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^kr[F ~ ]. This, and the fact that the Pfc's satisfy the other properties of the projectors Zfc, leads to the identification Pfc = Zfc. We have thus another closed expression for the eigenprojectors of a Bell matrix: (Zfc[B])nr = Mnk[F} Bfc r [F<- 1> ] .
(15.64)
It follows then that N
N
0 * (Pk)nl = £ > ? *nk[F] Bfc 1 [F < - 1 > ]
ff<<> = BU9} = £
fc fc AT
JV
= 5 > n f c [ F ] B f c x ^ - ^ ^ u ) ] = Y.BnktfiF]
Bfc 1 [F<- 1 >(u)] .
fc fc
Consequently,
= E S T,^k[g\F]F<-l> = E S 5 > f c [F<_1> ( ^ • n>l
'
fc
n>\
'
fc
Continuous
225
iterates
The continuum i-iterate of g, g
(15.65)
appears in this way as a consequence of the Schroder equation. From g
(15.66)
and Rkn = Zfc[B]„i = Bnfc[F] Fk A first trivial check is given by £ f c > 1 Rk{x) = F<~1>{F{x)) = x. Another simple check involves the Stirling numbers. Consider as given the function g[x) = ln[l + a (ex - 1)]
(15.67)
and consider the Schroder equation e9^
- 1 = a (ex - 1) .
The solution is ex-l.
F(x) =
Well, F < " 1 > ( a ; ) = ln(l + a:), so that F f c < - 1 > = s[l) (see Eq. (13.37)). As 1
°°
n
><«»' = E^i". we find Rkn = 4 X ) ^ f c ) = (-)k~Hk oo
- 1)! B nfc {l} ; „
n=k
The continuous iterate is then gKt>(x)
= ln[l + a< (e x - 1)] .
(15.68)
Equations (15.59) and (15.60) can be formally checked. For a < 1, for example: in a~ng
226
Projectors and
iterates
The same happens, when a > 1, for the other expression, ang<
n>
(x)
=
We see from these considerations that, given any mapping / for which the inverse / < _ 1 > exists, a closed expression for the continuous iterate of g(x) = f~1[af(x)] is at hand. The Schroder equation can also be used to identify the eigenvectors of a Bell matrix. Taking matrix elements in (15.61), we obtain an eigenvalue equation, £ > n j [ 5 ] Bjr[F] = g\ B n r [F],
(15.69)
j
from which we see that the eigenvector related to the r-th eigenvalue g\ has, in its j'-th row, the element (
(15.70)
The eigenvector v^ of B, corresponding to the fc-th eigenvalue, will be a vector of components V(k)s = Bsfc[F]. Checking directly, [B[
= J2M™[9] «
= Mrk[F o g) = Brk{giF}
V(k)s = J2Mrsl9] s
B.fc[F]
= g* Mrk[F] = g* v{k)r .
Notice that F is fixed by the first eigenvector: Fs = B s l [F] = v^s. As B[g] is not normal, the vectors v (j.) will not constitute a linearly independent set. Their very special character lends to Bell matrices some of the properties of normal matrices. The above B [ F < _ 1 > ] has some analogy to the unitary diagonaUzing matrix of section 1.4. This is, however, only an analogy. If we try to use B [ F < _ 1 > ] to diagonalize B[g], we do get a diagonal matrix, but with the first eigenvalue gi everywhere in the diagonal.
Chapter 16
Gases: real and ideal
Statistical Mechanics of real gases provides some of the best applications of Bell polynomials. The whole formalism of cluster decomposition can be recast, advantageously, in their terms. Though, as a rule, the approach illustrates our frequent assertive that Bell polynomials are informationorganizers and not creators of information, the approach gives much supplementary insight. Perhaps the main point is the presentation of all the results as properties of a certain basic matrix A — we call it the "Lee-Yang matrix" — which lies somewhat behind the scene, but must exist. All physical quantities (cluster integrals, pressure, density, virial coefficients) of the theory of real gases appear as invariants of this matrix under similarity transformations, a fact which brings to light the existence of a large group of invariance, which remains also implicit in the usual presentation. Furthermore, the Bell formalism gives closed expressions for all the relations involved. And, as sometimes a better organization of known information does produce something worthy, some recursion relations turn up and a better understanding comes forth in diagrammatics. For instance, we obtain a trivial proof of the Matsubara formula, according to which the contribution of all graphs to the grand canonical partition function is equivalent to the exponential of the contributions of the sole connected graphs. It should be said that, on the other hand, these physical applications give a much more intuitive view of the polynomials themselves. In the grand canonical formalism, the set of canonical partition functions contains the same information as the set of cluster integrals. The relationships between them show in which sense the polynomials are organizers, not creators of information. Let us then retrace the well-known trivium of the subject: the microcanonical, the canonical and the grand canonical ensembles. 227
Gases: real and ideal
228
16.1
Microcanonical ensemble
The microcanonical ensemble is, in Statistical Mechanics, the ensemble describing an isolated system, for which the energy is constant (see, for example, [99]). Energy is supposed to fluctuate in so small a range that its distribution is taken as a very narrow Gaussian centered at a value E. We shall use below the extreme limit of a narrow Gaussian, a Dirac delta function. Each microphysical state, describing the detailed situation of all the constituents (their position, momentum, spin, etc.) is supposed to have the same probability. The collective, macroscopic behavior of a system of N particles enclosed in a volume V is described by the three variables N, V and E. An average is taken over all the microscopical states, weighted by the probability of each state. As these probabilities are supposed equal a priori, the average over the states become simply a summation of their number. Discrete variables are easily taken into account, as summations over the values they can attain. For the continuum variables, summations reduce essentially to an integration on configuration space, times an integration on momentum space. This means the volume of the total phase space, an integration constrained by the requirement of constant energy or, in the relativistic case, of constant energy-momentum.
16.1.1
Phase space
volumes
High-energy physics makes essential use of the microcanonical ensemble. Phase space volumes play an essential role in the calculation of crosssections. In the absence of dynamical effects, only inert phase space remains, so that its volume is also important in assessing the meaning of experimental data. We shall consequently adopt the jargon and procedures of particle physicists [100], as well as their generous and smart use of delta functions. Calculations of relativistic cross-sections for processes given by some scattering matrix T lead to integrals of the type
IN(P) = J
n
d3Pi
N
64(P-J2Pi)T,
(16.1)
Li=l
where P — (E,p) is the total 4-momentum and Pi = (ej,pj) are the indi-
Microcanonical
ensemble
229
vidual 4-momenta of the N particles involved. What is usually called phase space volume for a system of N particles is, in high-energy physics jargon, the expression
ntf\p,v) = A J
n
d3Xid?pi h3
N
* 4 (P-£P<) t=i
N
-»'\h*j
N
3
j Y[d
4
Pi « 5 ( P - £ P » ) -
(16-2)
i=\
These expressions are clearly classical, that is, they suppose a Boltzmann statistics for the particles, though indistinguishability finds a partial allowance in the Gibbs factor (1/iV!). This is indicated by the upper index (B). Statistics is of course implicit in the normalization of the scattering matrix. We can, however, bring the normalization down to the phase space calculations. The use of Bell polynomials will allow (1) to write the quantum phase space volumes in terms of the above classical, Boltzmann expression; (2) to find a recursion relation [lOl] for the iV-particle phase space i?jv volume in terms of integrals of phase spaces Rk, with fc = l , 2 , 3 , . . . , (7V-1). A much used trick in Statistical Mechanics comes from noticing that, when going from the classical to the quantum description, the whole integration
AH /
n
d3Xjd3pi
h3
is replaced by taking the trace. In the present case, JV
5\P - ^Pi)
N
N
= S(E - £ > ) < 5 3 ( P - £
N
P i
) = <53(P - X > ) t r ( £ - H0),
j=l
i=l
with H0 the free Hamiltonian operator. Then, RN(E,-p,V)
=
tv%8(E-Ho),
(16.3)
230
Gases: real and ideal
where the tags P and N are reminders that the trace of the formal operator is to be taken for N particles at fixed total momentum P = Yli=i P»- When the whole system is at rest, N
RN(E,0,V) =-fa J
n
d3Xid3pi
N
6(E-H0)63C£pi).
(16-4)
In the laboratory system, we retain only
RN(E,V) = ^J
N
n
d3Xid3Pi h3
S(E - H0).
(16.5)
Now, it is possible to obtain the phase space volume for different statistics if the trace is taken with conveniently normalized state-representing kets. This points to the fact that the role of the symmetric group S/v in Statistical Mechanics is actually summed up in the ket normalization. This is much simpler than taking into detailed account the intricate real nature of the configuration space of identical particles (summarized in section 16.4). The state of a system with n free particles is represented by an eigenket \pi,P2,P3, • • • ,Pn > of the momenta. For Boltzmann statistics, the ket normalization is given by
,Pn >
= *3(p'i -Pi)* 3 (P2 ~P2)63(p'3 -P3)
... S3(p'n -Pn).
(16.6)
This comes directly from the normalization < p'k\pk > = <^3(Pfe - Pit) for one-particle kets and from the fact that the total ket for a system of n free Boltzmann particles is a simple direct product of individual kets, \Pl,P2,P3, • • • ,Pn > = \P1 > \P2 >\P3> •••\Pn> , reflecting the complete independence of each particle. For bosons (upper sign) or fermions (lower sign), the ket is a sum over all permutations, \Pl,P2,P3,--' ,Pn > = ^T ^ ( ± ) a f c | p a i > \Pa2 > \Pa3 > " • ' \Pan >, (16.7) where the sign factor indicates the permutation parity. The first examples are \Pi,P2 >= 57 [\Pi > \P2 > ±\P2 > \Pi > ] ;
Microcanonical ensemble
231
|P1,P2,P3 > = 37 [|Pl > |P2 > |P3 > ± |P2 > |Pl > |P3 > ±
(16.8)
|Pl > |P3 > |P2 > ± \P3 > \P2 > |Pl >
(16.9)
+ |P2 > |P3 > |P1 > + |P3 > |P1 > |P2 > ]•
(16.10)
For fermions, Pauli's principle is automatically enforced: t h e kets vanish if, for example, p\ = p2. Let us see in detail t h e normalizations for t h e n = 2 a n d n = 3 cases, in which t h e essentials already come out. Normalization for t h e m is < p'i,P 2 |Pi,P2 > = ^
3
( P i - Pi)<53(p2 - P 2 ) ± <*3(P2 - pi)<S 3 (pi - p 2 )];
- p'2)63(p3
3
- p'3) ± 63(Pl - p'2)63{p2
3
3
3
±<5 (pi - P i ) 5 ( p 2 - P' 3 )^ (P3 - P'2) ± S (Pl - p'3)6 (p2 +S3(Pl -p2)63(p2-p3)63(p3-p>1) + 53(Pl -P'3)63(p2
- p'1)53(p3
" P'3)
3
- p'2)6 (p3 - p i ) -Pi)«53(p3 - P 2 ) ] -
Now two (long) remarks are of interest. T h e first is t h a t t h e Hamiltonian for free particles is diagonal in this m o m e n t u m basis, so t h a t we should consider only t h e diagonal cases. For n = 2, we will only have < P i , P 2 | p i , P 2 > = 5?[* 3 (pi - P i ) £ 3 ( P 2 - p 2 ) ±<5 3 (p 2 -Pi)<5 3 (pi
-Pi)](16.11)
For fermions, this is t h e determinant < P l , P 2 | P l , P 2 > = 2l
<53(px - P i ) 3
S (P2
-Pi)
<S3(P1 -- P 2 ) <* 3 (P2-- P 2 )
(16.12)
For n = 3,
i[<5 3 (pi - p i ) 5 3 ( p 2 -p2)63(p3-p3)±63(p1-p2)63(p2 -pi)S3(p3-p3) 3 3 3 3 3 3 ±5 (px - p i ) J ( p 2 - p3)6 (p3 - P2) ± £ (pi - Ps)<^ (p2 - p 2 )^ (P3 - P i ) +<53(pi -p 2 )<5 3 (p 2 -p3)S3{p3 - P i ) + <53(pi -p 3 )<5 3 (p 2 - P i ) 6 3 ( p 3 - P 2 ) ] For fermions, this is again a determinant:
3!
<53(Pi - P i ) <*3(P2 - P i ) <*3(P3 - P i )
<*3(Pi - P2) S3{p2 - P2) <^3(P3 - P 2 )
S3(Pl - p 3 ) S3(p2 - p 3 ) S3(P3 - p 3 ) (16.13)
232
Gases: real and ideal
Notice that for bosons the expressions are the same, but always with the positive sign. If we take a determinant and write for it the Laplace expansion as above, but taking always the positive sign, what we have is another object, called a permanent. Thus, we have determinants for fermions and permanents for bosons. A widely used trick reveals something about the real meaning of the deltas in the above purely formal manipulations. It goes as follows: we recall the definition of the Dirac delta, < * 3 ( P i - P 2 ) = /'d 3 xe i < P l - p a >- x . The equality p i = p 2 would give, of course, an infinite value for a true delta. But we notice the / d3x and say that, actually, we only integrate over the volume of the system, so that
^ 3 (Pi-Pi) = jd3x = V. And then we use this systematically. For example, < Pi,P2\Pi,P2 > = a [V2 ± VS3(Pl - p a )] and R2(E,P,
V) = t r P 5 ( £ - Ho) = / d 3 P l d 3 p 2 < Pl,p2\S{E = / d3pid3p25(E
= £ [ d3Pld3p25(E
-ex-
-ex-
- H0)\pi,p2
>
e2) < Px,P2\pi,p2 >
e2) [V2 ± V53(Pl - P2)] • (16.14)
In consequence the deltas, used in this way, are actually symbols of certain objects which tend to real Dirac deltas at very large (mathematically, infinite) values of V. Experiments using such results must prepare conditions in which the total volume is very large with respect to the other ("microscopic") scales involved. Here ends the first remark. The second remark: all these expressions will be integrated over all the momenta. They will appear only in the integrand of / n » = i ^3P* anc ^ w i u consequently be symmetrized over the pj's. This means that, in the above products of S's, it is not important which particle is which. What remains is the way in which the particles associate. There are terms in which each
Microcanonical
233
ensemble
particle couples t o itself; t e r m s in which a first particle couples with a second a n d t h a t second couples t o t h e first, forming a 2-particle cycle; and, in t h e TV = 3 case above, there are also 3-particle cycles: a first particle couples t o a second, t h a t one couples t o a third which t h e n couples to t h e first. W h a t we want t o say is t h a t only the cycle structure imports, only distinct cycle structures will give different results after integration. Under the integration sign, t h e last expression for < Pi,P2,p3\pi,P2,P3 > gives the same result as
- pi)53(p2
> = 5f[<53(Pl -Pl)$3(P2
- p3)S3(p3
-P2)S3(P3~P3)
- p2) + 2<53(pi - p2)S3(p2
- Ps)S3(p3 - Pi)}-
W i t h t h e same trick previously described concerning t h e S with vanishing argument,
3V253(p3
> -
P2)
+ 263(Pl - p2)53(p2
- P3)S3(p3
- Pi)] • (16.15)
We can represent the above results (always with the proviso t h a t the expressions only hold under the multiple integration symbol) in t h e compact notations
< Pl,P2,P3\Pl,P2,P3
(16.16)
3 > = h 8 ±35152+253
,
(16.17)
etc. T h e symbol 5k represents the product of k deltas forming a cycle. It has t h e general form 4 = <53(Pi+l - Pi+2)S3(pi+2
- Pi+3) • • • S3{pi+k
-Pi+l)-
For example, S3 = 63(pi+i - pi+2) S3(pi+2 -pi+3) S3(pi+3 -pi+i). These expressions are cycle indicator polynomials which, as we shall see, are Bell polynomials of a certain type. This spurs us into doing a short parenthesis on p e r m u t a t i o n s and cycles.
234
Gases: real and ideal
16.1.1.1
Cycle indicator polynomials
Let us recall some well-known facts about permutations [102]. We have seen their matrix representation in section 3.1. A general permutation P of particles with labels xi,x2,. •. , X N - I , X N is indicated by X\
X2
...
Zw-l
xpi
xP2
...
xPN_x
X
N
XpN
and can always be decomposed into the product of disjoint cycles, those particular "closed" permutations of type p
i
x
x
2
•••
Xr_ \
Xr
•*'2
"^3
...
thy
"1
l
This is a cycle of length r, or an r-cycle. We may attribute a variable tr to each r-cycle and indicate the cycle structure of a permutation by the monomial t™1^2 • • • Cr> meaning that there is a number ni of 1-cycles, a number n2 of 2-cycles, etc. Permutations of the same cycle type, that is, with the same set v = {rij}, go into each other under the action of any element of the symmetric group S^: they constitute conjugate classes. To all permutations of a fixed class will be attributed the same monomial above. In this sense, such monomials are invariants of SN- The total number of permutations with such a fixed cycle configuration (that is, the number of members of the corresponding conjugate class) is
Cv =
N]
•
(16.18)
The iV-variable generating function for these numbers is called the cycle indicator polynomial [73] for SV, written
cN(t1,t2,...,tN)
= TiTi"rtN
'.
* ? ' % ' • ••«?••,
( 16 -!9)
where the double-prime in the second summation is a memento of two conditions: the summation is over all values of the rij's satisfying the conditions J2i=i nj — m a n d J2j=i Jni = N. These are just the same conditions (13.7)
Microcanonical
235
ensemble
and (13.8), and simple comparison leads immediately to N
CN{h,t2,...,tN)=
£
Bjvm{(j - 1)! *,-}•
(16.20)
C o m m e n t 16.1.1 We can, as a test, calculate the number p(N,m; tations of type {ni} and whose number of orbits is fixed, £2 ni — rn:
{n*}) of permu-
m=l
ii
p(N,m;{ni})=Y:
M N
'
It is obtained by putting all tj = 1 in Bjv m - Recalling the relationship (13.35) between Bell polynomials and Stirling numbers, p(N, m; {m}) = MNm{(j
- 1)!} = \s^\
.
(16.21)
Thus, the number of JV-permutations with m orbits is the modulus of the first-kind Stirling number (Cf. Comment 13.1.8).
In view of (16.20), the recursion relation (13.89) leads to C j V + l ( * l , * 2 , - •• )*jV+l) = 2_j /yy _ u\\ * f c + 1 ^N-kihih, fc=0 ^
• •• ><JV-fc)i
'•
(16.22) with Co = 1. C o m m e n t 16.1.2 P r e s e n t a t i o n There is another means of introducing the symmetric group. Introduce the elementary transpositions (alluded to in Comment 3.1.1)
( (
Xl X2
X2 Xi
X3 X3
X4 X4
... ...
XjV-2 X;v_2
Xl Xl
X2 X3
X3 X2
X4 X4
... ...
XjV-2 X/v-2
x
X
X
X
(
Xl
X2
X3
X4
...
x
Xjv-l
Xl
X2
X3
X4
...
X/v-2
N-2
x
X
x
X
N-\ N-\ N-\ N-1
X
N
N N N N
X;v x
N-l
\ /
;
\ I J
;
\ J
Notice that s\ exchanges the first and the second particles, «2 exchanges the second and the third, and so on up to SJV-I> which exchanges the two last particles. Every permutation can be obtained as a product of such elementary transpositions, which are for that reason called generators of the symmetric group. They satisfy the product
236
Gases: real and ideal
rules s;Sj-|.iSj = Si+\SiSi+\;
(16.23)
SiSj = SjSi for \i - j \ > 2;
(16.24)
SiSi = I .
(16.25)
The last rule has a very clear meaning: exchanging twice two particles of a fixed pair is the same as doing nothing, that is, applying the identity transformation. Waiving this condition would lead to quite a distinct group, the braid group we have found in Quantum Phase Space (Chapter 10). The Statistics of particles whose exchanges are governed by such an infinite group is significantly different from the Statistics (bosonic and fermionic) of particles whose exchanges are ruled by the symmetric group (see section 16.4). The rules above characterize completely the JV-th symmetric group. Any group whose generators satisfy them is isomorphic to SAT. Discrete groups can be introduced in such a way, by • defining the generators, elementary members of the group such that every other member is a product of their powers in some order; • imposing constraints on some of their products. This is called a presentation, presentation".
and the group is said to have been "introduced through a
Going back to the normalization problem: in the general case, the normalization of the iV-particle state is given by < Pi,P2,
• • • ,PN\PI,P2,
••• ,PN
>
N
= & C H ( ± r ^ - } = jv7 £
BNm(±y-1{(j
- 1)! 5j} . (16.26)
Relation (16.22), with the convention (0 particles|0 particles) = l, leads then to a recursion relation for the normalization factors: (pi,P2,---
,PN\PI,P2,---
,PN)
N-l
= N E W "
1
" ^ -
,Pr) - (16.27)
r=0
We insist that these expressions are not to be taken at their face value: they only hold under the multiple integration sign. Notice also that, once it is understood that < pi|pi > = <53(pi - p i ) = S3(0) = V, what we have here is a recursion relation: normalization of any n-particle state can
Microcanonical
237
ensemble
be obtained from the 1-particle normalization, in particular (16.16) and (16.17). For example, (16.16) is (pi,P2\pi,P2) = \ p i ( P i b i ) ± S2] = \ [V2 ± V83(Pl - p a )] . The phase space volume, or the microcanonical partition function, is then •
RN(E
.P.^) = M /
N
N
Y[d3Pi
N
*(£;-5>)*(p-;£>) i=\ i=i
N
x J2 (±)N~m ®Nm{(j ~ 1)! & .
(16.28)
m=l
The integration variables appear in the arguments gj = (j — 1)! Sj of the Bell polynomials. A detailed consideration of the low-order cases will convince the reader that it is possible to introduce carefully the integrations "inside" the arguments. The result is the Prakash-Sudarshan formula, an expression for the phase space volume of a quantum system of N particles, bosons (upper sign) or fermions (lower sign), in terms of the Boltzmann phase space volumes. In our notation, it reads N
/)
"
< ( ^ ) = ir£(±)iV-mv£ m=\
N
^E(±yv-mE' f
N
& E(*)
Cv
R{*\P,V)
(16.29)
llj=i3
TV!
n , = i ^ ! j 4n,-
RgHP,V)
(16.30)
N
JV m
"
£
R£HP,V) MNm{ j\j - 1)!} . (16.31)
m=l
Now, using tj = (±y xSj in (16.22) leads to a recursion formula for the quantum phase space volumes. With the convention Ro(P,V) = S(E)S3(p)
238
Gases: real and ideal
the formula reads N
RN(P, V) = jj£
r
J^i)*"
1
fc=i
16.1.2
Towards the canonical
/
d3
lRN-k(P
J
- k, V) .
(16.32)
formalism
The canonical partition function of an ideal quantum gas of N particles, to be examined in the next section, can be obtained as a Laplace transform of RN(E, P = 0, V) in the energy: QN((3,
V) = J dE e-pERN{E,
V) ,
(16.33)
where /? = 1/kT is the inverse temperature in natural units, with k the Boltzmann constant. Expression (16.32) leads then to JV-l
QN(J3,
V) = ±J2
W * ^ i P + 1)0, V] QN-i-k(P, V).
(16.34)
fe=0
This gives, in the non-relativistic limit, some expressions we shall meet below, such as
QN((3,*0 = & & £
{ (n
*L,2
fc=o ^
'
QM>n
(16-35)
F
( 16 - 36 )
and N-l
QN{P,
v) = ± ^ J2(N -k)
b
*-k
QI°(P>
)>
fc=0
where [see Eq. (16.84)] b
i =
(±V' _1
{
-JJ^
( 16 - 37 )
for bosons (upper sign) and fermions (lower sign), and \=\l
H
\ m (16.38) y V 2-nmkT ' is the non-relativistic mean thermal wavelength for a particle of mass m. These are particular cases of a much more general expression, to be seen in the next section. Let us here only say that the canonical partition function
The canonical
ensemble
239
of a real non-relativistic gas of N particles contained in a d-dimensional volume V is a cycle indicator polynomial for the symmetric group 5JV, with the (combinatorially meaningless) variables tj = j bj^j giving the contribution of the j-th order cluster integral: QN((3,
V) = ±CN
{jbj^y
(16-39)
where in the general case bj is the j-th cluster integral, in principle computable from the interaction potential between the constituents [20]. For 3-dimensional ideal quantum gases, statistics is accounted for by an effective interaction for which the j-th cluster integral is (16.37). We shall meet later the canonical partition function for a Bose/Fermi gas in the 2-dimensional case [Eq. (16.122)]. Quantities describing a gas of N identical particles must be invariant under the transformations of the symmetric group SN- We are quite used to this idea. We shall later be able to exhibit a certain matrix A (to be called the "Lee-Yang matrix") to whose characteristic polynomial the grand canonical partition function is related. The virial coefficients are then related to traces of powers of A and the partial canonical partition functions to the minors of A. Ultimately, everything of physical meaning can be written in terms of traces, and is consequently invariant under transformations of the unitary group U(N), which appears then as an underlying symmetry.
16.2
The canonical ensemble The canonical ensemble represents the system as immersed in a thermal bath, which fixes the temperature.
The temperature, conse-
quently, replaces the energy as the preserved quantity. The macroscopic variables are N, V and T.
As said in the previous section, the canonical partition function for a gas without interactions can be seen as a Laplace transform of the microcanonical partition function: QN((3, P , V) = tiNe13"0 = f dEe-0EttN5(E
- H0).
Gases: real and ideal
240
We can use the momentum eigenstates and take the trace at fixed total momentum P . This would give
QN{0, P,V) = JdEe-?E RN(E, P, V). In the standard treatment of Statistical Mechanics, the physical system as a whole is supposed at rest. In that case,
QN(p, V) = QN(J3,0, V) = JdEe-PBRN(E, 0, V). In the presence of interactions, the free Hamiltonian is replaced by the total Hamiltonian H and, in the rest system, RN(E,V) =
RN(E,0,V)
= jftj
[ n i l i ^ a ] S(E-H).
(16.40)
Thus, QN(0,
V) =
f dEe-VERN(E, V) = -faj dEe-W j [uti £*£*] 5{E - H), or
QN(/3,V)
= ^J'[nit&g*: -0H
(16.41)
This is the semiclassical expression of the canonical partition function. For the general quantum case with interactions between the constituent particles, QN(P,V) = tre~PH'. In the usual non-relativistic, semiclassical approach, the Hamiltonian is N
JV-1
N
" = £2m £ ; +E E "(*-*). 2=1
i=l
j=i+l
where U{vij) = U(ri — r^) is the potential between particles i and j . Abrupt and/or divergent potential behavior are smoothed down through the introduction of the Mayer functions fa = /(r»j), which are such that e-mn,)
=
!+
/(r..).
The
canonical
ensemble
241
The only effect of the independent integrations on the momenta is to engender factors in the (cubes of) the thermal wavelength, so that QN(P,V)
=
1
N\X3N
/ [Ul^r
n^1nf=i+i[i+/(^)]-
(16.42)
Expanding the product, a succession of terms of the form ^2YlYlfijfik frs ... appears, on which the integrations are to be performed. In this way QN(P> V) is written as a sum of contributions of subsystems. There will be one term taking isolated particles into account, there will be terms taking into account all subsystems with 2 particles, terms taking into account all subsystems with 3 particles, and so on up to a term in which the TV particles are considered. There is a diagram corresponding to each kind of linkage between particles. Each particle is represented by a point, and a line is drawn linking r and s if a factor frs is present. We shall see a simple example below [see Figs.(16.1) and (16.2)]. A diagram is connected when we can go from a particle to any other by following a succession of links, and disconnected otherwise. We call a k-cluster any subset of fc particles, which can be united to each other by some succession of common indices or not (a cluster can be connected or not). The procedure is systematized with the introduction of the cluster integrals, defined as
K=
ni\^-DvJd3rid3r2---d3rn n ^ i E w * -
(i6-43)
bn accounts for contributions of the subsystems of n particles. We have consequently to separate the N particles into subsets: V\ isolated particles, 2/2 particles which constitute pairs, v^ particles forming a fc-cluster, etc. Each cluster of fc particles will contribute a factor k\X3(-k~^Vbk to the expression of QN in (16.42). As there are Vk of such sets, the set of all k-clusters will contribute a factor [fclVA 3 ^ - 1 ^]"*. How many ways are there to separate N indistinguishable particles into the subsets above ? If the particles were distinguishable, there would be N\ permutations between them. The particles inside each fc-cluster are k in number. They are indistinguishable, so that we have to divide N\ by fc! As there are Vf- of such clusters, we actually have to divide by (&!)"*. But particles can also be exchanged in block, by exchanging entire fc-clusters, which are by themselves identical. Thus, we must still divide by v^\ The
242
Gases: real and ideal
desired number is consequently N\
(16.44)
nti(fco^! C o m m e n t 16.2.1 In Quantum Field Theory, this number is well known as the number of Feynman diagrams "topologically equivalent to each other".
We have then QN(P,V)
=
N\ 3N
rJV
N\X
ITCLiW*"*!
TlLi(w\*e>-»bky*.
The sum S{Vfc} i s meant to take place over all the possible sets of numbers z/fc, that is, on all the solutions of the equation N = ^2k=1 kvy.. The number of clusters will be m = X)/t=i vk- We recognize the expressions and conditions defining Bell polynomials. We can, as seen in Comment 13.1.1, separate the summation into two parts, so as to have Y^'tv i = E m = i IC/v y C o m m e n t 16.2.2 Braving the danger of boring repetition (of what has just been said and, for example, of Comment 13.1.2), we can now reinterpret the coefficients in the detailed expression of the Bell polynomial in (13.6). First, the factor N ! / ( r j . y ! ] " J ) counts the number of ways to have N identical particles distributed in m clusters, with Vj clusters of j particles. Division by j ! means that the j particles in each of the Vj clusters are identical, so that their permutations must not be counted. The total number of distinct configurations of this kind is (j!)"-', for each j . Configurations differing by permutations of entire clusters of "size" j must also be excluded. There are i/j\ of such for each j , wherefrom the further division by the factor FT. VJ\
Finally, the general expression for the canonical partition function in terms of the cluster integrals is (with v = p-) a complete Bell polynomial: N
^—,(vhr(vb2r...(vbNrN
(16.45)
m=l{1/i} l l j = l " j N
m Yl
N M
Nm{jlvbj}
4 ^ B ,
m=\
m=l
= iV!Y" $ ;
t
v^Tbktk
(16.46)
L fe=i
T,?=1bkzk .
(16.47)
As the cluster integrals appear always multiplied by v, we shall sometimes use the simplifying notation *i
vbj
V A3
K
The canonical
ensemble
243
And, once Bell polynomials have been identified, we can use all their machinery to get some results. For example, now we have not only N
QM
N
V) = j±T £ ®Nm{jl b*} = ± £ BNm [Y,?=lK **] (16-48) m=\
7n=l
but also its inverse [from Eq. (13.71)], n
K = h E(-) f c _ 1 ( A ; - !) ! B « ^ ' ! QiY
(16.49)
fc=i C o m m e n t 16.2.3 For relativistic systems, potentials lose their meaning. Interactions are, for them, described by the S-matrix. The relationship between the bk's and the S-matrix elements have been extensively studied by Dashen, Bernstein, Ma and Rajaraman (see [103]).
There is a remarkable kinship between expressions (16.48), (16.49) and the pair of equations (14.4), (14.5). This fact suggests the presence of a matrix behind the scene. It will be seen in section 16.5.2 that an N x N matrix A actually exists such that tr(Ak)
= {-)k~lk
QN(p,V)
=
b*k, and detA.
The analogy also helps to understand the meaning of "sub-partition functions" Qj in a gas with N > j particles, which will be particularly important in the grand canonical formalism. From the discussion following Eq.(14.6), Qj is the sum of all the partition functions of the subsets with j particles. The Fredholm formalism (section 16.6) will show them as the minors of a Fredholm determinant which coincides with the grand partition function. And here comes one of the best illustrations of the meaning of Bell polynomials which Physics can provide. Imagine a gas with N particles. The partition function QN counts all the ways by which the N particles can be exchanged, or are interacting with each other, or still the manners in which they can be symmetrized or antisymmetrized. The particles are grouped into subsets ("clusters") of vk particles, with the vk satisfying (13.7) and (13.8). At a given moment, there will be v\ isolated, noninteracting particles; v-i sets of particles which are interacting two-by-two; Vj, are interacting three-by-three, Vj are interacting j-by-j. Such a situation,
244
Gases: real and ideal
characterized by the set of partitions (see Comment 13.1.1) {i'k}, is called a "configuration". The number m = Ylj=i vi ls the number of clusters in that configuration. The possible cases for N = 3, Q3 = ji.{v3 + 3&i 2b2v2 + 3\b3v}, are shown in the diagrams of Figs.(16.1) and (16.2). Points joined by
:vv Fig. 16.1
Three-body clusters: left, m — 3, v\ = 3; right, m = 2, vi = 1 and 1/2 = 1.
V7VV Fig. 16.2
Three-body clusters: m = 1, 1/3 = 1.
a line represent particles in binary interaction. A set of points directly
The canonical
ensemble
245
or indirectly linked to each other constitutes a cluster. The number m appears clearly as the number of clusters in a configuration. The upperleft configuration of Fig.(16.1) has 3 "clusters" of 1 particle each. This is what V\ = 3 says. In the upper-right configurations the particles interact in pairs. There are thus 2 clusters in each case, one with 1 particle (so that ^i = l) and another with 2 particles (so that v-i = 1). Finally, in the configurations of Fig.(16.2), the 3 particles are joined by interactions: the diagrams are connected. They form only one cluster (m = 1) of 3 particles (v3 = 1). Putting what has been said above in other words, for an arbitrary total number N of particles each configuration {i/k} will be a solution of conditions (13.7) and (13.8). Figure (16.2) shows also that a closed (doubly-connected) 3-cycle is counted in the same configuration as the three connected but open diagrams with 3 particles. The latter are not really cycles, but 63 counts all of them together. Thus, though we speak of cycle decomposition, in the virial approach the cycles are actually lost in the integrations. Complete cycle decompositions are recovered in the Fredholm formalism (section 16.6). The graphs with m — 1 correspond to the case in which each particle interacts with every other, that is, to the irreducible contribution of the iV-body interaction. All the particles are interacting and the graph is completely connected. The contribution of the completely connected graphs to QN will consequently be the m = 1 term in (16.48): (c) _
Q JV
v bN.
(16.50)
When two particles are exchanged, it is as if they were interacting for everything concerning graphs. There is an effective interaction ("exchange interaction") which simulates the effect of the quantum correlations induced by symmetrization or antisymmetrization. For an ideal non-relativistic gas of bosons or fermions, with the 6/s given in (16.37), 00
N
(+')k~i
(16.51)
9 L k=\ N Nm m=l
^fiCV 'A 3
fc=l
jfeS/2
(16.52)
(16.53)
246
Gases: real and ideal
The parameter g is the number of values taken by some extra, "internal", degree of freedom (such as the spin). The relativistic case will be examined in section 16.2.1.1. The case m = N has only one configuration, that is, only one solution of the conditions (13.7) and (13.8): v\ = N, Vk^i = 0. Each particle is a cluster by itself, there is no exchange at all. The corresponding contribution is
7^ B ™ [gZZi (±)fc-1(* -1)! Qi(kP)tk] =w.l9 Qi(P))N , that is, just what would come up for a Boltzmann gas. In the opposite extreme case, m = 1, the only solution is vpi = 1, Vk^N = 0. This means that all particles together form a single cluster. It is actually much simpler to get the cluster integrals from the generating function of the QJV'S, which is the grand canonical partition function (see Comment 16.3.1). From the "recursion" formula (13.89) for the Bell polynomials with QQ — 1, we get JV-l
QN(P, V) = ^ ] T (N - p) bN.pQp.
(16.54)
p=0
Some of the first examples are:
Qi=v ; Q2 = £ [hv2 + 2b2v] Qz = i r { M 3 + 3\hb2v2 + 3\b3v};
Qi = %{hv4 + 12b2v3 + \2b\v2 + 24b3v2 + 4\b4v}. Notice that &i = 1 in all cases. We can easily check that vb2 = i { 2 ! Q 2 - Q2},
vb3 = ^{3!Q 3 - 3!QiQ 2 + 2Q\},
etc., which illustrates Eqs. (16.48) and (16.49). Also the fact that these relations are only an information rearrangement is clear: the sets {Qp} and {bj} contain the same information.
The canonical
247
ensemble
Comparing (16.46) with the identity (14.19), we find Widom's determinant form* of the canonical partition function,
QN(P,V)
v -2b2v 3b3v
N\
(-^^NbNV
1 v -2b2v
0 2 v
0 0 3
••• ••• •••
(-) N (AT-l)6 N - 1 v
0 0 0 . 0 v (16.55)
This can be seen as a solution of the recursion equation (16.54). C o m m e n t 16.2.4 The role of the symmetric group S/v, already discussed in the context of phase spaces, is also exhibited in the above cluster decomposition of the canonical partition function QN, which happens to be an invariant polynomial of SN [78]. The decomposition has been so built as to be invariant under the exchange of particles. Thus, QN(/3,V) is a multi-variate polynomial (in the variables 6„) which is invariant under the action of Sjv •
Once the partition function is given, it is of interest to introduce a thermodynamic potential, a function of QN(/3, V) from which physical quantities (internal energy, specific heats, etc.) are obtained as derivatives. For the canonical ensemble, the potential usually taken is the Helmholtz free energy F, which is given by QN(P, V) = exp[—@F], or F
= E-TS=-^
lnQN(p,V).
The internal energy is then given by E = — ^ ln<3jv(/3, V), the pressure by P = kw 16.2.1
ln
QN(P,
Distribution
V), etc. functions
In the process of calculating cluster integrals it is sometimes necessary to take integrations into the arguments of Bell polynomials, or to extract them. As anyhow no higher order integrations have ever been achieved for the simplest interaction potentials (the record is Boltzmann's calculation of the 4-th virial coefficient for a gas of hard spheres), we shall use the solvable, simple case of the exchange interaction to illustrate the procedure. We shall actually consider the distribution functions, whose non-triviality measures * An expression of this type was first found by B. Widom for the ideal Bose gas case [104].
248
Gases: real and ideal
quantum correlations between particles in a gas. Relativistic gases are more willing to exhibit such effects. We shall therefore consider their case [79]. 16.2.1.1
Relativistic gases, starting
Some preliminaries are in order. When considering a gas of relativistic particles, it is extremely convenient to change to adequate units. In particular, there are two lengths of major interest: • Given a particle of mass m, its static Compton length A c = —%
(16.56)
will be a natural unit of length. • At inverse temperature (3 = 1/kT, thermal wavelength is given by A3(/3) = 2 TT2 Pmc2
(the cube of) its relativistic
e
(" ^ )
•
(16-57)
Here K2 (x) is the modified Bessel function of second order. To ease some calculations, and to obtain the non-relativistic and the ultra-relativistic limits, it is useful to recall the asymptotic behaviors of both ^ ( x ) and
• for x »
1,
* ( x ) - \2xJ f i ) , " « - (Vi + M 8a; + -
*w(5r«~K+-) • for x << 1,
K2(x) « A
X
The canonical
249
ensemble
• as well as the differential relationship between them: x ^S^L = -2K2(x) - xKx(x). (16.58) ax If we are interested in high-temperature behavior, it is also natural to use the variable r — (kT/mc2) as the temperature variable. For example,
• The non-relativistic limit gives the usual expression: \NR(P)
=A= hy ^
= y ^ A
c
.
(16.60)
• The ultra-relativistic limit turns out to be Au«G3) = ir2/3 Phc .
(16.61)
In the absence of interactions, the Hamiltonian operator depends only on the momenta and consequently its exponential can be written, in the momentum basis, e-^
H
= J d3p1d3p2...d3pN\pi,P2,--
,PN > e - " E £ » £ i ( p )
,PN\.
It is only natural to take the trace in the same diagonal momentum basis {\pi,P2,---
QN(p,V)
,PN >}•
= tre-PH
= / d3pid3p2...d3pN
= / d3Pld3p2
• • • d^pNe-13*:^^
,PN\e~pH\pi,p2,---
,PN >
,PN\PI,P2,---
,PN
> •
We can then use (16.26) and introduce carefully the integrations in the arguments of the Bell polynomials. A personal contact with the first lower cases is of great help to get the gist of how the cyclic S's work to simplify the calculations. It is, however, necessary to have beforehand some knowledge
250
Gases: real and ideal
of the kinds of integrals turning up. The relativistic energy of a particle of 3-momentum p is e(p) = \ / p 2 c 2 + m2c4, and the integrals are of the type /d3pe-n/3[(PV+m2c4)V2_mc2]
—
/l
3
J
It is better to consider the integral
/(r,P) = ~
f d3p e~*p
r
2 2
e-0[(P
c +™2c*)^-mc2]^
(16 62)
which will be important in the ensuing discussion of distribution functions. Its value is 1 * * . ^ l + (p) / ( r . # = TTT^V , • , r v,2 A3 y / L . _ ^ ( ^ ) ! + (ftfc) K2((3mc*)
a
(16-63)
•
A is the relativistic thermal wavelength (16.57) and r = |r|. Notice that / ( r , 0 ) = S3(r). Of course, what really appears in the partition function is simply f(0,nP)=
1
A3(n/?)
Once in possession of the result above, the integrations inside the Bell polynomials can be performed, leading to the canonical partition function
Q»
•
(16 64)
-
*•
A recursion formula follows then from Eq. (16.54): JV
Q»
\m-l
x 4 ^ y QN-^V>®
•
(16 65)
-
We shall have more to say on relativistic ideal gases in the context of the grand canonical ensemble (see section 16.3.1.1).
The canonical
16.2.1.2
251
ensemble
Quantum correlations
It is sometimes interesting to use some other basis to take the trace of the exponential operator e~!3H. The most usual alternative is the configuration space basis {|ri,r2, • • • ,r^ > } . In that case, ^ J d3rid3r2
QN(J3, V) = t r e - ^ = j
d3Pld3p2
.. .d3pNe-p^^uip)
\(n,r2,-
. •. d3rN x
• • ,rN\p1,p2,
• • • ,PN)\2 • (16.66)
This is what must be done when we want to calculate the AT-point distribution function Fjv(ri,r2, • • • jfjv), which is such that FN(ri,r2,
• • • ,rN)d3rid3r2
. ..d3rN
is the probability of finding a particle in a region of volume d3r\ around r\, another particle in a region of volume d3r2 around r2, etc. In that case, it is enough to refrain from integrating on space in (16.66), and to normalize to 1: FN(ri,r2,---
,rN)
I d3Pld3p2...d3pNe
N\h3NQN 0
^^ei{p)\(ri,r2,---
,rN\Pl,p2,
• • • ,pN)\2
.
(16.67) Use of the S function definition allows to write the cyclic 5's in the form Sj = / d 3 r i d 3 r 2 . . . d 3 r j e - * [ p i ( r i - r ^ + P 2 ( r 2 - r i ) + - + p ^ ^ - r ^ l ) ] . Once this is introduced in Eq. (16.26), the integrations / d3rf. can be extracted from the Bell polynomials to put the partition function in the form N
/
d3nd3r2
. ..d3rN
J2
® N m { ( ± ) J _ 1 ( j - l ) ! / i 2 / a s • • • fn}
,
m=l
(16.68)
252
Gases: real and ideal
where each / y stands here for / ( r , — Tj,f3), given by (16.62). The distribution function can then be immediately read: 1
N
= -—-^mNrn{(±y-1ti-iy.f12f23.
FN(r1,r2r--,rN)
../,•!>. (16.69)
The Fredholm formalism provides a determinant form for this expression (see Comment 16.6.1). Again a recursion formula follows from Eq. (16.54), with the convention Fo = 1: FN{TI,T2,-
••
,rN)
V N = T7 ^2(±)m TV
V12/23 • • •/mi QN-mFN(rm+i,rm+2,-
m=l
• • ,rN) . (16.70)
From this TV-point distribution, correlations can be calculated. The most usual, the pair correlation function, is given by v(r) = g(r) — 1, where g(r) is defined by the TV —> 00 limit of l
TV: g(r1-T2)
= N(N-l)
fd3r3d3r4..
.d3rNFN(n,r2,-
• • ,rN).
It is here that recursion (16.70) shows its usefulness. Not very informative by itself, it greatly eases the calculations of the integrals. To compute the above expression, we separate the terms including only r i and v2 and use the trivial results f d3r3d3r4 .. .d3rNFN-i(r2,r3,fd3r3d3r4...d3rNFN-2(r3lr4,--and the non-trivial d3r3d3r4 • • • d3rkf23(0)f3i(P)...
• • ,rw) = £ ,rN) = 1
/«(/?) = f12[(k - 1)0].
/ •
The result is
«c>-A; {vm+/(r' « E M - ' * * - * ' . <- - w} • (16.71)
The grand canonical ensemble
16.3
253
The grand canonical ensemble
The grand canonical ensemble represents a system S immersed in a far larger system which, besides playing the role of a thermal bath, is also able to exchange particles with S. There are, consequently, three macroscopic variables: the volume V, the temperature T and the chemical potential fj,.
The grand canonical partition function for a gas of identical non-interacting quantum particles with chemical potential \i is given by the trace E(J3,V,n) = tr [ e - / 3 ^ ^ - ' J ) " ' | ,
(16.72)
where each hi is the occupation number operator corresponding to the energy level e,. The thermodynamic potential for the grand canonical ensemble is the Gibbs potential J = — pV, such that E(f3, V,[i) = exp[—(3J]. We shall use the occupation number representation. Calculating the trace above means, in that case, simply summing over all the possible sets {rij} of occupation numbers at each fixed level. Thus, E(J3, V,n) = J2 (nonin2 . . . \e~p £«<<«-">** \nonin2
= F y TlO
l Til
y
1
' "
e -/3(eo-M)n 0
g-^ei-^m
...)
e -^(£2-^)«2
Tl2
= I I 5Z e_/J(e'~M)n = I ] E e-K'-ri" . i n
(16.73)
e n
We see that, in this non-interacting case, each level contributes an independent factor. The system can also have internal degrees of freedom, which will likewise contribute separately. Suppose there is a single degree of freedom taking g possible values. The partition function will, in that case, be
s(/3,v;/i) = IJ
E
e -/3(e-M)n
(16.74)
254
Gases: real and ideal
The type of statistics appears in the summation, which is over the possible values of the occupation number n, from n = 0 up to the maximum number of particles allowed in each state: 1 for fermions, oo for bosons. The product can be transformed into a summation by using the formal identity n { . . . } = ri[exP(M---})]=exp^ln{...}. To treat bosons and fermions at the same time, we adopt as always the convention upper signs for bosons, lower signs for fermions. It is also eventually convenient to use the fugacity variable z = e13^. The above formulas lead to
S(/3,V»
1
=e{»XMW-'T
}
=e{^^M^"-"\}.
(16.75)
Expanding the logarithm and collecting like terms, we find S ^ M / M - e x p L I
£ f ) £ ± ^ e-«-")*}. i fc=i J
(16.76)
We might wish now to change J2e m t o a n integral over the momenta. In principle, the prescription is £ ) e —» * ^3 p, but a preliminary step would be necessary, as this rule gives zero weight to the zero-energy states. These states must be treated separately first. If this is done, they are given by the factor in curly braces of the expression
',M={£z}'**P
~<±>(V,
e
k=l k/3e
The prescription can now be applied. The sum £) e e~ is just the particular function Q\(k(3, V), so that we recognize the canonical partition function (16.51). We arrive in this way at E( ± )(K,/3, M ) = ( 1
^ » Y, zNQ" • (16'7?) w=o As long as the zero-energy terms can be neglected, the grand canonical partition function appears as a generating function for the canonical partition functions in terms of the fugacity: T
oo
-(/?, V, /x) = 1 + £ JV=1
ZNQN(/3,
V).
(16.78)
The grand canonical
255
ensemble
The same procedure can be applied to a gas with interacting constituents, with the same formal results. In the non-relativistic case, the partition function E(/?, V, /i) can be written, in terms of the cluster integrals bj, as oo
j(
IV
V ] T bjZ3 N=l
m=l
(16.79)
j=i
Notice that we have, on one hand, oo
n ln~(/?,V,/z) = nX = l> & ;
on the other, -ik
N = E(-) - (fc-D!f[ f^z QN fc 1
ZN
ln~=ln
1+J2 QN N=l
JV=1
fe=l
Y,{-)k-\k-l)\Y,^Bnk{j\Qj} it=i
0 0
n=k
77
n
)fe 1(fc 1)!Bnfc{j !Q }
= £^5> ~
~
' '' -
Comparing the two expressions for InH, we get just (16.49),
fc=i
16.3.1
The Mayer
formula
In the cluster approach to Statistical Mechanics [20] of gases and liquids, the pressure and the density are given by pV fcT
In H =
vy^bjZ^; 3=1
(16.80)
256
Gases: real and ideal
N(z) =nV = z—InE = v^jbjzj.
(16.81)
3=1
Clearly, the concentration is given by n(z) = z-j^ ur> a n e Q u a t i ° n which can be integrated by homotopy [94] to give jff- = fQ dt 2i|£i. C o m m e n t 16.3.1 As an example, let us give these functions for the case of ideal quantum gases. These come up in terms of Bose and Fermi functions, denned as
Then, In It is immediate to recognize the formal cluster integrals: 6
(16"84)
"= ^75- '
The "virial equations" (16.80), (16.81) can be thought of as a parametric form of the equation of state, which can in principle be found by inverting (16.81) to get z = z(n\3), which is then introduced into the pressure expression. The resulting "virial equation" is a series* pV = l + ]Tafc+1(nA3)fc, NkT
(16.85)
where the a^'s are the "virial coefficients", related to thefc^'sby the Mayer formula [78]
ak+i = J2(-y{kil^l{k1_)[)l
B
*( 2 | k' 3 ! 6 »-->'
<16-86)
which we shall prove in the following. The closed expression for the equation of state in terms of the computable 6fc's is, therefore
fc>i
j=i
v
'
v
'
(16.87) t For a strong criticism of this approach, see [2l].
The grand canonical
257
ensemble
It is simpler to prove (16.86) by analytical methods (see, for example, [77] and references therein). We shall give here a more combinatorial proof, which is rather lengthy. We need beforehand two combinatorial identities [22], [72]:
r—j
D-J'-'QCT')-®
(fori
-
<16 89)
-
C o m m e n t 16.3.2 Only to get a feeling of how such beautiful and esoteric results are proved, let us examine (16.88): it is the same as £ " = , • C - i ) = (")• Multiply the lefthand side by x ' - 1 and sum over j : £ " = i £"=.,• ( ^ V " 1 = £ » = 1 £ J = 1 ( J l J ) ^ - 1 = £ ? = i ( l + XT"1 = EfcloC 1 + * ) * = ^-li+l) = i K 1 + * ) " " !]• ° n t h e "ght-hand side, we have ]C?=i ( " ) x J _ 1 = x 2Z?=i (") x J > a n t I * n e s a m e expression obtains. As it holds for an arbitrary x < 1, the coefficients must coincide.
It is convenient to introduce the two formal series \3
°° r=l
9{z)
°° f r=l
= I T = ziLf{z) = 5r =>l * r = E fK •
<16-91)
The equation of state (16.85) is A(g) = - £ — = 1 + ^ NkT
an+l9" ,
(16.92)
and is in principle obtained by elimination of the fugacity z in the parametric form (16.90), (16.91). The direct procedure would consist in inverting (16.91) to get the function z(g) and then insert the result in (16.90). As Y^=i an9n = 9-A(g) — f[z(g)], the composition would lead to 1
"
an = — 5 3 r brMnr(z1,z2,... n
- r=i
,z„-r+i) •
(16.93)
258
Gases: real and ideal
The coefficients zT of the series inverse to (16.91) are given by Lagrange's formula: zi = 1 ;
Zn = J2(~YnJ
Mn
~^
In
9{t)
(16.94)
This approach does not lead to the closed form (16.86). Let us begin with a method going back to Kahn [105]. It gives the inverse to what we want, that is, it gives the bk's in terms of the an's. The Lagrange formula (13.86) gives, for the series g(z) in terms of its inverse, n-l
»„_!,,• In *(*)'
9i = 1 ; 9n = ^2{-)3nJ
(16.95)
j=i
We see that what matters is the function In ^-.
9{z)
= z±f{z) = zfznz)
' At
As dA
^
'
it comes out that In
z(t)
£ i•3- ^+ a1 , + 1 f .
(16.96)
j=i
The expression Pj+i —
Y~
a
(16.97)
j+i
has an importance for itself, and is called the "virial coefficient for the connected graphs". As gn = n n! bn, Eq. (16.95) becomes n-l
b
» = A £(")''«'" Bn-1.>
In
z{t)
j=\
1 ^ — ZJ^-^^ 3= 1
/ 3' 4' rt-i,j ( 2! a2, — a3, - a 4 ,..
E
259
The grand canonical ensemble
=h
{-j!/w^
I>:FV B ^
(16.98)
J=I
What is necessary now is to obtain the inverse to these expressions. We notice, to begin with, that (16.96) can be put into the form a
n
n+l
3>nl
In 1 +
Z(t)
= 22 B„J
In
(f"1)
% i [ l n ( l + u)]
= E(-r1o--D! i„,,(f,f,...)
(16.99)
3=1
Use of the multinomial theorem gives
Mnj [in (*1 - l)] (-)» =^(?,f,...)=ni5: r t ( f c - j ) ! ( n + j)!
'n+W [*(*)] •
(16-100)
It is now that the combinatorial identity (16.88) comes in handy. We use it, after taking the last result into (16.99), to arrive at (n + 1)!
»(_)*-! (j_i)!
n
.J B n + j j [*(*)]•
(16.101)
.3=1
We use now the Lagrange formula (13.86), which in the present case reads
9{ty
lato get, after an inversion of the summation order,
in+1)1 n
_ ^
k
9(t)
fc=l
(16.102)
260
Gases: real and ideal
The form of the last summation can be modified, by using the generating function (13.36) for the second-kind Stirling numbers, into fe-i
where use has been made of the second combinatorial identity (16.89), which applies because k — 1 < n. It is enough to look again to (13.36) to recognize the above expression as n f c _ 1 . Consequently, a
"+!
_
B
(n+1)! £ f c = l ( ~ )
'
fc=l
"fc
In
9(t)
n
Bjk [ln(l + «)].
t
j=k
Using (13.50), we find finally 1
yv
(" + l ) ' t t
v
(n +
j-1)1
(n-1)!
B n ^ {(fc + l)!6 fc+ i} ,
(16.103)
which is (16.86). C o m m e n t 16.3.3
Take the grand canonical partition function for bosons as oo
1
ze-P^
'
Suppose a very special system, for which the energy levels are efc = fee, with t some constant. Referring to Comment 14.2.2, we see that E B ( 1 ) = n i t L i y ^ F = E ^ L o P C " ) 1 " ' leading to the interpretation Qn = p(n)e~kl3e. The canonical partition function appears as the number of partitions of n (see Comment 13.1.1) weighted by the Boltzmann factor.
16.3.1.1
Relativistic gases, continued
In the relativistic case, interaction potentials lose their meaning. Unless we are willing to use the involved 5-matrix formalism mentioned in Comment 16.2.3, all we can do is to study the non-interacting gas. The integrals
The grand canonical
ensemble
261
appearing are those already seen, and the result is
•S<±>(V,/M={ ^
}expLKg(±1)J_1
z
3=1
•3—3
V
A (j/3) j
(16.104)
The pressure comes immediately, n-l
p=f£lnZ(±HV,P,ri=9kTJri{±)
(1,6.105)
A3(n/3) z",
n
n=l
where we have dropped the zero-energy terms, which vanish in the thermodynamic limit. It is also easy to obtain the number density, n =
9_ V
zJ^lnS'^W/J./i) V,0
n-l
X
\-z z 1+2
(±) A3(n/?)
+ * £-
(16.106) where the zero-energy contribution may resist the thermodynamic limit in the bosonic case. To obtain the average energy, however, use must be made of the relation (16.58). Including the masses, its expression is
^\nZ^(V,P,n)
E •
V.z
W
+ 9 3^ 3 ^ 'hc (3
£
^
^
^
l
(
^
)
.
(16.107)
n=l
It is comforting that the virial equations can be given the same aspect of the usual, non-relativistic expressions, provided A(n/3) is everywhere replaced by A3(n/3) and the effective cluster integral (±l)n"1 n
Bn
A3(/3) A3(n/3)
(16.108)
is used instead of bn. We find indeed pV kT
9
s 3 II ***; = AA3
=
(16.109)
*:=i oo
n
(o) + ^ E f c s ^ f e k=i
(16.110)
262
Gases: real and ideal
The virial equation has then the form P ^--J2Ak(A3n)k-\ nkT
(16.111)
fc=i
the Mayer formula being now Ak+1 = ^(-y^±10-^Mkj(2lB2,3W3,...).
(16.112)
All the above expressions tend, of course, to the previous non-relativistic formulas in the due limit. On the other hand, the ultrarelativistic limits are 87T ^ f r ) * - 1 , c3h3P4 ^ j 3=1
TIUR = n(0)UR
87T
^
+ g -^j^
^
( ± )
i-l
p
z
'
EUR = 3pc/flV. They lead to the well-known formulas for the photon gas (g = 2, z = 1) in terms of Riemann zeta functions, ^
= ^ 3 ^ 4
C(4) , n 7 = n ( 0 ) 7 + ^
^ <(3) ,
and for the massless neutrino gas (g = l,z = 1), "" = 4 ^ W £ 3 C(3) • We have mentioned in the previous section that relativistic gases more easily exhibit quantum effects. Indeed, quantum degeneracy is present when the number of particles inside a cube with sides equal to the thermal wavelength, the "degeneracy coefficient" d = nA3 is of the order of 1. In the ultrarelativistic case, d= g Y^Li —^—• z " - ^ o r photons and massless neutrinos this gives 2£(3) and §C(3). As £(3) « 1.2, such gases are always extremely quantal.
The grand canonical
16.3.2
Only connected
graphs
263
ensemble
matter
Recall that the contribution of the completely connected graphs to the canonical partition function QN will be given by Eq. (16.50), Q£ = vb^. Let us call E^ the contribution of such connected graphs to E. As there is no sense in speaking on connectivity for zero particles, we define oo
oo
S
(16.113)
N=l
This is just In S(/3, V, /x). We arrive thus at a version of an important result, the Matsubara formula [106]: E(/?,V»=e-c)^'v^).
(16.114)
There is a graphic treatment of the problem, in which of course Q}ff is given by a totally connected diagram. The theorem is then stated as: The contribution of all the graphs is the exponential of the contributions of the connected graphs. This is an important property of the several diagram techniques used in Physics. There is an analogue for the Feynman diagrams of Quantum Field Theory [107]. Here, a remarkable consequence is that ^.=E^(P,V,n).
(16.115)
For an ideal non-relativistic Boltzmann gas, E^c\f3, V, /x) = vz = -p-z. As in that case z = nA 3 , it follows that S(c) = N. For a gas without interactions, the contributions of the "connected" graphs to the grand canonical partition functions is just the number of particles. Introducing in a similar way the contribution of the connected graphs to the expectation value of AT, we have NV=z±
InSW =z I
dz
dz
In I n S - ^ "
6
^ " = - ^ .
J^n K *n
^{C)
Thus, for the ideal gas, i V ^ = 1. We see in this way how the connected graphs come to provide corrections to the ideal gas law: the equation of
264
Gases: real and ideal
state takes the form PV _ _JV_
16.4
(16.116)
Braid statistics
We have repeatedly said that particle exchange, for a gas of JV particles in ambient space, is ruled by the symmetric group S/v. This is true if by "ambient space" we understand our usual home-space, the 3-dimensional Euclidean space E 3 .
Things are different if the
particles are somehow confined to a 2-dimensional space: the group responsible for the exchange of particles is, in that case, the braid group Bjv- As mentioned in 11.3.1, this is an infinite group, which contains SJV as a subgroup and leads to a different statistics. This kind of statistics is believed to be of interest in the study of superconductivity, as electrons in a superconductor are forced to stay at its surface.
Take two strings. It is possible to interlace them so as to get back the initial position of the extremities. The difference with respect to the simple permutation of the extremities is that there are infinite ways of getting back to the initial positions. What would be the identity transformation in the permutation group can now be any transformation in which one of the strings winds around the other any complete number of times [70]. Each way of coming back to the original situation is different. As with permutations, a phase appears in the wave function as a result of an exchange. A transposition of two particles is given by \p2P1 > = 0"i|piP2 > = e^|pi£>2 >• However, while performing twice the same transposition of Sjv leads back to the initial wave function [s\ = I, see Eq. (16.25)], doing twice the same braid elementary transformation only doubles the phase: cr\\piP2 > = el2
The braid group can be introduced by a presentation (see Com-
Braid
265
statistics
ment 16.1.2). Its generators satisfy, as mentioned in section 11.3.1, <7i<Ti + lCTj = ai
+
l(Ti<Ti+r,
(16.117)
(Tjcr,' = ujCTi for \i — j \ > 2 ,
(16.118)
which shows a lot in common with the symmetric group: these are exactly the same conditions (16.24). The difference comes from the absence of condition (16.25).
The presence of the braid groups in 2-dimensional spaces is a topological effect. Quantum Mechanics of a given system is controlled by the group of classes of closed loops (the fundamental group) of its configuration space [109]. Quantum Statistical Mechanics will, of course, take that in consideration [53]. The configuration space of a system of N identical particles will have a non-trivial fundamental group [llO], that is, will be multiply connected. Let M be the configuration space of each particle in a gas. The configuration space of a gas of N distinguishable particles contained in M will be the set MN, the cartesian product of manifold M by itself N times. M is usually a box in E 3 , whose volume V is taken to infinity at the thermodynamic limit. When the particles are indistinguishable, two points x and x' of MN are equivalent if the sets {xi,X2,... ,x/v} and { x ' i , x ' 2 , . . . ,x'jv} differ only by a permutation, a transformation belonging to the symmetric group SN, and the configuration space becomes the quotient MN /SN- If M = E 3 , and is in consequence topologically trivial, the fundamental group, indicated ni[MN /SN], is just SN- MN will be the universal covering of the real configuration space and the iV-particle wave function will belong in the carrier space of some representation of SN- Things are more involved when the particles are also impenetrable. In that case, coincident positions must be excluded. This is done by introducing the set Djv = {xi,X2,... ,XJV
such
and taking its complement in FNM
that
x» = x^ for
some
i,j}
MN, =
MN\DN.
This would be the configuration space for N distinct impenetrable particles, as a classical hard-sphere gas. The fundamental group of this space, PN = TTI[FNM], is the pure braid group of N strings. If the particles are impenetrable and indistinguishable, both the quotients above are to be put
266
Gases: real and ideal
together. Let B^M be the space obtained by identifying all equivalent points of FfjM, its quotient by Sjv: BNM = FNM/SN
=
[MN\DN]/SN.
The fundamental group of this configuration space, B^ = TTI[BNM], is the full braid group, or simply braid group. Now, what is important for us here is that the combinatorics for the braid statistics is the same as that of the symmetric group. This means that a recursion relation similar to (16.26) will hold and be instrumental provided we are able to find the normalization for some initial state. In the case, a detailed examination [70] shows that, for the 2-particle case, Eq. (16.16) is replaced by
•
(16.119)
The angle <j> is, in superconductors, somehow fixed by the medium. We see that bosons and fermions are recovered when
Actually, this is not quite so. Recall that the fundamental group is a group formed by (classes of) loops. Every loop which can, in a punctured space, be reduced to a point in a continuous way is related to the identity. In a 3-dimensional space, point-holes are of no great importance for these loops, as they can be displaced around them. It turns out that, for a space of dimension > 3, the braid group reduces to the permutation group: BNM — S?jM. Consequently, braid statistics is of interest only for 2-dimensional configuration spaces. The exponent 5/2 in the formula above comes from the kinematics of a 3-dimensional space. In a 2-dimensional kinematics it reduces to 2, so that the effective configuration integral is actually =
(cos|^
r
Condensation
267
theories
The recursion relation (16.26) becomes then N
,PN\PI,P2,---
,PN>
= jTi ^2(cos<[>)N-mBNm{(j
-ly.Sjj.
m=\
Higher order normalization can be calculated starting from (16.119). A 3-particle state, for example, will have
(16.121)
Usual quantities change from 3-dimensional to 2-dimensional kinematics. Volume V is replaced by surface area S, the cube A3 of the thermal wavelength becomes the square A2, etc. The canonical partition function, whose expression for a Bose/Fermi 2-dimensional gas would be
QMS) = ifrCWpf11 J} = & E B ^{^F" °'"1)! £}' (16.122) becomes N
B
wg)4E "-( ( c o s r l
(j_1)!
i^l
(16 123)
for a braid gas. 16.5
C o n d e n s a t i o n theories
As more detailed knowledge is obtained on a subject, it tends to earn a proper name. Second order phase transitions are nowadays commonly called "critical phenomena". Condensation is the paradigmatic example of first order phase transition, and sometimes lends its name to the whole class. Here, we shall use "condensation" in its old, less well-defined, sense. We shall actually consider only two specific theories, because they provide an intuitive, "physical" vision of the different elements of a Bell polynomial. As an appetizer, let us give a short account of Mayer's model for condensation, which has a lot of heuristic interest.
-
268
Gases: real and ideal
16.5.1
Mayer's
model for
condensation
Let us derive the canonical partition function (16.46), N
1
QN(P,V) = — Y , ( ^ )
(V\m
BNm{klbk},
with respect to the volume, to obtain N
1 1 1 V d_ InQiv = 7 5 - T » $ 3 "»BjVm{j' ! ^T3} = <"»>• JJV, QN Nl ^ AJ dV T As m is the number of clusters for each configuration, this expression gives a kind of "average value" for this number. This relates, by Eq. (13.109), volume variations to the homogeneity in the virial coefficients: V
d QV
1
N
V
lnQN
N,T
(16.124)
m=l
We now take a bold and risky step: we suppose that fluctuations are negligible, so that the results of different ensembles can be used simultaneously. That assumed, we find d
w
kT
= (m)
lnQN N,T
and arrive at the curious general equation of state pV = (m) kT.
(16.125)
This is the Boyle-Mariotte equation with (m) instead of N. It is as if we had always an ideal gas of clusters. Of course, as the signs of the 6fc are not necessarily positive, the "weights" in the above "average" are not positive and (m) is no real average after all. But the expression is too suggestive to be simply put aside, and Mayer used this line of thought [l 11] to obtain a heuristic theory for condensation. With the above "equation of state", (m) relates to the connected graphs: =(»)
V A3
£;
JV=0
(16.126)
Condensation
theories
269
Using the detailed form of the Bell polynomials (13.6), we find in the same way, and with analogous interpretation, that
We get then JV
(m) = ]T (Vj), 3=1
and arrive at Jg- lnQ^r = jsz3'. Consequently, V_ (Vj) = ^z* A3
b3 .
(16.127)
Pressure, the Lagrange multiplier associated to volume preservation, is obtained by taking the homogeneous derivative V-^. This leads to an analogue interpretation of the {vk)'- they can be seen as variables conjugate to the bk in the same sense that pressure is conjugate to the volume. Knowledge on condensation, involving large clusters, requires knowledge of virial coefficients of very high orders. The result above is usually obtained by supposing that, for N > > 1, only one term dominates the summation (16.46) for QM- That is, a single configuration, or set of numbers Vj, all of which very large, is such that 1 f V i v m { k\ bK 3 QN = TT, k A N\ BiVm The set {i/j} which maximizes InQjv is looked for, with the condition Ylj=i 0 vi = N being imposed via a Lagrange multiplier. Of course, the "average" is also here unreliable: the (VJ) of (16.127) might be negative. For example, if we calculate the average energy from (16.46) for an ideal gas of bosons or fermions, we get {E)=U=\kT
(m)
(16.128)
(actually, this expression holds whenever the bk's are temperature-independent). Thus, from u = | kT ^- we see that JV~^m^ measures the breaking of energy equipartition (which holds true when (m) = N). We can also
270
Gases; real and ideal
relate the specific heat with the fluctuations of (m): Cv = | k(m) + | k [(m2) - (m) 2 ] = - ^ - + \ k [(m2) - (m) 2 ]. (16.129) When (m) — TV in the canonical ensemble, we have the law of DulongPetit. The deviation from Cy = U/N comes from the fact that particles behave in a collective way, in clusters, and we have less agglomerates than particles. What is the behavior of bj, and consequently of (UJ), for large values of j ? Well, let us suppose that bj « Cj b> for large j ' s , with b a general constant and Cj some coefficient. This can be reasonably justified [20], [112]. Under this assumption, (i/,-) = p- zj bj ss ^ Cj (bz)i. Then, for z < 6 - 1 , other numbers of large clusters will be small; but for z > b"1 this number will tend to become larger and larger. There is thus a change of regime at some critical value zc = b~1. The critical volume is given by 1
.j
00
1
00
- = j? £ JV* = ^ £ J w-' •
(16-13°)
As soon as z > zc, the series ^ = p- YlTLi jfyzi grows quickly and the specific volume falls down. Let us here only retain some qualitative aspects. The picture emerging from Mayer's approach is that of a free gas of clusters, of a mixture of gases: one gas with vi really free particles, another with 1/2 pairs, still another with 1/3 triples, etc. In the canonical partition function (16.46), we sum over m in Bjvm{fc! bk}. We have seen that pressure gives a measure, however faulty, of the average value of m. With an analogous interpretation, (i/j) measures the partial pressure related to clusters of order j . 16.5.2
The Lee-Yang
theory
The Lee-Yang approach [113] establishes clearly the relationship between phase transitions and the thermodynamic limit. It involves a detailed account of the order in which the limits are taken, and stresses the great generality of the problem. The details of the interaction are relegated to a secondary role. The physical values of the fugacity z = e ^ , of the canonical partition function QN{P, V) = Ylm e~®Em and of the grand canonical partition
Condensation
theories
271
function oo
l+Y,zNQN((3,V)
S((3,V,fi) =
N=l
are real positive numbers. Stability conditions require the two-particle interaction potential to include a hard core of some radius a. For a finite volume V, the summation above is actually restricted by the number M = Nmax corresponding to maximal packing, M « V/vo, with vo <x a3 the hard-core volume of each particle; for N > M, the total potential is necessarily infinite positive and QN>M(P,V) = 0. As a consequence, the grand canonical partition function is a positive polynomial of order M, monotonously increasing with z: M
3(/?, V,z) = J2
ZNQN(P,
V) .
(16.131)
The meaning of the partition functions QN appearing here in a gas with M > N particles has been discussed in section 16.2, below Eq. (16.49). As the coefficients Qjv are all positive and QQ = l,it follows that E > 1. Furthermore, • There are no zeros on the positive real line of the variable z; • In E(/3, V, z), being the logarithm of a polynomial, is a well behaved function of z; • it has furthermore well behaved derivatives in the three variables and, for finite V, will present no singularity; • the relationship between phase transitions and the thermodynamic limit comes handy: only when V —> oo and M —> oo can a singular behavior show up. We shall not be interested here in the limiting behavior. We shall only recall that the theory examines P{z,T)
= kT lim d- InS)
(16.132)
V—>oo V
and a m
v(z,T)
(77 . , " ) = lim
in the complex plane of the variable z.
— — (— InS)
(16.133)
272
Gases: real and ideal
Consider then the series to be finite, and take for the grand canonical partition function the polynomial E^M\z) = ^2N=0ZNQNIts roots will be indicated by r*: M
E2 (M) (r fc )
=
£
r"
Qff^t
V)
= 0.
(16.134)
JV=0
Let us recall the Weierstrass representation (14.30) for a polynomial, M E(M){z) =
M
,
..
N
J- z QN(f3,V) = A ( l - - ) • AT=0
fc=l
^
rfc
(16.135)
'
The roots will depend on V and (3, r^ = rk((3, V). By the above considerations on the good behavior of lnS, there are no roots on the real positive axis. They will be either real negative or complex. In the last case they will come by pairs with opposite imaginary parts. The particle number M will tend to infinite with V, so that the number of roots will go to infinite in the thermodynamic limit. When we approach the thermodynamic limit, it may happen that some root (simultaneously with its complex conjugate) touches a point in the physical real positive line. Their imaginary parts go to zero. The signal for a phase transition lies in this "pinching" of the real axis by a zero or more. The "pinched" point zc on the real line is said to be a critical value. P(z,T) will have a singularity at zc. For the lattice gas, Lee and Yang have shown their "third" theorem, the circle theorem: in the thermodynamic limit, all the roots of H = 0 display themselves on the unit circle of the complex fugacity plane. We may speculate on the generality of this result [114]. It is known that, for a 1-dimensional hard "sphere" gas (the Potts gas), the roots crowd on the negative real axis, but no phase transitions are expected in this case (van Hove theorem*). Let us go back to Eq. (16.135). Expanding the logarithm in
fc=l
V
'
8=1
t The van Hove theorem, as well as other classics on the subject, can be found in Frisch and Lebowitz [115].
273
Condensation theories we find M
V
v
jb
=
i
- E r ^-
( 16 - 136 )
Consider t h e alphabet r = ( r i , r 2 , . . . ,rM), whose letters are t h e roots of E((3,V,z). As usual, also its reciprocal will play a n important role. Recalling t h e power-sum symmetric functions Pj[r*] = Sfc=i( — )~"' r fc"'> ^ is clear from t h e above t h a t v j bj = {-y-ipjlT']
.
(16.137)
Also, N(z)
=v ^
= 1
3bjZi
MM
M
v
= - E E ^ " = E(-)J'+1wtrizi =zp[r*>-*]>
( 16 - 138 )
where use h a s been m a d e of (14.11) in t h e last step. O n t h e other hand, from (16.131), M QM
= M\ E
r B
M
\ r J
^ i -0' -1)! E fc" \ -
• m=l
I
fc=l
(16-139)
J
Consequently, M
M
QM = (-) I ] r *' •
(16-14°)
fc=l If t h e circle theorem holds, QM = 1- Consider now t h e diagonal matrix AM=diag
[-rf1,-^1,-^1,... ,-r^1]
(16.141)
and its powers. We have clearly QM = detAM
(16.142)
and v j bj = (-y-1
tr A{j .
(16.143)
This suggests a n interpretation of t h e exponent k in A ^ : when we take the fc-th power of AM, we are taking into account interactions between u p
274
Gases: real and ideal
to k particles: two-by-two interactions given by 2wfe2 = — tr A M , threeby-three interactions given by 3v&3 = tr A M , etc. Thus, at least as long as M is finite, there is a "Lee-Yang matrix" A M whose characteristic polynomial is related to the grand canonical partition function. The characteristic polynomial determines a (normal) matrix only up to the (unitary) transformations leading to its diagonalization. Thus, the whole description, based on the partition function, will have an underlying invariance under such transformations, which constitute a group U(M). Summing up: a matrix A with the characteristics given above should exist for each real gas. However, the description of the system is not really dependent on A. It depends only on its similarity class. If the circle theorem holds for the system, A must be unitary in the thermodynamic limit. In the next section [Eq. (16.163) and at the very end], we shall see how the grand partition function can be assimilated to a Fredholm determinant.
16.6
The Fredholm formalism The formalism of the Fredholm integral equations, and in particular its finite, matrix version, provides a most detailed view of the decomposition into cycles. It helps to see, for example, which particulars are lost by the virial decomposition and by Bell polynomials in general. The grand canonical partition function can be assimilated to a Fredholm determinant, and the partial canonical partition functions to the Fredholm minors.
We recall now the main lines of Fredholm theory. § The theory is concerned with a current type of integral equations and, as such, finds its place in many chapters of Physics. Applications to semiclassical quantization of chaotic systems have deserved special attention recently (see, for example, [116]). As usual, this compact will have a physicists's bias, with little concern on the deeper mathematical questions like convergence, or the spaces the involved objects belong to. The presentation given is purely descriptive and utterly simplified. It mixes the continuous and the discrete points of view, § The author is grateful to M.A. Ozorio de Almeida for calling his attention to reference [116] and, more generally, to the possible interest of the Bell formalism to Fredholm theory.
The Predholm formalism
275
but has in mind, ultimately, finite N x N matrices. Suppose an equation of type X = J + AKX.
(16.144)
We shall in what follows take this as a matrix equation. Nevertheless, if taken as a symbolic expression, it might as well represent a Predholm integral equation of the second kind, X(ar) = J(x) + \ f dx'K{x, x') X(x') .
(16.145)
X and J are members of some Hilbert or Banach space, on which K is a well-behaved kernel. Objects J and K are supposed to be known and defined on some finite domain. In case (16.145), if J and K are wellbehaved enough (say, continuous, or square-integrable), then the Predholm alternative holds: • either there exists a unique, equally nice solution X(x), or • the homogeneous equation (J = 0) has a solution. In the last case, (16.145) becomes an eigenvalue equation. The set of eigenvalues for which the solution is not unique is discrete. The main result of Fredholm theory is that the results we shall report, talking on matrices, are valid also for a large class of integral-functional cases. Given (16.144) we can write, at least formally, (16.146) The resolvent operator or matrix R(A) (the name has been previously applied in another sense, see sections 1.1 and 1.3) is defined by X=(I + R)J,
(16.147)
so that 00
R(A) = £
AK A" K " = j — ^
.
(16.148)
n=l
Everything works as if, in the equation, X was repeatedly replaced by its own expression, in an infinite recurrence. The above comment on eigenvalues can now be restated as:
276
Gases: real and ideal
The inverse (I — AK) discrete set {Afc} .
x
exists for all values of A except a
This means that the spectrum of K is discrete. The inverse of a matrix is always some matrix divided by its determinant [see Eq. (14.46)]. The expression (I - AK)" 1 = - £ - N D(A)
(16.149)
introduces the Fredholm determinant £>(A) = det(I - AK).
(16.150)
This determinant can be written as oo,N
£>(A) = 1+ £
A D
(16.151)
" "
n=l
with
Dn = — j —
dsi
ds2--
•**3lSl
-R-S1S2
**S2Sl
K-S2S2
^S„Sl
**SnS2
Ks
dsn
Ks (16.152)
In the discrete version with N x N matrices [see Eq. (14.7)], it can be directly verified that
N
N
££-£
S l = l So — l
N
-"•S2S1
-"-S2S2
Ks
•R-SnSi
^SnS2
Ks
Sn = l
= £
Mnmii-y-'U
~ 1)! trK''}.
(16.153)
m=l
Consequently, (-) £>n = M " £
B n - { ( - ) J _ 1 ( j - 1)! trK''}.
(16.154)
The Fredholm
277
formalism
C o m m e n t 16.6.1 There is an obvious analogy with the canonical quantum ideal partition function. We see from (16.68) that QN = (±)NDN if we identify above KriTj with the function fri,rj there defined. The distribution function, which is the integrand of QN normalized t o one, will be fr1ri FN(r\,T2,
• • • ,TN)
=
fr\rN
frir2 /r 2 r2
JT2TN
N\QN
All this follows from the definition (16.150) using (14.2). We have extended the summation down to m = 0, which is consistent with the convention Do = 1 and is irrelevant for n ^ 0. A few lower-order examples:
2!
R2
[(tr K)2 - tr K2]
D,=
2!
tr K trK2
1 trK
= det (2)K ;
[(tr K ) 3 + 2tr K 3 - 3(tr K 2 )(tr K)]
3!
3!
1-3
i—1
Dn
*v ;
trK trK2 trK3
1 trK trK2
0 2 trK
=
-det(3)K
These expressions are identical to those of the Lie algebra invariants (14.53), and also reminiscent of several other formulas previously found. It should be borne in mind however that, in the trace computations of those formulas, say Eq. (14.20), the indices run from 1 to N. D3, for example, is not a determinant unless N = 3. Notice D(X)
1 + Zj>i
*Dj
2-i ~^\ 2—i
d-Dj}.
(16.155)
The relations between minors, determinants, traces of powers and symmetric functions have been discussed in section 14.2, but the expressions (16.152) and (16.154) help clarifying the minors appearing in many expressions of Chapter 14. We shall see that, in the same line, they will shed light on the meaning of the partial canonical partition functions turning up in the grand canonical formalism.
278
Gases: real and ideal
It comes not as a surprise, in view of the "recursion" relation (13.89), or more specifically of Eq. (14.16),
- = ;£<->-<
PkCn-k
fc=i
that one of the main ingredients of the Fredholm formalism is the property 1 n Dn = - - Y,Pi Dn-j > (16.156) J=I
7
where pj = tr K- . The Fredholm first minor is of fundamental importance. It is defined as N
D(x,y;X) = l + ^2\nDn(x,y)
,
(16.157)
n=l
where Dn(x,y)
(-)" / dsi I ds2 • • • / ds n
**-Xy
-"-XSi
-"-X52
-"•siy
**-s\s\
**-siS2
S2V
S2S1
-"-S2S2
**-sny
**snsi
Kx
K,
*^snS2
(16.158) D(x,y;X)
is important because the theory shows that R(x,y;X)
= Rxy(X)
1 D(x,y;X) D(X)
.
(16.159)
The first cases of Dn(x, y) are: Dl(x,y)
D2(x,y) = ^ -
=
(Kxy[(ti
^-[(tiK)Kxy-Klv];
K ) 2 - tr K 2 ] + 2Kzxy - 2(tr K)K2xy)
.
If we move the first row and column to the last positions in (16.158), it becomes
279
The Fredholm formalism
Dn(x,y)
= •*»-SlS2
- ^ 5 l 5
•^52Sl
-'*-S2'S2
XV S 2 S n
•^5
•*^3nS2
-"-SlSl
(_)»
N
N
EE-I
N
n
5l
-**XSl
n
• #,„,„ ^x«„
-*^X52
-Ksiy -f^s2y
"•SnV
l*-xy
And if we now put x = y = sn+i, we find Dn(sn+i,Sn+l)
=
(-rf E E - E Si=ls2 —1
s
n
=l
Ks
KS1S Ks->s
Ks
Ks
Ksn+1S!
KslSn
Ks1sn
K-SOS-n
KsS2S„_|_1
Ks K.Sn +
K-sn+1s2
+
i
L
l
s
"
K, K. (16.160)
To get Dn+i it would be enough to add a factor ( - ~^ J and sum over Sn+i= - ^ r S S n + 1 D n (sn+i,Sn+i) = A H - I , which can be rewritten as Dn = --~y]Dn-i(x,x)
.
(16.161)
As long as K is well-behaved, D(X) is an entire function (an absolutely convergent series) of A. In the N x N matrix case, D(\) will be an iV-th order polynomial. The similarity of the above expressions to those we have found in Statistical Mechanics of real gases will not have escaped the reader. Indeed, recall the expression of the grand canonical partition function for the M-particle real gas: M {M
E \z)
= det [I + zAM] = J2
ZN
QN
•
(16.162)
JV=0
A M is the matrix (16.141) which shows up in the Lee-Yang theory, whose eigenvalues {—1/r-;} are related to the zeros r, of 'E^M) on the complex fugacity plane. We obtain a complete analogy with the Fredholm case by
280
Gases: real and ideal
putting K = A and z = —X: (16.163)
£>(A) = S(-A).
The analogy between Dn and the canonical partition function is also evident. Indeed, Eqs. (16.137) and (16.54), which are vjbj
= (-y-1
tr AjM
'
N 3= 1
clearly relate to (16.156). It would then be possible to trace the contribution of each term to the pressure (16.80) and the density (16.81) pV_ kT
v
Y^,hix
In
J=I
d N{z) =nV = z — InZ = v^2
j bJzj
It can be easily seen that the summands in (16.152) provide a very detailed cycle decomposition, including eventually asymmetric interactions. Most details are lost after summations are applied and, consequently, erased in the cluster integrals. This is an opportunity to see in detail what is retained and what is lost. Because it is the lowest order exhibiting all the main aspects, let us examine in detail the case N = 4:
^ = i££££ 4!
Ks Ks Ks KK
Ks Ks Ks
Ks Ks Ks Ks
Ks Ks K,S3S4 Ks
This is to be assimilated to the contribution Q4 in a gas with JV particles. Let us drop V V \ V V and look at the summands. A diagram indicating the particle associations will be drawn below each term in the expansion of the above determinant. We shall consider the terms three by three. A convention must be established. We shall suppose that Ktj means, as in Field Theory scattering matrices or Green's functions (and as our stochastic matrices), a transition j —>• i. Let us also recall that, in M^m, m
281
The Fredholm formalism
is t h e number of clusters a n d Vk is t h e number of clusters with k particles, so t h a t m = ^2k v^. Now, let us examine t e r m by term: K-sysi
s
"32»2
3a3
s
1 •
4a4~^~
s
l
s
3
l ^ *
s
2
a
3
.•
a 2
3
a
4
s
3
4
s
2 '
\
s
a 2
a
\"
1
2
s
4s3
3
•
t
s
a
4
•
3s2
2
4
/"
4 • —> • 3 <- • 3 The first term accounts for the non-interacting particles (y\ = 4, m = 4). After summation, it will give (tr K ) 4 . The next two will both give tr K ( t r K ) 3 after summation, but they are differently arrowed. This detail, which would account for possible differences between Ky and Kji, is blurred out by the summations. They correspond to v\ = 1, 1/3 = 1, m = 2. 4 •
•
s
-**3iai-
4
a
s
3
4
3
a
3
s
4
1 .
. 2
4 a
a 3
•
^3\31**3334**3433**3232
2
•
4 a
s
" a j s ^ " 3 4 S 4 " S 3 S 2
1 * <->
2
a 3
1 *
•
4 a
a 3
2
s
3
2
Three cases with binary clusters: vi = 2, 1/2 = 1, m = 3 in each term. Summations will lead to (tr K ) 2 t r K 2 in each case. ~
s
a
l
2
s
a
2
s
l
1 a
3
s
s
3
o
s
4
4
—
^
a
l
a
a 2
4 a
s
2
2
s
1 a
a 3
s
3
3
3
i
t
4 a
- •
3
4
«-
s
4
l
—
3
"
3
l
a 2
2
a
l a
2
s
4
3
4
3
3
3 "
<-
3
3
l
a 2
><: a 3
4 a
<-
a 3
The first has again a binary, u\ = 2, V2 = 1, m = 3 terms, leading to (tr K ) 2 t r K 2 . The two other are completely connected diagrams, 1/4 = 1, m = 1, with the summations producing tr K 4 . +^3la
2
'^
3
2
3
1 a
4^
3
3
<-
4
3
3
3^
4
3
l +
**3ls2-"3334**3433-'*323l~'~
a 2
1 •
<-t
•
^3l32^3A3A^3ZS1^323Z
2
1 a
/*
<-
a 2
N T
4 a
a 3
4 a
•(->
a 3
4 a
a 3
The first and the third are distinct contributions with one triad and one isolated particle. They are thus of type 1/3 = 1, m = 2, tr K t r K 3 . The middle term has two binaries, so that 1/2 = 2, m = 2, (tr K 2 ) 2 .
+^
3
l
3
3"
3
2
3
l"
1 •
3
3
3
2^
->
3
•
\ 4 a
4
3
4~'"
^
3
1
3
2
3^
3
2
3
2^
3
3
S
4^
l a
4
4
. 3
4 a
•
3
4
3
1 +
^
3
2
l
3
3 ^»234 ^3432 ^
l a
•
3
3
3
1
2
55
\ ->
a 3
4 a
. 3
Two cases with triads, and the last containing two binaries. ~"
3
l
3
3
a
2
s
4 "
s
l a
s
2 •
4 4 a
3
5<
s
4
2
s
l
—
"
3
1
3
3
3
3
1 a
4 "
3
-t
4
3
3
2 •
2 2
3
1 ~~
3
1 33 ^ * 3 4 3 4 * * 3 3 3 1 * ^ 3 2 3 2 l a
!K
4 •
3
3
4 a
->
. 2
NJv. a 3
4 a
•
3
282
Gases: real and ideal
The last case is a binary, the first two are connected diagrams, with m = 1. — "sjs4^a2a3^S4s2'^s35l
— - f ^ 5 1 s 4 ^ s 2 s 1 - ^ 3 3 5 2 ^ 3 4 3 3 — Ks1s4Ks2S2Ks3s3K34s1
1 a
-+
a 2
la
i
U
a 3
4a
t 4 a
<-
a2
la
a2
t a3
&
t
4a
a3
The clockwise cycle is, of course an m = 1 case, as is the third term. i"ajS4"s233"33S2^54Si
la
n 4a
a2
'
s
l
a
3
s
3
3
->
t
s
n a3
s
4
1 a
4a
4
s
2
s
2
3
l
^
s
ls4
a 2
la
a3
4
a
4s3
S
3S1
S
2S2
a2
T
\
a
<-
a 3
It should be noticed that the summands show only simple cycles. Each particle has an incoming line and one outgoing line. No terms of the type 1 • -> • 2
U 4 •
X
t • 3
turns up. Such terms, with one particle coupled to 3 or more others, do appear in the usual virial graphs. The above formalism provides a real cycle decomposition, precisely that division of the symmetric group into cyclic factors which, as we have said, should exist. The summation of all the contributions above gives i {-6tr K 4 + 8(tr K)tr K 3 - 6(tr K) 2 tr K 2 + 3(tr K 2 ) 2 + (tr K ) 4 } , as it would be expected from (16.153). This decomposition: (i) is really cyclic; (ii) is arrowed: would work even if K\j ^ Kji. All this suggests that, besides the existence of a general invariance under a unitary group, the formalism of real gases indicates the presence of a Fredholm equation with the Lee-Yang matrix in the role of kernel. That equation would have a unique solution for all values of the fugacity which are not zeros of the grand canonical partition function. The meaning (and interest, if any) of these solutions is as yet unknown. It is tempting to examine the circle theorem from this point of view. In the thermodynamic limit M —> oo, the matrix K = A would become unitary. There are two interesting points. One is a general result on the eigenvalues of a Fredholm operator: the eigenvalues have no finite accumulation point. Another seems more physical: for a phase transition to occur, it is enough that one single pair of roots come close to pinch the real positive axis in the limit [21].
Appendix A
Formulary
The formulas below have been used in checking the text results and can be instrumental in completing the examples and comments. The assortment is by no means exhaustive on any topic. A few of them may even be new. Readers interested in more detail should consult the sources indicated at the end.
A.l
General formulas (1) Useful finite and infinite sums sums of powers
{->
y
fc=j
'f-1
\ N -j + l
fc=o
I
\l x = l
J
(whenever the infinite series in the middle member makes sense) in particular, for x ^ 1,
l-x
'
^
fc=0 fc=l
the iV-th roots of the identity obey N
N-l
fc=l fe=0 283
xk = x
l-x
284
Formulary
the Kronecker identity iV k=l
combinatorial identities
X!
fc!(z-fc)! general binomials (r-ra!)U!
r = non-negative integer
S
(~) ^FT^T
r = -\r\
= negative integer
r s com
^ - r(r-»+i)r(a+i) general factorial
'
P l e x numbers
binomial formulae r (i + t r = Er=o(")* (i-*)- m = E~ 0 ( m T 1 )* r «! = r(x + i),; (*)
(2) Functions and series OO
e x = lim (l+x/m)m
n
n=0
arccosz =
7T
2
z
Z3
OO
/
\
n
_ 1
= V ^-; ln(l + a:) = V ^
- + ... ; arccosfl—2x) = 2-Jx-\ 3! ^
7i
—
n=l X3/2
3
arcsinz = 7r/2 — arccosz = > „ , . , . . „ . , rz ' ^22k(k\)2(2k + l)
1
X5/2
20 T
.
h...
General
285
formulas
(3) Gamma function \)nz
1 • 2 • 3 . . . (n -
T(z) = lim
n->oo z(z + \){Z + 2)...(z
+
7l-l)
= - TT(1 + i )( 1 + £ r 1 = / dte-w-1 z 1=1
n
n
J0
T(z + l) = z T(z) ; r ( z ) r ( l - z) =
~^— s i n -KZ
T(n + 1) = n! ; T(l) = T(2) = 1; T(l/2) = ^
;
Stirling formula In AT! = AT In N - N +
o(l/N).
(4) Gauss sums
Efc=oe t w f c = E f c = i e l - f c
= ^ J ^ - T -
(5) Gauss integrals / ° ° dxe~a*2 = l J l OO
;
r
dxe-a*2+P*
=
efl^J^
/
/
du e - Q [ ( x - ^ 2 + ^ - z ) 2 ! = 4 / — e - Q ( x - 2 ) 2 / 2 -oo V2a (6) Bernoulli polynomials Bn{x) and numbers Bn -£=1 = ^n=0Bn(x)
~
(for |t| < 27T) ; B0(x) = 1; B„ = B„(0)
B„(a: + 1) - B „ ( i ) = m " " 1 ; £ „ ( z ) = £ £
(fc) 5 * 1 ""*
286
Formulary
Bo = l;Bi = - | ; B a = | ; B 4 = - ^ ; f t » + i = 0 (for n = 1,2,3,...) Euler-MacLaurin formula £/fc = /
dxf(x) + I[/(n) - /(0)] + E 7 ^
I ^ " 1 ' " /o 2fc_1) l •
(7) Euler polynomials En(x) and numbers En 2ext -f^Y
= Zn=oEn(x)
cosh*
tn - (for \t\
= EZoEn ^"=0
7t
= 2"£„(l/2)
-7 (for |t| < TT/2) n!
So = 1;E2 = - 1 ; £ 4 = 5;£ 6 = -61;£ 2 n+i = 0 (for n = 0,1, 2,3,...). (8) sum of fc-powers of integers
B t + i(JV + l)-.Bfc-n(l)
So = N ; Si = N(N + l ) / 2 ; S2 = N(N + l)(2N + l ) / 6 ; ft = N2(N + l ) 2 / 4 . (9) modified Bessel function Kn(x): for x » 1
extreme cases
*.w-(sr«-0+5+-) for a; < < 1
K2{x)*\;
K,{X)^X*
X
287
Algebra
(10) Riemann's zeta function 00 1 £(z) = V^ — = TT
1 -
1
— (p = all prime numbers) V
roo
tz~^
some values: C(0) = - 1 / 2 ; C(-2n) = 0; 2n
™2n-
(11) Dirichlet series
*>(*) = ££i°*T* Bose function is the case gn{z) = D{n), with Oj = zJ". (12) Inverse and identity series [ 5 <" 1 > o g)(x) = fo o A. 2
1
ff<-
>](2;) = e(x).
Algebra (1) characteristic equation of an N x N matrix M A M (A) = det(AI - M) = Uk=i (A - Afc) = 0 (2) Cayley-Hamilton theorem AM(M)=n^=1(M-Afc)=0 (3) components of a matrix
(4) function of a matrix F(M) = E f = 1 F(Xj)
Z.M.
288
A.3
Formulary
Stochastic matrices (1) stochastic N x N matrix M: defining requirements (i) 0 < A f 0 6 < l , V 0 , 6 = 1 , 2 , . . . J V
(ii)EliM o 6 = l property: M has at least one eigenvalue = 1; if there is only one state with unit modulus, M is totally regular the discrete finite Chapman-Kolmogorov equation is = Za=l Ec"=l MacMcb = 1.
Za=l M\b
(2) Markov chain a column vector p is a probability distribution if p > 0 and Yla=i P° =
l
evolution equation (Markov condition)
P (n+1) « = E £ i Mab p<»>«, average of / on a distribution p /p
=
l~ia faPa
detailed balancing Mab pb = Mba
pa
a state a is positive-recurrent if it is recurrent and its expected return time ta = inf{n | Mnaa > 0} is finite; a chain is positiverecurrent if all its states are; for a totally regular positive-recurrent chain holds the ergodic theorem: i
" -
1
_
lim - V / pW = /„(„., = Y,U k=0
a.
P^-
289
Circulant matrices A.4
Circulant matrices (1) general form: Aij = a,i-j, or
\
a\
ajv
CLN-I
a-i
ci2
ai
aw
a3
ajv_i
ajv-2
aiv-3
.
.
.
ajv
/
(2) cyclic convention a^v
=
flo and
a/v+ r = a r = a,—^
(3) relations between eigenvalues Afc and entries ay.
(4) with w = el(2*/N\ (5) projectors
t h e eigenvectors are U(fc)s = ^ 7 ^ -
[P(k)]rs
= Hk)rU*{k)s = Mrk M fc -/ = i
W-"^
(6) convolution JV 2
(a* )r
TV
= J2asar-s
= (a*a)r
; (a*m)r = X ] < ( m _ 1 ) °r
s=l
(a * 6)„ = X)m=l
a
mK-m
(7) t h e powers of A have entries ( A m ) r s = ( a * m ) r _ s (8) relations between powers A
fc - L j = l
w J f l
j
. Oj
-]vZ^fc=lw
(9) shift operator U: Urs = 5r,s+i
; (UA)ij =
Ai-ij
J
\
290
Formulary
any circulant is of the form
(10) projectors and shifts
P(fc) = £Et^- f c "U" ; U"=E£LX"P« (11) product of two circulants A and B :
AB = E^=1(a*&)nU" (12) the V operator: V = Udiag, or (
u
0
0
0
w2
0
0
0
0
\0
0
•••
w3
0
•••
0
0
w4
0
0
0
0
0 0 0 0
\
wN )
0
(13) Weyl realizations of the Heisenberg group V m U™ = ujnm U™ V m (14) U and V operators
UK) = \vk+i) ; v|ufc) = \uk-i) (15) the Schwinger operators S (m ,„) = ei*mn entries ( S ( m , n ) ) r s = properties
U m V n = w^
UmVn
&r-.,mu>n^-mM
S(0,0) = I , S ( m , „ ) = S ( m , „ ) = S ( - m , - n ) • ( S ( m , n ) S ( n s ) ) S(fe,/) = S(TOi„) S(NtP)
= ( - ) p S ( 0 , P ) ; S(p,N) ^(m,n)
=
S(pm,pn)
(S( r>s )S( fej ;))
= ( - ) p S ( p , 0 ) ; S{N,N) i t r S( m > n ) =
= (-)JVS(0,0)
NSno6mo
Circulant
matrices
291
general matrix A is written — JV / j •™(m,n)^(m,n) (m,Ti)
A
(m,n) = t r [ S j m n ) A ]
notation m = (m1,m2), n = (ni,n 2 ), r = ( r i , r 2 ) , 0 = (0,0), - m = ( - m i , - m 2 ) , m + r = (mi + n , m 2 + r 2 ), m x r = (mir2 - m 2 r i ) , m • r = (miri + m2r2) C
—
ei(ir/W)mim2Tjmiym2
A = lEAras» (16) in terms of the cochains Sm \vk) = eia^k^
\vk+mi)
; a1(i;m) = J(2fc + m,)m2
S r S m = e 2iQ2 ( m - r ) S m S r = eia^m^
Sr
a 2 ( m , r) = — (mir2 - m2r{) = — (m x r). product of two operators is AB = ^
E
m
E „ ^ m S n e i Q 2 ("•«") S m + n
££
_
\ ^
J
4m^P-me«"2(p,m\
(17) Weyl-Wigner transformations Aw(q,p)
= j dadb
i{aq+bp) e
A(q, p) = I da db ei{acl+bp)
A(a,b)
A{a, b)
292
Formulary
Weyl prescription for the commutator t A > B ] = TX E E ^ p
m B P _ m
2isin[a 2 (p,m)]S F
m
continuum versions [S a ,Sb] = 2isin
[A, B]p = j
a x b >a+b
f da I dh A(b) B(a - b) i
a x b
(18) Matrix differential geometry: exterior derivatives defined by dM(X) = X(M) = adxM
= [X,M]
(P+ l ) ( d E ) ( e i , e 2 , . . . , e p + i ) = eiE(e 2 ,e 3 ,.. . , e p + 1 ) e 2 E(ei, e 3 , . . . , e p + i ) -\
h ( - ) p e p + i E ( e i , e 2 , . . . , ep)
-H([ei, e 2 ], e 3 , . . . , e p + 1 ) + E([ei, e 3 ], e 2 , . . . , e p + i) -E([ei,e 4 ],e2, . . . , e p + i ) + . . . + ( - ) p E ( [ e i , e p + i ] , e 2 , . . . ,e p ) +E(ei, [e2, e 3 ] , . . . , e p + i) H
h (-) p E(ei, e 2 , e 3 , . . . . , [ep, e p + i])
natural Poisson bracket { A , B } M = i[A,B] (19) in basis S m the derivative algebra is generated by the operators e„ — ad(iS m ) = ad i S m (for all S m •£ S 0 ) the general commutators are [e r ,e m ] = 2sin[a 2 (r,m)] e m + r . structure coefficients C p r m = 2sin[a 2 (r,m)] S^+T
Bell polynomials
293
(20) quantum symplectic form
« = \ Ep, m > r s P c*>mr e™ e* = £ P i m sin[a 2 (m l P )] s p em e*-m components
P
hypermatrix it = (fi r m ) has an inverse r'j
'
_ _ L2 OJ c i — _ i p«*2(i,j) c i + j _ J _ — N jyT e >-> — jy2
i(ir/AT)ixj o i + j >->
e
property: rU
_
e 2J«2(i,j)
jji
(21) the (AT — 1) generators {
for
<Ti
2 = 1,2, ...,iV -
|i — j ' | >
1
2
for N = 3, this is equivalent to the Yang-Baxter equation rfjk
rtib
rtca
nij
jyck
nab
with R^mn = B3''mm this is the same as B12B23B12 = B 2 3 B 1 2 B 2 3 .
A.5
Bell polynomials (1) Faa di Bruno formula f\9(x)] = £ ~ = i 7t E L i Bnfc(5i,52, • • .,gn-k+i) B ldxn i - / [ 0 ( z ) ] ] x = o = E f c = l » f c ( 9 l ' 3 2 , - - - , 9 n - f e + l )
fk /fc
(2) Multinomial theorem -ifc
E ^ l f
^'
=E~=fc^T
Bnfc(Sl.52,-",5n-fe+l)
294
Formulary
(3) Complete Bell polynomial Y„[u;g] = YTj=iui
^nj(9i,92,---,gn-j+i)
(4) Notation B„fc(gi,g2, • • • ,5n-fe+i) Bno[g]
= =
Bnfc[s(z)] = Bnfeb] = Bnfe{g.,} 5n0
(5) Properties »iVm[5] = f m—MX)}™) \ -
/ JV
= —i ®Nl[9m} m -
B„l[s] = ffn ; B n2 [ 5 ] = | J™"* ( .) 9j 9n-j
B„„[g] = 9? ; »n,„-i[9] = ( 2 )
j=l
m
=l
N^/
Bjvm[e(a;) = 1 ] =Bjvm(l,0, . . . , 0 ) = <5jvm mB n m [g] = YX~=m-x {nk)9n-k *k,m-i\g] (6) Stirling numbers: generating functions first kind c(x - l)(x - 2)...(x - fc + 1) = ©fc! = E"=o «fc ^ '
TJtMi + ^ E S U S ^ second kind
** = EjU ^ ' ^ - W* - 2)... (a: - j + 1)
Bell
295
polynomials
fc! \e
~ 1> — 2^n=k n! °n
relations between the two kinds
E nk=m
(fc) c ( m ) _ ^ n S
" ^fc
—
q(k)
(m) _ c m
JL,k=m °™ *fc
—
°n
(7) Stirling numbers: relations with Bells B nfc [ln(l + x)] = Bnfc(0!, - 1 ! , 2!, - 3 ! , . . . ) = B»fe{(-r1(j-i)!} = #
)
Bnfc(0!, 1!, 2!, 3!,...) = B nfc {(j - 1)!} = |*W | = (-)"+ fc s «
B„fe[eu - 1] = B nfc (l, 1 , 1 , . . . ,1) = B„ fc {l} = Snk) (8) Bell number
"(«) = E L i ^
; "(" + 1) = ELo (fc) ^ («)
(9) Reciprocal series (1 + * ( * ) ) - - 1 = £ ~ = 1 £ E ^ - ) * * ^ B m ,[ fl ]. (10) Logarithmic Bell polynomials Ln[9) = E ( - ) " _ 1 ( f c - !)' Bnfctff] ; ln[l +g(x)} = £
^L„[y]
fc=l
(11) Power Bell polynomials k
^ f c ) [ff]= ( " * * )
Bn+fc,fc(1.2ffi,3ff2,...)
fc! B,^] = £ (JV^-PWM ; L„b] = £ ^
P«M
296
Formulary
^
(50 + E ~ ! %ti)m = £ ~ = 0 £ Y.U
i
®N,m-k\9) •
(12) recursive relations
4E; Y
" —1 /•
(>
= 1
3
-E;:„'W[3]
n[w;ff] =
E"=lBnj(ugi,Uff2,---,Uffn-j+l)
=
E;Zo(n-p1)9n-pEPm=0^m[u9]
_ i\
P
"—1 /
= H ( n )ffn- P X^ M m ] B P m ^ p=0 \ P / m=o
=
_ i\
Sl(n j5r„-pYp[M;5] p=0 \ P /
(13) Leibniz rule
B^[F(.) OWl - ( « S S ( » ^ [ f l " - « l f if n>2* 0 for fc < n < 2k .
(14) Lagrange inversion formula n" - l*
1
fc(n
+ fc-l)! (n-l)!
fc=l
9n
-
^-
gi2^k=l\
1 >
)
ff2 ff3 ff4 Jn
(n-l)!
-llfcV25i,35l'4Sl
JB
"- 1 - f e \(fe+l) S l f c + 1 /
= ^EZ;i 1 (-)*»* !>n-l,fc
.(9 < - 1 > ! = ^ C I ' . ) E ; : O ™ B . - „ , »
git
-"•(#)]•
Bell
A.5.1
Orthogonal
297
polynomials
polynomials
(1) Hermite polynomials Hn(x) (with the conventions of [22]) e
-t2+2tx_1=E°o=i£tfn(x)
Hn(x) = Y„[l; 2tx - t2} = E ; = i Mnj[2tx - t2} = E"=i ®nj(2x, - 2 , 0 , 0 , . . . ,0) = 2xHn-1(x)
- (n -
l)Hn-2(x)
(2) Legendre polynomials Pn{x) 1
_ V°°
P
tn
(T)
i = £~=i££^iEWj!^W}
exp Vl-2tx+t2
(3) Gegenbauer polynomials Cfi{x)
C - 1 m TCI - rr)
JL
C%(x)=
{
T m=[N/2] . ..
-jl i (1 — a — my.
N
J2
PN(X) = ±
m=[N/2]
^-BNm(2x,-2,0,0,...,0)
1 2m (2m - 1)!
l(2x,-2,0,0,...,0)
N
h AT
^
; m
vi m=[JV/2] Z^ 2
-: MNm\2x t - t2}
(2m-1)!
[
J
(4) Chebyshev polynomials of the first kind Tn(x) l-t2 1 - 2 / + ft ~ Tm[Tn{x)\
1 = 2
°° 5 Z Tn{x) tn ; T„(x) = cos [narccosx]
= [Tm o Tn}(x) = Tmn{x) = Tn[Tm[x)\
T n m(i) = \TnoTnoTn...oTn]{x)
=
T<m>(x).
;
298
Formulary
When defined,
B[T<m>] = BlTnoTnoTn...oTnj
= Bm[Tn]
m times
A.5.2
Differintegration,
HI
derivatives
T~1
of Bell
J2,
d_ -®nk{g] = f j
Tn-q
polynomials ™
B„-i,fc-l[ff]
dgi n—m+1
d
.7 = 1
A.6
^
Determinants, minors and traces d e t X = exp{tr[lnX]}
det [I A + z A] = J2 ^ T ^ j=0 1
det A = ^
E
% m {(-) f c _ 1 (fc " 1)' tr(A fe )}
m=0
V-JV
E^= 0 BiVm{(-) f c - 1 (fc " 1)! tr(A fc )}
tr(A») = £ 9 -
EZ=i(-) f c _ 1 (fc - 1)' BnibO'! det,-A}
det,- A = i E i = o % - { ( - ) f c _ 1 ( f c - 1)! tr(A fe )}
N -*
J " 5„
N
„ = 1 1 i = l1 52
N
4
4 rl
-^•5251
' S2'S2
4
4
AS1
s
•"•SI
s
^
Sj
i
3
„ Sj—1 .—1
299
Determinants, minors and traces
A.6.1
Symmetric
functions
S [ x ; t ] = E n = 0 ^W1
N
= n l l ( l + *nt)
N
P[x;z] = YtPM*'-1 = E 3 ^ 1 ~ x^~' = -Tz
lnS[x;
~Z
j=\
J'=I
l
_ 1 = n;=i (i -\~*,-*) = w^
H [x;<] = ^NE ;=„ w
N
p[x, z] = £ e ^ = £ 1 f > z)» = J ; 3 Pn M z» _7 = 1
n=0
n=0
j—1
c™[x] = i E ? = i B„,{(-) fc -Hfc - 1)! PfcM}
Pk[
X] =
( - ) fc-i Jk~[y. E ( - ) J _ 1 ( J - 1)! Bfej-{n! e n [x]} (for k > 1)
n!e„
1
Pi P2 P3
Pi P2
Pn
Pn-1
0 2 Pi
0 . . 0 . . 3 . . .
0 0 0 0
• Pi
E;=0 ( - r J pn-^ = Eiu E;=0 ( - r J *r^ = o
0.1
1 ai
0 2
as
a-2
ai
O.N
&N-1
ai N
EIWH'--1^-!)!^}
0 . . 0 0 . . 0 3 . . 0 . 0 . ai
300
Formulary
0 2
1 9i -92
9i -92 93/21
Nr _, BjVmfe} =
0 0 3
9i
=1 (
\N-1
v 1
det A = -^
A. 6.2
(IN
(
(N-iy.
^ 1
trA tr A 2 tr A 3
1 tr A tr A 2
trA^
trA^-1
\N
gjv-i
(N-2)
0 2 trA
0 0 3 tr A 2
0 . . 0 . . 0 . . 0 • • 9i 0 0 0 0 tr A
Polynomials
• expressions P(z) = CLNZN + a,N-izN~l
h aiz1 + a0
H
= aN Uj=i(z ~ rj) = ( - ) aivejvW £[r*,z *N
= (-)NaNeN[r]
£j=o « * [ ' V = aN E ^ - r ^ i v - i M *
p(o) n;=i(i - z/ri) = {-)NaN eN\v} n;N =1 (i - z/rj) T
N
N
= a0 Y[(l-z/rJ)=aN j=l
£ 1=0
l
zN-l
— '
£ > . » • { - ( * ~ 1)! P*M} r=0
• coefficients a.j = (-)N~jaN a
i = «iv
feS
ejv-j [r] ; a0 = P(0) = (-)NaN
eN[r]
EiL"o J '^-i,r{(-) f c - 1 (fc - I ) ' Pk[r]}
Viete's formula a
i = °JV feS)T ^ = 7 ' B i v - J > { ( - ) f c - 1 ( f c " I ) ' PfcW}
Determinants,
301
minors and traces
• discriminant N-l l JV-1 2
r r
A(P) = det(rf "') = {-)"p[r] =
N-l N
Characteristic
polynomials
and
„N-3 l „N-3 2
r
r
r
N-2 N
r
A.6.3
„JV-2 l „JV-2 2
r
r
N-3
T
N
classes a k
A A (A) = det[AI - A] = Zk=o
^
= nf=i(A - A,-) = E j L o H ' ^ - ^ M = d e t [ A I - Al = Ef=o * " " ' * # E'TO=o%m{(-)fc-1(* - I)' tr(Afc)} =E
.7=0
AJV_J
'^r E B^{(-)fc_1(* - 1)' PfcW} = jE= 0 A ^ ( - y e,[A] m=0
(1) Lie Algebras Invariants multilinear invariant symmetric tensors Va%..an
= M ^ E B ™ { ( " ) f c _ 1 ( f c - ! ) ' t r ( J 0 l Ja 3 • - •/„*)} TV. t—' m—1
(2) Characteristic Classes if F is a matrix curvature 2-form, the fc-th Chern class is 7r*(cfe)
= ( ^
= 0 fc (F) = efc[iF/27r]
E ™ = 0 B f c m { ( - ) r - 1 ( r - 1)! t r ( F ' ) }
Chern total character oo
N
.u
ch(E) = tr(exp [ - F ] ) = £ - — fc=ov ;
trF* = £ fc=0
fc-th Chern character
.-* (27T) fcfe!
tlF
=
! fc!
Pfc[ F/27rl
*
-
Pfc [iP/2»]
302
Formulary
relations between classes and characters (for j , fc > 0) C
^ E ) = 77 £ JJ
(-)"-\k
B
^ { ( - ) f c _ 1 ( f c - 1)' W chk(E)}
m=0
- 1)! chk(E) = 1
^ ( - ) ' _ 1 0 ' - 1)' Bfcj-{r! 3=0
A. 7
Bell m a t r i c e s (1) Fundamental property B[g}B[f]
=
B[fog]
(2) Eigenvectors of B[g] B
[ # ] *>(fc) = fll *>(*) 5 [ B [ p ] U(fe)] r = 3 l «(fc)r
«(fc)» = B i f c [F] wit/i F[ff(a;)] = B_1
[ffl w(fc)
ffiF(a;)]
=9ikV(k)
(3) Projectors (a = #i) (see also §A.7.1) (Z f e ) n r = B„fc[F] B f c r [F<- 1 > ] = vWn
B f c r [F < - 1 > ]
N
Z,[B] = 53[A_1]ifc B>fefc fe=i e-oi^U-fc)(_)jejv_.[a] E fifc-i Er=lE.=0°"(-),eiV-.M (4) arbitrary power of B
clN)(t) = Zl1a*ATl}
cr(E)}.
Bell
303
matrices
(5) continuum iterate function
A.7.1
Schroder
equation
(1) for a function F , given g(x) and a constant c, the equation is F{g(x)} = c F(x) (2) matrix version B[g] B[F] = B[9lF] (3) relation to projectors (Z f c [B]) n r =B n f c [F]B f c P [F < - 1 > ]. A.7.2
Fredholm
theory
(1) equation X=J+AKX=
I 3 1 — AJv
(2) Fredholm determinant oo.JV
D(\) = det(I - AK) = 1 + ^
\nDn
n=\
Dn = - n- f*•—' > K ' Dn^ = ^n ! *—' T B nm {(-)>- 1 0' - l)!trK''}. j=l
m=0
(3) Resolvent and Green's function 00
R(A) = £A"K» 71=1
=
\K _ _ _ ;
1 (I-AK)-1 = — x
N. '
304
Formulary
A.8
Statistical mechanics
A.8.1
Microcanonical
ensemble
(1) double signs: upper, Bose; lower, Fermi (2) ideal quantum gas: normalization of the iV-particle state N
,PN\Pl,P2,---
x
,PN >=
]^T ^ l & N m ^ y
- 1
^
-IV-
$j}
N-i J r N r
"jvE^^
^ ' P 2 ' " " >Pr\Pl,P2,---
^ ~
,Pr)
r=0
(3) phase space volume N
N
N
RN{E,P,V) =-fa J H
«=i
AT
*£(±) N—m
dU -1)'
*j} •
m—l
Prakash-Sudarshan formula (b, / , B: bosons, fermions and Boltzmann particles) N
N
R%f)(P, V) = ±Y1 W"'™ E m=l
R
™](P> V) E^{
33U - I)'}
m=l
recursion relation
iJjv(P,V) = %$
^(i)^ fc=i
A.8.2
Canonical
1
/
d\RN-k{P-lq,V).
^
ensemble
(1) ideal quantum gas: partition function QN(P,V)
= JdE e~pERN{E,V)
formal cluster integral for bosons and fermions: bj = i—*-,•5/2
Statistical
305
mechanics
(2) real gas: semiclassical partition function QN
n
(/W = M /
d3Xjd3pi
-0H
h3 N
N\\\
N
JV-1
3
l[d J _i=l
3N 3N
r n n [!+/( «)i i=l j=i+l
J
above, # and the Mayer functions fy = /(r^-) are
^ = ES i=l
+
E E ^('i-'i) ! /(r«) = e"^'«) - 1
i = l j'=z+l
cluster integral (thermal wavelength: A = v/^rmfcT ) .
6n =
.
n!A3("-DV /
n—1 n
d3rid3r2
•••
d3rn
/ij 5 P 1=1 J=2 + l
notation: u = -^, bj = vbj = jgbj (3) relations between partition function and cluster integrals QN(P,V)
=
~YN
A3
bkx
\Yl
JV-1
7rH(N-P)b*f-pQp
= p=0
fc=l
JV
N
B
Q/v(/3, ^) = i E ^«0'! &;> = i E B » - " m=l
m=l
£*:**
.*;=!
^^^ELiH^Hfc-i^BnfciJiQ,}. A.8.3
Grand canonical
ensemble
(1) expressions for the partition function (fugacity: z — e^M) Z(J3,V,ri =tr[e-Wtto-ri*i]
=Y[
£< -j3(e-M)"
= ex P { 3 £ £ £^ 1 i ± i pe-^-^}
306
Formulary
= {11;-z}«P{pET=1^i^E«e-^} neglecting zero-energy terms see, v,u) = i + ZN=I *NQN(P, v)
Matsubara's formula
(2) finite approach (M particles) AM = diag [-r1 \ -r2 1,-r31,... , -rMl] QM = det A M vjbj = ( - ) ' - 1 tr A J M with D = Fredholm determinant with K = A, M
s (M) (z) =
M
j - 2ivgiv =
d e t [ I + zAM ]
N=0
fc=l
,
-,
= J ] (l - - J = D(-z) ^
rfc
^
(3) virial approach density and pressure related by v °° | p = In 2 = v]Th*
^) °° ; ^ ( * ) = nV = z ^ l n S = w 5 ^ J V '
J> = 1
.7 = 1
virial equation i& =
1
+ Efc> 1 «fc+iM 3 ) f c
Mayer's formula "k+i=ZU(-)\k+mk% A.8.4
Ideal relativistic
quantum
(1) parameters Compton length Ac =
he
» fcj (2!6 2 ,3!6 3 ,...)gases
Statistical
307
mechanics
cube of relativistic thermal wavelength -0mc2
A6(f3) =2^
2
a^„2 (3mc
/
N3
ftc
2
K2((3mc )
2
\mc
an integral: -fa' f(r,/3) = ± Jd3pe-i
1 A (/?) 1 + ( l ^ ) 2
2 2
e-P[(P
c +m2ci)i/2-mc2}
K2 [Pm^y/l + (^-c)\
3
K2(f3mc*)
(2) canonical partition function QN(V,(3)
=^ L
i ^ S QAT-mW/J)
= ilELiB^{(±)fe-1(fc-l)!A^j} (3) distribution function (with / ^ = / ( r , — Tj,(3)) 1 M Q A/
* m=l
AT
J2(±r~1fuf23...fmiQN AT T71=l
(4) pair correlation function
(5) grand canonical partition function
A3(j/J)
308
Formulary
(6) two-dimensional gas Bose and Fermi statistics
Qs(ft5) = l C ^ | } = l^l Sm {(£(,-l)!|} braid statistics
' m=l
I
J
)
Sources Besides Gradshteyn & Ryzhik [22], Riordan [72], and the site [74], we have extensively used Abramowitz & Stegun [119], Prudnikov-BrychkovMarichev [120] and Karatsuba [l2l].
Bibliography
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Index
adjoint of a matrix, 5 algebra defined, 5 Hopf, 120 infinite, 123 Lie, 5 algebra invariants, 200 alphabet, 5, 153 eigenvalue, 196 letters, 5 noncommutative, 186 of roots, 191 reciprocal, 193 with one missing letter, 208 word, 5 alphabets relations between, 191 alternating function, 192 annihilating polynomial, 4 approximants Pade, 195 area-preserving transformation, 124 asymptotic distribution, 29 equilibrium, 29 attractor of a Markov chain, 54
average on a distribution, 29 basis matrix, 93, 106 Bell matrices eigenvectors, 180, 226 group of, 167 introduced, 165 not normal, 168 Bell number, 159 Bell polynomials and orthogonal polynomials, 178 partition function, 243 complete, 153, 154, 177, 199 definition, 151 derivatives, 181 logarithmic, 164 matrices of, 165 notation, 153, 154 partial, 153 power, 165 properties, 155 recursion relation, 177 special cases, 160 bialgebra, 120 Boltzmann function, 70 Borel transform, 155 Bose function, 256 315
316 bracket Moyal, 103, 114, 119, 124, 133, 137, 146 Poisson, 100, 103, 111, 124, 132, 133 quantum, 129, 138 braid, 146 equation, 103 gas, 265 group, 140, 236, 264 presentation, 265 statistics, 264 and superconductivity, 264 calculus fractional, 180 canonical ensemble, 239 partition function, 260 Casimir operators, 201 Cayley-Hamilton theorem, 4, 6, 84, 197 Chapman-Kolmogorov equation, 28 character of a group, 123 defined, 118 characteristic class, 203 equation, 3 matrix, 3 polynomial, 3, 196 forms, 196 Chebyshev polynomials, 179, 220 Chern characters, 203 classes, 203 circle theorem, 272 circulant and Fourier transformation, 84 application of, 182 basis, 88 defined, 83 determinant, 84
Index projectors, 87 semi-, 169 stochastic, 53, 96 class characteristic, 203 Classical Mechanics, 99 cluster expansion, 255 integral, 241 cofactor, 198 and minor, 198 composition function, 151 condensation, 267 Lee-Yang theory, 270 Mayer theory, 268 connected diagram, 155 graphs, 155, 263 continuous iterate, 211 continuum time in Markov chains, 60 convolution, 87, 90 twisted, 102, 114, 115, 123 coordination number, 75 correlation quantum, 251 correspondence principle, 102, 112, 117 critical phenomena, 267 cycle decomposition, 245, 280 cycle indicator polynomial , 234 cyclic group, 85 Darboux theorem, 128 delta Kronecker, 85 derivation, 129 derivative cohomology, 131 detailed balancing, 67, 68 and equilibrium, 70 determinant, 183
Index and characteristic polynomial, 197 and fermions, 232 and traces of powers, 185 circulant, 84 Fredholm, 276 Vandermonde, 193 diagram connected, 155 differential forms matrix, 130 differintegration, 180 discriminant of a polynomial, 195 displacement operator, 89 dissipation, 66 distribution equilibrium, 29 probability , 26 distribution functions, 251 doubly-stochastic matrix, 50, 96 duality Fourier, 122 Pontryagin, 122 Tanaka-Krein, 122 dynamic quantity classical, 99 and quantum, 113, 117, 140 quantum, 107 eigenvalues, 4 alphabet, 196 of a Fredholm operator, 282 of Bell matrices, 219 of normal matrices, 17 of stochastic matrices, 47 symmetric functions of, 187 eigenvectors of Bell matrices, 226 of circulant s, 86 of stochastic matrices, 46, 62 ensemble canonical, 239
grand canonical, 253 microcanonical, 228 entire function, 279 equation braid, 103 Chapman-Kolmogorov, 28 characteristic, 3 Liouville, 127 master, 61 Maurer-Cartan, 129 Schroder , 222 secular, 3, 197 Yang-Baxter, 103, 120, 143 equation of state Boyle-Mariotte, 268 Mayer, 268 virial, 256 equilibrium, 46, 54 and detailed balancing, 70 asymptotic, 29 distribution, 29, 53 unstable, 66 ergodic theorem, 67, 74 eternel retour, 66 evanescent root, 53 Faa di Bruno formula, 153, 161, 181 Fermi function, 256 Feynman diagrams number of, 242 Fibonacci numbers, 164 formula Faa di Bruno, 153, 161, 181 Lagrange, 173 Leibniz, 172 Matsubara, 263 Mayer, 256 Newton, 191 Prakash-Sudarshan, 237 Viete, 194 Wronski, 169
318 Fourier duality, 122 Fourier transformation and circulants, 84 and cyclic groups, 85 and quantum groups, 122 double, 95 operator, 113 fractional calculus, 180 Fredholm alternative, 275 determinant, 276 minor, 185 first, 278 theory, 185, 245, 274 free energy Helmholtz, 247 Frobenius spectral theorem, 47, 49 fugacity, 254 function Bose, 256 composition, 151 entire, 279 Fermi, 256 Heaviside, 195 inverse, 157 of a Bell matrix, 207 of a circulant, 91 of a matrix, 8 fundamental group, 265 fundamental theorem of symmetric functions, 187 gas braid, 265 ideal, 245 quantum, 248, 256, 260 real, 243, 256, 267 relativistic, 248, 260 ultrarelativistic, 262 Gegenbauer polynomials, 178
Index generating function Chebyshev polynomials, 179 cycle indicator as, 234 for Bell numbers, 160 for Bell polynomials, 153, 177 for conditional partition numbers, 211 for Hermite polynomials, 178 for Legendre polynomials, 178 for number of partitions, 153 for Stirling numbers, 158 for symmetric functions elementary, 187, 208 homogeneous, 187 power-sum, 188 Gegenbauer polynomials, 178 manipulations, 162 generators of the braid group, 141, 265 of the symmetric group, 235 Gibbs potential, 253 Gibbs-Di Marzio law, 34, 39, 78 glass binary, 72, 77 Golden Ratio, 164 grand canonical ensemble, 253 partition function, 188, 253, 260 graphs connected, 155, 245, 263 group braid, 140, 264, 265 compact, 201 complex linear, 203 cyclic, 85 Heisenberg, 104, 108, 111, 116, 123 Lorentz, 202 of permutations, 26, 234 orthogonal, 202 Poincare, 202 rotation, 201 semisimple, 201 symmetric, 26, 234
319
Index unitary, 203 Hamiltonian approach to Markov chains, 61 harmonic analysis, 122 harmonic oscillator, 221 Heaviside function, 195 Heisenberg group, 104, 108, 111, 116, 123 Helmholtz free energy, 247 Hermite polynomials, 178 Hermitian matrix, 5 Hermitian conjugate of a matrix, 5 homotopy integration by, 256 Hopf algebra, 120 hypermatrix, 49, 136 hypersensibility to initial conditions, 66 ideal gas, 245 idempotent number, 163 imprimitivity, 48 index, 71 index imprimitivity, 48 initial conditions hypersensibility to, 66 insensibility to, 56 interpolating polynomial Lagrange, 11, 12 invariants Casimir, 201 Killing, 201 of a Lie algebra, 200 of a matrix, 199 inverse of a function, 157 of a matrix, 9, 198 of a series, 173 involution, 193
Ising model and circulants, 87 and detailed balancing, 70 stochastic approach, 58 iterate continuous, 211, 215 Jacobi identity, 5 Kerner model, 23 Killing-Cartan metric, 201 Kronecker delta, 85 Lagrange interpolation, 11, 12 series inversion, 171 Lah number, 163 law Gibbs-Di Marzio, 34, 39, 78 Lee-Yang circle theorem, 272 matrix, 274 phase transition theory, 270 Legendre polynomials, 178 Leibniz formula, 172 Lie algebra, 5 derivative, 128 Lie algebra invariants, 200 Liouville canonical form, 128, 138 equation, 127 Lissajous curve, 221 logarithmic Bell polynomials, 164 logistic map, 166, 219 extreme, 222 Lorentz group, 202 Lyapunov exponent, 218 main root, 53, 56
320
Markov chain, 28 cyclic, 51 ergodic, 67 irreducible, 51, 67 non-cyclic, 51 positive-recurrent, 75 recurrent, 66 reducible, 51 regular, 52 reversible, 70 totally regular, 51 condition, 28 process, 28 master equation, 61 matrix adjoint, 5 basis, 106 Bell, 165 characteristic, 3 circulant, 83, 182 degenerate or not, 4 differential geometry, 127 eigenvalues, 3, 196 functions of a, 6, 207 Hermitian, 5 Hermitian conjugate, 5 imprimitive, 48 invariants, 199 inversion, 9, 172, 198 Lee-Yang, 274 linear basis, 93 non-negative, 25 irreducible, 47, 49 normal, 5, 17 positive, 25, 47 primitive, 48 projectors of a, 6 projectors of a normal, 17 reducible, 26, 27, 50 completely, 26 reducible stochastic, 29 Schwinger basis, 105 spectrum, 3
Index stochastic, 27 doubly, 50, 96 regular, 52 totally regular stochastic, 51 unitary, 5 unitary basis, 93 Maurer-Cartan equation, 129 Mayer equation of state, 268 function, 240 theory of condensation, 268 virial formula, 256, 262 metric, 134 microcanonical ensemble, 228 minor and cofactor, 198 Fredholm, 185, 278 model Ising, 58 Kerner stochastic, 35 Lee-Yang, 270 Mayer, 268 Potts, 272 Moyal bracket, 103, 114, 119, 124, 133, 137, 146 multi-phase system, 62 multinomial theorem, 152 Navier-Stokes equation, 219 Newton formula, 191 no-fluctuation condition and detailed balancing, 72 noncommutative geometry, 100 normal matrix, 5, 17 number Bell, 159 Fibonacci, 164 idempotent, 163 Lah, 163
321
Index operator displacement, or shift, 89 optical theorem, 60 orbit of a distribution, 55 orthogonal polynomials, 178 Pade approximants, 195 partition, 152 conditional, 210 partition function canonical, 260 for a braid gas, 267 for a real gas, 243 grand canonical, 188, 253, 254, 260 recursion formula, 246 Widom, 247 Pauli matrices, 109 periodicity, 63 permanents and bosons, 232 permutation, 27, 234 group, 26 Perron theorem, 47, 49 phase multiple, 62 phase space, 99 quantum, 109 volume, 228 phase transitions, 267 theory, 270 Mayer theory, 268 Poincare group, 202 map, 218 Poisson bracket, 100, 103, 111, 124, 132, 133 quantum, 129, 138 polynomial annihilating, 4 characteristic, 3, 196 Chebyshev, 179, 220
cycle indicator, 234 Gegenbauer, 178 Hermite, 178 in general, 193 Legendre, 178 minimal, 5 orthogonal, 178 Pontryagin duality, 122 positive-recurrent chain, 75 state, 75 potential Gibbs, 253 Helmholtz, 247 Potts model, 272 power Bell polynomials, 165 Prakash-Sudarshan recursion formula, 237 prescription Weyl, 102, 112, 119, 146 presentation braid group, 265 defined, 235 symmetric group, 235 probability distribution, 26 projector, 6, 8 circulant, 90 closed expression, 16, 208-210 for degenerate matrices, 14 for nondegenerate matrices, 10, 12 of a Bell matrix, 213, 224 of a normal matrix, 17 quantization, 120, 125 quantum degeneracy, 262 quantum group, 120 Quantum Mechanics, 17 discrete, 92, 99 Weyl-Wigner picture, 101
322
random walk, 78 general, 96 simple, 97 rank of a Lie algebra, 200 of a matrix, 200 reciprocal series, 161, 169 recurrent Markov chain, 66 state in a Markov chain, 63, 66 reducible matrix, 27 regular stochastic matrix, 52 relativistic gases ideal, 260 real, 243 resolvent of a matrix, 4, 9 other sense, 275 retour eternel, 66 reversibility, 67, 70 reversible Markov chain, 70 root evanescent, 53 main, 53 rotations group of, 13, 201 rule Leibniz, 172 Schroder equation, 222 Schwinger basis, 105, 106, 109, 115, 119, 144 on Quantum Mechanics, 108 secular equation, 3, 197 semi-circulant matrix, 169 series generating functions, 162 group of, 157 invert ible, 157 Lagrange inversion, 173
Index non-invertible, 168 reciprocal, 161, 169 truncated, 154 unit, 219 virial, 256 shift operator, 89 spectrum of a matrix, 3 of a stochastic matrix, 45 star, or twisted product, 101, 115, 116, 118, 119, 138 state aperiodic, 67 independence of initial, 56 positive-recurrent, 75 recurrent, 66 transient, 51, 66 Statistical Mechanics, 227 canonical ensemble, 239 grand canonical ensemble, 253 microcanonical ensemble, 228 Stirling numbers, 158, 171, 225 meaning, 159 stochastic circulant, 96 matrix asymptotic, 53 defined, 27 superconductivity and braid statistics, 264 symmetric functions, 186 complete homogeneous, 187 elementary, 186 fundamental theorem, 187, 192 power-sum, 188, 273 symmetric group, 26, 234, 247 generators, 235 presentation, 235 symplectic form, 127, 132, 138 structure, 132 Tanaka-Krein duality, 122
323
Index theorem Cayley-Hamilton, 4, 6, 197 Darboux, 128 ergodic, 67, 74 Frobenius, 47, 49 Lee-Yang circle, 272 multinomial, 152 optical, 60 Perron, 47, 49 van Hove, 272 thermal wavelength non-relativistic, 238 thermal wavelength relativistic, 248 time reversal, 70 totally regular stochastic matrix, 51, 53 traces, 183 relations to determinants, 185 transformation area-preserving, 124 braid, 264 canonical, 124, 128 cyclic, 62 Fourier, 81, 86, 109 double, 95 general, 122 operator, 113 group, 121, 200 permutation, 264 rotation, 13 similarity, 17, 87, 199, 239 Weyl-Wigner, 102, 109, 112, 138 transience, 63 transient state in a Markov chain, 66 transposition elementary, 26 turbulence application to, 219 twisted convolution, 102, 115 twisted, or star product, 101, 115, 116, 118, 119, 138
unitary basis, 93 matrix, 5 Schwinger basis, 105 vacuum multiple, 62 van Hove theorem, 272 Vandermonde determinant, 193 vector field, 128 vector space dual to, 18 Viete's formulae, 194 wavelength thermal, 238 Weierstrass expression for a polynomial, 194 Weyl, 108 prescription, 102, 112, 119, 146 Weyl-Heisenberg group, 103, 116 Weyl-Wigner transformations, 102, 109, 112, 125, 138, 139 Widom partition function, 247 Wigner density, 113, 115, 116 function, 101, 113, 116, 118 word, 5 Wronski formula, 169 Yang-Baxter equation, 103, 120, 143
SPECIAL MATRICES OF MATHEMATICAL PHYSICS Stochastic, Circulant and Bell Matrices This book expounds three special kinds of matrices that are of physical interest, centering on physical examples. Stochastic matrices describe dynamical systems of many different types, involving (or not) phenomena like transience, dissipation, ergodicity, nonequilibrium, and hypersensitivity to initial conditions. The main characteristic is growth by agglomeration, as in glass formation. Circulants are the building blocks of elementary Fourier analysis and provide a natural gateway to quantum mechanics and noncommutative geometry. Bell polynomials offer closed expressions for many formulas concerning Lie algebra invariants, differential geometry and real gases, and their matrices are instrumental in the study of chaotic mappings.
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