated in glass and liquid theory, hence we propose the name "effective volume" for Y. Notice that our new functions have fewer dependencies than those of thermodynamics, e.g.

whereas

Similarly

as a consequence, there are no analogies of the Maxwell relations of classical thermodynamics. For a thermodynamic system

but since P = 0 = T in our powder, there is no value in an identification of X in these terms. The general problem of pressure in the resistance of a powder to

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compression is more advanced than anything at the level of this paper. This is because whereas the exertion of pressure will decrease volume irreversibly, there comes a point when the volume can no longer decrease by rearrangement, but requires deformation of the grains. This involves stress finding (multiply connected) "percolating" pathways through the solid, and this requires a different kind of analysis which is downstream from the present problem. Here we can omit a discussion of the pressure because we always assume that frictional thresholds are such that the powder can rest in the specified configuration. For simple frictional forces (F*£ n,N), and hard particles, the absolute gravitational field, or confining pressure does not matter. In a stable configuration the particles must be touching, so that these states are a subset of measure zero of the total phase space of a hard particle gas. Of itself this is no problem - it is only if states with different numbers of degrees of freedom exist that there is a problem because, since this is a purely classical problem, there is no natural scale analogous to Planck's constant, h. For frictionless particles stable states are local volume, minima, and so form a discrete set of states - the only remaining possibility is that the minima should have different weights. Two simple criteria for stability are possible. The simplest local criterion for stability is that no single particle is free to move. A global criterion is obtained if we use the Maxwell condition: the number of constraints greater than or equal to the number of variables. For spheres this implies that the average number of contacts per particle is at least 6. If the particles are not frictionless, then the true condition will lie between the local and global criteria. In particular states with fewer contacts will be stable, and maximising the entropy will select those states which minimise the number of contacts, whilst still being stable. To illustrate these ideas we study the packing of a simple one species powder. The crude model illustrates overall features, rather than detailed study of particular local environments. 2. The volume of a simple powder studied by the compactivity concept We start with the highly artificial, but instructive model of a powder in one dimension. Now powders obviously cannot exist in one dimension as gravity will always fully compact them, but we can still learn useful lessons. If the grains are rods of length a, whose midpoints are xn, with xn<xn_1, then clearly trivially

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1085

However we can also write W as a sum of local volumes,

The volume exclusion implies that

If we put as the stability condition that each rod touches its neighbour, so that

where

then the problem is trivial; there is one configuration, and e = 8(V— No). We can produce a model for a real - two- or three-dimensional - powder, if we consider our one-dimensional system as a section of the actual powder. The grains need not be touching in the section, but can have a range of separations up to a maximum b. O then becomes

and the integral for 5 is

Using the canonical ensemble we have

so that

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and

In two or three dimensions, this form using the particle coordinates is intractable: we have taken the hard particle fluid form, and added some difficult conditions. To do anything we must choose a smaller set of variables, with the same dimensionality as the set of stable states, and so making the stability and compatibility conditions simpler. By doing this we also bring in a Jacobian. The simplest problem in three dimensions is the compactivity of a simple powder of uniform grains of the same material in approximately spherical form. Each grain will have neighbours touching it with a certain coordination and angular direction. There will also be a certain number of near neighbours which do not touch. If we want to set up an analogy with the statistical mechanics of alloys or magnetism, we want to consider that each grain has a certain property, and this property interacts with its neighbours, e.g. in an AB alloy there are A and B type atoms with interactions vaa, vab, vbb. Suppose we take the coordination of a grain as such a property, i.e. suppose we assign a volume vc to any grain with c neighbours. Clearly this is not a comprehensive description of the powder, but we can suppose that

where nc is the number of grains with c neighbours, and vcc, the refinement of the volume function when there are ncc, pairs of neighbouring grains with one with c and one with c' neighbours. A problem arises: there must be some labelling of which grain is which, unlike the situation where a lattice exists and can be referred to. This is not as serious as may at first sight appear, for given the overall density one can consider the site of each grain as the distortion of a lattice which has the correct mean coordination (which is still of course to be discovered). But even cruder one can label the grains according to say a simple cubic lattice with the right lattice spacing, i.e. one can say this particular grain is the /th in the with row of the mh column. One only needs a label: the quantification of distance and volume comes from the formula for W. At the level of this paper, this labelling issue will not actually arise and we discuss it here only to assure the reader that there is no real problem. There are still compatibility conditions in this formulation: clearly one cannot have a grain with coordination 4 next to a grain with coordination 12. It is difficult to quantify these conditions, and we will not attempt to do so here, but hope to return to it in a later paper.

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1087

Suppose then we label our grains i meaning that their true positions can be deformed affinely onto lattice points rt. Suppose that grain i has a coordination c, so that

The final form of the integral will then be

where •/({£(/•,)}) expresses the combined effect of the Jacobian from the change of variables, and the stability and no-overlap conditions. We will not attempt to evaluate J in this paper, but work through the very simplest examples: firstly that of one kind of grain, and with the volume depending only on the coordinations of the individual grains. Since we are illustrating a point rather than being realistic we can be even simpler and say that there are just two types of coordination c0 and Cj which leads to an Ising model,

or

The two limits of V are M>0 corresponding to X — 0 and having the maximum density and N(v0 + v^/2 which corresponds to X = °° and is the lowest density. Although this is a very simplified model, it clearly will be related to the real problem of an array of c's and rc's and compatibilities. There will be a maximum density possible which will correspond to the highest coordination but then the other extreme is that of all the (stable) coordinations being equally likely. This crude analysis does not address the subtleties of sphere packings in three dimensions which arise because almost thirteen spheres can touch a central sphere; so that for small clusters one can achieve higher densities than the 0.7405 of face centred cubic. (Perhaps the demon we have forbidden might

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be able to beat f.c.c. in the large but not in extensive operations.) In practice the reproducible maximum density obtained by packing for example ballbearings is lower [7] (0.6366, "random close packing"). A large literature exists discussing the details of sphere packings (see Gray [8], or Cumberland and Crawford [9] for a review), although there is no satisfactory intrinsicrather than operational - definition of the random close packed state. The general result will remain that the highest density concentrates on a particular type of coordination whereas the lowest has all coordinations.

3. Mixtures of particles As another example, at the simplest level, we consider a mixture of powders of type A and type B. Let us focus on the fact that when an A is next to an A there is a contribution to the volume which is different from an A next to a B and a B next to a B. Thus although the nature of coordination number matters as in the example above, we just concern ourselves with the nearest neighbour quality. This one aspect is enough to throw interesting light on the separation and miscibility of powders. Suppose the number of A type at rt is m'A (=0 or 1), and similarly for B, so that

Then following Bragg and Williams we write

we have

We now quote Bragg and Williams who give = tanh v
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until for

Thus the A and B grains are miscible for vl\X<\, but, for vl\X> 1, the powder forms domains of unequal concentrations, until at A"= 0 the material separates into domains of pure A and pure B. If A and B are particles of different sizes, then we expect v < 0 and that the powders will be miscible for all X - at least as long as the pure phases are expected to be disordered. The well-known phenomenon where large particles rise to the top in a vibrated powder is a purely dynamical phenomenon [10]. The analysis here is relevant if we invert the powder at intervals, so that there is no extrinsic bias, or if we consider particles of the same size, but different shapes (e.g. cubes and spheres). An interesting computational study of this problem has been given by Barker [11].

4. Discussion This paper argues that, although detail of local structure is needed for a complete theory of powders, there are large scale behaviours which fit into the structure of statistical mechanics in the sense that analogues exist of Gibbs type integrals and relationships. The compactivity provides a useful way to characterize theoretically states of a powder with different densities, although, unlike the case with energy and temperature, the volume is the easier experimental quantity. We have illustrated the problems by some rather trivial examples, but have in hand some more detailed and significant examples for future publications.

Acknowledgements R.B.S. Oakeshott would like to thank British Petroleum for the generous award of a Research Studentship.

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S.F.E. thanks the Enrico Fermi International School of the Italian Physical Society for an invitation to a lecture on this problem, which much clarified his thinking [12].

References [1] M. Shahinpoor, Powder Tech. 25 (1980) 163. [2] M. Shahinpoor, Bulk Solids Handling 1 (1981) 1. [3] C.B. Brown, Proc. U.S.-Japan Seminar on Continuum-Mechanical and Statistical Approaches to the Mechanics of Granular Materials (Gakujutsu Bunken, Fukyu-Kai, Tokyo, 1978), p. 98. [4] R.L. Brown and J.C. Richards, Principles of Powder Mechanics (Pergamon, Oxford, 1970). [5] Advances in the Flow of Granular Materials, vols I, II, M. Shahinpoor, ed. (Trans Tech Publications, 1983). [6] B.J.B. Briscoe and M.J. Adams, in: Triboloty in Particulate Technology, MJ. Adams, ed. (Adam Hilger, Bristol, 1987). [7] G.D. Scott and D.M. Kilgour, J. Phys D: Appl Phys. 2 (1969) 863. [8] W.A. Gray, The Packing of Solid Particles (Chapman and Hall, London, 1968). [9] D.J. Cumberland and R.J. Crawford, The Packing of Particles, Handbook of Powder Technology, vol. 6 (North-Holland, Amsterdam, 1987). [10] A. Rosato, H.J. Strandburg, F. Prinz and R.H. Swendensen, Phys. Rev. Lett. 58 (1987) 1038. [11] G.C. Barker and M.J. Grimson, Proc. of Royal Society of Chemistry Food Group meeting: Food Colloids (1988), to be published in a special volume. [12] S.F. Edwards, in: Proc. Enrico Fermi School of Physics (to be published by the Italian Physical Society).

23

BUILDING A THERMODYNAMICS ON SAND Jorge Kurchan Laboratoire de Physique et Mechanique des Milieux Heterogenes, Ecole Superieure de Physique et Chimie Industrielles, 10 rue Vauquelin, 75231 Paris CEDEX 05, France Abstract

This is a brief review of the thermodynamic description of dense granular matter, with an emphasis on the connection with old and new ideas in the field of glasses.

23.1

Introduction

In the late 1980s, Sam Edwards made the proposal that one could construct a thermodynamic description of dense granular matter, the 'atoms' being the grains, and the phase-space the set of static mechanical configurations in which they can be arranged (Edwards 1994; Edwards and Oakeshott 1989; Oakeshott and Edwards 1994; Higgins and Edwards 1992; Mounfield and Edwards 1994; Edwards and Grinev 1998). The assumption is that these blocked configurations are reached with equal probability: one can thus compute expectation values, by-passing the details of the dynamics. This intuition is all the more remarkable because equi-probability properties are the exception and not the rule in out of equilibrium situations, even in cases with slow dynamics. If this assumption holds for granular and glassy matter—and we now know that it does, at least approximately—it is because of properties that are specific to these systems and that we still do not yet fully understand. Static granular matter is constituted of particles each of which is immobilised by the forces exerted by its neighbours and by gravity. In a high-density situation with very slow compaction or shear the system is always close to one of these blocked configurations, and indeed in some cases we may think of the dynamics as a walk from one TV-particle blocked configuration to another. The assumption is then that the configurations have equal probability to be dynamically accessed, at least if we make the restriction to those having the known total volume V and—whenever relevant—energy E. If we accept for the moment that a flat probability distribution holds, or is a good approximation, all the elements are in place to construct a statistical mechanics: we start from the configurational

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Building a Thermodynamics on Sand by Jorge Kurchan

entropy defined as the logarithm of the number of configurations S(E, V), and define a compactivity X~l = 8S/8V and a temperature T^1 = 8S/8E. We can also work in the canonical ensemble, if we wish to. More recently, it has become clear that this idea is closely related to a phenomenological construction for glasses originating over sixty years ago. Glasses are out of equilibrium systems that undergo a slow relaxation known as aging, completely analogous to granular compaction, a connection that has been appreciated for some time (see, e.g., Ch. 7 of Struik f 978, and Edwards and Mehta f 989). Tool (f 946) proposed that the state of a glass can be described in terms of the temperature of the bath, plus an additional, fictitious, temperature, roughly equal to that at which the system fell out of equilibrium, i.e., when the glass was born as such. This temperature had a more operational than first-principles definition, and was used to interpret and fit the experimental data: the point to retain at this stage is that this implies a generalization of thermodynamics, as applied to a non-equilibrium situation (an equilibrium system has, of course, only one temperature). In the absence of a first-principles justification based on the microscopic dynamics, Tool's proposal was never wholly accepted—or rejected, and a similar situation happened years later with Edwards' one. The situation has changed in the last few years, following the realisation that a number of models provide a mean-field caricature of glasses; see, e.g., Kirkpatrick and Thirumalai (1987) and Kirkpatrick and Wolynes (1987). When their out of equilibrium dynamics was analytically solved (Cugliandolo and Kurchan 1993; for a review see Bouchaud et al. 1997; for an outline of a more general solution, see Cugliandolo and Kurchan 1999), it was found that indeed the glassy regime was characterized by two temperatures, the bath temperature plus an effective one Teg—the latter having all the properties of a true temperature, and measurable with any thermometer coupled to the slow motion (Cugliandolo et al. 1997; for a recent review see Crisanti and Ritort 2003). Thus, Tool's intuition is realized, and becomes precise in this framework. On the other hand, one can show that the observables obtained dynamically are exactly given by the flat averages over blocked configurations (in this case the local energy minima) having the corresponding energy. Furthermore, the effective temperature defined by the dynamics exactly coincides with the one calculated a la Edwards as above (Monasson 1995; Nieuwenhuizen 2000; Franz and Virasoro 2000; Kurchan 1997), Having a precise definition and a limit in which Tool's and Edwards' ideas hold strictly and can be seen to be related, it became natural to check with simulations more realistic systems—especially since, as we shall see below, the theory indicates what exactly to look for. This program was mostly followed for glasses, but there are by now quite a few computations for 'granular' models. As a result of these numerical investigations, it has become established that in some cases these ideas indeed apply, perhaps as an approximation. From the first-principles point of view, we still do not know why the distribution should be flat, or almost. Principles invoking maximum entropy within restricted subsets of phase-space have been invoked in the past (see, e.g., Jaynes

'Granularizing' Simple Glass Models

377

1979), and argued for on the basis of information theory. Of course, information theory cannot be a substitute for an underlying dynamical reason, playing the role of a Liouville theorem, that justifies the preservation of the measure. 23.2

Compact granular matter is an athermal glass. 'Granularizing' simple glass models

The slow compactification of granular matter was recognised long ago to be closely analogous to the aging of glassy systems (see, e.g., Ch. 7 of Struik 1978, and Edwards and Mehta 1989). If one considers granular matter as simply a system of hard particles, polydisperse both in size and in shape, then its settling under gravity or under the pressure exerted by a piston would be just the situation in any aging glass, but at essentially zero temperature and extraordinarily strong fields and pressures. The only difference is indeed the presence of static friction, which is absent in a system of microscopic particles interacting with conservative forces, or in colloidal suspension in a liquid. In practice, other important differences appear when we consider driven systems. In that case, the peculiarity of granular matter is not so much that the microscopic temperature is negligible, or that the interactions are not elastic (both things are true for colloidal glasses), but rather the fact that almost all of the energy in play is supplied by a non-thermal source, one that is not itself in thermal equilibrium, such as vibration or stirring (see Knight et al. 1995). In this sense, a strongly driven colloidal glass is in fact close to compact granular matter. This can be roughly quantified as follows: if the system is in contact with a thermal bath of temperature T, and in addition to another non-thermal source of energy, such as shearing or tapping, the motion is athermal if the fluctuations in time of the energy per particle are larger than kT. This means that a granular system driven this way cannot be described as in thermal equilibrium, even in the high-energy/low-density 'fluid' phase. On the other hand, in the low-energy, dense regime, the system would not equilibrate, even if in contact only with a legitimate thermal bath for glassy reasons: the relaxation takes longer than realistic times. Granular matter can also be studied in the more traditional, static 'sandpile' form (see Chapter 25 by Bouchaud and Gates in this volume), with the important characteristic feature of the prominent role played by static friction. Even in the absence of static friction, glasses formed after sudden quenches to very low temperatures or violent compressions share problems of inhomogeneity, arching, and history dependence that are typical of sandpiles. All this suggests that if one has a model for glasses that, although simple, is believed to capture the essentials of glassy physics, one can obtain a model that may do as much for granular matter by introducing the missing elements: static friction, strong drive and/or very rapid and deep quenches. This granularized glass model may then be compared to the original glass version, to see what new features arise.

378 23.3

Building a Thermodynamics on Sand by Jorge Kurchan The assumption

Suppose we have a dense system of particles. If Edwards' hypothesis holds, we can compute physical magnitudes without the need to follow the TV-particle dynamics. If the interparticle potential is E, the partition function reads:

where the sum runs over the Q(V,E) blocked configurations of energy E and volume V: this restriction is the only place in which the actual dynamics enters.1 In fact, if there is no static friction, we can restrict the sum over the blocked configurations by writing:

and computing the equilibrium values for (/?,/? aux ). The limit /?aux —> oo picks out the static configurations, and tuning (3 we get the flat sampling over those of energy E(f3). (/?aux is just an auxiliary variable, not to be confused with the fictive or effective temperature.) If we have no analytic means to do this computation even approximately, we have to resort to simulations. Note, however, that they can be performed with any method guaranteed to converge to the canonical measure: we are by no means forced to use the true dynamics of the system, which is by assumption slow. This reduction to an equilibrium calculation thus helps us by-pass the solution of an extremely complicated dynamics. Let us mention here a weaker form of the assumption, which seems to work surprisingly well for a class of systems. In eqn. (23.1), the external drive enters only in /?, which fixes the energy, while the factor Q(V,E) is independent of the drive. In Dean and Lefevre (2002) it was proposed that even giving up the flat measure assumption, a form (23.2) can still hold, with Q(V,E) now a system-dependent factor not equal to the number of blocked configurations. This weaker assumption is the easiest to check by measuring the energy fluctuations in a driven system (Dean and Lefevre 2002). 23.4

Caveat I. Flat distributions are not generic out of equilibrium

To see what a flat distribution could give us if it always applied, consider the case of fully developed turbulence. To fix ideas, let us imagine we simulate the hydrodynamic equations with a cellular automaton on a lattice (Prisch et al. 1986) with appropriate stirring forces, e.g., in the boundary conditions. The 1 I am assuming here that every configuration is counted once, even if in hard systems with static friction one configuration can correspond to several possible interparticle forces. If one decides to weight each configuration with the volume of solutions in force-space, one runs into arbitrariness; see Ch. 25 by Bouchaud and Gates in this volume.

Caveat II

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particle positions and velocities take on a discrete set of values; the system jumps discretely from one of these configurations to another. If it were true that within the set of, say, given energy, momentum and angular momentum configurations, all configurations visited dynamically were equiprobable then we could reduce turbulence to a statistical-mechanical problem. We would have a notion of entropy (in terms of the given energy) and hence a temperature. Even in practical terms, we would be able to use any of the very efficient equilibrium simulation strategies to obtain information on hydrodynamics. Needless to say, this does not work. Perhaps the problem is that turbulence is not really slow dynamics? Consider instead a three-dimensional Ising system on a lattice, with a weak, random, time-independent magnetic field on each site. If we quench a high-temperature configuration to very low temperature, the dynamics consists of the slow growth of domains with essentially up and down magnetizations, respectively. At long times, the domain configurations are stable to reversal of a few spins, only activated jumps through higher energies allow the domains to grow. This seems indeed to correspond to our idea of slow dynamics: rare activated passages through ever higher barriers. Here, again, we can test a flat-measure hypothesis. We investigate the configurations that are stable to one spin flip, and have a given total energy, or, equivalently, a total domain wall surface. It turns out (Barrat et al. 2000, 2001) that typically these configurations are made of islands of one magnetization, immersed in a surrounding sea of the opposite magnetization: they have a net, extensive magnetization. This is in contrast with the dynamically obtained configurations of the same energy, which are composed of domains of positive and negative magnetization of essentially the same size, and hence have no net magnetization, up to finite-size fluctuations. In conclusion, a flat measure does not reproduce dynamics. What about specifying more macroscopic values? In the random field Ising case, for example, we could restrict the flat measure to given energy and zero total magnetization. This amounts to fixing its k = 0 component, and the question that immediately arises is: Why stop there? On the other hand, if we impose all k components, the calculation is not giving us much extra information. 23.5

Caveat II. Convection currents, shear bands, inhomogeneities, insufficient relaxation

The remarks above show that flat measures (or maximum entropy principles, for restricted ensembles) do not necessarily reproduce the results obtained from dynamics—even in some homogeneous systems with slow dynamics. The kind of problems we now address are of a different nature: they concern the systematic inhomogeneities. These problems can be present in glasses, but they may be expected to be more severe in granular matter. Consider first the case of tapped granular matter. In a system with particles of two different sizes, one has the familiar 'Brazil nut' effect, whereby large particles

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tend to migrate to the top. Although such particle-size segregation obviously does exist in equilibrium systems, the Chicago group has argued (Knight et al. 1993) that in granular matter it is a consequence of convectional currents—and that these are themselves unavoidable, even in infinitely wide containers. If this is so, granular size segregation falls outside the scope of a pseudo-equilibrium measure, since it is then the consequence of closed current loops. In the context of slow sheared matter, a problem also arises: the profile of velocity tends to be nonuniform, with the velocity gradient concentrating in narrow 'shear bands! To the extent that the bands are thin (a few grains wide) one can hardly invoke a macroscopic description, even locally. Banding is not always present in colloids, but it seems difficult to avoid in granular matter, the more so the harder the grains and the higher the shear rate. There is also the question of history dependence. If a statistical description applies, history can only enter through the knowledge of the final volume and/or energy; or, in general, other macroscopic restrictions we impose on the set of configurations. In fact, we know that even in cases in which Edwards' assumption holds when the system is prepared in a gentle manner, it does so much less if the system is subjected to a violent quench that can perpetuate arches and inhomogeneities. Now, much of what is interesting in the static, sandpile context is history-dependent (see the chapter in this volume by Bouchaud and Gates), and to that extent lies outside the scope of a statistical description. 23.6

Encouraging news from the analytic front

After all this discouragement, it may seem wise to abandon the flat probability idea. Let us now turn to the more positive evidence. The approach we shall describe consists of treating the TV-particle equations of motion within an approximation scheme involving infinite resummations of the perturbative expansions, or alternatively self-consistent closure approximations. The reader may be surprised by the introduction of these pre-renormalisation group relics, but it should be born in mind that in the glass problem, unlike most well-studied equilibrium problems, we do not yet understand even the nature of the underlying order. 23.6.1 Closure approximations Suppose one has a system of many variables with energy E((f>~), and a dynamics given by:

where fa(t) is an external forcing, and the last two terms define a thermal bath of temperature T: r/a is a white noise, (r/a(t) r/b(t')} = 1^/T S(t — t')J 0 &. The linear term has been shown explicitly for convenience. If we wish to make a description of the dynamics that is as complete as possible, but staying at the level

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of one- and two-point functions, we should consider the ordinary two-time correlation functions Cab(t,tw) = (4>a(t)4>b(tw)} and also the response functions: (a(t}rib(tw})/T = Rab(t,tw) = 6((j>a(t))/6fb(tw). A systematic way to do this is to start with the Schwinger-Dyson equations (for a review see Bouchaud et al. 1997);

and to express the S's and D's as sums of all two-line irreducible diagrams where the bare propagator [the inverse of the left hand side in eqns. (23.5)] has been substituted by the exact two-point functions of the C"s and fl's ('mass renormalisation'). The simplest of such schemes is the Mode Coupling approximation, which consists of keeping the diagrams with two vertices: it has the additional simplifying feature that the S's and D's become ordinary functions (as opposed to nonlocal functionals) of C"s and fl's. In the high-temperature phase and in the absence of external forcing /0 = 0, whatever the initial condition the system equilibrates, which manifests itself by the fact that correlations and responses become stationary (they depend only on time-differences), and obey the fluctuation-dissipation relation:

[In fact, (23.6) implies (23.7): the reader will suspect that there is an underlying symmetry.] In some systems, one finds a transition temperature below which, starting from a typical initial condition, the system never manages to equilibrate: the two-point functions never become stationary and the fluctuation-dissipation relation (23.7) does not hold: this is the glassy regime. Within this regime, one can solve exactly the long-time (yet out of equilibrium!) asymptotics. (See, Cugliandolo and Kurchan 1993; for a review see Bouchaud et al. 1997; for an outline of the more general solution, see Cugliandolo and Kurchan 1999.) For a subclass of systems, as soon as the system is below the critical temperature, the motion becomes the superposition of fast vibrations and much slower (so-called)

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structural rearrangements. As time passes, the structural rearrangements become slower and slower: this is the phenomenon of aging. Dynamic transitions of this kind are sometimes referred to as random first, order or mean-field glass scenario, the terms 'random' and 'mean-field' serve as a reminder of the fact that the present approximation scheme is exact for models with fully connected, quenched random interactions. The relation between fluctuations and dissipation out of equilibrium is important. It is conveniently characterised as follows: consider the integrated response Xab(t,tw) = $1 Rab(t,r}dr. In equilibrium, the fluctuation dissipation theorem relation (23.6) implies:

A plot of Xab(t,tw) vs. Cab(t,tw), where the times enter as parameters, yields a straight line of slope —1/T. Consider now the same plot for the out of equilibrium solution, at long times: It turns out that for the class of systems having two timescales as described above, the same plot (with now t and tw entering as independent parameters) yields a curve like that in Fig. 23.1, composed of two straight segments: one of gradient -1/T as before, and one with gradient, say, — 1/Teff. Within the present approximation scheme, one can also show that any other pair of observables a', b' would yield a similar curve, with the same T eff . The remarkable thing about this curve (at least as obtained in the models we know how to solve, such as the mean field case) is that one gets one curve for two parameters. Choosing different (large) tw, one retraces with subsequent t the same broken line, independent of tw. In more realistic models, e.g. Lennard-Jones glass, the same thing seems to happen, although there may be an extremely slow drift of the curve for vastly different tw. The two fluctuation-dissipation behaviours are in correspondence to the two timescales mentioned above: the fast vibrations happening on microscopic times are associated with the temperature T, and the slow, structural relaxations with the

FIG. 23.1. Fluctuation-dissipation characteristic for a simulated particle system. Data by L. Berthier and J.-L. Barrat (see text).

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effective temperature Teg, which is unambiguously defined and common to all observables. One can furthermore show [Cugliandolo et al. (1997); for a recent review, see Crisanti and Ritort (2003)] that Teg is indeed what a thermometer with very slow response coupled to any observable would read. Having found a temperature with a thermodynamic meaning describing the structural motion, the next step is to make contact with Edwards' assumption. This can be done as follows: at each level of approximation, one can compute within the same approximation the number of metastable states2: it turns out that the quantities observed dynamically at each energy are reproduced exactly by the flat average over the metastable states of that energy. Furthermore, if one computes the configurational (or Edwards') entropy as the logarithm of the number of metastable states, the temperature obtained from derivation of this quantity exactly coincides with the effective temperature obtained with the dynamic solution and no extra assumptions. Edwards' proposal is then seen to hold strictly, but with metastable states substituting blocked configurations. In general there are many blocked configurations for each metastable state, the one-to-one relation materialises, if at all, at strictly zero temperature. I shall come back to this essential point. 23.6.2

The athermal situation

So far we have only considered systems in contact with a good thermal bath, and have found that there may be a transition to a low-temperature regime in which they are, however, unable to thermalise. What happens if we now consider the same kind of system, but now subjected to a non-thermal source? Several possibilities have been tried: strong vibration or 'tapping' [via the term fa(t) in eqn. (23.3)] (Berthier et al. 2001), nonlinear friction terms and nonthermal random noise (Kurchan 2000). The answer, still within one of these approximation schemes, is the following: in the high-energy regime, the system is fluid, and yet non-thermal due to the nature of the source: the system cannot be described by any temperature T. In the low-energy compaction situation, the high-frequency motion is again non-thermal, and depends on the details of the driving mechanism. What may come as a surprise is that for weak enough driving, the structural motion still retains the thermodynamic form with an effective temperature Teg. In a plot like the one in Fig. 23.1, one still obtains at high densities a straight line with gradient — l/Teg for the long timescales, but nothing simple for the fast timescales. In other words: if the driving mechanism is arbitrary, but such that the system still evolves with two widely separated timescales, then the slow timescale behaves just as in the thermal case, although the fast motion does not. 2 There are several ways of counting the number of metastable states in the literature. A method that can be directly implemented at each level of approximation consists of counting the number of low-lying eigenvalues of the evolution operator, a strategy proposed by Gaveau and Schulman; see Biroli and Kurchan (2001).

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23.6.3 Intrinsic limitations of the approach The analytic approach within closure approximations was important in that it reintroduced the idea of effective temperatures in a clear setting, it established its relation with Edwards' approach, and, more generally, it made the connection between landscape and dynamics explicit. It also clarified the relation between aging and the rheology of driven systems, and contributed to further sharpening the connection between glasses and granular matter. Perhaps more importantly, it pointed out what to look for in simulations and experiments, something not at all obvious in glass matter, where the actual order is hidden. There is, however, an intrinsic limitation of the theory. In writing the closure approximation we are implicitly resumming an infinite number of terms in the perturbation development. Indeed, the 'mass renormalization' procedure mentioned above allows one to add more and more families of diagrams systematically. However, certain nonperturbative effects ('activated processes') cannot be captured this way. These effects are always important, because they are responsible for smearing the dynamic transition into a dynamic crossover. More to the point of this paper, they are also responsible for rendering the lifetime of metastable states finite, except at strictly zero driving energy. Now, precisely one of the conclusions of the analytic approach was that, whenever there is any form of energy source, the relevant objects to consider are the metastable states, rather than the energy minima. This was unambiguous within the closure approximation, within which they are absolutely stable. Once activated processes are included (and of course in real life!), the states acquire a finite lifetime, and what we consider as a distinct metastable state depends on a minimal lifetime we choose as a criterion. There is no absolute way out of this intrinsic arbitrariness, although pragmatic strategies have been followed. What features of the dynamics survive beyond this approximation scheme is at present not clear, and is the subject of active research. The effective temperature idea has been tested in realistic models: in fact, the Figure 23.f is the response vs. correlation curve obtained by Berthier and Barrat for a threedimensional dense system of particles interacting via a Lennard-Jones potential. The two-temperature behaviour seems well reproduced, but in the absence of analytic results we cannot know if only as an approximation.

23.7

Towards realistic models and experiment

The evidence in support of a thermodynamic description for more realistic situations falls into two classes. Firstly, there are the experimental and numerical tests of effective temperatures both in aging and in driven systems. This is somewhat weaker than a full thermodynamic description, because there could be (and most probably are) relaxations with different properties happening in even longer timescales than the one measured. Second, there are studies of (often somewhat simpler) systems for which a full comparison between out of equilibrium values and those obtained with a flat assumption is made.

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23.7.1 Effective temperatures Soon after the analytic results, the existence of effective temperatures was checked numerically in aging (Parisi 1997; Barrat and Kob 1999) and later sheared (Berthier and Barrat 2002a, 2002b; Ono et al. 2001) systems of particles. Figure 23.1, taken from Berthier and Barrat (2002a, 2002b), shows a typical response vs. correlation curve obtained in a sheared system of particles interacting via a Lennard-Jones potential. In that same work, correlations and responses were measured for several observables, and the agreement of the respective effective temperatures is striking. From the experimental point of view, there are by now several indications that the fluctuation-dissipation relation is different for the slow modes, although not yet many details on exactly how for each timescale. Grigera and Israeloff (1999) measured the effective temperature of Glycerol glass, with a result that seems to be in agreement with the theory. Subsequent experiments of the Lyon group (Bellon and Ciliberto 2002; Buisson et al. 2003) have found unexpectedly high effective temperatures, but these have an interesting explanation (see 'Counter-examples'). More recently, measurements have been performed in vibrated granular matter (D'Anna and Gremaud 2001; D'Anna et al. 2003), but do not as yet correspond to the time-density domain where one expects Edwards' entropy to be the relevant quantity. There have also been numerical and experimental results for spin-glasses, but this corresponds to a phenomenology different from that of structural glasses, and is outside the scope of this article.

23.7.2 Tests of the flat measure hypothesis The literature on different checks of the flat measure hypothesis is by now rather extensive, so this can only be an incomplete and arbitrary account. Several models with slow dynamics have been considered, ranging from the very schematic to the quite realistic (Brey et al. 2000; Barrat et al. 2000, 2001, 2002; Lefevre and Dean 2001, 2002; Dean and Lefevre 2001a, 2001b; Lefevre 2002; Berg and Mehta 2001, 2002; Berg et al. 2002; Colizza et al. 2002; Makse and Kurchan 2002; see also Behringer 2002). In a few cases, the flat measure has been shown analytically to reproduce exactly the dynamic observables in some limit (see, e.g., Brey et al. 2000; Leuzzi and Nieuwenhuizen 2001), though in general one has only a numerical check. For example, for the Kob-Andersen model, the structure factor obtained from compaction dynamics and with the flat measure were compared, with very good agreement. It seems, however, that this agreement can only be approximate, since the relevant entities in this problem are ergodic components rather than blocked configurations (Biroli, G., private communication; Biroli et al. 2003): this is one instance of the metastable state vs. blocked configuration problem mentioned above. In cases in which the correspondence is not exact, it has also been proposed that the measure should be further conditioned by more and more macroscopic

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observables (other than energy and volume) in order to obtain better approximations. This had been discussed in a different context already (see Jaynes 1979), although in the case of glassy dynamics we have more reason to believe that the approach is promising. Some numerical experiments readily lend themselves to be reproduced in the laboratory. In Makse and Kurchan (2002; see also Behringer 2002), a system was studied consisting of particles interacting with rather realistic granular interactions, including static friction, subjected to shear. The predictions from the flat measure are in good agreement with the dynamics. Another example is the model of sheared foam in Ono et al. (2001), where special attention was paid to shear-rate dependence. A particularly interesting possibility was explored by Lefevre and Dean (2002), who studied a system whose Edwards measure has a phase transition. They found that the 'tapped' dynamics reproduces the transition, with the tapping intensity as the control parameter. When the system is such that the transition is first order, they obtain a hysteresis loop (in the tapping intensity) ending in the spinodal points obtained with the flat measure.

23.8

Inherent structures

Let us now turn to investigations presented in a slightly different context: the mention is separate because, though closely related from the physical point of view, this rather large body of work tends to be scarcely interconnected by cross-referencing from the rest. Using blocked configurations as a means of analysis of liquid systems has an important precedent in the chemistry literature (Stillinger and Weber 1982). The idea is to classify all configurations of phase-space according to the basins of attraction under a zero-temperature dynamics. Thus, a single blocked configuration (here called inherent structure) labels all the configurations that lead to it in a purely relaxational evolution. The total number of configurations is factored into number of inherent structures (yielding a definition of configurational entropy) times the average number of configurations per inherent structure (yielding the vibrational entropy). The notion that an inherent structure contains the qualitative information of the basin it represents is bolder than one can imagine at first sight: in most of the contexts in which this mode of analysis is applied, the system has a large thermal energy, way above the barriers separating basins. Indeed, unlike the case of a gently tapped granular system, the dynamics is smooth, and one does not observe any jumps between blocked states. To the best of my knowledge, the first application of this construction within the out of equilibrium, glass regime was made by Kob and Sciortino and by Tartaglia (Kob et al. 2000; Sciortino and Tartaglia 2001), who proposed that the flat measure over metastable states found within closure approximations could be substituted in finite-dimensional systems, at least operationally, by a measure

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over inherent structures (supplemented by a vibrational motion around them)— thus going back closer to Edwards' original idea? In this way, they gave a pragmatic response to the problem mentioned above that in finite dimensions and at nonzero temperature there is no absolute definition of metastable states, since they have a finite lifetime. The numerical results obtained for Lennard-Jones glasses are encouraging. However, here too the analytic computation teaches us something: in solvable mean-field disordered models we have an unambiguous definition of state, and one can check that the inherent structures and states are not generically equivalent—so that the strategy yields results that are approximate even in that simple solvable case. The inherent structure strategy has been also applied in the context of granular matter in a series of works by the Naples group (Coniglio and Nicodemi 2001; Coniglio et al. 2001; Fierro et al. 2002), where the connection with Edwards' original assumption is also discussed. 23.9

Counter-examples

Counter-examples are important because, at the very least, they show what should go into the theory. Even within closure approaches, the two-temperature scenario is not generic: one can have many temperatures, but still one for all observables at each timescale. It has been taken for granted that structural glasses possess only two widely separated timescales, but this is becoming increasingly dubious: if it should turn out that structural glasses develop several timescales that become more and more separate as the system ages then more than two temperatures are to be expected. A thermodynamic construction inevitably fails whenever there is no observable-independence of the effective temperature at a given timescale. Several instances have been reported: two interesting examples are the ferromagnetic coarsening in the zero-temperature Ising model (see Corberi et al. 2002, and references therein), and the trap model (Bouchaud 1992; Bouchaud and Dean 1995), and there is an extensive discussion of the latter situation (Fielding and Sollich 2002; Ritort 2003; Sollich 2003). As mentioned in the introduction, a flat-measure hypothesis fails to reproduce the form of the domains in the activated domain growth of a ferromagnet with random fields (and presumably the closely-related wetting phenomena) (Barrat et al. 2000, 2001). It is also clear that in all cases the hypothesis works only to 3 There is, however, a difference in the sampling prescription: Kob et al. proposed, instead of making a flat sampling of blocked configurations, to explore them as follows. One considers an equilibrium situation at a given (high) temperature, and then performs a quench to zero temperature. The energy (or volume) attained is reproducible, up to finite-size fluctuations. Given a dynamic situation at energy E (or volume V), the statistical ensemble of inherent structures considered is the set obtained starting from a temperature T—itself chosen so as to yield the required target E (or volume V). This sampling is not strictly equivalent to a flat sampling (see Barrat et al. 2000, 2001). It is indeed more practical for simulations though rather intractable for analytical calculations.

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the extent that the relaxation becomes really slow, and this is less so, the more heavily driven the system is. Experimentally, an interesting result that does not quite match the theoretical picture was obtained for aging colloids and polymer melts by the Lyon group (Bellon and Ciliberto 2002; Buisson et al. 2003). In the latter case they have observed effective temperatures that are many orders of magnitude higher than the glass temperature, contrary to expectations from the models. It turns out that the origins of this behaviour are strong, short pulses of noise, which tend to be more frequent at short times. These fluctuations seem to be quite common in structural glasses, and perhaps have a counterpart in granular matter (perhaps related to D'Anna et al. 2003): they are very probably not captured by the theory as it stands. 23.10

Conclusions

This brief account, with its title, caveats and counter-examples, has been intentionally cautious. I hope the reader still shares the excitement of these ideas involving basic principles pertaining to much of the solid matter that surrounds us, to which Sam Edwards has contributed in a major way. Acknowledgments

I wish to thank G. Biroli and D. S. Dean for useful discussions, and J.-L. Barrat and L. Berthier for the data in Fig. 23.1. References

Barrat, A., Colizza, V. and Loreto, V. (2002). Phys. Rev. E 66, 011310. Barrat, J.-L. and Kob, W. (1999). Europhys. Lett. 46, 637. Barrat, A., Kurchan, J., Loreto, V. and Sellitto, M. (2000). Phys. Rev. Lett. 85, 5034. Barrat, A., Kurchan, J., Loreto, V. and Sellitto, M. (2001). Phys. Rev. E 63 513001. Behringer, R. (2002). In Nature 415, 594, News and Views. Bellon, L. and Ciliberto, S. (2002). Physica D (in press); cond-mat/0201224. Berg, J., Franz, S. and Sellitto, M. (2002). Eur. Phys. J. B 26, 349. Berg, J. and Mehta, A. (2001). Europhys. Lett. 56, 784. Berg, J. and Mehta, A. (2002). Phys. Rev. E 65 031305. Berthier, L. and Barrat, J.-L. (2002a). Phys. Rev. Lett. 89, 095702. Berthier, L. and Barrat, J.-L. (2002b). J. Chem. Phys. 116, 6228. Berthier, L., Cugliandolo, L. F. and Iguain, J.-L. (2001). Phys. Rev. E 63, 051302. Biroli, G., Fisher, D. S. and Toninelli, C. (2003). To be published. Biroli, G. and Kurchan, J. (2001). Phys. Rev. E 64, 16101. Bouchaud, J-P. (1992). J. Physique (France) I 2, 1705.

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Bouchaud, J.-P., Cugliandolo, L. F., Kurchan J. and Mezard, M. (1997). In Young, A. P. (ed.) 1997. Spin Glasses and Random Fields. World Scientific, Singapore. Bouchaud, J-P. and Dean, D. S. (1995) J. Physique (France) I 5 265. Brey, J. J., Prados, A. and Sanchez-Rey, B. (2000). Physica A 275, 310. Buisson, L., Ciliberto, S. and Garcimartin, A. (2003). Europhys. Lett, (in press); and cond-mat/0306462. Colizza, V., Barrat, A. and Loreto, V. (2002). Phys. Rev. E 65, 050301. Coniglio, A., Fierro, A. and Nicodemi, M. (2001). Physica A 302, 193. Coniglio, A. and Nicodemi, M. (2001). Physica A 296, 451. Corberi, F., Castellano, C., Lippiello, E. and Zannetti, M. (2002) Phys. Rev. E65, 066114. Crisanti, A. and Ritort, F. (2003). J. Phys. A (Math. Gen.) (in press); condmat/0212490. Cugliandolo, L. F. and Kurchan, J. (1993). Phys. Rev. Lett. 71, 173. Cugliandolo, L. F. and Kurchan, J. (1999). Physica A 263, 242. Cugliandolo, L. F., Kurchan, J. and Peliti, L. (1997). Phys. Rev. E 55, 3898. D'Anna, G. and Gremaud, G. (2001). Nature 413, 407. D'Anna, G., Mayor, P., Gremaud, G., Barrat, A. and Loreto, V. (2003). Europhys. Lett, (in press). D'Anna, G., Mayor, P., Barrat, A., Loreto, V. and Nori, F. (2003). Nature 424, 909. Edwards, S. F. (1994). In Granular Matter: An Interdisciplinary Approach. Mehta, A. (ed.). Springer-Verlag, New York. Dean, D. S. and Lefevre, A. (2001a). Phys. Rev. Lett. 86 5639. Dean, D. S. and Lefevre, A. (2001b). Phys. Rev. E 64, 46110. Dean, D. S. and Lefevre, A. (2002). cond-mat/0212297. Edwards, S. F. and Grinev, D. V. (1998). Phys. Rev. E 58, 4578. Edwards, S. F. and Mehta, A. (1989). J. Phys. (Parts) 50, 2489. Edwards, S. F. and Oakeshott, R. B. S. (1989). Physica A 157, 1080. Reprinted in this volume. Fielding, S. and Sollich, P. (2002). Phys. Rev. Lett. 88, 050603. Fierro, A., Nicodemi, M. and Coniglio, A. (2002). Europhys. Lett. 59, 642. Franz, S. and Virasoro, M. A. (2000). J. Phys. A 33, 891. Frisch, U., Hasslacher, B. and Pomeau, Y. (1986). Phys. Rev. Lett. 56, 1505. Grigera, T. S. and Israeloff, N. E. (1999). Phys. Rev. Lett. 83, 5038. Higgins, A. and Edwards, S. F. (1992). Physica A 189, 127. Jaynes, E. T. (1979). Where do we Stand on Maximum Entropy?. In Levine and Tribus (1979). This, and related articles can be downloaded from http://bayes.wustl.edu/etj/nodel.html (article 37). Kirkpatrick, T. R. and Thirumalai, D. (1987). Phys. Rev. B 36, 5388. Kirkpatrick, T. R. and Wolynes, P. (1987). Phys. Rev. A 35, 3072. Knight, J. B., Fandrich, C. G., Lau, C. N., Jaeger, H. M. and Nagel, S. R. (1995). Phys. Rev. E 51, 3957.

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Knight, J. B., Jaeger, H. M. and Nagel, S. R. (1993). Phys. Rev. Lett. 70, 3728. Kob, W., Sciortino, F. and Tartaglia, P. (2000). Europhys. Lett. 49, 590. Kurchan, J. (1997). http://www.itp.ucsb.edu/online/jamming2/; and in: Liu, A. J. and Nagel, S. R. (eds.) (2001). Jamming and Rheology: Constrained Dynamics on Microscopic and Macroscopic Scales. Taylor and Francis. Kurchan, J. (2000). J. Phys. Cond. Matt. 12, 6611. Lefevre, A. and Dean, D. S. (2001). J. Phys. A L213. Lefevre, A. (2002). J. Phys. A 35, 9037. Lefevre, A. and Dean, D. S. (2002). Phys. Rev. B 65, 220403R. Leuzzi, L. and Nieuwenhuizen, Th. M. (2001). Phys. Rev. E 64, 011508. Levine, R. D. and Tribus, M. (eds.) (1979). The Maximum Entropy Formalism. M. I. T. Press, Cambridge, MA. Makse, H. A. and Kurchan, J. (2002). Nature 415, 614. Monasson, R. (1995). Phys. Rev. Lett. 75, 2847. Mounfield, C. C. and Edwards, S. F. (1994). Physica A 210, 279. Nieuwenhuizen, Th. M. (2000). Phys. Rev. E 61, 267. Oakeshott, R. B. S. and Edwards, S. F. (1994). Physica A 202, 482. Ono, I. K., O'Hern, C. S., Langer, S. A., Liu, A. J. and Nagel, S. R. (2001). Phys. Rev. Lett. 86, 111. Parisi, G. (1997). Phys. Rev. Lett. 79, 3660. Ritort, F. (2003). cond-mat/0305376. Sciortino F. and Tartaglia, P. (2001). Phys. Rev. Lett. 86, 107. Sollich, P. (2003). cond-mat/0303637. Stillinger, P. H. and Weber, T. A. (1982). Phys. Rev. A 25, 978. Struik, L. C. E. (1978). Physical Aging in Amorphous Polymers and Other Materials. Elsevier, Amsterdam. Tool, A. Q. (1946). J. Am. Ceram. Soc. 29, 240.

24

REPRINT THE TRANSMISSION OF STRESS IN AN AGGREGATE by S. F. Edwards and R. B. S. Oakeshott Physica D: Nonlinear Phenomena, 38, 88-92 (1989).

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Physica D 38 ( 1 9 8 9 ) 88-92 North-Holland. Amsterdam

THE TRANSMISSION OF STRESS IN AN AGGREGATE S.F. EDWARDS and R.B.S. OAKESHOTT Catcndish Lahiratnn; Camhn,_lKL' CB3 (IIIK. IK

We consider the question ol how stress is transmuted in aggregates. The maeroscop.c state of a powder is summarized by its compactiviu ,\'— (t'j I '/ti,S I. In aggregates wheTC the particles liavc stuck together ( e g. lipid crystals in margarines, or in ilocculated co'.loids) the material has a fractal structure on which stress is carried. In a non-cohesive powder we consider the possibility thai the stress will be can led b> domes of particles, g i v i n g a dimensionality tor the set of stressed particles between 2 and 3. FinalK we :>bsencthat there is a coupling bet ween the structure (described by us compacliviu X \ and t h e distribution of stress.

1. Introduction

There is an increasing interest in theoretical physics in the problems of assemblages of panicles which arc sufficiently numerous and are assembled under well-defined circumstances, so that it is reasonable to expect the physical properties of the assembly to be predictable ( a n d interesting). Whereas in problems dominated by thermal behaviour one can expect ergodicity. an aggregate can be perverse, i.e. a Maxwell Demon can put an aggregate together in a way which has no particular relationships to a physical law. The stones in the Great Pyramid follow the instructions of the Pharaoh, but if they were poured out of a gigantic hopper we could expect to be able to predict the shape of the resulting heap. It will be assumed in this paper that systems which are formed by extensive operations, i.e. by shaking or stirring, or aggregation according lo some explicit rules, will be predictable and capable of having their specification solved. This is the anologue of the ergodicity of thermal systems. So suppose w-e have a powder and apply stress to it, e.g. suppose a powder fills a vertical cylinder and it is compressed. How docs it rearrange itself internally to sustain the stress, and how is that stress transmitted? A variant of this is lo consider a horizontal cylinder filled with powder with a piston at one Essays in honour of Benoit B. Mandelbrot Fractals in Physics - A. Aharnnv and .1. Feder (editors J

end but free to move at the other, remote, end. If the piston moves lo compress the powder, how is stress transmitted through it'.' The reason that this topic is offered as part of the celebrations of Professor Mandelbrot's 65th birthday is that the dimensionality of the transmission is not clear. We conjecture that the pattern is fractal; but has extra complications in that the fractal dimension may depend on ihc extent of the compression consequent on the stress. This paper does little more t h a n state the problem and discuss the framework for a solution. However, it is of such technological importance as well as being a challenge just as physics, it is certain to gain more attention in future.

2. The description of an aggregate In the presence of an appropriate background, or with particles forming strong inter-particle bonds, aggregates can take up configurations ranging from the low density found in DLA clusters, to that of a space filling powder. Although there will be reference to DLA-lypc clusters, more attention will be paid to space-filling systems and it is here that a simple description should be possible and will be attempted. The key fact is that a powder can be formed with a variety of densities. In the best known case of hard 0167-2789/89/503.50 © Flscvier Science Publishers B.v. (North-Holland Physics Publishing Di\ision}

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S.I-. L'dwai-ds and R By. Oakashon / Transmission of ureas in a*i aggregate

spheres there are two limiting densities of random, loose and close packing (0.6 and 0.6366). Making the assumption that the density is sufficient to characterize the powder completely, one may make a fruitful analogy with normal statistical mechanics. There the energy has a value E, so that it H is the Hamiltoman

gives the microcanonical probability that the system is found with £, and i' is the normalization

k serving to give the entropy S the dimension of energy. Further one defines temperature T=(dE/dS) and can transfer to the Canonical ensemble with

Suppose now that there is a function W which plays the role of H in the powder in that the volume V is the value taken by W for the particular configurations of Ihe grains. Then if we define /. to be a constant which gives entropy the dimension of volume

89

where we eall }' the effective volume; it has the role of free enerev and

Thus, just as it is easier to think of a temperature in a thermal system rather than its energy, we propose a compactivity X, linked to an effective volume }'. which is easier to handle than the density of the powder. Just as in a thermal system one can then go over to a temperature gradient, and a heat source of a point or plane giving rise to such a temperature gradient. We can now think of an injection of stress into a system at a point or over a plane, which leads to a gradient of X emanating from that point or plane. The effect of A" is most apparent in the coordination distribution it entails. For example, a simple calculation of the distribution of coordination number «c shows that this reaches a maximum at A'= 0 and is most uniformly distributed at X=oo. Thus the most highly close-packed system has A = 0 and just as one cannot be more packed than that, X cannot be negative. The lowest density corresponds to X= CD, where all configurations are equally probable, subject, however, to coordinates which imply a solid, i.e. not having so slow a coordination as to have unconnected or unstable material. A typical formula at the simplest level of approximation applied to a model of only two possible coordinations is

leading to

The interesting step is to go to the canonical ensemble, defining a compactivity A'hy

where X=0 gives Nv,, and A~=cc \N(vu + Vi). Although this is a gross oversimplification, it gives one a flavour of a true description.

3. The emergence of structures Section 2 considered a three-dimensional spacefilling aggregate and we return to this in section 4.

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S.I'. LdnardsandR.lS.S. Oaknhnll / Transmission ofyress in an a-igregali'

<)0

Stress can be earned by precursor structures, and indeed such structures appear cenlral in understanding systems which are just emerging as solids. For example a margarine has a balance of oils and lipid crystals, the latter forming chains which when of sufficient density make the margarine a solid. For a low density of lipid crystals the material is not a solid. The lipid crystals have some fractal structure in space which goes over into the usual material for a high enough content. This range of materials is very similar mathematically to a flocculating colloid, and the same transition of particles-'aggregatc into fern-like fractal • orthodox amorphous solid appears in a host of everyday materials. Brown and Ball [ 1 ] have given a theory lor this process. They consider a typical diffusion-limited cluster which has a fractal dimension dt. In addition another exponent aUcm can be defined in terms of the scaling behaviour of the electrical conductivity of the cluster. Brown and Ball are then able to deduce an clastic spring constant for the aggregate provided they use the longest and shortest length scales R and a of the aggregate. The constant is given by

For a mono-disperse system, by numerical simulation £„,.„, = 1.066 ±0.07 , whereas a poly-disperse system has rfdmr.= 0.96 ±0.033 . When an assembly of these clusters is formed by packing them together with a density p. the resulting modulus is given by

which agrees well with the experiments of Buscall, and of Buscall, Stewart and Sutton [ 2 ], who were able to measure G and control p by ultra-centrifuging (sec fig. 1). It is clear that stress is transmitted in these rnate-

Fig. 1. Elastic shear modulus 0 versus density p tor tully flocculated 0.3 urn acrylic spheres (crosses), 0.33 u.m polystyrene spheres (asterisks). 0.5 urn polystyrene spheres (stars I. 0.96 iim polystyrene spheres (circles). (From the Ph.D. thesis of Vv'.D. Brown [3].)

rials in one-dimensional paths which arc branched and are characterised by a fractal dimension. The crystals or floes stick together and can sustain an arbitrary stress tensor. For a powder the situation is not so clear for one has to have stress spread amongst several neighbours to maintain stability. Nevertheless one can imagine a locus of stress in this way. Suppose a powder is stressed by the application of a force in some way (e.g. at a point or plane). Mark all particles where the stress exceeds a certain level. What locus do the marked particles take up? This is effectively possible using photoelasticity as in the work of Dantu [4]. One can imagine a picture as in fig. 2, which is uncovered by removing overlying panicles. In this picture the stress pathway resembles that of the packed floe aggregates, or the lines of lipid crystals in the margarine. Although a (racial index is involved, the picture is made of lines related by some topology. There are, however, other possibilities.

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S.F. Et/H-urds and R.BS. Oakcshotl / Transmission of stress in an aggregate

91

Fip. 3. SlresvPoming domes in powdct.

Fig. 2. Possible line of stress transmission in the highlighted section of the powder.

4. The dimensionality of stress transmission At this point we reach the most difficult problem of our discussion. We argued above that if we were to mark those grains sustaining a stress above a given level, one might expect a series of branched lines in space resembling the physical lines of lipid crystals in a margarine with high oil content or one of the numerous other forms of percolation diagrams such as the path of an electric current in dielectric breakdown. These all have the familiar form of starting at a point and progressing in a directed manner in a treelike growth. There are very familiar examples of a quite different structure however. Consider for example a funnel, i.e. a body of diminishing cross section. Everyone is familiar with the problem that a powder flowing through such a body is liable to jam, i.e. form a dome which is able to sustain the stress caused by the weight of the powder above it, and transmits the stress to the container. The dome of powder grains holds just like the dome of a building or arch of a bridge. This process could be a central one in the transmission of stress through a powder, i.e. a point injection of stress produces a dome where the injection point has the role of a capstone to the dome. The picture would then be a series of domes terminating at a dome below them, i.e. in a section like fig. 3.

This can be regarded as a two-dimensional transmission of stress in contrast to the one-dimensional model above. We consider some consequences of these models below, but must also make the obvious comment that when the powder is at its m a x i m u m close packing and even more so if bonded, the transmission of stress must be adequately described by a three-dimensional transmission of stress as in a continuum theory of elasticity, although the powder problem remains non-linear since the material can support only compressive and not tensile stresses. Thus in addition to the kind of lingering in a directed DLA assembly, the fingers themselves have dimensionality which can be between 1 and 3. It is reasonable to suppose that this dimensionality will depend on the compactivity so that there will be an A'-dcpcndent index associated with the transmission with X=0 associated with high dimensionality and A' = oo with low. The precise form of this law has yet to be resolved and offers an important problem. We can, however, suggest some crude consequences of the two-dimensional stress flow as againsl the one-dimensional form discussed in the last section. To do this, consider the pressure exerted by a conical heap of powder as in the diagram in fig. 4. The lines drawn in this diagram are at an angle steeper than that of the angle at repose of the powder. If the powder becomes filled with domes or, in the two-dimension section shown, arches, then one can expect the stress to be transmitted down in such a way, from arch to arch, that fb. on the base supports the mass of powder formed in the section inclined at the 'arch'

396

92

,<>./•". Ed\\wdsandR.B S. Oakesholl / Transmission ofslrcts in an a'^regu

Fig. 4. Lines of stress in fully arched pyramid.

Fig. 5. Pressure distribution under 2D pyramid.

angle to the vertical. Two examples arc shaded in. Thus the pressure distribution can be expected to be as in fig. 5 with the total pressure equating lo the gravitational force of the powder. The maxima occur at +Xi, where the angle which is postulated to arise from the arching defines a line from apex to base. A similar analysis is found in Trollopc [ 5 ]. Something like this is found by Briscoe, Pope and Adams [6] but as one would expect is much smoother than the crude diagram above. Rather complex experiments on the dynamics of powders forced through Lheir angle of repose are being reported (Jaeger, Liu and Nagel [ 7 ] , Evesque and Rajchenbach [ 8 ] ) and offer a fascinating challenge, but even simple experiments still require explanation and the purely static situation offers a fractal problem which has yet to be resolved.

The problem above is one of many which we can characterise this way. There is a compaclivity X in a powder which has a role like temperature in a thermal system, i.e. an inhomogeneous system which ib extensively created will have a single A throughout, jusl as a gas of particles of different mass will have inhomogeneous distribution in space under gravityeven though there is a single temperature. This compactivity can then have a weak and slow variation X(r,t). When a point source of stress is introduced, a variation in X is induced, but unlike T(r, t) which will satisfy a differential equation, Xmay fluctuate as in a discharge phenomenon. This fractal form can of course give rise to a simple average behaviour described by a differential equation, but need not, and particular studies so far have not resulted in simple solutions of equations such as Pick's equation. However, quantities like < A'>, <-O'> can be studied and it is hoped to define a proper calculus of this problem in due course. References [ I ] W . D . BrownamlR.C. Ball,J.Phys. A 18 (1985) L517-L521. [2] R. Buscall, Colloids Surf. ( 1 9 8 2 ) 269-283; R Buscall. R.F. Stewart and D. Sutton, Fillr. Sep. 21 (1984) 183-186. [ 3 ] W.D. Brown, Ph.D. thesis, Cambridge ( 1 9 8 6 ) . [4] P. Dantu, Ann. Poms Chausees 4 (1967) 193. [ 5 ] D.H Trollope, in: Rock Mechanics in Engineering Practice, K.G, Stagg and O.C. Zienkiewicz. eds (Wiley, London, 1969). [6] SJ. Briscoe, L. Pope and M.J. Adams, Powder Technol. 37 (1984) 169. [7J ll.M. Jaeger. Ch. Liu and S. R. Nagcl, Phys. Rev. Lett. 62 (1989)40-43. [8] P. tvesque and J. Rajchenbach, Phys. Rev. Lett. 62 (1989) 4^.

25

GRANULAR MEDIA: THREE SEMINAL IDEAS OF SIR SAM J.-P. Bouchaud1 and M. E. Gates2 1

Service de Physique de 1'Etat Condense, CEA, Orme des Merisiers, 91191 Gif-sur-Yvette CEDEX, France 2 School of Physics, University of Edinburgh, JCMB King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, Great Britain Abstract

We discuss the statics and dynamics of granular media, emphasising three linked insights of Sir Sam Edwards. These are: the idea of 'force propagation' in models of rigid grains; the thermodyamic analogy for the statistics of jammed states; and the idea that granular assemblies arrest in a state of marginal coordination number, just enough to ensure mechanical stability. 25.1

Introduction

Although granular materials are made of classical particles of macroscopic size, they exhibit many interesting and sometimes counter-intuitive properties, which are relevant both to the academic community and for industrial applications (Jaeger et al. 1996; Herrmann et al. 1997; Duran 1997; Duran and Bouchaud 2002; Bouchaud 2003). Whereas the grain-level physics is reasonably well understood, a full understanding of granular media requires a proper statistical treatment of collective effects. The presence of strongly nonlinear interactions demands concepts and methods only now emerging from the study of disordered systems in statistical mechanics. A key feature of granular media is the existence of a large number of microscopically different metastable states that are macroscopically equivalent. This is common to a wider class of materials in arrested states, including structural, colloidal, spin- and vortex-glasses. Even the macroscopic properties of a static powder can depend strongly on the conditions that prevailed prior to arrest. For this reason, stress patterns in dry granular media exhibit some rather unusual features when compared to either liquids or elastic solids. For example, the vertical normal stress below a conical sandpile does not follow the height of material above a particular point. Depending on the way the pile is prepared, it shows a central stress minimum (or 'dip') underneath the apex of the pile when the pile is built from a point source (Smid and Novosad 1981; Brockbank et al. 1997; Vanel et al. 1999), and a broad, flat

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Three Seminal Ideas of Sir Sam by J.-P. Bouchaud and M. E. Gates

maximum when it is built layer by layer from a uniform 'rain' of grains (Vanel et al. 1999). The experimental work of Smid and Novosad (1981), showing the dip, caught Sam Edwards' attention soon after it appeared, and led him to create some of the first models of arching (Edwards and Oakeshott 1989a; Edwards and Mounfield 1996a, 1996b; Mounfield and Edwards 1996), notably the paper with Oakeshott (Edwards and Oakeshott 1989a) reprinted here. Arching can arise through the creation of chains of strong contacts or 'force chains' (Dantu 1967), whose mechanics, as recognised by Sir Sam, may be quite unlike that of conventional elastic media. This seminal idea helped promote a large number of further theoretical and experimental studies into the physics of force chains, whose unusual properties include large stress fluctuations and high susceptibility to external perturbations. The dynamics of slowly driven granular systems also exhibit unusual features when compared to either liquids or solids. The way these systems very slowly compact when vibrated, the unusual dependence of their density on the system history, etc., again has instead strong similarities with the properties of glasses. In both cases, the dynamics appears to proceed by a succession of hops between microscopically different arrested states. The understanding of granular assemblies thus requires a proper description of the statistical properties of these so-called 'blocked' configurations. Sam understood the importance of this issue very early on, and proposed two crucial simplifying ideas. The first idea is that the state of a system of perfectly hard grains at rest is generically marginally stable, in a sense specified below. This allows intergrain forces to be calculated without recourse to elastic theory—an idea closely linked to the concept of force chains mentioned above, and developed in several of Sam's papers (Edwards 1998; Edwards and Grinev 1999a, 1999b, 1999c, 2001a, 2001b; Blumenfeld et al. 2001). The second idea is that, when gently tapped, a granular system explores all its blocked states with a uniform probability. This allowed Sam to propose an analogue of the microcanonical ensemble of statistical mechanics, with powerful consequences (see Edwards and Oakeshott 1989b, reprinted in this volume; Edwards 1993; Mehta and Edwards 1989). It also allowed him to initiate new phenomenological theories of granular compaction and relaxation (Mehta and Edwards 1990; Edwards and Grinev 1998). All three simplifying insights just described have had a profound influence on the field, and on its perception amongst the physics community. This is not because these insights reveal the deepest truths about granular materials— on the contrary, we fear it will be some time yet before these are laid bare! Rather, their influence stems from the way they dramatically sharpen the physics questions involved. Sam's work has offered bold hypotheses, which somehow seem to capture a large part of the phenomenology—stimulating many attempts, by theorists and experimentalists alike, to develop, verify or disprove the underlying ideas. [We like to think this is also true of our joint work with Ravi Prakash and Sir Sam on granular avalanches (Bouchaud et al. 1994), which for reasons of brevity we do not discuss further here.] We now review in more depth the three

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seminal ideas of Sir Sam outlined above and explain further their physical context and significance. 25.2

Statistical mechanics of granular matter

25.2.1 Force chains and arching granular assemblies Experimental measurements on the state of stress in a granular aggregate are beset with difficulty. For example, local stress fluctuations are found to be large, at least at short scales, and sometimes also on length scales much larger than the grain size: repeatedly pouring the very same amount of powder into a silo results in fluctuations of the weight supported by the bottom plate of 20 per cent or more (Brown and Richards 1966; Vanel et al. 2000). Weak perturbations of the packing can sometimes cause large-scale rearrangements (Combe and Roux 2000; Dauchot 2003). Quantitative experiments were performed in Brockbank et al. (1997), Liu et al. (1995) and Blair et al. (2001), where the local fluctuations of the normal stress deep inside a silo or at the base of a sandpile were measured. It was found that the probability distribution for stress is rather broad, decaying exponentially for large stresses. This behaviour was also found in numerical simulations (Radjai et al. 1998) [and more recently in arrested states of other systems (O'Hern et al. 2001), including compressed emulsions (Brujic et al. 2003)]. Qualitatively, all these features can (in granular media at least) be plausibly attributed to the presence of force chains—chains of compressive interparticle contacts that can focus the stress field into localised regions. The orientational distribution of these chains may (or may not) be highly anisotropic. Sam Edwards (Edwards and Oakeshott 1989a) was among the first in the physics community to realise that the micro-heterogeneity of stressed granular media required an approach quite different from the classical continuum ideas of elasticity and plasticity extensively developed in the soil mechanics literature (Wood 1990). For conical sandpiles, Edwards proposed a picture of force-chain arching, based on discrete contacts among totally rigid particles, whose anisotropic arrangement reflected the natural geometry of the pile. He showed how this simple idea could explain the central stress minimum in sandpiles, by shielding the centre of the base from the gravitational forces exerted above it. The granular texture was treated as homogeneously anisotropic—a kind of mean-field description. In Edwards and Oakeshott (1989a), the assumption of a continuum description in terms of a stress tensor was bypassed, with the vertical force distribution on the base (tantamount to the vertical normal stress) found directly. The role of continuum mechanics for general problems in granular statics may be doubted on general grounds: standard 'triaxial' tests used to determine the (macroscopic) relation between stresses and deformations show erratic hysteresis, which only seems to converge towards a well defined curve after a large number of deformation cycles have been imposed (Brown and Richards 1966), taking us far from the physics of a freshly poured sandpile. [This limiting curve might not even exist in the absence of friction (Combe and Roux 2000).] Nonetheless, much of the

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Three Seminal Ideas of Sir Sam by J.-P. Bouchaud and M. E. Gates

work on arching subsequent to Edwards and Oakeshott (1989a) has attempted to reformulate Edwards' ideas in continuum terms. The basic problem stems from the fact that the equilibrium equations for the stress tensor among rigid particles are not sufficient to determine the stress. (For example, in two dimensions the stress tensor has three independent components, but there are only two equilibrium equations.) Some constitutive assumption is needed, and the standard procedure until quite recently has been to use elastic or elastoplastic theories from soil mechanics (Wood 1990; Cantelaube and Goddard 1997; see also de Gennes 1999). But so far, no systematic coarse-graining procedure exists to connect grainlevel equations to either an elastoplastic or an alternative theory at large scales. A major difficulty is grain-level indeterminacy: many different configurations of contact forces could, in principle, satisfy local equilibrium. This leads to several conceptual problems involving, e.g., the need to define a reference state relative to which elastic displacements are defined (Gates et al. 1998b). The absence of any obvious deformation field from which the stress tensor may be constructed has motivated an alternative, 'stress-only' approach (Bouchaud et al. f 995; Wittmer et al. f996; Wittmer et al. f997; Bouchaud et al. f997; Claudin 1999), described further in section 25.3.2. The initial idea behind some of these models came to us after a seminar we attended, given by Sam at the Cavendish Laboratory in the spring of 1993. These models suppose that some (history dependent, average) relations between the components of the stress tensor should prevail at large scales. Any such relations should be determined by the global statistical features of the packing but should not depend on its microscopic details. Until recently, candidates for the required 'stress-only constitutive relations' were constructed purely by ansatz. But, much as random collisions between molecules give rise to well-defined hydrodynamical equations, an appropriate coarse-graining of the grain-level force balance equations should lead, on large length scales, to these missing equations. Sam Edwards has played a role in recent formal developments leading towards this goal, starting from a grain-level statistical mechanics of force balance (Edwards 1998; Edwards and Grinev 1999a, 1999b, 1999c, 2001a, 2001b; Blumenfeld et al. 2001; Ball and Blumenfeld 2002; Tkachenko and Witten 1999, 2000). 25.2.2 Marginal coordination of granular packings Many of the developments just referred to are based on the hypothesis, long advocated by Edwards, that rigid grains in contact have the lowest coordination number consistent with mechanical integrity of the system as a whole. (The relevant value depends on dimensionality, on whether friction is present, and on whether particles are spherical or irregular.) Such packings are often called 'isostatic,' and a relatively wide class of nonfrictional granular packings are now known to have this property (Tkachenko and Witten 1999, 2000; Moukarzel 1998a, 1998b; Roux 2000). Sam's assumption of marginal coordination offers a possible resolution to a central difficulty involving the indeterminacy of the static equilibrium state,

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which exists already at the grain level (see the simple case discussed in section 25.2.3 below). Without such an assumption one must calculate in detail the microscopic history of each contact in order to determine the static forces within a given packing. The assumption made by Edwards is motivated by the idea that grains relaxing towards static equilibrium will tend to stop moving as soon as they are just stable, and not evolve further (Blumenfeld et al. 2001; Gates et al. 1998a). [The term 'fragile' has been used to describe such states, in which even small perturbations can require reorganisation of the packing (Gates et al. 1998a; Tkachenko and Witten 1999, 2000).] Intriguingly, a similar statement is known to be exact for mean-field spin-glasses, where the equilibrium states reached dynamically are marginally stable (i.e. the spectrum of the matrix of second free energy derivatives terminates at zero eigenvalue) (Kurchan and Laloux 1996). Though evidence supports this picture for frictionless grains, the extent to which frictional packings can be described by this type of (isostatic and/or fragile) theory remains doubtful (Moukarzel 1998a, 1998b; Roux 2000; Gates et al. 1998a). Silbert et al. (2002a; see also 2002b) report molecular dynamics simulation of monodisperse grains with a specific form of contact dynamics and a certain energy dissipation coefficient at each collision. For frictionless spheres, they found a coordination number n = 6, as expected for an isostatic packing (Moukarzel 1998a, 1998b; Roux 2000). However, for static spheres with a nonzero friction coefficient, Silbert et al. found that (a) the fraction of fully mobilized frictional contacts is negligible; and (b) as a sequence of simulations is done with harder and harder grains, n saturates to a value significantly above the isostatic value (with friction, this is n = 4). The observed n varies with the coefficients of friction and of restitution, with greater damping leading to lower n (perhaps approaching the isostatic limit for large damping). This dependence on the details of the dynamics indicates that no universal statement about the validity of Edwards' marginal coordination hypothesis for frictional granular packings can be made. In all likelihood, it is a limiting approximation which is, nonetheless, close enough to reality in several classes of packings to capture important aspects of the phenomenology, and to serve as a starting point for more complete approaches. 25.2.3 Thermodynamics without temperature: the Edwards ensemble The existence of large static fluctuations confirms that many different packings, and many different assignments of the contact forces for any given packing, are compatible with the local mechanical equilibrium of each grain. To obtain reproducible results, averaging is required. In the case of sand piles, one must repeat the construction of the pile several times, and use a pressure gauge that averages over a sufficiently large number of grains, before a smooth stress profile is found. For packed beds, one can instead vibrate the packing so that it explores a sequence of static mechanical equilibrium states ('blocked states') having the same macroscopic geometry, and average over these. Sam Edwards' work has provoked theoretical and experimental study of the following questions: With

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what statistical weight will the different blocked states arise in a given experiment? To what extent do these weights depend on the dynamical pathway that led to that ensemble of blocked states concerned? Is an ensemble of 'native' packings (e.g. sand repeatedly poured afresh into a box) similar to the ensemble of packings obtained under repeated low-amplitude shaking or 'tapping'? The simplest ansatz, proposed by Sam more than ten years ago, is to postulate that all blocked states with a given volumetric density are equiprobable (Edwards and Oakeshott 1989b). This 'microcanonical' assumption defines what is now called the 'Edwards ensemble' and it turns out that several models of jammed systems do obey, either exactly or to a good approximation, Edwards' prescription (Berg et al. 2002; Dean and Lefevre 2001; de Smedt et al. 2002; Coniglio et al. 2002). This is a first and important step towards a 'thermodynamic' description of nonequilibrium, dissipative systems (Barrat et al. 2000; Maakse and Kurchan 2002; Berthier and Barrat 2002; Barrat et al. 2002a, 2002b); see the article by Kurchan in this volume. For a granular ensemble generated by tapping, Edwards' approach assigns a temperature-like role to the tapping strength. In many cases, this is a useful intuitive guide and recent experiments (Nowak et al. 1998; D'Anna et al. 2002) qualitatively confirm the analogy. Several caveats are in order, however. First, tapping is a long wavelength excitation, whereas thermal fluctuations in most materials are dominated by short wavelengths. Although tapping excitations can cascade down in length scale through collisions, important ingredients of any true thermal ensemble, such as detailed balance and a fluctuation-dissipation theorem, cannot be relied upon here, since they stem from microscopic reversibility, which applies at the atomic level in normal statistical mechanics but not at the grain level in granular mechanics (as would be required for the analogy to be complete). Nontrivial clustering patterns induced by dissipative collisions might therefore obviate simple thermal analogies for the statistics of blocked states. Secondly, one must distinguish between at least two types of tapping. Very gentle taps are insufficient to change the packing geometry, but do change the contact forces for each grain. In this case, tapping induces a random walk in 'force space' at fixed geometry. The Edwards hypothesis in this case assigns a uniform weight for all force configurations satisfying the conditions for static equilibrium (forces and torques on each grain add up to zero, and—if friction is present—a Coulomb inequality is satisfied at each contact). At larger tapping amplitudes grain motion is possible, inducing a random walk both in force space and in packing space. The uniform measure of the Edwards ensemble is then taken to extend to the particle coordinates as well as the forces. We do not know conclusively yet whether Edwards' prescription is correct in either, neither, or both of these cases. Third, to describe Edwards' prescription as a 'uniform' or microcanonical measure is potentially ambiguous. In practice the ensemble is usually created by applying a set of delta-function constraints (to ensure the static equilibrium conditions are obeyed) to a flat measure in the space of Cartesian contact forces.

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In other words (assuming gentle tapping and a fixed grain geometry), we write ff for the contact force on the a-th contact of the i-th grain, and rf the position of the contact point. Denoting as /x the static friction coefficient, and using N and T to refer to the normal and tangential components of the force, the Edwards measure is:

This assumes, before the constraints are applied, an a priori measure on the space of contact forces that is uniform, in Cartesians. This choice is reasonable, but not obvious, since uniformity would be sacrificed under any nontrivial changes of variable in the way forces are described. The same complaint could be raised about the microcanonical ensemble in thermal systems, except that here the priority assigned to a uniform measure on the canonical phase space (of coordinates and momenta, not forces) is justified, ultimately, by Hamilton's equations and Liouville's theorem, which underlie the thermodynamic description. An instructive and simple instance of Edwards' measure concerns a single disk of mass M resting in a wedge under gravity (Ertas and Halsey 1999). There are two contact points and four unknowns: /i^, /i^ and /2,Ar, /2,T, where fa^ > 0 means that the force pushes upwards. Mechanical equilibrium imposes three constraint equations, leaving one degree of freedom only; and it is easily seen that fi^T = h,T = IT and fi^ = fa,N = IN- The Edwards measure therefore reads:

where ?/> is the opening half-angle of the wedge. From this result, one can compute the distribution of the ratio r = /T//W, describing the relative mobilization of the frictional forces: P(r) oc (sin ?/> + r cos ?/>) for —/x < r < n with P(r) = 0 outside this interval. One could test this simple prediction by repeatedly tapping spheres made of different materials, and investigate the relevance of the tapping mode and the contact dynamics on the statistical ensemble of forces that one generates. This would be quite a valuable starting point for discussion of more complex multi-grain situations. So far, however, such tests have not been made. 25.3

Some related developments

We now discuss various other developments that have, during the last decade or so, been strongly influenced by (or built directly upon) the ideas of Sir Sam Edwards summarized above.

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25.3.1 The discrete scalar model Sam Edwards' view of granular materials lies at the heart of the simplest 'scalar' model for the statistics of forces in granular materials. This highly stylized (though surprisingly rich) model was proposed in Liu et al. (1995) and Coppersmith et al. (1996). In effect, the model retains one component of the stress tensor, namely the 'weight' w (or vertical normal stress) and, accordingly, only considers force balance along the vertical axis. We suppose grains reside on the nodes of a two-dimensional lattice, labelled by integers ( i , j ) in the horizontal and downward vertical directions. The equilibrium equations are then:

where WQ is the weight of each grain, and q±(i,j) are 'transmission' coefficients giving the fraction of weight which the grain ( i , j ) transmits to its right and left neighbour immediately below, such that q+(i,j) + q_(i,j) = 1 for all grains. These q±(i,j) are distributed to allow for various sources of randomness. Within this model the idea of Edwards' ensemble can easily be worked out, since each 'blocked' state corresponds to a particular choice of q+(i, j ) for every grain [with
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Claudin et al. 1998), perhaps because arching effects are absent in this scalar model. A generalization of the (/-model, allowing for arching, was suggested in Claudin and Bouchaud (1997), which dynamically generates some sites where q+ = I and J))/2- If v is small, the local weight is smoothly varying, and the discrete equation (27.3) can then be written in the following differential form:

where x = ia and t = jr are horizontal and downward vertical variables corresponding to indices i and j, with a and r lattice-related parameters, comparable to the grain size. [The vertical coordinate has been called t because it is analogous to time in a diffusion problem, as can be seen from eqn. (25.4).] Here, p is the density of the material (we set the gravity g = 1), and DQ a 'diffusion' constant, which depends on the underlying lattice geometry. The standard q model, outlined above, takes the mean value (v} to be zero: there is no preferred direction for force propagation. In many cases, however, this will not be true. Consider, e.g., a sandpile built by pouring from a point source: according to Edwards and Oakeshott (1989a) (see also Wittmer et al. 1996, 1997), the history of the grains imprints a certain fabric or 'texture' to the contact network, which we can now model as a non zero value of (v}, directed outwards from the central axis of the pile. Suppressing the effects of local disorder, we can model this in two dimensions by writing v = VQ sgn(x), in eqn. (25.4). For a constant density p = po and for DQ = 0 (representing the zero disorder limit) the weight distribution is then the following:

where c = l/tan, with is the angle subtended by the slope of the pile to the horizontal. For DQ ^ 0, the above solution is smoothed by diffusion, but in any case, the local weight reaches a minimum around x = 0. These results, though differently obtained, are essentially the same as the ones that Edwards (Edwards and Oakeshott 1989a) put forward to relate the sandpile 'dip' (Smid and Novosad 1981) to the arching of force chains. Equation (25.4), with the same form of v but now with noise added, can be obtained naturally within an extended (/-model, with an extra rule to allow a grain to lose contact with one of its downward neighbours when the shear stress is too large (Claudin and Bouchaud 1997). This generically leads to arching; in the sandpile geometry

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there is a probability threshold for local sliding, beyond which a nonzero mean velocity arises and (25.5) is recovered (Claudin and Bouchaud 1997). Because it refers to shear stress, this now goes some way beyond the original scalar model. Such ideas can be taken much further within a fully tensorial framework, as discussed below. 25.3.2 Continuum closure schemes for granular stresses Sam's work on arching and on marginal coordination has inspired ourselves and others to propose a number of closure schemes for the stress equations in granular media, in an attempt to bypass the need for any underlying description in terms of strain variables. [The problems inherent in strain-based descriptions were outlined in the introduction; see Gates et al. (1998b) for details.] Some aspects of these schemes have been justified in detail for specific (albeit isostatic) packings (Edwards 1998; Edwards and Grinev 1999a, 1999b, 1999c, 2001a, 2001b; Moukarzel 1998a, 1998b; Blumenfeld et al. 2001; Ball and Blumenfeld 2002), but much of the work remains purely phenomenological, e.g., Gates et al. (1998b), Wittmer et al. (1996, 1997) and Gates et al. (1998a). One way into the problem is via a vectorial analogue of the (/-model discussed above (Bouchaud et al. 1997). We now consider the case of three downward neighbours (see Fig. 25.1). Each grain transmits to its downward neighbours not one, but two force components: one along the vertical axis t and one along x, which we call respectively F t ( i , j ) and Fx(i,j). Now suppose that a fraction p of the vertical force travels through the middle 'leg' of the tripod, with fractions q± = (1 — p ) ( l =p e)/2 through the left and right legs (so that p + q+ +
FIG. 25.1. Each grain transmits two force components to its three downward neighbours.

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The resulting balance of the horizontal force on the grain at ( i , j ) now imposes a specific value for e, namely e = Fx(i, j')/[(l —p) tan ^Fz(i, j ) ] . Hence any nonzero horizontal force Fx necessarily leads to q+ ^
where CQ = (1 — p) tan 2 1(1, and a is a lattice parameter comparable to the grain size. We have kept the second order diffusion terms, which were the only ones left in in the symmetric scalar model described above. Here they play only a secondary role, in smoothing out the (singular or cusped) solutions that arise in their absence; these are discussed below. Comparing eqn. (25.6) with eqn. (25.4) shows that the bias term introduced by hand in that equation arises naturally in a fully vectorial approach, whenever there are nonzero local shear forces. Eliminating (say) Fx between the above two equations now leads to a wave equation for Ft (up to the aforementioned smoothing, diffusion term), where the vertical coordinate t is timelike and CQ is the 'wave speed' (Bouchaud et al. 1995). Accordingly, forces no longer propagate vertically downward, but along two rays at nonzero angles ±y, such that CQ = tan tp. Note that tp ^ 1(1 unless p = 0; the local rules for force transmission, not the underlying lattice structure, control the characteristics of the wave equation (within limits that are set by the lattice). The wave structure of the resulting equation is correspondingly universal. The above equations can be reformulated in terms of classical continuum mechanics as follows. With stress tensor components a^, the usual stress continuity equations read

Suppressing diffusion effects, we now identify the local average of Ft with att, and that of Fx with atx- Then using atx = &xt (as usual) we see that eqns. (25.6) and (25.7) are equivalent to (25.8, 25.9) combined with a linear relation between the vertical and horizontal normal stresses:

This relation between normal stresses was postulated in Bouchaud et al. (1995) as the simplest closure relation among stress components, obeying the correct symmetries and avoiding all reference to strain variables, that one can possibly

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assume. A more formal route towards extracting continuum relations of this type from a microscopic (off-lattice) statistical mechanics of grain contacts forms part of the subject of Sam Edwards' recent and ongoing work; see Edwards and Grinev (1999c). As suggested by the vectorial (/-model, the resulting wave speed CQ should depend on the local geometry of the packing (friction, shape of grains, etc.) and thereby on the construction history of the granular assembly. It is natural to associate the characteristics (in the usual mathematical sense of this word) of the resulting wave equation with the local orientation of 'force chains' within the medium (Bouchaud et al. 1997). Only for simple, spatially homogeneous constructions, such as a uniform sand-bed created by sieving, will these be uniformly oriented on the mesoscopic scale, and, even in those cases, local fluctuations will usually be present as well. A key experimental test of these ideas lies in measuring the response function to a perturbing force. The simplest situation is that of a wide uniform bed of sand, of depth H, with a localized overload on the top. The additional weight at the bottom then defines the response function; for the wave equation in two dimensions this is the sum of two peaks localized at x = ±coH. (Each peak is broadened to have width ^H1/"2 by the diffusive terms in Eqs. (25.6, 25.7) but is a (5-function in their absence.) This is notably different from an isotropic elastic body, whose response function shows a single peak whose width scales as H. However, for anisotropic elasticity, a two-peak response function can arise (Goldenberg and Goldhirsch 2002; Gay and da Silveira 2002; da Silveira et al. 2002; Otto et al. 2002), but the width of the peaks also scales as H, not as H1/"2. This means that the issue of the shape of the response function is influenced, but not solely determined, by whether the underlying stress equations are hyperbolic (wavelike) or elliptic (elastic-like) in character. Comparable remarks on all these points apply in three dimensions. Obtaining an accurate experimental response function is rather difficult: the perturbation must be small enough not to disrupt the packing, but large enough to lead to a measurable signal. A very sensitive technique, based on a lock-in detection of an oscillating perturbation, has recently led to precise and reproducible results (Reydellet and Clement 2001; Serero et al. 2001). For strongly disordered packings, these show a single-peaked response with a width growing as the height H of the layer. A double-peaked response has, however, been reported (Mueggenburg et al. 2002; Geng et al. 2001, 2002) under conditions (involving ordered packings and/or frictionless grains) that might be expected to favour the marginally stable state. The finding of a single-peaked response for disordered frictional packings contradicts the predictions of hyperbolic (wave-like) propagation that emerge from the isostatic/fragile picture. On the other hand, one can investigate perturbatively the role of disorder on these hyperbolic equations. The two-peak structure of the response function is preserved on large length scales, although the peaks are diffusively broadened (Claudin et al. 1998), in qualitative agreement with numerical simulations (Eloy and Clement 1997; Breton et al. 2002; Head et al. 2001). However, extrapolation to strong disorder suggests that the large-scale equations might undergo a transition from hyperbolic

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to elliptic behaviour; the latter could recover a single-peaked response without recourse to any elastic description. Partial support for this view is provided by recent work on directed force chain networks in which the statistical mechanics is developed directly at the level of force chains, rather than particles (Bouchaud et al. 2001; Socolar et al. 2002; Socolar 2002). Turning now to the sandpile geometry, an important issue is that of the dip: when and why does the vertical normal stress beneath a conical pile exhibit a minimum under the apex (so that the maximum lies on a ring, at a radius proportional to the height), as opposed to a maximum instead? Despite its twopeaked response function, the simplest ansatz, Eq. (25.10), leads to a central stress maximum in this geometry. This is observed experimentally when building a pile from a uniform rain of grains, whereas piles made by pouring from a point source show a central minimum instead (Vanel et al. 1999). To explain this, one can invoke Sam Edwards' concept of arching (Edwards and Oakeshott 1989a) in which force chains carry load preferentially away from the central axis. Since, in the poured case, one expects anisotropy in the local fabric of the packing, the characteristics of the relevant wave equation need not be symmetrically disposed about the vertical direction. The simplest such ansatz is (Wittmer et al. 1996, 1997):

with v ^ 0. Note that x = 0 is a singular line across which the directions of propagation change discontinuously. (This singularity remains present in three dimensions, albeit disguised when written in polar coordinates.) For one particular choice of i/, this closure is equivalent to demanding that the major principal axis of the stress tensor 'locks into' the fabric of the packing, in the sense that it is fixed by the construction history alone (Wittmer et al. 1996, 1997; Gates et al. 1998b). The resulting 'fixed principal axis' model is, arguably, the closest tensorial analogue to the original arching model of Edwards and Oakeshott (1989a). It predicts a central stress minimum for poured conical piles that closely reproduces the experimental data (Smid and Novosad 1981). The model has been criticised on various grounds (Savage 1997a, 1997b), not all of which are cogent (Gates et al. 1998b). We are left with an intriguing contrast between the success of these simple closure schemes at predicting force distributions in both poured and sieved sandpiles [including, crucially, the prediction that they are different, which preceded the experiments of Vanel et al. (1999)] and their apparent failure to predict the single-peaked response function in a granular bed. As explained above, closures resulting in hyperbolic equations, such as the wave equation arising from (25.11), appear natural in this context, and the simplest of these invariably predict a two-peaked structure. In confirmation of that, numerical simulations of anisotropic packings of frictionless beads indeed display a diffusively broadened two-peak response function (Head et al. 2001). However, for isotropic isostatic packings (Ayadim et al. unpublished), a single-peaked response is instead found,

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which closely resembles that seen in experiments on frictional granular beds; in elliptic (strain-based) models; and in the chain splitting model for high disorder (Bouchaud et al. 2001; Socolar et al. 2002; Socolar 2002). In these isotropic cases it is seen by simulation that the force chains are short, strongly branching, and randomly oriented (Ayadim et al. unpublished). It appears that the concept of linear force chains, on which the simplest wavelike picture depends (Bouchaud et al. 1997; Gates et al. 1998a), becomes more valid only as packings are made more anisotropic. The force chain concept is therefore best kept distinct, in principle, from the concept of marginal coordination, although the two ideas still appear to be related at some deep level. The results just outlined suggest that the much vaunted dichotomy between elliptic (elastic-like) and hyperbolic (force-chain-like) continuum models of granular media (Savage 1997a, 1997b; Gates et al. 1998b) is far more subtle than was realised at first. It is quite possible, given the remarks at the end of section 25.2.2, that a satisfactory description will have to confront both types of physics explicitly. 25.3.3 Slow compaction and dynamics An intriguing experiment (with obvious industrial applications) is that of compaction under gentle tapping. The packing, initially very loose, progressively compacts at a rate that decreases with time; the decay of the occupied volume is strongly nonexponential. It can be fitted by an inverse logarithm of time, or by a stretched exponential, with fit parameters that depend on tapping amplitude. (Both forms were previously used to study the volume or energy decay of glasses after a temperature quench.) More complex experimental protocols have also been tested. For example, one can change, in the course of the experiment, the amplitude of the tapping and reveal interesting memory effects, again similar to those found in glasses and spin-glasses (Kovacs 1963; Kovacs et al. 1979; Nowak et al. 1998; Josserand et al. 2000). A now classic experiment (Nowak et al. 1998) is to start from a loose sample and increase slowly the tapping amplitude F, waiting for the density p to apparently saturate at each amplitude. One finds that p(T) increases with F. But when F is reduced back to zero, the density retains a high value. This high density branch of the /o(F) curve is reversible, whereas the low density branch followed initially is not. A closely analogous hysteresis is seen in spin-glasses under a magnetic field: when the temperature is increased the (zero-field cooled) magnetization increases, but it does not follow the same path on the way back. The temperature at which the two branches separate defines the (rate-dependent) spin-glass transition. Similarly in the granular case, there appears to be a tapping amplitude beyond which the two density branches meet. Fet us call p* the maximum density that can be reached in a tapping experiment, and define a free volume parameter $ = p* — p. The simplest possible relaxation equation for $ is obviously d$/dt = —7$, giving exponential decay if 7 is constant. However, the dynamics here is self-inhibitory, in that the free volume itself is what allows further compaction. Indeed one expects 7 to vanish as $ —s- 0. Supposing a power-law, 7 ~ &13 with (3 > 0, one obtains a powerlaw relaxation at long times: $ ~ (t + to)~ 1 /' 3 • For large /? this approximates a

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logarithmic decay. Arguments for large /?, based on the need for congregation of many 'quanta' of free volume to create a void large enough for particle rearrangement, have been given in the glass literature (Nowak et al. 1998; Struick 1978). A closely related argument suggests, instead, that the probability of such void formation is, for $ 0. One possibility, suggested in Falk and Langer (2000) and Lemaitre (2002), is that each grain has a number of possible positions proportional to the free volume, which leads to s($) = ln($/wo), where VQ is a microscopic 'quantum' of free volume. Using this form for s($), we find:

showing that the Edwards' compactivity vanishes for $ —> 0. Now, consider a small region immersed in a large container with grains, which acts as a reservoir of free volume. Exactly as in thermodynamics, one can show that the probability that a free-volume equal to u is found in this region is given by p(u) oc exp(— u/X). The probability for a hole of the size of a grain v to appear is therefore: p(u = v) ~ exp(—$* 2 / ( I>), where $* = p*v. Assuming that the grain motion takes place when a sufficiently large hole appears, one finds that the rate of compaction has the same Vogel-Fulcher form as given above. 25.4

Concluding remarks

We hope we have demonstrated the role of three seminal ideas of Sir Sam in provoking a decade of work by many physicists on granular materials. Sam is adept at stealing the gold, but once found, he does not keep it to himself: the coins fly everywhere, and are picked up by all manner of people. The authors are

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among those who would like to thank Sam personally for this great generosity of spirit. Acknowledgments

We also want to thank our closest collaborators, Ph. Claudin and J. Wittmer, for sharing with us their ideas on these matters. Discussions with E. Clement, D. Levine, J. Socolar and M. Otto have also been very useful. References

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Brujic, J., Edwards, S. F., Hopkinson, I. and Maakse, H. A. (2003). Faraday Disc. 123, 207. Cantelaube, F. and Goddard, J. D. (1997). In Behringer and Jenkins (1997), pp. 231-234. Gates, M. E., Wittmer, J. P., Bouchaud, J.-P. and Claudin, P. (1998a). Phys. Rev. Lett. 81, 1841. Gates, M. E., Wittmer, J. P., Bouchaud, J.-P. and Claudin, P. (1998b). Phil. Trans. Roy. Soc. Lond. A 356, 2535. Claudin, P. (1999). Ann. de Physique 24, 1. Claudin, P. and Bouchaud, J.-P. (1997). Phys. Rev. Lett. 78, 231. Claudin, P., Bouchaud, J.-P., Gates, M. E. and Wittmer, J. P. (1998). Phys. Rev. E 57, 4441. Combe, G. and Roux, J.-N. (2000). Phys. Rev. Lett. 85, 3628. Coniglio, A., Fierro, A. and Nicodemi, M. (2002). Eur. Phys. J. E 9, 219. Coppersmith, S. N., Liu, C.-h., Majumdar, S., Narayan, O. and Witten, T. A. (1996). Phys. Rev. E 53, 4673. da Silveira, R., Vidalenc, G. and Gay, C. (2002). Preprint cond-mat/0208214. D'Anna, G., Mayor, P., Gremaud, G., Barrat, A. and Loreto, V. (2002). Extreme events driven glassy behaviour in granular media. Europhys. Lett. 61, p. 60 (2003). Dantu, P. (1967). Ann. des Fonts et Chaussees 4, 144. Dauchot, O. (2003). Private communication and manuscript in preparation. Dean, D. S. and Lefevre, A. (2001). Phys. Rev. Lett. 86, 5639. de Gennes, P.-G. (1999). Rev. Mod. Phys. 71, S374. de Smedt, G., Godreche, C. and Luck, J. M. (2002). Eur. Phys. J. B 27, 363. Duran, J. (1997). Sables, Poudres et Grains. Eyrolles Sciences, Paris (1997); Springer, New-York (2001). Duran, J. and Bouchaud, J.-P. (eds.) (2002). Physics of Granular Media. Comptes Rendus de I'Academic des Sciences (Special Issue) 3, 129. Edwards, S. F. (1993). In Granular Matter, An Interdisciplinary Approach. Mehta, A. (ed.). Springer, NY 1993. Edwards, S. F. (1998). Physica A 249, 226. Edwards, S. F. and Grinev, D. V. (1998). Phys. Rev. E 58, 4758. Edwards, S. F. and Grinev, D. V. (1999a). Phys. Rev. Lett 82, 5397. Edwards, S. F. and Grinev, D. V. (1999b). Chaos 9, 551. Edwards, S. F. and Grinev, D. V. (1999c). Physica A 263, 545. Edwards, S. F. and Grinev, D. V. (2001a). Physica A 294, 57. Edwards, S. F. and Grinev, D. V. (2001b). Physica A 302, 162. Edwards, S. F. and Mounfield, C. C. (1996a). Physica A 226, 1. Edwards, S. F. and Mounfield, C. C. (1996b). Physica A 226, 25. Edwards, S. F. and Oakeshott, R. B. S. (1989a). Physica D 38, 88. Reprinted in this volume. Edwards, S. F. and Oakeshott, R. B. S. (1989b). Physica A 157, 1080. Reprinted in this volume. Eloy, C. and Clement, E. (1997). J. Phys. (France) I 7, 1541.

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Ertas, D. and Halsey, T. C. (1999). Phys. Rev. Lett. 83, 5007. Falk, M. L. and Langer, J. S. (2000). M.R.S. Bulletin 25, 40. Gay, C. and da Silveira, R. (2002). Preprint cond-mat/0208155. Geng, J., Howell, D., Longhi, E., Behringer, R. P., Reydellet, G., Vanel, L., Clement, E. and Luding, S. (2001). Phys. Rev. Lett. 87, 035506. Geng, J., Reydellet, G., Clement, E. and Behringer, R. P. (2002). Green's function measurements in 2D granular materials, preprint cond-mat/0211031. Physica D 182, 274 (2003). Goldenberg, C. and Goldhirsch, I. (2002). Phys. Rev. Lett. 89, 084302. Head, D., Tkachenko, A. and Witten, T. (2001). Eur. Phys. J. E 6, 99. Herrmann, H. J., Hovi, J. P. and Luding, S. (eds.) (1997). Physics of Dry Granular Media. NATO ASI, 25. Jaeger, H. M., Nagel, S. R. and Behringer, R. P. (1996). Rev. Mod. Phys. 68, 1259. Josserand, Ch., Tkachenko, A., Mueth, D. M. and Jaeger, H. M. (2000). Phys. Rev. Lett. 85, 3632. Kovacs, A. J. (1963). Adv. Polym. Sci. 3, 394. Kovacs, A. J., Aklonis, J. J., Hutchinson, J. M. and Ramos, A. R. (1979). J. Poly. Sci. 17, 1097. Kurchan, J. and Laloux, L. (1996). J. Phys. A29, 1929. Lemaitre, A. (2002). A dynamical approach to glassy materials. Preprint condmat/0206417. Liu, C.-H., Nagel, S. R., Scheeter, D. A., Coppersmith, S. N., Majumdar, S., Narayan, O. and Witten, T. A. (1995). Science 269, 513. Maakse, H. A. and Kurchan, J. (2002). Nature 415, 614. Mehta, A. and Edwards, S. F. (1989). Physica A 157, 1091. Mehta, A. and Edwards, S. F. (1990). Physica A 162, 714. Moukarzel, C. F. (1998a). Phys. Rev. Lett. 81, 1634. Moukarzel, C. F. (1998b). Granular matter instability: A structural rigidity point of view. Preprint cond-mat/9807004. Mounfield, C. C. and Edwards, S. F. (1996). Physica A 226, 12. Mueggenburg, N., Jaeger, H. and Nagel, S. (2002). Phys. Rev. E 66, 031304. Narayan, O. (2001). Phys. Rev. E 63, 10301. Nowak, E. R., Knight, J. B., Ben-Nairn, E., Jaeger, H. M. and Nagel, S. R. (1998). Phys. Rev. E 57, 1971. O'Hern, C. S., Langer, S. A., Liu, A. J. and Nagel, S. R. (2001). Phys. Rev. Lett. 86, 111. Ostogic, S. and Panja, D. Cond-mat/0403321. Otto, M., Bouchaud, J.-P., Claudin, P. and Socolar, J. E. S. (2002). Phys. Rev. E67, 031302 (2003). Radjai, F., Wolf, D. E., Jean, M. and Moreau, J. J. (1998). Phys. Rev. Lett. 80, 61; and references therein. Reydellet, G. and Clement, E. (2001). Phys. Rev. Lett. 86, 3308. Roux, J.-N. (2000). Phys. Rev. E 61, 6802. Savage, S. B. (1997a). In Behringer and Jenkins (1997), pp. 185-194.

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see also Savage, S. B. (1997b). New Scientist, 2083, p. 28. Socolar, J. E. S. (2002). Discrete models of force chain networks. Discrete Gout. Dyn. 3, 601 (2003). Socolar, J. E. S., Schaeffer, D. G. and Claudin, P. (2002). Eur. Phys. J. E 7, 353. Serero, D., Reydellet, G., Claudin, P., Clement, E. and Levine, D. (2001). Eur. Phys. J. E 6, 169. Sexton, M. G., Socolar, J. E. S. and Schaeffer, D. G. (1999). Phys. Rev. E 60, 1999. Silbert, L. E., Ertas, D., Grest, G. S., Halsey, T. C. and Levine, D. (2002a). Phys. Rev. E 65, 031304. Silbert, L. E., Ertas, D., Grest, G. S., Halsey, T. C. and Levine, D. (2002b). Phys. Rev. E 65, 051307. Smid, J. and Novosad, J. (1981). Proc. of 1981 Powtech Conference. Ind. Chem. Eng. Symp. 63, D3V 1-12. Snoeijer, J. H., van Hecke, M., Somfai, E. and van Saarloos, W. (2002). Effect of boundaries on the force distributions in granular media. Phys. Rev. E 67 030302 (2003). Snoeijer, J. H., Vlugt, T. J. H., Van Hecke, M. and van Sarloos, W. (2004). Phys. Rev. Lett. 92, 054302. Struick, L. C. E. (1978). Physical Aging in Amorphous Polymers and Other Materials. Elsevier, Amsterdam. Tkachenko, V. and Witten, T. A. (1999). Phys. Rev. E 60, 687. Tkachenko, V. and Witten, T. A. (2000). Phys. Rev. E 62, 2510. Uuger, T., Kertesz, J. and Wolf, D. E. cond-mat/0403089. Vanel, L., Claudin, P., Bouchaud, J.-P., Gates, M. E., Clement, E. and Wittmer, J. P. (2000). Phys. Rev. Lett. 84, 1439. Vanel, L., Howell, D. W., Clark, D., Behringer, R. P. and Clement, E. (1999). Phys. Rev. E 60, R5040. Wittmer, J. P., Gates, M. E., Claudin, P. and Bouchaud, J.-P. (1996). Nature (London) 382, 336. Wittmer, J. P., Claudin, P. and Gates, M. E. (1997). J. Phys. (France) I 7, 39. Wood, D. M. (1990). Soil Behaviour and Critical State Soil Mechanics. Cambridge University Press.

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CHAPTERS ON THE EDWARDSIAN APPROACH TO RESEARCH

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26

THE CASE FOR EDWARDSIAN RESEARCH IN SOLID MECHANICS: A SERMON J. S. Langer Department of Physics, University of California-Santa Barbara, Santa Barbara, California 93106, U.S.A. I have been asked to write a sort of 'sermon' about the Edwardsian style of research. I like this assignment because it gives me a chance to make some philosophical remarks about Sam's impact on modern physics and, at the same time, express my opinion about a topic that has been much on my mind recently. Specifically, I want to write about solid mechanics, a crucially important part of engineering physics where I think an Edwardsian approach is badly overdue. And I want to be clear from the beginning that I believe there are many other areas in modern science, typically regarded as being narrowly disciplinary, where a more open-minded, free-wheeling Edwardsian approach will be necessary in the future. First—some ancient history. In the fall of 1955, I was a beginning graduate student at Peierls' Department of Mathematical Physics in Birmingham, England, where Sam Edwards was a junior member of the faculty. Peierls' scheme for me seemed to be that, in a period of three years, I should learn all of theoretical physics by working on one project after another in various fields. So, after learning quantum electrodynamics by helping Gerry Brown compute the Famb shift in heavy atoms, I was assigned to work with Sam and the late Paul Matthews on what in those days passed for elementary particle field theory. With their help and guidance, I wrote what turned out to be an utterly uninteresting paper about how one might explain an anomaly in pion-nucleon scattering by invoking interactions with the newly discovered 'strange' particles. The useful result of that exercise was not that it advanced particle theory, which it didn't, but that it helped me learn a style of investigation that has been enormously productive in mainstream theoretical physics. By 1957 when my paper appeared, and by which time Edwards and Matthews had published better papers on related topics, we knew that: • The powerful methods of quantum field theory could be extended to other fields, especially in nuclear and condensed-matter physics. • The best way to deal with an apparently complex physical situation is to start with the simplest possible model that is consistent with known symmetries and any other fundamental principles that can be brought to bear on it.

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• The perturbative techniques that had been so successful in quantum electrodynamics were not going to work for the most interesting, strongly interacting systems. We would have to develop new, nonperturbative methods. Each of these themes became a component of the Edwardsian style. Sam always has been a master builder of models, especially field theoretic models that capture just the essential features of physical situations. And he has couched his theories in mathematical languages, most notoriously function-space methods, that sometimes exasperate rigorous critics but which work beautifully for his purposes. An additional, essential ingredient of the Edwardsian style has been chutzpah. In a way that I especially admire, Sam has had the guts to venture on to other people's turf and the wisdom to do that so successfully that those people have paid serious attention to him. His most successful excursion of this kind, of course, has been into the area of polymeric materials. I know from painful personal experience that this element of his style cannot easily be imitated. In the spirit of chutzpah, I propose to devote the rest of this sermon to a discussion of some foreign turf that has attracted Sam's attention recently, but not yet in a way that has got him into enough trouble. The area I have in mind includes granular materials, soils (quite literally 'turf') and, by simple extension, deformation and failure—including fracture—of amorphous solids. I am thinking primarily about materials that undergo what Sid Nagel has called 'jamming' (Liu and Nagel 2001). Deformation and fracture have proved to be particularly dangerous areas for physicists like me. They traditionally have been the property of the engineering community which has, for about a century, been extremely successful in applying phenomenological rules to practical problems; but there are some puzzling internal inconsistencies that pervade all of conventional solid mechanics, including theories of crystalline materials. I am convinced that these puzzles will have to be resolved if this field is to meet modern technological challenges such as those that arise in the development of complex composite materials or in the search for materials that perform well under extreme conditions of temperature or stress. I also suspect, as I said at the beginning, that we shall discover puzzles of this general nature in phenomenological descriptions of other strongly nonequilibrium phenomena that occur in geology, polymer science, and especially biology. If so, then an Edwardsian style of research will prove to be increasingly necessary thoughout much of modern science. My questions about solid mechanics are the following. What are the fundamental distinctions between brittle and ductile behaviours7 A brittle solid breaks when subjected to a large enough stress, whereas a ductile material deforms plastically. Remarkably, we do not yet have a fundamental understanding of the distinction between these two behaviours. Conventional theories of crystalline solids say that dislocations form and move more easily through ductile materials than brittle ones, thus allowing deformation to occur in one case

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and fracture in the other. But the same behaviours also occur in amorphous solids; thus the dislocation mechanism cannot be the essential ingredient of all theories. Moreover, some materials are ductile when loaded slowly and brittle when subjected to sudden stresses, which implies that a proper description of deformation and fracture must be dynamic, i.e., it must be expressed in the form of equations of motion rather than phenomenological rules and flow criteria. What is the origin of memory effects in plasticity7 Standard, hysteretic, stress-strain curves for deformable solids—and even for qualitatively different kinds of substances such as soils—tell us that these materials have rudimentary memories. Roughly speaking, they 'remember' the direction in which they most recently have been deformed. When unloaded and then reloaded in the original direction, they are hardened and respond only elastically, whereas, when loaded in the opposite direction, they deform plastically. The conventional way of dealing with such behaviour (Lubliner 1990) is to specify phenomenological rules stating how the response to an applied stress is determined by the history of prior loading; but such rules provide little insight about what is actually happening or what might be the nature of a more satisfactory theory. As is often recognized in the literature, these memory effects indicate clearly the need for internal state variables that can carry information about previous history. All too often, however, the plastic strain itself is used as such a state variable—a procedure that violates basic principles of nonequilibrium physics because it implies that a deformed material must somehow remember all of its prior history. How can breaking stresses be transmitted to crack tipsl Solids generally deform plastically under high stresses such as those in the neighborhood of a crack tip. Those stresses, supposedly, must be large enough to break the bonds between neighboring molecules; therefore, except possibly in special cases such as cleavage fracture, they must be more than large enough to deform the material and blunt the tip. If the material near a crack tip always flows before it breaks, how can cracks ever propagate in a brittle manner? The currently most popular answers to this question invoke hardening via dislocation entanglements (Fleck and Hutchinson 1996); but, once again, the same phenomena occur in noncrystalline materials where dislocation mechanisms cannot be relevant. This question, in my opinion, implies a deep connection between fracture dynamics and plasticity, a connection that will need to be taken seriously in trying to answer the final question on my list. What is the origin of instabilities in brittle fracture! One of the most significant recent developments in fracture dynamics has been the experimental demonstration that fast brittle cracks undergo material-specific instabilities (Fineberg et al. 1991, 1992; Gross et al. 1993). Fracture surfaces frequently are rough; they may even be fractal. We now know that this roughness occurs because fast cracks are unstable against bending away from their directions of propagation, dissipating energy in the form of tip-splittings or side-branches. We still do not know what mechanisms control fracture stability. In some respects, the present state of the theory of dynamic fracture resembles that of solidification theory almost half a century ago, before Mullins and Sekerka had identified the

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diffusive instability that underlies dendritic pattern formation in crystal growth. Just as the understanding of morphological instabilities in crystal growth has led to progress in solidification processing, a basic understanding of instabilities in fracture is likely to be essential for predicting crack speeds, crack arrest, and the like, and thus for determining the strengths of materials. During the last four or five years, my colleagues and I (Falk and Langer 1998, 2000) have been developing what I would like to think is an Edwardsian approach to this collection of problems. We have found a relatively simple theoretical framework that, at least qualitatively, accounts for the basic phenomena of amorphous plasticity including yield stresses, work hardening, strain softening, strain recovery after unloading, necking instabilities, and the like. I think that we are close to some understanding of the brittle-ductile puzzle, and perhaps are even catching glimpses of a new picture of crack advance in deformable materials (Eastgate et al. 2003). Clearly, we have not (yet) achieved an Edwardsian success in this effort; but maybe we could make faster progress if Sam would risk the ire of the engineers by helping in it. As I shall point out, there are some ingredients here that are close to his heart. The new ideas in the work to which I am referring are largely due to Michael Falk and, more recently, Leonid Pechenik, Anael Lemaitre, and Lance Eastgate. Falk (Falk and Langer 1998; Falk 1998) started with molecular dynamics simulations of shear deformations in two-dimensional, amorphous, Lennard-Jones solids. He found an ingenious way of visualizing the irreversible molecular motions in these materials and discovered that, roughly as postulated by Cohen, Turnbull, Spaepen, Argon and others (Turnbull and Cohen 1970; Spaepen and Taub 1981), those deformations are localized in so-called 'shear-transformation zones' (STZ's). The molecules in these zones undergo shear rearrangements in response to applied stresses. Falk discovered that they behave somewhat like two-state systems. That is, they can transform only a finite amount in one direction before they become jammed but, once they have done so, they can transform in the opposite direction in response to a reversed stress. These STZ's can also be created and annihilated during irreversible deformations of the material. The first step in formulating an Edwardsian kind of theory based on this picture is identifying appropriate order parameters, i.e., determining the physically relevant internal state variables and their symmetries. The STZ theory, in its present form (which is evolving as I write these words) contains two of these variables, a scalar field A that is proportional to the density of STZ's, and a traceless symmetric tensor field A^j that describes their spatial orientations. The second step is to use these fields to write down the simplest possible equations of motion that capture the essential features of the physical model. The full STZ theory consists of equations of motion for A, A^, the strain rate, and the stress. Even in their simplest versions, these equations are intrinsically nonlinear and are quite impervious to perturbative analyses. (We don't yet need functional integrals, however.) Surely Sam shares my amazement that, these days, we solve such equations almost instantaneously on our laptop computers.

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To convey some sense of how the pieces of the STZ theory fit together and what they mean, I must now be more specific by exhibiting—if not fully explaining—these equations of motion. I specialize to the simplest, so-called 'quasilinear' version of the theory, and consider only the case of pure shear deformation (no compression or tension). The plastic strain rate (more accurately, the plastic part of the rate of deformation tensor) is:

Here, Sij is the deviatoric stress, i.e., the traceless part of the stress tensor, expressed in units chosen so that the yield stress will be unity. The factor CQ is a material-specific constant; and r is a time-scale that, for the moment, we may assume to be a constant. In this form, the quantity s^ — A^/A looks like an effective stress, with A^/A playing the role of what is sometimes called the 'back stress' or 'hardening parameter' (Lubliner 1990). The STZ interpretation of eqn. (26.1) is that the right-hand side is the balance between forward and backward transitions in the direction of the stress, and that the term — Ajj/A accounts for the fact that the backward transitions occur more frequently when the STZ's are oriented in the forward direction. The equation of motion for the state variable A—the scaled density of STZ's—is

where F(s, A, A), is proportional to the rate of annihilation and creation of zones, and A = 1 is the scaled density at which the annihilation rate exactly balances the creation rate. We assume that F(s, A, A) is proportional to the rate per zone at which the plastic work is dissipated—a new and I think important idea, largely due to Pechenik (Langer and Pechenik 2003)—and find

for any tensor The equation of motion for A^—the STZ version of the 'back stress'—is

where

The first term in parentheses on the right-hand side tells us simply that A^ increases with the plastic strain; the second term accounts for the rate at which this effect is diminished by the annihilation of zones. To understand the behaviour of these equations and, especially, the role played by AJJ, it is easiest to look at a uniform system subjected to a constant, uniform shear stress in the (x, y) plane. Let

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A^y = 0. Equation (26.2) tells us that the stable stationary value of A is unity; thus, for simplicity, we set A = 1 in eqns. (26.1) and (26.2), and find

and

At s = 1, which is the effective yield stress in this theory, these equations exhibit an exchange of stability between the non-flowing steady-state solution with e = 0 and A = s (for s < 1) and the flowing solution with £ ^ 0 and A = l/s (for s > 1). That is, the system becomes 'jammed' or 'hardened' in the direction of the applied stress for s < 1. On the other hand, for s > 1, new STZs are created as fast as existing ones transform and, as a result, there is a nonzero steady-state plastic strain rate at fixed stress. Specifically,

which is essentially the 'Bingham law' in conventional plasticity. In summary, these equations of motion predict that a transition between jammed states and viscoplastic flow occurs via an exchange of stability at a characteristic yield stress that has been scaled here to s = I . (In the unsealed theory, the yield stress is a function of the nonlinear coupling parameters.) It must be emphasized, however, that yielding in the STZ theory actually occurs at all stresses and, at any given stress, the amount of yielding depends strongly on the internal state of the system. In these dynamical features, we are departing strongly from engineering conventions. While this minimal, quasilinear version of the STZ theory successfully accounts for many of the qualitative features of solid plasticity, it falls well short of broader goals. For example, it does not retain all of the memory effects that were present in the original, fully nonlinear version (Falk and Langer 1998). That original version, however, was based on a specific model of the STZ's that I now think needs not to be taken so literally. The STZ picture more generally implies that an unloaded system 'remembers' the direction in which it previously was deformed because, at zero stress and sufficiently low temperatures, the zones do not rapidly lose their earlier orientations. In the quasilinear theory, however, the state variable A, which carries the orientational memory, always relaxes to zero in a time r after unloading, as may be seen by setting s = 0 in eqn. (26.6); and T is the only time-scale in this version of the problem. What is missing [and was provided in a special way in Falk and Langer (1998)] is some mechanism that will cause the internal degrees of freedom in the system to slow dramatically during and after unloading. More generally, we must ask what physical mechanisms will introduce new time-scales in STZ-like systems.

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There is other evidence for the existence of multiple, competing time-scales in deforming solids. For example, in creep tests where samples are held at constant stress, many materials initially appear to come almost to equilibrium at a fixed strain but, after some delay time, begin to deform and ultimately fail. Another example of multiple time-scales is stretched-exponential stress-relaxation in glasses. Although I shall demonstrate my own chutzpah by offering some speculations in the next few paragraphs, I certainly do not know what these mechanisms will turn out to be. I do think, however, that we must look for them in an Edwardsian mode by considering broad classes of phenomena—nonequilibrium processes in amorphous metals, granular materials, foams, colloidal suspensions, etc.—and by trying to discover general principles rather than by inventing different phenomenological models for each special case. I conclude this sermon by mentioning two ideas that I think may be useful to pursue in the search for new fundamental mechanisms. The first arises from fairly recent work by Sam Edwards. Falk and I were thinking about an important paper on granular materials by Mehta and Edwards (1990) when we constructed the first version of the STZ theory. Falk had carried out his molecular dynamics simulations at about 0.3% of the glass temperature for his Lennard-Jones materials; thus, the STZ transitions that he observed were not driven by ordinary thermal activation over energy barriers. Mehta and Edwards had recognized that, in a situation where thermal kinetic energies are negligible, as they are when the 'molecules' are macroscopic objects like grains, the interesting thermodynamic variable is the volume rather than the energy. Falk and I therefore guessed that the activation factor governing the rate of STZ transitions would have the form e x p ( — A V * ( s ) / V f ) , where AV*(s) is the magnitude of a local volume fluctuation needed to enable a transition at stress s, and Vf is the free volume, the intensive variable thermodynamically conjugate to the volume V. That is,

where S is the entropy and kp is Boltzmann's constant. There remained—and still remains—the question of how to determine the attempt frequency, i.e., the fluctuation rate that multiplies the exponential for non-thermal systems. The first of the ideas that I want to raise, therefore, is that the free volume might be given the status of an independent, dynamical, state variable in theories of jamming. In Falk and Langer (1998), Vf was an intensive variable, roughly analogous to temperature as defined by eqn. (26.8), and was assumed to be a constant. Lemaitre (2002) has suggested that this dilational 'temperature' should obey an equation of motion that supplements the equations for A and A^. In Lemaitre's theory, Vf increases as the system undergoes plastic deformation, and decreases spontaneously, even at zero shear rate, as the fluctuating system settles toward closer packing of its constituents. Because the activation rate is strongly sensitive to Vf, this mechanism may account for the multiple timescales described above. Lemaitre has shown that it even can produce stretched

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exponentials. I believe that this is a promising approach, but it still provides no theory of an underlying fluctuation rate for granular materials. The second idea that I want to mention might solve the latter problem. There is increasing evidence that the nonequilibrium states of sheared foams or granular materials are meaningfully described by an effective temperature that characterizes slow transient states of disorder (Cugliandolo et al. 1997; Sollich et al. 1997; Berthier and Barrat 2002; Ono et al. 2002). If such a nonequilibrium noise temperature can in fact be defined and computed, then it should be possible to compute transformation rates in STZ-like models. Pechenik's idea (Pechenik and Langer 2003), mentioned in connection with Eq. (26.3), might help here, because it provides a means for identifying the rate at which energy is being dissipated in these processes as opposed, say, to being stored reversibly in the internal degrees of freedom. Perhaps this heat content can systematically be associated with an effective noise temperature and thus with an attempt frequency for activation rates. In any case, the plasticity and jamming problems are forcing us to ask fundamental questions about nonequilibrium physics. The fact that these questions arise at all, it seems to me, implies that we need to take a new look at the foundations of solid mechanics and a variety of related fields. Physicists, chemists, mathematicians, and especially engineers will have to play active roles in this endeavor. We all shall have to be open minded about new ideas, tolerate each others' mistakes, and be willing to risk mistakes ourselves. I believe strongly in that style of research, and I trace my attitude in substantial part to lessons I've learned over many years from Sam Edwards. Acknowledgments

The research upon which this 'sermon' is based was supported primarily by U.S. Department of Energy Grant No. DE-FG03-99ER45762. It was also supported in part by the MRSEC Program of the NSF under Award No. DMR96-32716 and by a grant from the Keck Foundation for Interdisciplinary Research in Seismology and Materials Science. I would like to thank Lance Eastgate and Premi Chandra for critically reading earlier versions of the manuscript and for helpful suggestions. References

Berthier, L. and Barrat, J.-L. (2002). Phys. Rev. Lett. 89, 095702. Cugliandolo, L., Kurchan, J. and Peliti, L. (1997). Phys. Rev. E 55, 3898. Eastgate, L. O., Langer, J. S. and Pechenik, L. (2003). Phys. Rev. Lett. 90, 045506. Falk, M. L. (1998). Ph.D. Thesis. Physics Department, University of CaliforniaSanta Barbara. Falk, M. L. and Langer, J. S. (1998). Phys. Rev. E 57, 7192. Falk, M. L. and Langer, J. S. (2000). M. R. S. Bulletin 25, 40.

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Fineberg, J., Gross, S. P., Marder, M. and Swinney, H. L. (1991). Phys. Rev. Lett. 67, 457. Fineberg, J., Gross, S. P., Marder, M. and Swinney, H. L. (1992). Phys. Rev. B 45, 5146. Fleck, N. A. and Hutchinson, J. W. (1996). Strain Gradient Plasticity. In Hutchinson, J. W. and Wu, T. Y. (eds.), Advances in Applied Mechanics 33, Academic Press, New York. Gross, S. P., Fineberg, J., Marder, M., McCormick, W. D. and Swinney, H. L. (1993). Phys. Rev. Lett. 71, 3162. Langer, J. S. and Pechenik, L. (2003). Phys. Rev. E 68, 061507. Lemaitre, A. (2002). Phys. Rev. Lett. 89, 195503; and private communication. Liu, A. J. and Nagel, S. R. (eds.) (2001). Jamming and Rheology. Taylor and Francis, New York. Lubliner, J. (1990). Plasticity Theory. Macmillan Publishing Company, New York. Mehta, A. and Edwards, S. F. (1990). Physica A 157, 1091. Ono, I. K., O'Hern, C. S., Durian, D. J., Langer, S. A., Liu, A. and Nagel, S. R. (2002). Phys. Rev. Lett. 89, 095703. Sollich, P., Lequeux, F., Hebraud, P. and Gates, M. E. (1997). Phys. Rev. Lett. 78, 2020. Spaepen, F. and Taub, A. (1981). In Balian, R. and Kleman, M. (eds.), Physics of Defects, 1981 Les Houches Lectures, p. 133. North Holland, Amsterdam. Turnbull, D. and Cohen, M. (1970). J. Chem. Phys. 52, 3038.

27

A SCIENTIST FOR ALL SEASONS Sir Geoffrey Allen, F.R.Eng., F.R.S. Chancellor, University of East Anglia, Norwich NR4 7TJ, United Kingdom Polymer science

Sam Edwards came to the Department of Physics at Manchester in 1958 and quickly made a reputation within the Faculty of Science for his capacity to solve problems or draw analogies with apparently unrelated solutions. He was appointed Professor of Theoretical Physics in 1963. At this time Prof. G. Gee led a research group, of which I was a member, with strong interests in polymer melts, rubber elasticity, the glassy state of polymers and polymers in solution. Gee had a lifelong interest in the equation of state of vulcanized rubbers relating the force / to the strain A. For a lightly crosslinked network of 'phantom' chains that are allowed to intersect each other, this assumes the simple form:

being the analogue of the equation of state of an ideal gas, PV = RT. As wit real gases, real rubbers deviate from this simple law and at moderate extensions the equation approximates to

Gee tried to modify the ideal network theory by introducing empirical statistics for non-intersecting (real) chains. The starting point of the argument was to replace the Gaussian form of the root-mean-square distance r between the ends of an intersecting chain of n links of length /,

by an empirical result from computation.

where the value of a was about 0.18 ± 0.01. Unfortunately, an equation of state could not be extracted from this analysis. In desperation and with some trepidation Sam was approached. He listened politely and returned a few days later to say that there were inconsistencies

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which negated the treatment. As a throwaway line he added: 'By the way the asymptotic value of a for an infinitely long linear chain will be 1/5; the exponent is not ~1.2 it is 6/5!' We were very impressed because in those early days of computers there was considerable argument about appropriate methods of computation and approximations to be made in simulating long chains on small computers. Here might be an opportunity to establish one of the cornerstones of polymer physics. Sam promised to come back when he had cast the analysis in the appropriate form. And comeback he did, with a thirst to know more of the molecular aspects of polymers that gladdened our hearts. It was a time when high-powered theoretical physicists rarely were interested in such 'ill-defined' substances; in fact most physicists regarded polymers as materials unsuitable for precise measurement. Sam sensed an opportunity to deploy his pioneering theoretical techniques developed for the statistical physics of disordered metals. It so happened that Gee and Edwards were members, respectively, of the chemistry and physics committees of the newly formed UK Science Research Council. They often traveled together on the train to and from London, and so Sam received 'tutorials' from Gee on the thermodynamics of polymer melts, polymer glasses and polymer solutions. Gee had a comprehensive knowledge of the molecular implications of the thermodynamics and, as a former director of a leading rubber research laboratory, also of technological applications of polymers. My role in this induction process was to meet Sam in his office ostensibly to carry on the good work. More often, I was subjected to searching questions about the current state of knowledge in polymer science. I had to explain why my research programme had two strands, one shared with Gee covering essentially 'equilibrium' properties, the other being concerned with dynamic properties such as mechanical, dielectrical, N.M.R. relaxation, and flow, and, in addition, molecular vibrational spectroscopy. The unifying theme was to understand the relation between physical properties and chemical structure. Sam concluded that many of the experiments were too complex to provide useful insights. I found myself re-examining my portfolio in the light of his penetrating comments; soon we were envisaging experiments of 'simple specification' that might be more fruitful. There was a corresponding search for chemical methods to make polymer samples of well-defined structure, for use in these experiments. For my research group it was the age of enlightenment. Sam used his growing familiarity with the profile of polymer science, backed by a remarkable ability to delve into the extensive literature and extract the essence of a paper to judge its relevance to the current scientific and technological scene. He is blessed with a sense of proportion and a deep insight that enables him to spot something untoward; even more, to recognize a salient feature or new avenue of intellectual (or practical) promise. Without any fuss, he launched an attack on the theoretical basis of polymer physics. In doing so he brought the science of polymers and gels within the compass of the statistical physics of soft condensed matter.

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The thesis was that amorphous polymers were primarily entropy-driven systems. The result was a series of groundbreaking papers with his co-workers, which revolutionized the theoretical basis of polymer physics. This first set of papers dealt with: (a) a single polymer chain, (b) polymer chains with excluded volume (i.e. non-intersecting), (c) polymer solutions at intermediate concentration where chains overlap and become entangled, and (d) polymerized matter, particularly the melt or rubber state and crosslinked structures. In 1965 quantum field theoretical techniques and the form of the 'Edwards Hamiltonian' were introduced, the concept of polymer screening emerged and the Edwards screening length was discovered. The treatment of entanglements in and between polymer chains (1966, 1967) brought clarity to what had been a rather imprecise idea. Apart from underpinning the 'equilibrium' properties, a key advance in this early period, 1967, proved to be the introduction of the concept of a polymer chain in a melt or concentrated solution being enclosed in a 'tube' formed by the neighbouring molecules. This simple concept has had many consequences, the most far-reaching being the discovery by de Gennes that the wriggling of the polymer chain along the 'tube' (reptation) provides a basis for understanding flow in polymer systems. In 1970 he introduced field descriptions of polymer gels. Again this was brilliantly extended by de Gennes, to bring modern renormalization-group theories into use in polymer science. A new mathematical treatment appeared in 1971 when Sam developed the Replica Formalism, which enabled the closed formulae describing rubber networks to be written down. This work settled, finally, a thirty-five year-old series of acrimonious quarrels over the exact stress-strain relationship of simple networks. They differed by a factor of ^! The exact result for the modulus G of a network containing N effective crosslinks is now accepted to be,

a result derived by James and Guth in 1943. This innovation was used again in 1975, in collaboration with Anderson, to give the first solution of the disordered magnet problem, the spin glass. A computational advance in 1972, the Edwards-Lees boundary condition, enabled simulations of liquids, later including polymers and colloids, to be extended to describe shear flow. 1974 was a year of great productivity. Sam produced the first dynamical treatment of a polymer network, which provided an alternative method to the Replica Method. This was followed by the discovery of hydrodynamic screening of polymers. Then he performed the first calculation of the stabilization of colloid suspensions by polymers—of which more later.

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More major advances came in 1977, when the structure of the tube enclosing a polymer in the melt was quantified in detail. Coupled to de Gennes' reptation theory, this analysis resulted in the first comprehensive theory of polymer flow. The partnership with Masao Doi in this field is encapsulated in their classic text. By 1980, Sam had become interested in stereology and its application in the theory of powders. This has been his main interest over the past two decades. From time to time, however, he has returned to the theory of polymerized matter with considerable effect. In 1982, he worked on rod-like (i.e. liquid crystal) polymers, and was the first to derive the structure of the glass transition. This led on, in 1986, to the first theoretical derivation of the empirical WilliamsLandel-Ferry equation of glassification, until then an empirical law that unified the frequency dependence of glass transitions in amorphous materials. In this same period he continued to work on solution viscosity, producing eventually the first extrapolation formula from dilute to concentrated solutions. This is a convenient point at which to note that this massive foray into polymer physics began at a time when Sam was increasingly involved, through the Science Research Council committees, with national policies for science and, in particular, the leadership of the Science Research Council Physics Committee. From 1973-77 he was seconded, full time, to become a very successful Chairman and Chief Executive of the Science Research Council, responsible for government support of academic research and for special national facilities in all physics (including astronomy, space and particle physics), chemistry, mathematics, engineering and mainstream biology. Nevertheless, he continued to direct his research group, seeing his students regularly and attending major conferences. He famously carried his research notebooks into his various meetings and, when not in the chair, was observed to be furiously scribbling down mathematical equations! This could have equally disturbing consequences for his students who would be advised 'By the way, I was in a rather boring meeting the other day and I managed to make some progress with your problem!' Only once, I believe, was he outgunned. When Doi was confronted with this situation, Sam wrote out the form of the solution, indicating that the constant had a value approaching unity; Doi replied 'Yes Sir,' and promptly gave the exact form of the constant!! To witness this tour de force at close quarters and to interact with the members of his group was a real privilege. Sam has the gift for sensing what the form of the solution to a problem has to be, and then is adept at including ideas or solutions of earlier workers tackling what he considers to be a related problem in another adjacent field. Thus in dealing with networks, Ampere's work was often mentioned in discussion. In some discussions of crosslinked networks (vulcanised rubber) the Debye-Hiickel theory of electrolyte solutions was cited; the network junction-points being likened to individual ions in solution as centres of high potential. When one commented on this facility Sam would simply say T am a woodchopper! It is true that even on a new problem he could always see where he had to go, and would use any available theoretical tool to get there. If none was suitable, he invented one! He has kept up the pace in soft solids for forty years.

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As noted previously, Sam's influence on experimentalists was also powerful. Not simply because he gave forthright support to proposals for work related to his own interests, but also because his searching mind would cause co-workers to rethink experiments to reduce their complexity or to seek new techniques capable of more direct interpretation. In the first category, e.g., my group made model networks by controlled chemical routes designed to allow a crosslink to form only between two different chains, and thus minimize the formation of closed loops along a given chain. Simple swelling measurements were then used to probe the equation of state of these model systems. At this time, Sam developed the statistics of an isolated chain crosslinked internally to itself. The dimensions of the chain, as a function of m, the number of internal crosslinks (or knots!) was predicted to be:

where (rg) is the mean square end to end dimensions of the free chain. To synthesise model compounds the polymer chains had to be crosslinked in extremely dilute solution, to minimise overlap between chains and hence interchain crosslinking. At these low concentrations conventional chemical crosslinking reactions were ineffective and novel crosslinking chemistry had to be introduced. Even so, the timescales for reaction were protracted. Nevertheless, samples were produced by David Walsh in which the molar mass of the 'knotted' chains was substantially the same as the uncrosslinked material. Using solution viscosity measurements and small angle x-ray scattering we were able to demonstrate that the theoretical law was broadly obeyed. The satisfaction derived from these exacting but essentially simple experiments is enhanced by remembering that the Science Research Council rejected the original proposal to do the work as being 'too trivial! Sam was very quick to realise the potential of new techniques, too. As a result, he and de Gennes had an important role in supporting experimentalists promoting neutron scattering studies of polymers. At the time I began to collaborate with him in 1963, I was seeking a technique in molecular spectroscopy to complement our infrared and Raman studies of the torsional vibrations of side- groups such as — CHs in polymer chains or in simpler model compounds. A technique in which the activity of the normal modes was not governed by optical selection rules, so that we could observe the optically inactive torsional modes of symmetrical tops. Since the side- groups usually contained H, it was clear that a Raman-type experiment using a monochromatic neutron beam might detect the torsional frequencies as components of the inelastic scattered neutrons. I was hopeful, too, that the quasi-elastic broadening of the elastic peak might give information about the diffusive motions of the polymer chains. It happened that, quite independently, Peter Egglestaff was converting a neutron beam of the Dido reactor at Harwell for a similar group of experiments. By the end of 1996 a small band led by Bill Mitchell, including John White, myself and Peter Brier, made common cause to help Egglestaff develop a 'time of flight' apparatus for neutron inelastic incoherent scattering. Even with this low-flux instrument, we

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observed torsional frequencies in model compounds and polymers. Furthermore, the results were in accord with low-temperature loss peaks observed in our mechanical, dielectric and N.M.R. studies. Sam was not very interested in this work, though he helped enormously to move the project along by his inputs to the planning meetings at Harwell. Sam was more excited when we were able to demonstrate qualitatively the existence of quasi-elastic broadening of the elastic peak in scattering from a polymer melt, which disappeared when the main chain motion was frozen out in the glass. His growing interest in the dynamics of polymer chains in melts and in networks coincided with the UK neutron scattering community becoming a partner in the Institut Laue-Langevin and thus having access to superior time-of-flight instruments and a high-resolution back-scattering machine. Theory based on a 'bead and spring' model, introduced by Rouse, to represent the influence of interconnectivity on the motion of the polymer chain and using frictional drag on the beads to describe the effect of the surrounding molecules, predicted a scattering law in which the broadening is proportional to the fourth power of the wave vector. Zimm incorporated hydrodynamic interactions into the model to take into account the presence of solvent molecules in polymer solutions. This model predicted a third-power scattering law. Although there are complications arising from the influence at higher frequencies of the density of states spectrum from the propagating acoustical modes in the polymer melt, we were able to demonstrate that the fourth-power law was the best fit to the data. In a separate study of polytetrahydrofurane in solution in carbon dioxide the results supported the third-power law. No doubt de Gennes and Edwards et al. were delighted! Colloid science and food science

Sam's excursion into colloid science was stimulated by an invitation to join a Science Research Council Chemistry Committee panel in 1970 to review 'the present status of the science of colloidal dispersions in academy and industry! Following World War II, UK research in the field had become diffuse and was losing momentum. As chairman, it seemed to me a good idea 'to invite a theorist with general experience in condensed matter physics to comment on the current situation in colloid theory and as an outsider, give a new perspective on the subject and make some guesses as to the way it will develop! He joined a group of six physical scientists and a biophysicist, and duly obliged! It was an ideal framework within which to engage his eclectic powers. We reviewed the history of research into colloid dispersions and assessed the current state of knowledge and likely developments. Sam was in his element, absorbing information, asking questions and suggesting new leads. The panel decided that the final report, issued in January 1972, should include a special section written by himself on 'Theoretical Problems in Colloid Science! It began: 'As in all theoretical studies of real world situations two stages must be discerned. Firstly the 'pure' state in which experiments are designed to take place in artificially simple situations so that theories in the lowest number of unknowns can be

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developed and rigorously tested. Then one reaches the "applied state" in which, now confident of one's theories, one may either solve problems of complex geometry and composition etc. or alternatively investigate composition etc. from the observed results. Although these remarks are basic and trivial they are crucial for complex sciences like polymer or colloid science since theories cannot emerge until "simple" experiments are performed and the great difficulty is to perform simple experiments—for simple here means simple specification! Probably because of his familiarity with polymerized materials he tackled the problem of stabilisation of colloidal suspensions by adsorbed polymers. Soon he was proposing experiments of simple specification! In the aftermath of the acceptance by the Science Research Council of the panel's recommendation that a few selected research centres be established, Sam formed a small group of theorists in the Cavendish Laboratory which forged links with experimental colloid teams in other universities; notably with the group in the Chemistry Department at Bristol. These links were maintained for more than two decades and his interest in colloid science was to have important consequences in industry. After succeeding Sam at the Science Research Council, in 1981 I moved to Unilever to become Research and Engineering Director. One of my first tasks was to reorganise the management of the corporate research and engineering fund, which supported the company's basic research programme. Responsibility was taken from the four heads of the research laboratories and vested in four scientists—one from each laboratory. This group allocated the fund (some £30 million per annum) to proposals emanating from individual research groups which were deemed to be timely and promising for the future of Unilever technology. Being mainly a soaps, detergents, cosmetics and food company, colloid science loomed large in all business areas. To my great delight they proposed that Sam should join them as the fifth member. Thus Sam's horizons widened into biological science and food technology. Sam's modus operandi remained much the same in this new environment. The corporate research programme flourished. After five years, the chief executive committee increased it's fund as a proportion of the total research and development budget. Even the patent agents joined in the fun! Once a year the Research Director would dine with this group of five in Sam's rooms at Cambridge to survey progress, agree on a programme and (much later in the evening) a budget, which shortly afterwards would be presented for approval to the chief executives. Essentially, the programme focused on the science and technology of viscous liquids, powders and soft solids. At Cambridge the emphasis in the theoretical Polymer and Colloids Group moved towards colloids and, following the Unilever experience, Athene Donald was encouraged to set up a food physics experimental unit which has become the leading group in the UK. Rather than summarise Sam's output in the literature in this period, there is another aspect of Sam's contribution to the physics of soft matter worthy of note. As the decade of the 1990s approached, some of Sam's students left Cambridge to form their own theoretical groups in other universities. These include Mike Gates (Edinburgh, colloid science), Robin Ball

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(Warwick, condensed matter), Tom McLeish (Leeds, polymer fluids) and, still in Cambridge, Mark Warner (liquid crystal polymers). In the 1980s, for good measure, Sam found time to be Chief Scientist to the Department of Energy on a part time basis. He was, for some ten years, also a non-executive director of Lucas pic, a company that manufactured high technology components and systems for the automobile industry. Thus he held major positions, simultaneously, in Academy, Industry and Government! In 1990, remembering the aims of the original colloid science panel, we decided to try to launch a national research activity in colloid science at the industry/academe interface. Sam was the leading protagonist. After much cajoling, the Department of Trade and Industry launched in 1992 a £7.1m programme of 'precompetitive' research in Colloid Technology. Half of the money was provided by ICI (which subsequently spun off Zeneca), Unilever and Schlumberger in cash or kind. Sam chaired the steering committee. The Cavendish Laboratory, along with Bristol, Imperial College and Edinburgh, collaborated with the industrial groups in a structured programme to study the behaviour of concentrated colloidal dispersions encountered in foods, Pharmaceuticals, cosmetics, paints, detergents and advanced oil recovery. Over three hundred papers were published and results were presented at two open meetings and a major conference sponsored by the Faraday Division of the Royal Society of Chemistry in 1997. The proceedings of this conference have been recorded in Modern Aspects of Colloid Dispersions (R. H. Ottewill and A. R. Rennie, eds., Kluwer Academic Publishers, 1998). The impact on industry has been substantial. For example, this project made possible the rapid rollout of new food emulsion products, and developments in the science of shear thickening led to new formations of processed foods and cosmetic products. ICI gained advantage in rubber-toughening of thermoplastics and the formulation of surface coatings. For Schlumberger, advances in the understanding of colloidflocculated structures and new structural characterization techniques stimulated the development of a new generation of water-based drilling fluids now used in oil-well construction. Zeneca benefited from improved knowledge of the rheology of concentrated particle-plus-soluble-polymer systems in the design of protective coatings and formulation of suspensions. The programme demonstrated that these complex systems yield to investigations carried out by a strategic alliance between industrial and academic research and share a common science in the colloid physics of soft matter. With the current interest of the UK Government focused on the wealthcreation role of Universities, on reflection one observes that the Department of Trade and Industry Colloid Technology programme identified themes of ongoing importance to the manufacturing industry. Furthermore, the proposal was put to the Department of Trade and Industry 'not as a framework of linked research but as a cohesive cluster of tightly targeted objectives with detailed analysis of research strategy involving industrialists and academics! Because generic targets were defined at the outset, neither academic nor industrial research was constrained; thus there was a substantial amount of original research of lasting

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value in the output. Collaborative research in colloid science and technology is now continued under the aegis of the IMPACT Faraday Partnership and many of the key players in the Colloid Technology programme are involved. Nevertheless, the approach to collaborative research used by Sam's steering committee remains an exceedingly effective role-model for some of the component programmes. Sam Edwards has left his mark on Physics, Academe, Industry and Government. The 21st

Century

In the last decades of the twentieth century Sam's interest moved towards the properties of granular matter. An interest strengthened, one hopes, by discussions on powder technology in his activities at Unilever. By 1982 he had already introduced the modern theory of deposition of powders which, true to the established pattern, led to the now classic and fundamental extension of Kardar, Parisi and Zhang. Further theoretical work has shown that the PeierlsBoltzmann equation gives Kardar-Parisi-Zhang indices in good agreement with experiment. Certainly he has been stimulated more recently by the elegant work of Nowak, Knight, Povininelli, Jaeger and Nagel in Chicago, which showed that in a newly deposited powder external vibrations lead to a slow, essentially logarithmic approach of the packing density to a final, steady-state value. He has, once again, formulated an entropy-driven approach to the themodynamics of these systems in which 'compactivity' becomes the analogue of temperature. This, in turn, led to the derivation of the so-called 'missing equations of state' for granular stress, which previously were the subject only of speculation. And so in retirement the show goes on! It evolves from a steady pattern of a mixture of established and innovative mathematical techniques applied always to practical systems for which Sam has sought to understand the current state of play of experiment and application in dialogue with experimental scientists. Inevitably it has produced excitement, stimulation and deeper understanding of theoretical and practical importance to scientists in physics, chemistry and biology in academe and industry. Surely a model to be emulated by aspiring theoretical physicists. In the Book of Proverbs, chapter 29, verse 18, it is written:'Where there is no vision, the people perish! Sam has certainly kept us on the 'qui vive! Floreat Condensed Matter Physics.

EDITOR'S ACKNOWLEDGEMENTS Our first and foremost thanks are to Sam Edwards for his inspired leadership of the scientific community and for his crucial role in shaping our scientific lives. We thank the many eminent scientists who have contributed to this book. We express our gratitude to the staff of Oxford University Press, and especially to Senior Editor Sonke Adlung for his enthusiastic support of this project. We also thank Vicky Henderson of the Cavendish Laboratory for providing technical assistance, and Robin Marshall, F.R.S., of the Department of Physics and Astronomy at The University of Manchester for providing the photograph at the front of the book. We express our gratitude to Masao Doi and Thomas A. Witten for advice and encouragement. PMG also thanks for their hospitality the Department of Physics at the University of Colorado at Boulder and the Kavli Institute for Theoretical Physics at the University of California at Santa Barbara where some of the work resulting in this book was undertaken. His work was supported in part by the U.S. National Science Foundation through grants DMR99-75187, DMR02-05858 and PH9907949, and the U.S. Department of Energy, Division of Materials Sciences under Award No. DEFG02-91ER45439, through the Frederick Seitz Materials Research Laboratory at the University of Illinois at Urbana-Champaign. NG also thanks the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge, U.K., for its hospitality during the preparation of this book. His work was supported in part by the U.S. National Science Foundation through grants EAR-0221743 and DMR99-70690, and the U.S. National Aeronautics and Space Administration through grants NASA NAG8-1657 and NAG8-1760. DS thanks the University of Oxford for its support and provision of facilities, and his family for accepting the calls made on its time. DS also thanks for hospitality during the preparation of this book the Institute for Advanced Study at Princeton, the Ecole Normale Superieure in Paris, the Kavli Institute for Theoretical Physics at the University of California at Santa Barbara, the Center for Nonlinear Studies at Los Alamos National Laboratory and the Santa Fe Institute. His work has been suppported in part during the period of the book's preparation by the UK Engineering and Physical Sciences Research Council under grants GR/R83712/01, GR/S17208/01 and GR/S56870/01, the the U.S. National Science Foundation under grant PHYS99-07949, the European Science Foundation under programme SPHINX, the U.S. Department of Energy under contract W-7405-ENG-36, the European Commission under programme STIPCO and the Bell and von Neumann funds at the Institute for Advanced Study at Princeton.

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INDEX A. D. Dinsmore's Laboratory, Amherst 292 aggregates 392-3 amino acids 216 Amorphous Magnetism Conference, Detroit 175 amorphous materials statistical mechanics of 239 amorphous polymers 430 amorphous solid state 214—5, 221, 227, 233-4, 289 equilibrium statistical mechanics of 232 macroscopic character 293—5 microscopic character 291—3 numerical simulation 229 amorphous solidification transition 290 amorphous solids 239 Anderson localization 23, 26, 27 Anderson, Philip 24, 175-6, 179 anisotropic packings 409 anomalous dimension 127 anomalous scaling 74—8, 79 ballistic deposition 348 bare knot 160 bead and spring model 433 Bell, John 104 Berezinskii, V. L. 24 Bethe lattice 177 Bingham law 424 blob model 330-5, 341-2 block copolymer 134—9 melt 132-7 Bogoliubov transformation 103 Boltzmann equation 119, 121, 378 Boltzmann factor 89-90, 112-3, 134 Boltzmann transport equation 21 Boltzmann's formula 240 Borel resummation techniques 67 Born approximation 16—7 Borromean rings 147—8, 162, 264 boson fields 106 Bragg-Williams theory 173 Brazil nut effect 380 brittle fracture 421-2

brittle-ductile puzzle 420-1, 422 Brownian motion 37, 276, 330, 335-6 Brownian noise 325 Brownian particles 335, 341 Brownian path 145, 154, 155 Burgers equation 349 cage model 302, 303 Cahn—Hilliard equation 136 cavity method 177, 226 Cayley tree 177 central stress minimum 397, 398, 409 chain screening 104 chaperon molecules 216 Chern— Simons form 160—1 Chern— Simons gauge theory 162 closure approximations 380—3, 384 Cole— Hopf transformation 352—4 complex systems 218 conductivity tensor formula 11—2 configurational entropy 221, 227, 375-6, 383, 386 conformation space formalism 132 constraint release (CR) 323 contour integration 20 contour length fluctuations (CLF) 322 convection currents 379—80 convective constraint release (CCR) 324 Coulomb potential 120 coupled replicas 286—7 crystalline solids 239 Curie—Weiss theory 166, 168 Deam and Edwards crosslink distribution 284—6 Deam and Edwards model 280, 287-8, 295 Debye— Hiickle radius 117 Debye— Hiickle theory 431 deformation 304-6, 313-4, 420-1 deformation tensors 311 delta-function approximation 103 delta-function constraints 278, 288 diblock copolymers 136 diffusion bias 351 dilute solution theory of polymers 318

440

dip See central stress minimum direct interaction approximation

(DIA) 70, 71, 78-9 directed path in random media (DPRM) 352, 354, 355 directed polymers 354—5, 357—9 discrete hopping model 308 discrete scalar model 404—6 DNA sequencing 359 Doi-Edwards theory 321 Domb, C. 87 Donsker's invariance theorem 141 double gyroid structure 137—8, 139 double reptation 324 Drude conductivity 23, 25 dynamical friction 40, 42 Dyson phenomenology 23—4 Dyson, Freeman 23, 25

EA spin glass See Edwards—Anderson spin glass eddy-damped quasi-normal Markovian (EDQNM) 70 eddy-damping covariance 69 eddy-noise convariance 69 Edwards delta-function interaction 281 Edwards ensemble 401—3 Edwards formula 100 Edwards Hamiltonian 102, 134, 430 Edwards measure 403 Edwards model of the self-avoiding walk 125-7, 131, 139 Edwards solution theory 125, 126, 128, 131, 135 Edwards, Sam 23, 66, 67, 79, 175-6, 179, 419, 420, 428, 429, 431 colloid science 433—6 granular matter 436 homogeneous isotropic turbulence 68-73 QED 99 quantum field theory 159 rubber elasticity 275, 277-8 Science Research Council 431 Edwards—Anderson mean field theory 184 Edwards—Anderson model 181, 184, 187, 192 Edwards—Anderson spin glass 181—8 Edwards—Lees boundary condition 430

Index Edwards—Vilgis slip-link tube model 280 Edwards—Wilkinson equation 344—7, 348 Efetov theory 26-7 Einstein law 87, 89, 94 electrical conductivity formal expressions 11—21 equilibrating variables 282 equilibrium susceptibility 202 Eulerian time-correlations 72 EW equation See Edwards—Wilkinson equation exactly soluble model 181 excluded volume 100-1, 102, 103, 109-10, 128, 134 rubber 256-63, 271

EC susceptibility See Field-cooled susceptibility Fermi surface 20 fermion fields 106 Feynman path integral 99, 105 Feynman weights 99 Pick's equation 245, 248 field-cooled susceptibility 201 field-gradient nuclear magnetic resonance (FGNMR) 323 fixed principle axis model 409 flat measure hypothesis 385—6 flexible chain polymers 319 Flory formula 100 Flory theory 248-9, 271, 277, 287 Flory—Huggins theory 140 flow curve 320 fluctuation-dissipation relation (FDR) 187, 207-9, 385 fluctuation-dissipation theorem 207, 208, 228-9, 231 Fokker-Planck equations 345, 349, 357 force chains 399-400, 409 Fourier series 149 Fourier transform 13, 48, 91, 116, 119, 130, 131, 153, 293 four dimensional 35, 49 fragile glasses 213 framed knot 160 Fredholm determinant 121, 124 free energy 168-9, 171, 239, 240, 282, 284

Index Gauss invariant 278 Gauss linking number 159, 160, 161, 162 Gaussian chain 120, 244, 322 Gaussian correlation functions 76 Gaussian distribution 38-9, 88, 167, 193 Gaussian random velocity field 73—7 Gee, G. 428, 429 generalized free energy 203 generalized phase space distribution function in 35—7 general expansion in 47—9, 49—55 Gibbs formula 239, 240, 241, 242 Gibbs magnetizations 195 Gibbs measure 204, 205 Gibbsian statistical mechanics 72—3 glass aging 376, 382, 384 effective temperatures 385 structural rearrangement 382 transition temperature 228, 230, 431 glass state 213-4, 221-2, 223-4, 376 Goldstone fluctuations 294—5 Goodyear 214, 276 Gorkov equation 26 granular matter 329, 338, 342-3, 397 arching 398, 399-400, 409 compaction 377, 398, 410-1 dynamical behaviour 340 flat distribution 378-9 inherent structure 386—7 static behaviour 38—40 stress 397, 406-10 transmission 404 Green function 12, 37, 60, 92, 96, 105, 245, 248, 250, 251, 252, 259, 265, 315, 330 Green's lemma 146 Greenwood, D. A. 11 Gumbel's law 232 Hamilton's equations 403 Hartree approximation 100 Hebb's rule 220 Hermite polynomials 43—4, 45—6, 58, 62-5, 70 higher-replica sector (HRS) 289 Hilbert space 44 homogeneous turbulence 32, 35, 37 statistical mechanics 31 Hopf's functional equation 68

441

Hopfield model 185, 221 Hopfield, John 220 Hubbard—Stratonovich transformation 140, 289 impurity diagrammatics 23—5 independent alignment approximation 313-4, 322 independent alignment model 313 integer quantum Hall effect 26 Ising model 182, 371, 387 Ising spin glass system 226 isostatic packings 400, 409—10 James and Guth model 248, 250, 277, 287 Jones polynomial of knots 161—2 Kardar—Parisi—Zhang model 73 Kardar—Parisi—Zhang equation 347-52, 356, 360-1 Kauzmann temperature 187 Klein—Gordon operator 153 knots 145-9, 150, 159-62, 432 Kob—Andersen model 385 Kolmogoroff hypothesis 55 Kondo problem 24 KPZ equation See Kardar—Parisi—Zhang equation Kraichnan model 67, 73—4, 75—7, 78, 79 Kronecker delta functions 264, 265 Kubo, Ryogo 23 Lagrangian statistical mechanics 35—6 Lamb shift diagrams 154—5 Landau free energy 139-40, 290, 292 Landau theory of phase transitions 294 Langevin equation 332—35, 345 Laplace transform 155 Lemaitre's theory 425—6 Lennard-Jones glass 382, 387 Lennard-Jones materials 425 Lennard-Jones potential 384, 385 Levi-Civita antisymmetric tensor 161 Lifshitz, Ilya 24 linear diffusion equation 346 linear viscoelasticity 319 Liouville equation 32—5, 37, 68

442

Liouville theorem 377, 403 liquid-solid transition 295—6 Lyapunov function 219 magnetic susceptibility 193, 199-202 magnetization density 194 Markov chain 90 Markov process 103, 151 Markovian equation 73 Martin-Siggia-Rose field theory 68, 77 Maxwell demon 364 mean correlation theory 166—8 mean field thermodynamic equilibrium solution 183 meson theory 153—4, 156 mesoscopic physics 25 microcanical assumption See Edwards ensemble mode coupling approximation 381 modern dynamical spin glass theory 186 molecular rheology 318 monomer-monomer interaction 103-4, 131 Monte Carlo simulation 218 Mooney-Rivlin form 279, 287 Mooney-Rivlin term 239, 269, 271 Mott, Nevill 24 multifractality 75 Nambu—Goldstone mode 132 Navier-Stokes equations 32, 48-9, 69, 70, 71, 78, 79, 349 Navier—Stokes turbulence 73, 77—9 negative temperature 72 Nelson, David 357 neural networks 220 neutron spin echo (NSE) 323 nonequilibrium nonstationary dynamics 186—7 non-Gaussian path integral 69 nonlinear MBE models 351 non-linear
Index Fade resummation techniques 67 pair interaction energy 217 paramagnetic susceptibility 172—3 particle deposition 347 Peierls, R.E. 11, 23 percolation theory 290-1, 294 perturbation theory 14-21, 99, 102 perturbative expansion 78, 132 perturbative techniques 67, 420 phantom chain 239, 244-8, 263, 271, 277, 301, 428 definition 244 phonon gas 239 physical glasses 230—1 Pitaevskii, Lev 24 plastic strain rate 423 plasticity 421 Poisson's equation 113—4, 129 polymer chains 430, 431 polymer dilute solution theory 126, 132 polymer melt 304, 433 polymer perturbation theory 100, 112, 132-3 polymer single chain 150—1 polymer solutions 101—2, 109-11, 256 intermediate density 114—7 nonzero concentration 127—31, 134 semidilute solutions 131—4 osmotic compressibility 134 very dilute density 122—4 polymer solution theory 278 polymers Feynman diagrams 99, 102 quantum mechanics 104—6 relaxation process 306—10 statistical mechanics of 87-98, 278 powder 364 density 366, 392-3 entropy 365, 368 statistical mechanics of 364—5, 367 volume 366—8 compactivity 368—72 protein conformation 215—6 pseudo-potential 111-2, 113, 116, 118, 130 QCD vacuum 79-80 QED 99, 104, 153 quantum cryptography 105

Index quenched disorder 214, 225, 226, 282-4 quenched randomness 282, 283, 284, 285, 291, 356 random combinatorial problems 219 random deposition 348 random matrix theory 25 random systems 231—2 random walks 156 renormalization group 25, 26 replica formalism 430 replica method 224-7, 430 replica symmetry breaking (RSB) 17 182-3, 225, 231 full 226, 229 one-step 226, 229, 230, 232 replicas 282-4, 355-7 Reynolds number 51, 59, 66, 79 Riemann surfaces 148 Rouse model 304, 319-20 Rouse-Tube approach 323, 324 rubber 214 elasticity 279-80, 301-2 stress formula 301 entanglements in 263—71, 278 highly cross-linked 270 lightly cross-linked 270 mathematical models 243—56 statistical mechanics of 239—43 saddle-point method 353 scaling law 131—2 scaling theory of localization 25 Schrodinger equation 12—3 Schwinger, J. 99 Schwinger—Dyson equations 381 Schwoebel barriers 351 screening length 101—2, 430 self-afflne fractals 344, 353 self-affine roughness 346, 353 self-consistent field approximation (SCF) 89-91, 100, 101, 103, 104 self-linking number 160, 161 semiclassical equations 26, 27 semidilute solution theory 133, 140 semi-microscopic theory 275, 290 shear bands 379-80 shear-transformation zones (STZ's) 422-4, 425

443

Sherrington—Kirkpatrick model 182, 183, 184, 186, 197, 198, 217, 232 simulated annealing method 177, 184, 218 single polymer chain 117—22 SK model See Sherrington—Kirkpatrick model slip-link network model 302—4 small-angle neutron scattering 319 spin glasses 165-74, 179-180, 192, 196, 205, 214-5, 216, 217, 227-8, 232-3 numerical simulation 229 thermodynamic functions 168—71 solid-on-solid approximation 354 stochastic Langevin dynamics 69 stochastic stability 204-5, 209 Stokes' theorem 146, 149, 157, 158 stress relaxation 311—2 transmission of 392, 394-6, 421 one-dimensional 394, 395 two-dimensional 395 three-dimensional 395 structural glasses 212-4, 224, 226, 228-9, 387, 388 numerical simulation 229 surface diffusion 350 surface fluctuation 329 Langevin equation 335—42 numerical simulation 346—7 surface relaxation 347 symmetry-breaking systems 205 TAP 177 Tool, A. Q. glass state 376 topological constraints 243, 279 topological invariants 278 trap model 387 trivial knot 159 tube dilation 325 tube model 279, 302, 303, 313, 314-6 turbulence theory 66 turbulent distribution function 39—43 turbulent viscosity 42, 43 two body correlation function 245 two-parameter theory See polymer dilute solution theory

ultrametric space 218 ultrametricity 183, 188, 196, 197 uniform density assumption 250—6

444

Viana-Bray spin glass model 184 velocity correlation function time correlation 55—6, 71—2 Vogel—Fulcher law for granular media 411 vulcanization 214, 276—7 transition 288-91, 295 vulcanized rubbers 428 vulcanized macromolecular systems statistical mechanics of 280—7 WKBJ approximation 91 weak ergodicity breaking 228 weak localization 25 Wegner, Franz 26 Wegner-Efetov field theory 27 white-noise passive scalar model 73—6

Index Wiener integral 112, 151 Wiener measure 102, 287 Wiener process 126 Wiener—Khintchine theorem 336 Wigner, Eugene 23 Wigner—Dyson statistics 24, 26 Wilson loop 161, 162 Wilson, Kenneth 24

zero modes 76, 77, 78 zero-field-cooled susceptibility 201 zero-frequency viscosity 319 zero-mode approximation 26 zero-replica sector (ORS) 289 ZFC susceptibility See zero-field-cooled susceptibility