Selected papers of Morikazu Toda

dxdx

r dr J

J*-»*

(

=

ll=

r

where A denotes the area of the liquid-vapor surface and the partial derivative means that the valumes of the liquid and the vapor as well as the areas of liquid-wall and vapor-

we can easily calculate the energy shift, which is

The free energy of the total system is given by -

F=-ftriog2er"'" ,

(19)

and its partial derivative with respective to the area of the liquid surface tension of the liquid. During the deformation there occurs no vaporization nor condensation. The surface tension of the liquid under saturated vapor is given accordingly by

51

On the Theory of Quantum Liquids.

1955)

I

515

1 2;e-~/kT(cp~*{-!!:2;( 8~_ 8',)+2:: Xt/-ZI.I' !!!L lcp"dx ._-.l_. ___ ~ __ ~)" ax)" rtJ drLjr 2;e-e'.r- -- -- - -··- -- .

r = -- __ 21'

(20)

Here we have taken into consideration the fact that there are two surfaces, the upper and the lower sides of the film, each enlarged by the amount l'lJJ.. If we denote the density matrix of the film·vapor system by

p(xx') = 2; e- R / kT cp8*(X').p,,(x) ,

(21)

»

the expression for surface tension becomes

' -'

It is well known that surface tension may be attributed to the stress distribution near the surface. In fact for classical liquids one has the relation') (23)

where P and P T denote the pressure in the liquid and the pressure parallel to the surface. It is quite easy to see that an anologous relation holds in the case of quantum liquids. As a matter of fact from Eq. (11) we get, after integrating by part,

r=- ~)H(O-OT)dxdYdZ'

(24)

where

0= !(ozz+ov.+ozz) . We may write it in the form

r

=~(P'-PT')dz,

(25)

(26)

where P' is the macroscopic pressure in the liquid defided by Eq. (14) or P'=-Ho.z'+o .. ' +ozz' ) , (27) and (27')

§ 4. Discussion Experimentally we can except the quantum effect on'surface tension only in liquid helium and hydrogen if we are concerned with mole· cular liquids, because other liquids solidify before the effect becomes appreciable at low· est temperatures. It was remarked that the calculated value of surface tension turned out to be too high in the case of liquid helium. This was clearly due to the quantum effect.

- - - --. .. - - -- - -- - - - - -

In Fig . 1 we have the plot 0' T r"''=' - r vs cPo . Te'

with molecular diameter 0, depth of interac· tion potential cPo and critical temperature Te. We see the law of corresponding states of de Boer 8 ) exhibits the marked departure of liquid helium from classical liquids. Theoretical estimation of the quantum effect is rather difficult. We shall return to this problem in the next section, but here we shall be content with its order of magnitude. Since the main contribution will come from a few layers of molecules near the sutface, we may assume that only the molecules on the first layer of the surface are different in vibration from other molecules. We may use Einstein's model to descrive the vibration. The frequency of vibration tangential to the surface will not differ appreciably from that in the liquid, which we shall denote by". The frequency of vibration Vn normal to the surface will be smaller than v. Therefore at lowest tempe· ratures

_ _h.~ (~- ~)_..!.h(vn-v). 2m

az'

ax'

2

Thus we may except the effect to be of the order

Jr*=

(J2

cPo

(r-Tc1ass) ........ -kfJ/2f/>o .

For liquid helium the characteristic tempera, ture e=hv/k is about lS.s o K, and (l/k)cPolooK: the quantum effect will be Ar*-0.78, which is of the right order of magnitude as is seen in Fig . 1.

52

516

Morikazu TODA

L

As we approach the absolute zero of temperature the curve, suaface tension vs. temperature, becomes parallel to the temperature axis. This is in accordance with the third law of thermodynamics, because the derivative of surface tension or free energy is entropy itself.

H >

A _

§ 5. Surface Effect on Ideal Gasea

\\ K

Ne \

i fa

O

1

1

OS

04

(Vol. 10,

i 0-6

i 08

i 10

Consider an ideal gas contained in a finite volume. If the shape of the container is changed keeping the volume constant, we have the energy shift given by the kinetic term in Eq. (18). The boundary condition has much to do with the actual value. For instance, for the boundary condition 0 = 0 the surface tension of a Boltzmann gas is

T/Tc Fig. 1. Corresponding states for surface tension. Dotted curve: calculated by Eq. (35).

_h_ /~kTN 4 l / i r V 2m V

J

y -\ \%mhTl?) '

We know that the level density of an ideal gas consists of two part, i.e. g{E)=Vs /E+At, (29) v

Comparing the equation with Eq. (28) the constant t is obtained; namely, for the boundary condition V = 0

where the first term is proportional to the <32) volume V, and the second to the surface area A. s and / are constants; For the boundary condition d (29') normal) we can proceed analogously, and obtain The constant t can be determined using the . n 2m !32') expression for surface tension we have just ~ 4 k obtained. In fact from Eq. (29) the partition We may combine these results in the form function for one particle is seen to be

•HH^ 1

Z=\^-»i"g{E)dE (33)

{-l£fl<;l) , =

v

^

k

T

f

i

i

+

v h H v \ t ) -

(30) Surface tension is the derivative of the free energy F= — iVftTlog Z, and is given by 2 t (31) r=-

31

which is identical to Husimi's result. We can calculate surface tension of BoseEinstein gas and Fermi-Dirac gas using the level density given above. Wc get E

7

d

E

= -g— — \~ 4 "ft J, t'+W+l ' 3

(34)

5,1

1955)

On the Theory r,f Quantum Liquids. I

with

517

References

1) Born and Courant: Phys. Zeits. 14- (1916) 731. 2) Brillouin: Compt. Rend. ISO (1925) 1243. 3) Fowler and Guggenheim: " Statistical Thermodynamics", 1939. R. H. Fowler: Proc. Roy. Soc. A 1 5 9 (1937) 229. 4) J. C. Kirkwood and F. P. Buff: J. Chem. Phys. 1 7 (1949). 5) A. Harasima: J. Phys. Soc. Japan B (1953) 343. 6) " Recent Problems in Physics," Iwanami press, 1948. is shown in Fig. 1 in reduced units adjusting 7} Kirkwood et al: The Statistical Mechanical Theory of Transport Processes: J. Chem. Phys. the value of surface tension at 0°K and T,. 14 (1946), 180; 1 5 (1947) 72; 17 [1949) 938; 1 8 It will be of some interest to point out (1950) 817; 1 9 (1951) 1173; 2 1 (1953) 2050 ; 2 2 a possible surface effect on Fermi-Dirac gas, (1954) 783; 2 2 (1954) 1094. i.e. an analogue of de Haas-van Alphen effect. H- S- Green; " The Molecular Theory of Fluids," 1952. That is to say the pressure of an Fermi-Dirac gas oscillates in principle as the box is elong- 8) J. de Boer et al: Physica 1 4 (1948) 139, 119, 510; D. Ter Haar: Physica 1 9 (1953) 375. ated in one direction keeping the volume constant. Unfortunately the period of oscillation 9) K. Husimi: Proc. Phys. Math. Soc. Japan 2 1 (1939) 759, cf. H. Koppe: J. Chem. Phys. 1 8 is too small to detect. Only if the mass of (1950) 638. the particle were quite small and the temperature low enongh we may anticipate this effect. jc e— For Bose-Einstein gas a is zero below the condensation temperature T . Surface tension is continuous, but it has a kink at T„. To compare the result with liquid helium cohesion term is added and t

54

Reprinted from Transport Processes in Statistical Mechanics, ed. t. P r i gogine, Interscience Publ., 1958.

DIFFUSION IN VELOCITY SPACE AND TRANSPORT PHENOMENA M O R I K A Z U T O D A , Tokyo University of Education, Japan. Abstract This treatment aims at a method for obtaining a rough estimate of the transport coefficients and to clarify the nature of approximations involved in some of the current theories of transport phenomena. If we assume the well-known formula, due to Langevin, of Brownian motion to be valid for each molecule, we get the equation that gives the rate of change of the distribution function, which is identical with that obtained by Kirkwood in his theory of liquids. In this derivation we are led to the concept of diffusion in velocity or momentum space and its relation to the friction constant, Langevin's formula implies that the friction constant is independent of both the velocity gradient of mass motion and the temperature gradient. This is a weak point. But to this approximation we obtain the friction constant as a function of molecular force in the case of gases. For degenerate Fermi gas, the circumstance is a little different. In this case, however, we may speak of the diffusion of molecules on the Fermi surface.

I. I N T R O D U C T I O N The purpose of this treatment is to obtain a rough estimate of transport coefficients and clarify the nature of approximations involved in some of the current theories of transport phenomena, as well as to find the relation concerning diffusion in velocity space between the classical Boltzmann case and the case of degenerate F e r m i gas. If we assume the well-known formula of Langevin to be valid down to the B r o w n i a n movement of molecules themselves, we can find the equation that governs the rate of change of the distribution function, which is the equation obtained by K i r k w o o d i n his theory of liquid. In this derivation we are led to the concept of diffusion of molecules in velocity space or momentum space and its relation to the friction constant. I n Langevin's formula, the friction constant is independent of both the velocity gradient of mass flow and the temperature gradient. This is certainly a weak point. B u t to this approximation we can obtain the 1

148

55

VELOCITY SPACE AND TRANSPORT PHENOMENA

149

friction constant as a function of molecular force. A quite natural extension of the theory is obtained by considering the difference of diffusion velocities in each direction, that is to say in the direction of the velocity of the molecule and the direction perpendicular to it. Thus the diffusion coefficient in velocity space is dependent on the direction of diffusion, so that it is a tensor rather than a scalar quantity. With this modification we can see how the diffusion equation in phase space would look in the case of a Fermi system. For degenerate Fermi gas we may speak of diffusion of molecules on the Fermi surface. II. CLASSICAL STATISTICS If we assume that the Langevin equation applies to the Brownian motion of each molecule and that the fluctuating force results in the diffusion of molecules in momentum space, we obtain from the Liouville equation the diffusion equation in phase space:

01 + R.. .!...I +!... . (F/) = !... . {C

ot

m

or

op

op

(R.m - V)I +D· !...-/}' op

(1)

where 1denotes the single particle distribution function, m the mass of a molecule, rand p the position and momentum of a molecule, F the external force, Cthe friction constant, V the velocity of mass flow, and D the diffusion constant in momentum space. In general Cand Dare functions of the magnitude of the momentum p. Of course they are functions of temperature. And D will depend on the direction of diffusion and is expressed by a tensor. At equilibrium 1 Maxwellian distribution 1° is assumed, so that the element Dp, connected with the diffusion parallel to momentum p, must satisfy the equation

Dp

= CkT,

whereas the diffusion constant in ordinary space is given by the wellknown Einstein relation DO = kTjC. If we further assume that D is a scalar quantity equal to Dp, we obtain

CkT ,......, D ,......, (iJP )2/ i

,

where iJP is the change in momentum during the time interval i. A somewhat strange but interesting application of the approximate relation just obtained will be found in the problem of viscosity of liquid.

56

150

SYMPOSIUM ON STATISTICAL MECHANICS

F o r a molecular collision i n liquid we have (Afi) *•* V^TzmkT. If v is the frequency of oscillation of molecules, then r 2/v, so that £ <—' inmv. If we further make use of Stokes' formula £ = Gnarj, where a is the radius of a molecule, one finds for the coefficient of viscosity the relation i\ (2j3)mv/a, which is nearly identical to the formula due to Andrade. 2

Application to Gas. F o r a molecular collision (Ap~) ^ V2nmkT, and the time interval between successive collisions is T > Ijv ~ (Ijnna ) X V mjkT where / is the mean free path, v the average speed of molecules, n the number of molecules per unit volume, and er the diameter of a molecule. The friction constant is thus given b y C = oma^'V2nmkT, where a is a constant of the order of unity. A c t u a l l y , following the method of L o r d Rayleigh, M . S. Green had shown that <x = 3/8 for a heavy particle moving through gas. 2

3

To solve E q . (1) with D = CkT, for the gaseous state, we transform, as usual, its left-hand side for the transport phenomena. Thus we have 11/

2

\dV

!!

1e

5\31o ri g

d

I p

31

where we have put / — /°(l-i-oj), and u — pjm with E = p /2m. The right-hand side of the above equation can be written i n alternative forms, either z

k

d

B

p

d

d

d

dp

dp

mkT

dp

op

dp

which are to be compared with the case of degenerate Fermi gas (Section III). W e expand the left-hand side of the above equation as

and at the same time expand tp as

and thus obtain the relation between the coefficients of expansion a. and /?. F o r the case of laminar flow in the ^-direction w i t h the velocity gradient G = dVJdz along the z-axis, the only coefficient different from zero is K = GjmkT, and then we have /f^, = —Gj2mkTD. The B 2 1

57

VELOCITY SPACE AND TRANSPORT PHENOMENA

151

stress tensor turns out to be

so that the coefficient of viscosity is given b y JJ = nmkTj2D, In the same way for the temperature gradient along the *-axis, *iu = — {5/2)dT/dx/mT and a = dT/dx/2m kT . Then ft = (5l2)dTfdx/DmT and / J = —dTjdx/QDm'kT*. The heat flow is 2

2

3ll

u

3 1 1

.-T- f

m

/

e

5

\

5nkT(mkT)

kT

*= J ^ b - 2)

2

* * * = — 2

Thus the thermal conductivity x is given by x = (5/6)nk T/C- These values are approximately valid if we insert the above relation between £ and the temperature. The value of £ can be estimated from the relation £ = kT/D°, i n terms of the coefficient of self-diffusion D". One has thus

t = f

J^gJ


(1 -

2

cos 0) wgJ (g, 0) sin 0 rf0.

where e/ == (m/ikT) %g, g being relative velocity and I(g, 6) the crosssection. The last equation gives for a hard sphere a slightly different value from that of Green. The difference may be due to the fact that i n our case the reduced mass replaces the mass of heavy particle i n Green's treatment. T

III. F E R M I G A S In the case of F e r m i gas the distribution function assumes a Fermi distribution f° at equilibrium, so that diffusion parallel to momentum is connected with the friction constant by the relation (l-f*)D,=

&T.

However, for strongly degenerate F e r m i gas, diffusion i n momentum space takes place m a i n l y along the F e r m i surface. W e shall denote the diffusion constant along the F e r m i surface by D . We may put D ~ 0 for degenerate F e r m i gas. Then the fundamental equation simplifies itself into s

v

% + ~ • J - / + J ; (F/) = D f(l-.f) at m dr dp s

V* , x

(2)

58

152

SYMPOSIUM ON STATISTICAL MECHANICS

where we have put

f=f°+f°(i-f)xThe left-hand side of the above equation is for laminar fiow, mi u* \ a v . ( , - . ) - ( u u - . ) : - ,

/

/

T

where u is the relative velocity of molecule with respect to the mass flow V . Or, if the flow has gradient along the 2-axis, this term is j (\-j°)(mu lkT){dVJdz)Y _, where Y = sin 9 cos $ cos with the 2-axis as polar axis and angle <j> being measured from the a;-axis. T h e n we obtain t>

2

a

2 1

z

P

mv

2

X— G. * 6D kT where P = mv is the magnitude of momentum on the F e r m i surface and G stands for the velocity gradient of mass flow. After calculating the stress tensor P„= mj/u ti d u, we obtain the coefficient of viscosity S

3

x

j; =

l

Km j/»/60O . 3

s

F r o m the Uhling-Uhlenbeck equation we can verify E q . (2) for Fermi gas. The right-hand side of this equation is to be replaced b y the collision term

(I)

=

l ^ /

r

S

/

t

e

'

9

)

^

{

/

'

/

;

(

1

~

/

K

1

~

/

l

K

/

/

l

<

1

~

/

'

,

(

1

~

/

;

,

}

'

where, as usual, / and / ' denote distribution functions before and after collision, suffix 1 stands for the colliding molecule, / is the differential cross-section, g is the relative velocity of collision, Q is the solid angle of scattering, and dp is the elementary volume in momentum space. W i t h / = / + / ° ( l - / ) z we have t

D

0

(dfldt)

c

=

Aix'^+x'-x),

where A is an operator defined b y

A s the collision process takes place only i n the vicinity of the F e r m i surface and along i t , the end points of the momentum vector p, p', p,.

59

VELOCITY SPACE AND TRANSPORT PHENOMENA

153

and Pj lie on a circle. The scattering angle 0 is the angle between the end points of p and p' as observed from the center of this circle. Then A(x'~x)

= fj -J

tffc

0) sm

,«J

• / i M W ° > - * ) )

(z'- ) Z

2

with £ = [E-E )lkT, E = F*/2«*, £ = P / 2 m ; dS stands for the elementary area on the Fermi surface. O n the other hand, A {xi~Xi) ' approximately zero. This can be shown b y assuming that the scattering angle is so small that x\—Xi can be replaced b y (SxJds)g sin(0/2), where s is taken as a circle on the Fermi surface with its center at p. If we integrate first along s keeping 8 and g constant, Xi returns to its original value and the integral vanishes. Therefore if the scattering angle is small or if the scattering of small angle predominates 0

u

s

Mx'-x).

de \dt/c where

r de

1

r

dx

(

1

1

I

We expand %'—x i n the form X'-X

= *P • {V )-kApAp X

: (VV ), Z

where Ap = p'—p and V is the gradient operator on the Fermi surface. (Vx) as well as (VVyJ stand for the values at p. It can be seen that % is a complicated function of e. Integrating with respect to e we obtain an equation for the averaged value of x- That is

m \dtj,

=

D Vlx, s

2

where V | is the LapJacian operator on the Fermi surface ( V = V • V) and with the help of JJ^£ ( +£,)/( -r-l){e '+l)(l-^ 5

1

£

e

I

|f+E

)

')

60

154

SYMPOSIUM ON STATISTICAL MECHANICS

we have the diffusion

coefficient JT

3

5

m r r

12 F J \&k,%mt^m#& To see the magnitude of this quantity we m a y use the hard sphere model, which gives (kTf S

#

2

mP a,

7] '

n (py w — ——-60 W / {kT) a 2

The same k i n d of treatment was carried out for the diffusion of electrons in metals which results i n electrical as well as heat resistance.

4

References (1) J . G . Kirkwood, J. Chem. Phys., 14, 180 (1946). (2) E . N . da C. Andrade, Phil. Mag., 17, 497 (1934).

(3) M . S. Green, / . Chem. Pkys., 20, 1281 (1952). <4) M. Toda, J. Phys. Soc. Japan, 8, 339 (1953); 9, 440 (1954).

61

JOURNAL OF T H E P H Y S I C A L SOCIETY of

J A P A N , Vol.

17, No.

3, MARCH,

1963

Localized V i b r a t i o n and R a n d o m W a l k By Morikazu TODA, Takeyasu KOTERA and Kouzou KOGURE Department of Physics, Faculty of Science, Tokyo University of Education {Received December 9, 1961) It is shown that the problem of localized vibration of a lattice with an impurity can be treated as a problem of random walk, s-dimensional simple cubic lattice is considered. Examples are given for the special cases of 1-dimensional and 2-dimensional lattices with an impurity. Probability of steps of the random walk is interpreted by introducing a virtual time. § 1. Introduction It is well known that lattice vibration can be localized if a lighter atom is substituted in a perfect lattice". Detailed analysis for 1dimensional lattice which contains lighter impurities has been made by many authors '. In this case if the impurity is lighter than the others localized vibration is always possible. But, in 3-dimensional case, unless the mass of the impurity is small enough, the localized vibration can not take place". Though in 1-dimensional case special techniques of calculation can be used, in 3-dimensional case or in general s-dimensional case most of the analysis is based on the method of the Green function of forced vibration". In this paper another method is presented to treat the problem of localized vibration. This method is related to paths or random walk along the lines connecting lattice sites. 1

In § 2 description of the method is given, with attention to the convergent characters of the localized mode. § 3 is devoted to special examples of 1- and 2-dimensional lattices. In 54 we will discuss the probability of paths by introducting a virtual time to visualize the paths in the space of lattice sites. The problem we shall treat is the vibration in s-dimensional simple cubic lattice with an impurity at the center. Denoting each lattice site by a s-dimensional vector R with integer components ffi.ft, we express the displacement of the atom by a scalar quantity u(R, I). If the force constant of the harmonic force is denoted by k and the mass of the atoms except the impurity is m, the maximum frequency of the perfect lattice is given by <w"=4sr . with

04)

62

1962)

Localized Vibration and Random Walk

427

r=—.

tt-2)

m The equation of motion for the displacement u(R, t) is

1+

{ "0

J

0

j

" }' °"d?''

i = r

4 » - 2 » < « , 0 3

,

where m' denotes the mass of the impurity atom, and 1* is the unit vector along the lattice direction ft. To find a stationary solution we substitute H(ft,£)=e™'H(if) , (1.4) in Eq. (1.3), obtaining following equations for the time-independent part w'u(R) = £ {2u(R)-u(R+l )-u(R-W} r

f

.

(R*Q),

(1.5)

and m

-oSu{t»

= i{2a(0)-H(W-M(-W>,

(1.6)

T

§ 2. Method of PathB Instead of specifying the boundary conditions, we assume that the lattice spreads out in all directions to infinity. Eq. (1.5) can be written as u(R)=gZ. {u(R+U)+u(R~W} ,

(R^O),

(2.1)

where g s

=

r

( 2 2 ) !

z

2sr-a>* 2s(w,-2< ) U

We first solve Eq. (2.1) without considering Eq. (1.6) explicitly. After that we shall substitute the solution into Eq. (1.6) to determine the dependence of w or g on m'/m. To get the localized mode we may assume that w(Q) is not zero. Then the solution of Eq. (2.1) may be written as "(R)=5i

2 iT*, (2.3) mil p>ihil where the summation is carried out over all paths from 0 to R along the lines of steps, each connecting adjacent nearest neighbours and passing the same line as many times as we want, 5 is the number of steps contained in a path, and 3£ is a certain normalization constant. Counting the number of paths by a combinatory method we may easily write down the explicit form of u(R) as follows: *»=**<«.<+'".'•••••''.'

£

! „ <2.4) !

!

This series is convergent under the condition |g|uio. Under the same condition we can prove that u(R) approaches zero as |>?il -H-foH V\R.\ increases infinitely, and that the sum of (u\R)) over all lattice sites converges. As an example we take 3-dimensional simple cubic lattice. Our procedure of the proof can be easily generalized to the s-dimensional case. Denoting the lattice site by k), we assume that the only one impurity is situated on (0. 0, 0). Here we discuss the property of H(J, k) which is the scalar displacement of lattice site (i,j,k). From the general method we have !

(1.3

63 428

(Vo\. 17,

M. TODA, T. KOTERA and K . KOGURE

u(i j k)=lJIglil+Iil+lkl ~ ,

,

I ... . ft-O

{lil+lil+lkl+2(l+m+n)}! g2ll+m+n) 1!(lil+l)!m!(ljl+m)!n!(lkl+n)! .

The radius of convergence for g of the above series can be calculated as follows:

u(i: j, k)=lJIg'il+I;I+lkl ~ a(i, j, k; p)g'P ,

(2.6) Let C2.6)

p=o

then a(i, j k) satisfies the following inequality

b(i, j, k; p)
(2.7)

where

b(i, j, k;P)

(Iii + Ijl + Ikl +2P)!

[n ! [P ; l] ! [P;2] ! (i + [n) ! (j + [P;l]) ! (k+ [P;2]) !

c(lil + iii + Ikl ;P) _ Cli l+ Ii I+l kl +2p)! P' -[ ~J! [P;!.] ! [P;2] ! [lil+IiI;lkl+P] ! [l il+lil+ikl+P+l }[lil+ljl+ikl+P+2 } , (2.8)

and

[a] means the greatest integer which does not exceed a .

Both v(i , j k) and V(l il + iii

+ Ikj) which are defined vCi , j, k)=lJIglil+I;l+lk l 2:: bCi, j, k;P)g'p , p=o

VClil+ljl+lkj)=lJIg,il+IMlkl 2:: c(lil+IiI+lkl;P)g'P, p=o

)

(2.9)

are convergent for Igl <1/6, because

r

bCi, j, k;P)

:.rr:. b(i,j,k;k-1) r

c(lil+IiI+lkl;P)

6' ,

:..~ c(lil+ljl+lkl;P-1)

6'

As a consequence of the above facts we may conclude that the series u(i, j, k) is convergent for Igi < 1/6, and is divergent for Igi > 1/6. Farther, as

c(lil + Ijl + Ikl ;P) c(lil+ /jl+ Ikl-1 ;P)

lil+IiI+lkl+2P 6 [(lil+lj l+lkl+P+2)/3] < ,

we can see

I V(lil+ljl+lkl)1< 16gV(lil +IiI+ lkl -1) 1 . Consequently, for Igl<1/6 , the sum of [V(lil+li l+ lkj)]"s over all lattice sites is convergent. As we see from (2.7) that lu(i, j,k)I< IV(l il+ljl+lkj)I, we can also conclude that the sum of [u(i, j, k)]"s over all lattice sites is convergent. Thus we may conclude that such a vibrational mode, if exists, is localized. It is easily seen that u(R) given by Eq. (2.3) or (2.4) satisfies the following equation Wo'

-5l)u(RHw'u(R )=AoR .O 45

where

5l)

denotes the difference operator defined by

•

(2.10)

64

1962)

Localized Vibration and Random Walk ©u(R) = 2 { » C f f + l M ) + « ( - K - l » ) - 2 M ( f i ) } ,

429(2.11)

and A is a constant related to 91 by the equation A=-!^-
(2.12>

tSg

Eq. (2.10) means that u(R) is the Green function G(fl|0) of the forced vibration, in which a force with magnitude A is applied on the atom at the central site. To examine if there is a localized mode, we have to take Eq. (1.6) into account. Since u(l») and K( — 1„) are all the same and independent of p., we may write these as «(1). Then we get from Eq. (2.4) «(0)-31=2S£«(1).

(2.13)

Substituting this relation into Eq. (1.6) we get 1

9i M(0)

Zsr-im'ImW Isr—a)"

or ^ p a - . x i - ^ ) ,

-as.,

(2.u>

We note the fact that 9i/w(0) is positive irrespective of the sign of g. If, for a given value of £, this equation has a negative solution g whose absolute value is less than l/2s, a localized vibration can exist. As easily seen, for e>l there is no solution for g and no localized vibration. But for 0<e
= 3 t a

Then,

u , v K\j\m-r-iUQj\+Di2u (2 )" •=» (ij'i+i). «i ' S

where [o] =(a)»=a(ff4-l)(a+2)---(ff+»-l) , n

(«),=1 .

Writing Gauss's hypergeometric function as F(, ; ; ) we have «0)=WJFi(W+l,-U^-

:

l/l+liOfc)*) .

(3.2)

From the following relation t a n h z p / y + l . - ^ ^ ; n+1 ; sech'z)=e-"*(2coshz)" , we get (noting that g is negative)

(3.3)

65

430

M. TOD*, T. KOTERA. and K. KOGURE

«(iyi-D -

^ H / I + D "

~

(Vol. 17,

E

(

•

3

-

4

)

where — 2jf=sech z . This consequence can be obtained directly if we notice the fact that Eq. (1.5) for 1-dimensional lattice is a special case of PoincartVs difference equation. For the value of g for which the series of Eq. (3.1) is convergent, the ratio given by Eq. (3.4) is negative and is greater than —1. We get, from the equation of motion for j-0, Eq. (1.6), the frequency oi of the localized vibrational mode as a function of e-

<9M

'

(«<« -

(3.5)

This coincides with the well-kown result, (ii) 2-dimensional lattice Denoting the lattice site R by {j, k), we assume that the only one impurity is situated on the lattice site (0,0). Then we may write the amplitude ufj.k) in the following form: »,«=• m\(\]\ + m)\n\{\k\+n)\

(*^')(£!^')H^*'f

By the aid of the identity £

this double-sum can be con-

verted to a single-sum as Uti » = f e l ^ " i ± <*> " & mj

{(2P + l?l + |fe|)!}' „ p'.(p + \}\)\{p + \k\)\{p + lj| + |A|)!

S

( 3

7 )

g

This coincides with the result obtained by the method of the Green function". Writing w(0, 0) in the explicit form K(0,0)=9i |

a g*'=H 2

~£r-g

r

!p

we examine the property of this series. From the following inequality for g=—l/4 ^

= 1- i + ^ - > ^

—

ap-^-"

(3.9)

and the divergent character of the series 2 (1/p) we can conclude that «{0, 0)/3t increases infinitly as g approaches —1/4 from 0. This fact means that for lighter impurity localized vibration is always possible in 2-dimensional case. By the aid of the expansion formula for the modified Bessel function Jj{xj, u(j, k) in Eq. (3.7) is easily seen to be (3.10) with

u(; fc) = ( - l ) ' M j"drexp{-r(^-4r)>/i(2TrtM2rr), +

pl

A

m±M

v

/ w + w + 2

!

(ljl+lmH-l),(lj|+l),(!*l+l): 16iT ),

1/1+1*1+2

\i\+\k\+i

IJI+IAI+1 (3.11)

66

1962)

431

Localised Vibration and Random Walk

where «F ( , , , ; , , ; ) is the Pochhammer's generalized hypergeometric function. Eq. (3.11) agrees with Eq. (3.7). Then .F,( 16g*) converges inside of the unit circle in 16g* plane except at the origin; |16#*|<1, that is o^Xmt*. Especially 3

(3.12) (3.13) and (3.14) Eq. (1.6) gives then w'_

l/d-4g) 1

=

L _ JL( l - 4 g 2 VJ

**

V

(3.15)

l/l-ieg-'sin'p/ "

0

Performing the numerical calculation of Eq. (3.15) we can plot the dependence of the frequencies of the localized modes on the impurity mass (see Fig. 1). We have seen that there exists a stationary localized vibration when the mass of the impurity atom is smaller than that of the basic atoms. This effect disappears if the impurity atom is heavier than the basic atoms. When the mass difference is comparatively small, the frequency of the localized mode of the 2dimensional lattice is lower than that of the 1-dimensional lattice which was calculated by Montroll and Potts". However, as the impurity mass decreases, it rises steeper compared with that of 1-dimensional case.

Id lice

One dlmarrl icmal lull Ice

By a different method from that used in Fig. 1. The frequency of the localized vibration associated with a lighter impurity in a 2-dimen- § 2, we shall show below that as the distance sional lattice is plotted as a function of the from the origin of the lattice site approaches impurity mass m'. infinity, u(j, k) diminishes, and tends to zero. As a first step it will be shown that a(j,j)—*0 as From Eq. (3.11), M(J',;') can be written as follows: l

u(jj)^{4 r'^^ ^^^{\j\+\,

m+\,

S

w m ••

.

(3.i6)

Then, given any number j, we find the inequality such that #,(y\+j;,\jl

+ j,2\j\

With the aid of the relation

S

+ l;lSg' )< F,(lj\,\j\+±, l

2\j\ ; 16^) .

(3.17)

67

432

M . TODA, T . KOTERA and K . KOGURE

(Vol.

17.

we see that

Mi i)\<W4e)"» <

^

1

6

n(lii+{im

2

-^—(

l

V'""

1

^ ^VF^(i y-i -i6 .r'"' • +

g

Since the last term approaches zero as |j] approaches infinity, the function u(),j) approaches simultaneously zero in the same limit. Moreover, it is shown that |«G-+l,fe-l)|<|«(j,A)| .

l*|) .

K/-l,*+l)|<|
(l*tetfl),

and where we have compared the corresponding terms of the same order of g in each function by the use of the series expansion of Eq. (3.11). Thus the amplitude of each atom diminishes monotonically to zero as the distance from the origin of the lattice site increases. §4. Probability of Paths We have seen that the mode of localized vibration is related to the problem of random walk. In this section we shall clarify the relation in more detail in terms of the method of paths. We shall introduce a virtual time i to make it possible to follow the paths of random walk. The way of introducing such a parameter is not at all unique. In fact, we shall see, in what follows, some alternative ways, which lead to different interpretations of the same problem. Here, 1-dimensional case will be treated; extension to s-dimensional case is straightforward. First, let us consider the equation ^g(lU-

+ Xi <) ,

(4.1)

+

and define i = ^XjM^dT.

(4.2)

limrjfOe-^O ,

(4.3)

Under the assumption

we have Ui=g(!ii-, + u ,)+Xi(Q) .

(4.4)

it

If we put OB Eq. (4.4) coincides with Eqs. (2.1) and (2.10). Thus we can get the solution of Eqs. (2.1) and (2.10) by solving Eq. (4.1). Dividing the virtual time r into sufficiently small intervals of S each, we may rewrite Eq. (4.1) as 1

rj(t-+^) = A W+^(lj-iW+r - (r))-t-0(S ) ]

J

tl

-SWH'W'l+OW,

(4-6)

with ^i(0)=l .

Pi(l)=p (-l)=S s

g.

The last equation defines y (r) as the probability of the paths in (j, r) space. 3

(4.7) Taking S in-

68

1962)

Localized Vibration and Random Walk

433

finitesimal we get ^W=2G(M?»0)i, fO).

(4.8)

o

with the transfer function G(j'r|j„o)= 2 pi{i-h)PAh-h-<)---Ps(i>-U)

•

(4.9)

Making the Fourier transform and noting that jb(,0)=—£(4A/W)3/.«, we get Xfr)=-g^GUT\0,0)=-£Ola'

lit

•—•[' e-<'"e"™'-da m J_

,

(4.10)

r

or tii— \ r (r)e" rfr=—• — r I

—-r

3

Eq. (4.4) is satisfied if |g|
v*

(4-11)

equations including time r are "separable". For example, for 2-dimensional case we have ^•.i(r)=^(-D"' " W^rrW2r'-) , (4.20) . . , , , ,. . . . which leads directly to the expression in Eq. „ ^ H

defined by the equation of diffusion type ^

•.

^"^

c

a

nb

e

lll

w r j t t e n

^

fa

f

m

m t

Q

5

= Dtt>*.+A-.-2fc},

(4.12) E°- «- >.

D=-^-

(4.13)

01

with r=A3 and =-

r

,

-7 (0)-i+25r, ? ( - i ) = ? ( i ) = - 3 r . '«tm 6

l

f

A and M,' by

Consequently we have (Si(r)=2G(jV|/0)fl -<0) . .'4.23) (4.14) Thus the Fourier transform of G(;"r|/'0) leads to 2 ^ " ' ' " ' ' ' G O ' T I I ' T H ^ S e'"/7i(fl)* We assume that a, is large enough so that , i lime-^iW-O . (4.15) =\(\ + 2dr)-(e''+ Si Kiven by Eq. (4.17), ' i f. Therefore if we take irW-gJ (4.25) WOW**.. ' which coincides with Eq. (4.18). then, Eq. (4.16) becomes identical with Eq. (2.10). References The solution of Eq. (4.12) with the initial 1) E. W. Montroll and R. B. Potts: Phys. Rev. condition (4.17) is easily seen to be" 100 (1955) 525. J

l

+1

1

y

!

E

4

V

16

v e s

f

o

r

4

( 4 i l 7 )

6M=Ae<"I,(-2rT) f$&~JM m •

(4 18) t*- )

2 )

H o r i :

, o u r

P h y s

J a p a n

1

6 ( 1 9 6 1 ) 2 3

- ^ ' . Teramoto and S. Takeno: Prog. Theor. and Phys. 2 4 (I960) 1349. A. A. Maradudin, P. r, _ ,, . , , „, Mazur, E. W. Montroll and G. H. Weiss: Rev. Ui=A(.-iy"^ I&rMr , (4.19) . _ ^ 3) S. Takeno: Private communication, which is an alternative expression of us given j Mahanty, A. A. Maradudin and G. H. Weiss: by Eq. (1.11). Prog. Theor. Phys. 24 [I960) 648. Extension to s-dimensional lattice is quite 5) E. Teramoto: Prog. Theor. Phyg. 24 (I960) straight-for ward in these formalism as the 1296. 1B

E

M o d

4 )

P h T S

( 1 9 5 g )

m

l

69

Supplement of the Progress of Theoretical Physics, No, 23, 1S62

Statistical Dynamics of Systems of Interacting Oscillators Morikazu T O D A and Youzou K O G U R E Department of Physics, Faculty of Science Tokyo University of Education, Tokyo The dynamical behavior of systems of interacting harmonic oscillator is examined. Crystal lattices are examples of such systems. Description is made in terms of normal coordinates. It is assumed that energy is initially distributed canonically over the normal modes. The condition that the process is Markoffian is presented. As an example the behavior of one-dimensional lattice with a heavy isotope is discussed. The motion of the center of mass of a portion of a one-dimensional continuum is treated for comparison. Brownian motion of an oscillator is also discussed.

§1.

Introduction

One of the general problems in statistical mechanics is that of tracing the time evolution of systems. There has been a considerable number of investigations on this line using models of crystal lattices. ~ These models can be thought as special cases of systems of harmonic oscillators interacting by harmonic forces. Since the equations of motion of such models can be solved in principle, detailed investigations of systems of harmonic oscillators might shed light on the general feature of dynamical systems. One of the merits using such a model is that we can treat the system as a whole without omitting any term in the calculation. 1}

31

In §2 of this paper, we summarize statistical dynamics of a system of harmonic oscillators with special reference to the motion of a specified oscillator, for which the condition that the process be Markoffian is given. Examples of one-dimensional systems are given in § § 3 and 4. In §5 the theory of Brownian motion of an oscillator is discussed with reference to the treatment presented by the authors in previous papers.' ' 1,7

§2.

Description by normal modes

In this section we shall discuss the general feature of a system of coupled harmonic oscillators. The configuration of the system is specified by a vector u(t) whose j-th component stands for the displacement of the particle of the j-th oscillator. The equation of motion can be written in matrix form as

70

M. Toda and Y. Kogure

158

MK(t)+Ku(t)

=0,

(1)

where M is a diagonal matrix whose j-th element is the mass of the j-th part'.cle and K is a symmetric matrix with elements represent force constants. For brevity's sake we shall rewrite E q . (1) in a reduced form as x(i) +

Ax(t)=0,

(2)

with (3) and define momentum vector by p(rO = i ( r ) . The axes associated with the components of x (r) constitute an orthogonal coordinate system. B y an orthogonal transformation we can choose principal axes as the reference coordinates. Thus i n terms of normal coordinates X « ( r ) and their associated momenta Pn(.t) we can write Xi (t) = S c,nX (t), a

p (t) = 2 Ci„ P (t), t

n

(4)

jith (5) If we denote by J2 the frequency of the n-th mode, we have n

Xn(t) +S3lXn(t) = 0,

2

S2 = *£A*ctnC

,

n

(6)

and Xn (t) = X, (0) cos Qnt +P (0)

sin Qnt

n

Pn (t) = - Xn (_0)Sn sin Qnt +Pn (0) COS Qnt.

(7)

Thus the displacement and the momentum of a particle, say the 0-th particle, at time / , as well as those at ( = 0, are given by the initial values of Xn and Pn as 1 Pit)

P(0)

'

Xn(Q) = D

(»)

Aw

where we have omitted the subscript 0 : x(t) —xo(,t), etc., and D is a matrix defined by

ro« cosQnt ••• coaQ^ sin&nt •• — Can SIT! Q-,,t ••• CQnCOsGnt Cn ••• 0 1

D =

0

0

•-•

Con

(9)

71

Statistical

Dynamics

of Systems of Inter acting

Oscillators

159

W e shall now ask for the conditional distribution function for tbe 0-th particle. In other words, we ask for the distribution for x(t) and p(t) when the initial values x((i) and ^>(0) are given. In order to proceed, it will be assumed that initial values X,.(0) and P..(0) are distributed canonically. This distribution is characterized by a diagonal covariance matrix whose elements are 2

*X. - < X (0) > =

Sen

,

* = WO) r

s

(10)

> a*„ .

In classical statistics o->. = kT

(ID

First, the distribution Gaussian. That is, w m

function W(s)

a

= m

" (is

_ i

with the covariance matrix 2

D

1 / 1

for z turns out to be normal or

e x ( - -I-** s P

(12)

*),

given by 2

<x(r) >

< x ( r ) / > « > <x(_t)x{0)>

<xQ)P(0)>

<x(0)x(t)> <,p(0)x(t)> Con

= DdD — kT T

,—, 2-1

2

Con 2

2-1

0

sin £» t

COSfinf

^2 ton

21

sinifni

0 S Con

O

(13) Sal where S11. S B I S21 and S s i are 2 X 2 matrices. For given values of x(0) and p(0) the average values of x(t)

and £ ( r )

are 5(0

*(0)

Pit) and the conditional distribution function P(x(t), pit), t\x(0), normal distribution function with the covariance matrix

(14) p(G), 0) is the

72

160

M. Toda and Y. Kogure

\ /x(t)

I x(t) Pixit),pit),t\xiO),pi0),0)=H

,21-

(15)

3

W e define, in accordance with Rubin, ' the quantities

Sin (16) W e note that X(t)

satisfies the initial condition

X(0)=0, We

(17)

X ( i )= l-

have x(0) j l -

( )do\ +pi0) a

-x(0>X( )

Pit)

X(t)

(18)

+p(_0)X(t)

£

SPJ> J

2

2^X(T)^-X (r) -f^XC*)^] X(t)

jl-X(t)-^X(

f f

)rf

f f

1

)

-^X(rf)rf)

X(t) | l - X ( r ) 2

ft

2

l-X (r)-^X (i) (19)

Equations (18) and (19) are identical with Rubin's results. These equations are formally valid also in quantum mechanical case. The only difference in this case is that an is to be replaced by tfn^-^-ftfiaCOth

and accordingly Xit)

J'

(20)

by

• and

\ 2kT

Sn

ki

(21)

the initial value X ( 0 ) is to be changed accordingly. In

the following, however, we shall confine ourselves to the classical

case. It has been assumed that the initial displacements and momenta are normally distributed. T h i s leads to the consequence that the displacement and momentum x(t), pit) of particle 0 constitute a stationary Gaussian process. Next problem is to ask whether this process is Markoffian. The

73

Statistical

Dynamics

of Syste?ns of Interacting

Oscillators

condition that the stationary Gaussian process {xit), pit)) that the correlation matrix R(t — ti) satisfies the relation

161

be Markoffian is

}

Rit -t,)=R(t -t,)R(t -t ) a

i

where r j>i2^>ri.

'

(22)

1

The correlation matrix is defined as

a

1

i

Vpit^xit^/wxpty*

where, from S , , or terms of X(t), we have

< K O K * < ) >/*>

2

;

1

<x > = <x (t<)> = v' kT,

Xit)

i

(p*} = = kT.

In

^X't) (23)

U2

-u X(t)

\-^X{a)da

Further, since X(0) = 1, X(0) =0, we have R(At)

(0
= - »"*X(.0

At) At

tl

1 - » X ( 0 4ft A ' 3 3

in

l + X(0 + )At

v

At\

(24)

so that in the limit of small At, we have 1

R(_t)R(At)

X{t) + {X\Q +

)X(t)-vX(t)}At

= u

- ^X(t) v

+ |x(0

+ ) X(t)

+1 -v^XMdo^

At

/2

y' [Xit)+X(t)At) (25)

l-^Xio-)do-vX(t)At The condition that the process is Markoffian is R(_t + At) = Kit) R(At),

which

is satisfied if X\0 + )X(t)

-vX(t)

X(Q + ) Xit)

+

=

X(t), l-^X( )do-=Xit). a

These two equations are satisfied simultaneously if X(t) equation X(t)+fiX(t)

2

+
(26) is subject to the

(27)

where the "friction constant" 0 and the frequency w are determined by

74

162

M . Toda and Y. Kogure r J = - X ( 0 + ),

a>*=»,

(28) 2

F r o m the definition of X(t) and v, it is clear that (3 and a> are both positive. Thus, the necessary and sufficient condition that the process is Markoffian is that X(t) fulfills the damping equation given by E q . (27). W h e n this is the case, it is readily shown that the conditional distribution function P, with the covariances given by E q . (19), is the solution of the Kramers equation dt +PY-=4^1(PP Sx dp r

+'

lVKr

O 2

^ + < 1 ^dp £->

Q = &T.

(29)

T o show this we have only to note that the solution of E q . (27) is X(i)= —e-^smw.t , 2

(30)

2

where o», = a > - ( l / 4 ) £ , and substitute it in Eqs. (18) and (19). Then we obtain { x ( r ) , p(t)} and elements of covariance matrix S°> which are identical with those given by W a n g and Uhlenbeck solving E q . (29). The general feature of the process is therefore determined by the nature of X(t). From Eqs. (4) and ( 7 ) , if we take X „ ( 0 ) = 0, P»(Q) for all 11, we see that Xi(0) = 0 ,

MO) =0

for

t#0,

(31)

and * « ) = 2 ko»]*

x ( 0 ) = o, 2

P W = S \con | cosi? i = X{t), n

p(0)=l.

(32)

Therefore, X(t) is the solution of the equation of motion (2) for the initial condition that all the particles are at their equilibrium position and are at rest except for the particle 0 which is given unit momentum at i = 0 .

§3.

One d i m e n s i o n a l l a t t i c e w i t h a heavy

isotope

The problem of linear chain of atoms with an isotope has been extensively studied. In this section we shall apply the method described in the foregoing section to this particular system. In so doing we shall eventually find a connection between the lattice model and the model considered by the authors in previous papers.' This model, which we shall refer to as model S, consists of an oscillator interacting with a great number of surrounding oscillators. N o interaction is assumed among the surrounding oscillators. 1

W e consider here the linear chain of atoms of mass M labelled from

75

Statistical

Dynamics

of Systems of Inter-acting

— (7V+1) to J V + 1 with an isotope of mass M' at 0.

163

If the displacement

u,, the equation of motion for x's^VM

of the y'-th atom is denoted by

0 ' = 0 ) and x'^VM'un

Oscillators

u

}

can be written as

• *r X-s - -r

+ A'

a: . •/ x

f X-K t X t

= 0,

(33)

N

with the interaction matrix A' defined by

2r

-r

-r .

0

-r

-f

2r -f

o

ml

-r

-r

2r

(34)

—r —T

-r

2r ,

where

M'

VMM'' '

030

(35)

~M' '

and K stands for the force constant of the springs between atoms. W h e n M'^-M, we have r'<^T. This means that large mass of an isotope corresponds to weak springs attached to it. First we introduce new coordinate system parts of the matrix A':

"

, /

2

.

xjs

(x

(s=i,

( )

x)

which diagonalizes

m,

(36)

x=x, (cf. reference 1)) to get the simplified equations of motion of the form, x -\- >%x . + A x = [) (s= -N, !

a

l

t

x + «>(! x + S A,x, = 0. Thus x and x

t

- 1 , 1,

AO. (37}

constitute the model S. Frequencies and interaction constants

76

164

M . Toda and Y. Kogure

in this model are

W e have therefore

The motion of the isotope is determined by the function X{t), E q . (16) of the preceding section, in which the direction cosines cpn and the frequency i3 of the «-th normal mode are given by the simultaneous equations n

We first note that

2

2

to - J2

M'

2 ( A M 1)

2

» | - J3

i(AM-l) /here 1

yj

-

1

S

S m

•

-

-

-

—

(42)

2(/Y-H)

T o calculate the sum we note the identity KS . ., 1 sin 2(AM-1 ).r n [sinS 2(AM-1) - s i n " x J = —2 ». s i n- 2,,x — ,'

(4,i)

from which we get ,v

Y\ ,-i

1 — . .

its 2(AM-1)

5 l n

1 (N+1) - — — -— . , • a sinxcosxL ~

S

m

cos^(AM-1)^sin 2 (AM- l ) x

cos2x "j sin2xj

X

(44)

In our case 2

sin x = G*/4r, sin x cosx = (4r) " ' f l y ^ r - f l

2

.

(45)

77

Statistical

Dynamics

of Systems of Inter-acting

Oscillators

165

W e have therefore, for N^>1, 2

{(2r - fi*) - Q V A T - Q * c o t 2 ( N + l ) x ) .

= M r

(46)

Since a > i H 2 r M / M \ we have the relation

2

On the other hand, differentiating E q . (46) with respect to fi , we obtain, for A > 1

i£i ( - f - f i * )

8

M '

r

X c o t 2 ( A + \)

X

+~

2fi

-2fi 2 i/47

V 2

(N+1) (1 + c o t 2 ( i V + l)ar).

2

(48)

W e have therefore, for N^>1

Actually, uij forms quasi-continuum from u> = 0 to o> = 4r and the same holds for fi . T h e number of normal modes with frequency between fi and Q + dQ is f

f

n

^(£)rffi=

, 2(iV+l) " I 0

__^fi_ »/4r"~fi

(o
(ir-wj.

(50)

Therefore in general S A F(fi«) =

r (fi)/;(fi) F(a) dQ. 2

with

(51)

«

Thus we get

which coincides with the result of Takeno and H o r i . " W h e n Q ! > 1 , we can rewrite X(t)

as

78

166

M. Toda and Y. Kogure

w.x

2 p

sin (fir) ^

0

1

= y ( l - ^ )

Ci^ ),

i

(54)

i

where b =
t

1

(55)

U(p)=\ e-"x(t)dt. ja

If we choose the initial condition that x,(a~)=h(Q)=0, (.(=¥0), we have x(t)=X(t).

_r(0)=0,

i(0)=l,

The Laplace transform of X(t) is thus found to be

U(p) = ll\p*

+

» -22yf^\

(56)

a

For the model described above (cf. E q . (41)), s

2

S-rrf-- -=.\(2r+p )2N-pH4r+p )j: 3

X_.vj_.J_. -= 2AM-1 =» /> +«>? ^r4r+# 2

^

_

!

(AT—°°).

(57)

a

Thus, we have

which gives the same result as E q . (53) (cf. reference 5 ) ) . Before closing this section, we shall present a slightly modified system for which the calculation can be carried out rigorously. T h i s is a system of a chain of atoms subject to the same equation of motion as above except for the central isotope to which another spring is attached. Equations (33) to (46) hold for this model except that _g=-||-+*>*

(59)

79

Statistical

Dynamics

of Systems of Interacting

Oscillators

where to is a constant standing for the strength of the new spring. (47) is then modified as 4_3 - - ~ a >

(M-s

with s i n x = j2S/4r.

2

167 Equation

(60)

/

= _ . v 4 r - Q l cot 2 ( N + 1 ) x

Further, we find

= T r W 4£- (1 + -ot 2<JV+ JV+1 M ,

1 W

+

J3S(4r-_«)

M ' 1

M

_ J ( 4 r - _ 3 )

+

_.

(61)

[ ( ^ - - l ) _ 2 - ^

a

So that for Q ^ l , provided that a>,;!>ai , X ( r ) can be approximated by 9 X

(

0

=

;

* l

m

i"(j3 -_" j " 2

3

2

_

^

(62)

=—e""siniaf ,

when

is very large and b very small.

§4.

M o t i o n of the c e n t e r of mass

It is of some interest to compare the with the motion of the center of mass of uniform continuum. The argument of §2 T h e calculation is easily modified to examine of a portion of a one-dimensional lattice.

result of the preceding section a portion of a one-dimensional can be applied to this system. the motion of the center of mass

The displacement of the position x in the continuum at time t can be written as u(x, t) = ff-

S ? . ( 0 s i n ^

(0<x<0

(63)

where / stands for the length of the continuum, p its linear density, and q,(t) the K-th normal mode. W e have fixed tbe both ends x = 0 and /. The center of mass ii(t) of the portion of length 2a, from x=l/2 — a to x=l/2 + a, is given as a ( 0 = -=-\ la

Si/1-'

u(x,t)dx u

=—J4_ c-) a t ip * 5M

, ) / s

— KK

(64) \

I

/

80

M. Toda and Y. Kogure

168

and the velocity associated with it is 1

In-11/2

a V lp ««]d

I s

\n(^f-)q<(t).

(65)

In terms of the initial values q.(,0) and 9.(0) we have (66) where c denotes velocity of T>(0) are exq,(t) =the -mm-Ofs i nsound. ( - ^ ) +u(t), 9.(0)v(t), c o s (u(0) ^ ct) and , pressed as linear combination of q,(0) and
«(0

*(0)

vit)

!

(fi7)

«(o) vCO) where D is a 4 x « matrix. W e can use the method of §2, to evaluate the conditional distribution function for u(i) and v(t). W h e n the temperature is T , the covariances associated with q,(Q) and y«(0) are kT m

'

c:,. =
ffifi)

< ,(D)y,(0)> = 0. 9

The covariance matrix of the conditional distribution function for {n(t), v(t)} is calculated following the method of §2, to give <«<*)«<<)> s

<«(/)-(«)>

= <^W«(0>

F where, for the limit

(69) we have

81

Statistical

Dynamics

of Systems of Interacting

Oscillators

169

with T = C I / 2 , and A =

1-r

r
0

r>l.

— (2-r)r

(71) r
a c

t>l.

By inspection we see that the time dependency of given above is very similar to that of the covariance of heavy isotope given in the preceding section, when 2ap is replaced by the mass M' of the isotope. For t^i>2a/c, the position of the center of mass obeys the Einstein's law; (u(t) )oct. 2

§5.

Brownian motion of an oscillator

In this section we shall treat a system whose equations of motion are of the form given by E q . (37). Tbe model of such a system has been treated by the authors in previous papers and referred to as model S in the preceding section. In the previous papers, it was shown that the central oscillator of such a system exhibits Brownian motion provided certain conditions are satisfied. One of these conditions is that the interaction parameters are sufficiently small and another condition is that frequency spectrum of the surrounding oscillators forms quasi-continuum which extends from _ , = 0 to ™. Under these conditions we shall reconsider the motion of the central oscillator. If the number of surrounding oscillators with tbe square of frequency between oil and ufi + dto is denoted by G(oi )d(mi), we may approximate the sum in E q . (40) by (cf. reference 6 ) ) 2

^

At

^

2

S

A*(2)G(J2 )A

= -AHQ-)G(&)ncot^r

+ ^ [ A

where _? is replaced by sA.

where

A

2

(

a

m

; ) -AHQ)G(&)_

^ f ^ d M ) ,

„

(73)

1

J

a>i — S

Further we have

since J is a small quantity.

A

, O.P / ^ ) G (

Equation (40) yields

!

!

[/(£)G{fl )^]+(<»|-fi ) 2

s

'

therefore (75)

denotes the shifted value of ai . and is approximately given by

82

170

M . Toda and Y . Kogure

2

i

o) — to'i

2

with g(o,i)dws = G(w )d(^), XcU/A-dS, we find

vr.\

or g(m)=2w G{^). s

Since

2 ["" bO* sin fir , „ \ , _ , ^ —dQ, n: Jo b Q -V (_&% — Q ) Q

« ) =

a

a

a

L

v

cU/A-d(8r )=2Q

,-™\ (77)

S

where we have written i

bQ=A (Q)g(Q)n:/2Q.

(78)

s

If g(j2)ocj3 as in the vibrational spectrum in three dimensional substances, b turns out to be a constant provided A is independent of Q. If b is a constant we see that X(t~) satisfies E q . (27) and consequently, the motion of the centra] oscillator is Markoffian; that is, it exhibits Brownian motion. The distribution of the central oscillator satisfies the Kramers equation given by Eq. (29) with 0 = 2b. These results are in accordance with those in previous papers by the present authors. ' " 1

A s is well known, if the mass, the displacement and momentum are denoted by m, x and p respectively, Vm wx. and —p/v o> play equivalent roles in the motion of a harmonic oscillator with frequency to. Therefore our results are not changed by interchanging their roles. Actually, for a charged oscillator interacting with a radiation field interaction energy is of the form Jtpp: where p and p, denote momenta associated with the oscillator and tbe radiation field respectively. The coupling constant A and the frequency spectrum are ; 2 =

d

" t " ( m )"

=

*

k / c 3

f79

-

>

where in denotes the mass of the charged oscillator and c is the light velocity. In this case we have, in place of Kramers equation, the equation in which Vm IDX is replaced by —pjVo> and vice versa. That is. the distribution / of the charged oscillator is given by df at

,

d

P f m ox

a

9

/

op

r

( kT dx \ m

9

df . dx

,

r

\

with

The last equation is a counterpart of Kramers equation and is already given by one of the authors."

83

Statistical Dynamics of Systems of Interacting Oscillators References 1) 2) 3) 4) 5) 6) 7)

P. Chr. Hemmer, Det. Fysiske Seminar, Tronclheim No.2 (1955) , l. R. E. Turner, Physica 26 (1960), 269, 274. R. J. Rubin, J. Math. Phys. 1 (1960), 309; 2 (1961). 373. M. Tocla, J. Phys. Soc. Japan 14 (1959), 722. Y. Kogure, J. Phys. Soc. Japan 16 (1961) , 14; 16 (1962), 36. S. Takeno ancl J. Hori, Prog. Theor. Phys. Suppl. No. 23 (1962), 177, this issue. 1. M. Lifschitz, Nuov. Cim. Suppl. 3 Ser. X (1956), 716. M. Tocla, J. Phys. Soc. Japan 13 (1958), 1266.

171

84

JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN, Vol. 19, No. 9, SEPTEMBER, 1964

Some Properties of the Pair Distribution Function Morikazu

TODA

Department of Physics, Faculty of Science, Tokyo University of Education, Tokyo (Received March 31, 1964) Thermodynamics! consideration on the equation of state is combined with kinematical consideration on the two body distribution function. Thus, general relations between the equation of state and the distribution function are obtained without using statistical mechanics, and the fact that spacial correlation between molecules is responsible for the pressure equation is emphasized. The results are applied to classical systems and to quantum mechanical gases, and proved by statistical mechanics. § 1. Introduction The pair distribution function plays an important role in the theory of fluids . Internal energy, pressure, surface tension are rigorously expressed in terras of the pair distribution function and the interaction energy between a pair of molecules. If one replaces the pair distribution function appropriately by the two-body distribution function one obtains exact expressions which hold irrespective of the state of aggregation of. the system. Many attempts are made to derive approximate expressions for the pair distribution function. Finally Hiroike et al. found an exact method of calculating the pair distribution function and, at the same time, showed that the free energy of the system can be calculated exactly using the knowledge of the pair distribution function alone. However, this method requires solving a set of nonlinear integral equations and calculating an infinite series in powers of density. The purpose of this paper is to point out some general properties of the two-body distribution function and the pair distribution function. It is thermodynamics and kinematics rather than statistical mechanics. 11

Boltzmann constant and i is a quantity related to the chemical potential ft by the relation C=e"'« . (2.2) We have, therefore, 1

It is sometimes convenient to use the average volume v per molecule, and the average number density n defined by v = VIN, M=l/D=iW. (2.4) Eq. (2.3) is then written as

H

§2.

Equation of State in Thermodynamics and Kinematics a) We shall start with considering the wellknown thermodynamical relation

The expression for the compressibility is obtained. from

(3P\

(3C

_kr / a g T\3t>

(2.6)

where {dQdv) is given by differentiating Eq. (2.5) with respect to v.

°-f+*[#£)L(iX w By the use of this relation, the expression for compressibility reduces to ac Ut.r (2.8)

-#)-—(+:

.. 1

(2.1) where P is the pressure, V the volume, T the temperature, E the internal energy, N the average number of molecules in V, k the

C» (3'P

(2.8') where we have used Eqs. (2.4) and (2.5). One the other hand, we have

1550

85

1964)

Some Properties of the Pair Distribution Function

[^-«L-(1)-[*(#)L Combining Eqs. (2.8') and Eq.(2.9) we have an interesting relation

(2-10) From the thermodynamics of fluctuation we have :N-N)' (/V)'

kT/3v\ V'\dP) j

=

(2.11)

T

so that, comparing Eq. (2.8) with Eq. (2.3), we have (2.12)

1551

This equation holds irrespective of the state of aggregation (gas, liquid or solid) and irrespective of the statistics obeyed by the molecules of the system. It is seen from Eq. (2.17) that if BoyleCharles' law is violated, the two-body distribution function is no longer constant, but correlation in space exists. There are two effects in this respect. One is the effect of the potential energy between molecules. The other is the effect of quantum statistics. Even if the intermolecular interaction is absent (ideal gas), quantum statistics introduces spacial correlation between molecules. In Bose statistics molecules have the tendency to attract each other and in Fermi statistics they repel each other even if the intermolecular force is absent. Thus the deviation of the equation of state for a Bose or Fermi gas from that of a classical perfect gas can be attributed to the difference in the pair distribution function. This effect will be treated in §4.

b) Let p^Kr, r') be the two-body distribution function. This means that p (r, r') d'rd r represents the average of the product If the transition point is approached by of the number of molecules in the elementary increasing C, the fluctuation of density becomes volume d'r at r and that in d'r' at r' very large and Eq. (2.14) shows that iSfr>' (r. Foimally r')—n^d'rd'r' will have a singularity of the °«Hr, r ' ) = ( 2 S ' i ( r i - r ) i ( r r ' ) > „ , S-function type with respect to C. This (2.13) behavior of the distribution function will where ri stands for the position of the i-th ensure the constancy of the pressure during molecule. When p —« is integrated over the phase transition. c) From purely kinematical considerations, the volume V, we get Clausius derived the virial theorem. This {p*(r, r-)-n*\d>rd-r =N'-N-<,N) theorem is quite general and valid as well in quantum mechanics as in classical mechanics '. (2-M) The virial theorem can be written as du(r, ). PV = \K — ( - f ( f ) , , - i ) ' « dr, - w where we have used Eq. (2.11). (2.18) Eq. (2.10) yields therefore where K stands for the average of the kinetic energy of the system and u(r) represents the intermolecular potential energy between two = -^r^W'Kr. <-)-«W
3

!,

r

m

!

2

1

t

5

)

2

\[

{""Kr- r'\T,

C)-»1«V . = j^p<»(r, r')-n')d'rd'r' . (2-17)

(2.19)

86

1553

Moriki

This is a relation which is also valid in classical and quantum mechanical cases. d) In the case of fluids p'"(ri, n) is a function of the distance rn = |ri—fi\, andean be written as P«Kr,. r , ) = » ' j M ,

(2.20)

where g(r) is called the pair distribution function, which has the property Um£(r)=l. (2.21)

u TODA

(Vol. 19,

f) In general, thermodynamical quantities such as PV/T and n — N/V are functions of C or of the quantity (fugacity) e=c/J*(r>. (2.27) where we have introduced for convenience the de Broglie wave length, for the molecule of mass m, given by A(T)=hl(2nmkT)>". (2.28) We may formally expand PIKT as

In this case Eq. (2.17) reduces to

-£r=iwr)f.

^"-j^Jn-bjM-ltf'r.

(2.22)

(2.29)

We then have, as in Eqs. (2.3) and (2.8) n=T.Ib,i' ,

and Eq. (2.19) to

~ ( % )

(2.30)

- 4 - S » # .

(2.31)

and Eq. (2.10) can be verified easily. We have further = v\n*[g(r)-\)d>r

(2.23)

= V_/
J|(/>"'('-,r')-Ji')d'rdV

(2.32) e) We have used the formula for fluctuation, If we expand p'" in the same way as which is Eq. (2.11). But this expression gives P-"(rir,\T, f)=2 ajfxi. r,]T)(> . (2.33) the fluctuation of the number of molecules we have the relation between ai and bt, in the whole volume V which is assume to be very large, and gives no knowledge of the df\\ ^d^ ' ' ' ' ' ~ ' • functional form of g(r). (2.34) To seefluctuationin more detail we have It should be stressed that these relations to examine, for instance, tbe Fourier comare obtained without any knowledge of the ponent statistical properties of the distribution n = y\ Wr)-n}e-f'd>r . (2.24) function. So far, we have made use of thermodynamics, thermodynamics of fluctuwhere n(r) stands for the instantaneous number ation, kinematical definition of the distribution density at r and »=«(r) is its average. The function and the virial theorem which is Fourier coefficient of the spacial correlation considered as a purely kinematical equation. is given by Therefore we have not made use of the usual statistical mechanical consideration in phase (n(r.)-ji) {n(rt)~n) erW'-i'd'rd'r^y space or in terms of density matrix. For instance, in classical mechanics of an imperfect = N+ Vn' | \g(r)~ Dt-fcfr (2.25) gas we have the cluster integral of / atoms, Or, using Eq. (2.24). we get an extension of which is just equal to the bi in Eq. (2.34). Also in quantum statistics we have the Eq. (2.IS) conventional expansion formula of the grand •:'\ {g(r)-l\c-''d'r^V\n J -n . (2.26) partition function which has the form of Eq. (2.29). We may cite other examples, such If the | B / | ' is given by some method, the pair as the liquid drop model of condensation ', distribution function g(r) is given applying where the statistical mechanical interpretation the Fourier inversion formula to Eq. (2.26). of Eq. (2.29) is straightforward. For instance, for an elastic body |«/l is the intensity of phonon of the wave number f, § 3. Classical System and g(r) gives the correlation of the constituent a) For a classical system temperature is molecules. defined by r , i

, I

( T i r

) d

r

d

r ! = { !

l )

Vb

11

/

j

I

j

l

1

5

87

Some Properties of tke Pair Distribution Function

A-=J-WftT.

1553

(3.1)

4 s

In this case Eq. (2.19) reduces to

3 *ii(N-l)}

]r

(3.10) l!l

/> (r,,r,|r, Q T

,!,

1

= jj(r' (r.r'}-« lrfV
"• d*r ---d»r„

(3.2)

l

.

(3.11)

For a fluid we have, instead of Eq. (3.2),

where C/« = S2'ii('u) stands for the interiS J - l

action energy between molecules, (3.3) and in place of Eqs. (2.16) and (2.17)

m the mass of a molecule, and 8 is the grand partition function

1

C^(P-n*r) = - * r B J | ^ ( r ) - U d V ,

I

t

1

(3.12)

(3.4) (3.5)

^ =n-[ ^[B i£r()-)-l}rf r. ST Jo C J For an infinitely dilute (perfect) gas we have P=nkT, or

5= 2 ~Qx » i i iV!

.

(3.13)

with Q defined by , e-"»i"d>r - --d'rn . (3.14) N

l

• i - i

Integrating o<"(r,, r \T, Q) with respect to r and re, we get 2

(dPldQ ^kT(~)

^kTnlC. ,

T

(3.6)

where we have used Eq. (2.1). Integrating -we get, for a perfect gas. n=f(JX

(*-»0),

QN=N(N-1),

(3.7) ,l

III

,

1

where /(T) is a function of temperature, and Unyv,tfV = [[p' (r,)p (r )£; ridVi=(/V) , is not JJ is not necessary necessary here. But if one refers to JJ (3.15) ^statistical mechanics one sees that f(T) = 1/A\T) where A(T) represents the de Brogue l e averages. wave length defined by Eq. (2.28). On the other hand we have For a slightly imperfect gas we may use (3.16) Eq. (3.7) and g(r)=e-*< "' in Eq. (3.3). Then kT we get I

w

n

s

J~\r—d r=\(E-"»i"-l)d'r 3kT) dr J

h

e

r

e

w

e

h a u e

I

t a k e n

t

h

e e n s e m b

T

, (3.8)

tf=s£ In B, N'-Wy^S— . (3.17) of of where differentiations are such that T and V are kept constant. We have therefore

-which can also be obtained by integrating partially. At the same time Eq. (3.5) gives P=nkT-

fS<-

(3.9)

Therefore Eqs. (2.19) and (2.17) are the extension of the well known formulas Eqs. (3.8) and (3.9). b) If we refer to statistical mechanics and restrict ourselves to classical systems Eq. (2.16) can be derived using the expression for the distribution function in the grand canonical •ensemble' : 1

II

1

= -|^P' (r,,r )-n )dV,iiV , I

!

(3.18) in which f(5/3£ = £i?/dC since T is kept constant. Thus Eqs. (2.16) and (2.17) are verified using statistical mechanics as for classical systems, §*• Perfect Gas in Quantum Statistics As is well know, the number density of the molecules of a perfect gas is given by

8*

1554

Morikazu TOD A „=(

(4 1)

where i=p'l2m , (4.2) according as it obeys Bose (upper sign) or Fermi (lower sign) statistics. The pair distribution function g(r) can be calculated rigorously in this case, by using the quantized wave function lF(r) = V"" 2«*e "'*, and »(r) = if+(r)F(r), » = E a , « , . (4.3) * Then using the method given by Eq. (2.24) to (2.26) with the aid of Fourier inversion formula we obtain" I

i

,

»»(*<»•)-1) = =

± l | f e-""d'p

Eq. (4.4) expresses the fact that Bose particles attract while Fermi particles repe) each other. Therefore we obtain

l

8

W

-

1

|

f

' 4 i ( F ^ ? r r

and

_[d'p

-J

!

42 ( Jo e*'""TC «' Jo ^ n i j l n f l + Ce-""")^/^ Thus we get

(4.6)

(Vol. 19. • ^ = T jlndTCe-^W/A" .

(4-7)

which is the well-known expression for the equation of state. Acknowledgments The author is indebted to Emeritus Professor Takehiko Yamanouchi of the University of Tokyo for his continued interest and encouragement in the author's research fields. With the sincerest congraturation the author would like to dedicate this article to Professor Yamanouchi's sixtieth birthday. References 1) See. for example. J. O. Hinschfelder, C. F. Curtiss and R. B. Bird: MoUeular Theory of Gases and Liquids (Wiley, New York 1954). 21 See for references of the paper by T. Morita and K. Hiroike. Prog. Theor. Phys. 25 (1961) 537. which includes the references for the same but independent work by J. de Boer et al. 3) On mechanical and statistical mechanical derivation of the virial theorem, see M. Toda On the virial Theorem in Recent Problems in Physics (Iwanami. Tokyo, 194S). 4) L. Landau and L. Lifshiti: Statistical Physics (1951). 5) D. ter Haar: Statistical Mechanics (Rinehart. New York 1958). 6) T.L.Hill: Statistical Mechanics (McCnw H\[\, New York 1956). Certain parts of this text book should be corrected in view of the reference 3) above.

89

1965)

2095

Short Notes

J . P H I S . S O C . J A P A N 20

where a is an arbitrary constant. We then get

(1965) 2095 - 2 0 9 6

One-Dimensional Dual Transformation

!

H=Z ±- P?+ k^f-lQi+,-Qi)*, T

Q* .=0

i

+

(5) Morikazu T O D A where we have put Department of Physics. Tokyo University of Education. Tokyo _ L £ . (Received September 15. 1965) mi* a my In the theory of the lattice vibration of linear The system thus obtained consists of particles with imperfect crystals, we sometimes meet with different the masses m,*, m£*,---, TH.W" and spinas with the >* ™jv*i systems with the same frequency spectra. The force constants K,', Kt* . author found the following mechanical theorem in are respectively subject to the free and the fixed end conditions. This system is the "dual" of the this connection. Consider a linear chain of jV particles with original system, and viae versa. masses mi, ms,---, ms- if the force constant of The relation between the dual systems can be the spring between the (j-l)-th and the j-th particle summarized as is Kj, the Hamiltonian of the system is JL 1 " Ki (7) S= Z s — Z ¥l*i-tn-.$. (1) (ii) a fixed (free) end of a system corresponds We have assumed thefixedend condition for the to a free (fixed) end of the other and vice 0-th particle, ?fo=0, and the free end condition for versathe N th particle. The restriction thus imposed can The systems with the Hamiltonian (1) and (5) are be easily removed and the following results hold equivalent. Especially the frequency spectra of irrespective of the end conditions. the normal mode of vibration are the same, which The relative displacement can be checked with respect to special cases where ry=ui-u,-,, (2) the analytic solutions are obtained". The concept of duality can be generalized to the between the adjacent particle can be used as the generalized coordinate. If the conjugate momentum case of nonlinear force. The detailed account will be given elsewhere. is denoted by s;, we have (Bj-SF/Srf) =

m

1

m

11

m

a

n

d

r

!h


s

ri = Pi!a.

Si = ~aQi.

References 1) Progr. theor. Phys.: Suppl. No- 23 (1962). 2) to be published in Progr. theor. Phys, Suppl. (1966). (4)

90

Supplement of the Progress of Theoretical Physics, No. 36, 1966

H3

One-Dimensional Dual Transformation Morikazu T o d a

Tokyo

Department of Physics University of Education,

Tokyo

One-dimensional lattice with the nearest neighbour interaction is considered. If the interaction is harmonic, the system has its counterpart or the "dual" in which the mass of each particle is replaced by a force constant, the force constant by a mass and certain conditions are imposed. The systems thus defined behave the same and have the same frequency spectra. The concept of the dual systems is not restricted to those with harmonic forces. As an example, equivalence of a special anharmonic chain and a semi-relativistic chain is shown.

§1.

Introduction 0

In the theories of lattice vibrations one meets with similarity of expressions for the localized modes of a system and those of another. F o r example, the expression for the localized mode of a system with a single isotopic impurity resembles that of a system where a single force constant is changed.* In this case, a simple relation is seen to hold between these expressions. One of the aims of this paper is to make clear the similar relations which actually exist between systems which we shall call "dual" of each other. 5

In this paper the dual systems mass of a system corresponds to an of the other system and vice versa, are mutually equivalent in the sense

§2.

are defined in such a way that each interaction between nearest neighbours and the hamiltonians of these systems of canonical transformation.

Dual systems

Consider a chain consists of particles with masses mi, m%, ••; m . Between the particles j—1 and / , we assume the nearest neighbour interaction with the potential energy $ > ( b > w h e r e u, denotes the displacement of the j - t h particle. L e t p, — ?n,Ui be the momentum, then the hamiltonian H of the system is N

—

H=K+U

(2-1)

where K denotes the kinetic energy (2-2)

91

114

M- Toda

If the force f energy is

is acting from

right to the end particle N, the potential

ry=s^(«i-^-i)+/««.

(2-3)

W e may assume either fixed-end «o=0, &=?0 or free-end & = Q if / = 0 . Since and are canonically conjugate to each other, in terms of the Poisson bracket, we have [X„p>]=8„ .

(2-4)

t

N o w , the relative displacement ry=M/-«i-i

(2-5)

between the adjacent particles can be used as the generalized coordinate. Without loosing generality we may put u ~Q with 0

m=n,

u = r, + r , 3

Ui=rx+ri-\

2

i-r-j, •••

(2-6)

and consequently ,

K ^ ^ { , r ' + m.Xr + rd + --- + m (r + r +--- + hy+---} ft m

l

i

I

1

(2'7)

The momentum s, which is conjugate to r, is given by s, =

= m,(?\ + r + • • • + rj) + m, (.h + h + — + r ) + ••• t

+1

m

+ m„(r, + r + — +r ). 2

(2-8)

N

Therefore Sj—sjt, = m, ( h + h H s« = THN U

hf,) = w i , ,

(2 • 9) (2-9)'

N

or pi=Sj-S

M

0 = 1 . 2.-.N-1),

p„ = s„.

Thus the kinetic part

and the potential part

constitute the hamiltonian.

r and J , are mutually conjugate: s

(210)

92

One-Dimensional

Dual

Transformation

115

tr*.

(2-13)

It is now clear that we can exchange the roles of the generalized coordinates and momenta. That is to say, we may introduce the new momentum Pj and its conjugate coordinate Qi by r = P /a, J

i

S,= -aQ,,

(2-14)

[Q,, P.] = « / , . .

(2-15)

where a is a constant which can be chosen arbitrarily. If we choose a = l , the transformation from (r, s) to (P, —Q) means simply a rotation of (r, s) coordinate system by the angle —n/2. B y the transformation (2 14) we get

)+/— |+S-/-CO/ .-0>) +-^-
+

(2-16)

Actually in some special cases, the summand of the first and the second term can be interpreted respectively as the kinetic and the potential energy of physical systems. W e have thus obtained two systems, one given by Eqs. (2-1) to (2-4) and the other by Eqs. ( 2 1 5 ) and ( 2 1 6 ) . These systems are equivalent; they constitute a set of dual systems.

§3.

Ki,

5

Harmonic systems '

If the particles are joined by elastic springs with KN, the hamiltonian can be written either as

H =i S - ~ - ^ + Sy-i- ^d & * ^ * ^ * , - i 2mj

force constants i f , ,

«o=0

(3.1)

or as tf=S-^PH2^«?*.-Q,)*, j-i 2mJ i-i

Q

1

w +

,=0

where for brevity's sake the external force is assumed zero. masses m* and the new force constants Kf are such that -X-=^-, Jhja

*7=|p.

(3-2) T h e new

(3-3)

mj

Thus we have two systems equivalent to each other: T h e system with masses my and force constants Kj is equivalent to that with masses mf and force constants Kf, the displacements and momenta of the former being u, and p and of the latter being Qi and P . it

t

93

116

M . Toda

The relation between these equivalent, or dual systems can be summarized as ' 1

(i)

K mf

(ii)

a fixed (free) end of a system corresponds to a free (fixed) end of the other and vice versa.

1

= m Kf

= K m? = m Kf

1

2

2

= -=a\

(3-4)

Schematically, these relations can be expressed as (fixed) x ^ - m

^ - m

1

-^m (free)

1

w

*JlL *K^... %Kl_

m

m

m

x

(fixed)

The dual systems thus defined behave the same, and especially the frequency spectra of the dual systems are the same. Examples: (1) Simplest example is a mass m attached to a spring K. tically, the system

Schema-

(fixed) x - ^ - m ( f r e e ) is equivalent to the one ( f r e e ) m * - ^ - X (fixed) 2

if Km* = mK*( = a ).

T h e proper frequency of these systems is /


=v K*/m*

.

(2) Masses m, and m are coupled by a spring with the force constant K, and both ends are free; s

K m, wzj The dual system in this case consists of a mass m* and two springs Kf and Kf with fixed ends.

The relations to be imposed are m, Kf = Km*= m^Kf

(3 - 5)

The angular frequencies to are the same: „.

=

( - U - L W = \ Wi m / 3

(*? + * ? ) _ . m

(3-6)

M4

One-Dimensional

Dual

Transformation

117

(3) Localized mode of an infinite chain, with equal masses m and equal springs K except for an isotope with the mass ?n (^<,m), is given b y ' !

0

with OIL—V4K/m . T h e dual system consists of equal masses m* and equal springs K* except for a spring K*. T h e localized mode i n this case is given b y !)

2 § ^ - l V"K*

K* I /

with at = v 4iC""/m* .

(3-8)

T h e relation is

mK* = Km* = m Kt.

(3-9)

a

M a k i n g use of these relations we can easily verify that o)i = a>*,

(3 10)

w — w*.

T h e vibrational frequency spectrum of the dual systems are the same. W e may verify this using other examples, for instance, with respect to the maximum frequency a>r. or the optical frequency of a diatomic lattice, and the localized modes when the analytical expressions are known. T h e case of an infinite chain without impurity is a special case of the example (3), where we have only to put m — m, with the consequence that the maximum frequencies of the dual systems are the same: a> — u>*. (4) If there are two light isotopic impurities of mass m on neighbouring lattice points, the symmetric localized mode is given hy* a

L

a

}

T h e dual system i n this case consists of particles with the same mass and two neighbouring springs K* differing from others, and has the pulsating localized mode 4)

^ = „

?

^ . /

2

/ ^ - l

.

(3-12)

T h e relation between these systems (3-13) or K _ K* m m m* " m

_K* K*

(3-14)

9S 118

M. Tocla

implies that (Jh=wt and w = w* . other localized modes.

§4.

The same situation holds with respect to

Anharmonic systems

The equations of motion for the system defined by Eqs. (2·2) and (2 · 3) are . _ aR _ pj

Uj- -

-

-

ap;

aaR

1>j= -

Uj

-- ,

mj

=f/J~(Uj-Uj-1) -f/J~+1(Uj+1-UJ -fohN.

(4·1)

These are equivalent to those derived from Eqs. (2·11) and (2,12):

7-1= aR =

aS

- 1- (51- 52), m1

1

N - 1),

5j= -

aaR = rj

-f/J~(rj) - f

(j= 1, 2, "', N)

(4·2)

These equation may be written in terms of Q, and P j making use of Eq. (2·14). In the case of an infinite chain consisting of the same mass m, we have (a=l)

.

Pj =

1 m

- ( Q j-I

+ Q } + I - 2QJ, (4·3)

Therefore, if the second equation can be solved as

(4·4) where """ the inverse function of f/J~, is assumed to be single valued, then the equation of motion is reduced to

(4·5) As an example, we may note the anharmonic chain with nearest neighbour interaction

96

One-Dimensional

Dual

Transformation

119

f O W . J a v V + r ! -r >

(4-6)

0

where =

3

r„ is a constant. n , . / ? | r I. Since

F o r small r, ji^rjj = (K/2)?-

Q = 4' ( P ) = roKP/VP'

+ rl

and for large >-.

(4-7)

can be written as !

OiK*-Q*)P'=Q rl

(4-8)

the inverse of E q . (4-7) is

If we rewrite as mr„ = Mc/r,

r K=c,

(4 -10)

a

the equation of motion takes the form d

MQi

=- t2Q -Q - -Q,+{) Y

l

l

1

(4-11)

which is an equation partly relativistic. Thus it has been shown that the anharmonic chain with the interaction given by E q . (4-6) is equivalent to the semi-relativistic chain. If there is only a mass, the semi-relativistic oscillator with the hamiltonian H= i/MV

r

+ ? P - Mc> + ^-Q>

(4 • 12)

is changed by the canonical transformation Q-*P>

P—u

(4-13)

to the anharmonic oscillator with the hamiltonian M=

+ r,K {Vu^+rJ

-r„}

(4• 14)

where the parameters M, c and r are replaced i n accordance with E q . (4-10). References 1) 2) 3) 4)

c. f. Prog. Theor. Phys. Suppl. No. 23 (1962). E. W. Montroll and R. B. Potts, Phys. Rev. 100 (1955), 535. M. Toda, J. Phys. Soc. Japan 20 (1965), 2095. S. Takeno, S. Kashiwamura and E. Teramoto, Prog. Theor. Phys. Suppl. No. 23 (1962), 124.

97

JOURNAL OF THE PHYSICAL SOCIETY OP JAPAN, Vol. 22, No. 2. FEBRUARY, 1967

Vibration of a Chain with Nonlinear Interaction

Morikazu TODA Department of Physics, Faculty of Science, Tokyo University of Education, Tokyo (Received September 27, 1966) Vibration of a chain of particles interacting by nonlinear force is investigated. Using a transformation exact solutions to the equation of motion ate aimed at. For a special type of interaction potential of the form d(r) = 4-e-"'+Dr+const., (a, b>0) 0 exact solutions are actually obtained in terms of the Jacobian elliptic functions. It is shown that the system has N "normal modes". Expansion due to vibration or "thermal expansion" of the chain is also discussed. found by computer analysis that nonlinear system tj 1. Introduction Our aim here is to study a simple mechanical have "normal modes where a normal mode is model, a chain of particles. We assume that defined as motion in which each oscillator moves interaction is limited to the nearest neighbors, at essentially constant amplitude (energy) and at and that the interaction force is nonlinear. a given frequency". It is therefore anticipated Recent studies on nonlinear oscillator systems that nonlinear systems will admit analytic soluusing perturbation methods" and computers tions. Here we shall show that we have actually indicate that they have many properties in common analytic solutions. Thfise solutions reduce to the with linear systems. Thus J. Ford and J. Waters" normal modes in linear limit or in the limit of 11

98

432

Morikazu

small amplitude, and cover all the modes of the linear chain. Though the system treated in this paper is a particular chain, it seems that a wide scope of nonlinear systems is susceptible to analysis and solutions to the equation or motion may not necessarily be incredibly complex. Existence of normal modes implies nonergodic character of the system and the nonergodicity of nonlinear systems might have a wide range of validity. The complete solution to a nonlinear system is not just a sum of normal mode solutions. To answer the problems of an approach to equilibrium, of the capacity of a system as a heat bath, of heat conduction and of other interesting problems which have connection with the ergodicity, the behavior of the complete solution must be investigated. The complete solution which satisfies an arbitrary initial condition is not given at present. In § 2 the method of dual transformation -" is described. This method also serves in classifying the type of nonlinear interactions. Jn § 3 a system with a special type of interaction is treated. The normal modes are given explicitly. This system exhibit '-thermal" expansion, which is shown to be in agreement with the result of conventional method in the limit of small anharmonicity. 1

§ 2. Dual Transformation We are interested in the vibration of a uniform chain of particles with nonlinear interaction. The equation of motion for the n-th particle in the chain is H

mu,= - (S'( „ —n„-i)-+-(&'(""+1—u.) , (2.1) where in stands for the mass of the particles, 0(r) the interaction energy between adjacent particles as a function of r. the mutual displacement, and jj'(r) its derivative with respect to r. We consider displacement of particles along the chain, that is, londitudinal displacement. The mutual displacement r„ is defined by

(Vol. 22,

TODA

ticle, which we assume to be subject to a given condition. Though the equation of motion (1) does not include the boundary conditions explicitly, it does implicitly. We impose the boundary condition that the displacement of the 0-th particle is a given function of time, i.e., «o-r (/) (given) (2.4) and assume the force /(/) applied to the jV-th particle. The potential energy U of the system can be written as 0

tf=Sd(r„)+2r„/(0

(2.5)

and the kinetic energy as 2 (6

•HMWJW

->

We use the mutual displacements r. as the generalized coordinates, whose canonically conjugate momenta are given by s = dKldu (n=0,..-,N). In lerms of s„ the kinetic energy is n

(2.7)

31

K=-^~ E ( * . - ! . « ) »

(2.8)

where we have put s« = 0 . (2.9) The Hamilton's function is H(r„ i„)—K-\-U, which yields the canonical equations of motion of the form +1

r.=-r— = — (Zs.-s._i — J.+O . os„ m

(2.10) (2.11)

The set (s., —r.) can be regarded as that of general coordinate and momentum, which furnishes the dual transformation described in a preceding paper.* Consider next the case where eq. (2.11) affords single valued solution, which we shall write as 1

(2.12)

r. = —-x(s*)

r„ = u.—u„-i (2.2) In general ^ is also a function of /((). Equation We consider the chain to be of finite length and (2.10) then gives the equation of motion for the label the particles from 0 to JV £lfc=0,- • ,N). dual chain of the form The kinetic energy or the system is then given (2.13) ^-KsH) = -2s.+i.-i-r-J-+i by (2.3) Thus the equation of motion takes the form quite similar to the linear case. Alternately we 2 „~0 where for convenience we have not taken aside the may write eq, (2.13) as part responsible for the motion of the 0-th par-

^„)^-2S„+S„_,+S„

t l

(2.14)

99

1967)

Vibration of a Chain with Nonlinear faieractiott

4J3

with (2.15)

O a Therefore the equation of motion is (mass m=l)

If the applied force / is constant or if no external force is applied, we have

— V = — f>(2s„ —J,-,—J„ i) . +

r'(i,)i =-2j. + j„_, + j„ , . n

(2.16)

+

We see that the characteristics of the nonlinear interaction is now reflected in the functional form of x(Jr). Therefore we may classify the type of nonlinear interaction by the form of the function xThough the above treatment assumes finite length of the chain, we may treat travelling wave solution. Physically [his is accomplished by choosing appropriate input and output, or r (t) and /(f) at the both ends of the chain. c

(3.5)

«+». It is shown in Appendix I. Lhat the travelling uave solution for eq. (3.5) is of the form (3.6) where 1 —i + (3.7) [sn'^K/t) f" E Z(u)=l dn'udu u (3.S) Jo K In the above formula sn and dn represent the Jacobian elliptic functions : K and E are the complete elliptic integrals of the first and the second kind. These are all of the same modulus which we shall write k. The complete elliptic integrals are

I}-

11

§ 3. Nonlinear Chain As an example of nonlinear interaction we shall treat the anharmonic potential of the form t

^(r)=-^-e- '+or+const. (a,b>0) b

(3.1)

The two terms imply repulsive and attractive forces respectively. The parametersare so chosen as the minimum of the potential is just at r=0. or f[0) = 0 .

(3.2)

We assume the case of no external force, and then eq. (2.11) is •_

Mfr.)

=-a{l-exp(-MJ •

(3-3)

which yields

- = - > ( • +

s

f

i

Vl-^sin'fl ' E=£(£) = l

Vl-ft'sinWfl .

(3.9) (3-10)

The Z-function has the periodicity of 2K, Z( +2K)=Z(M) , (3.11) and v and X represent the frequency and the wave length respectively. The modulus k is responsible for the amplitude of the wave as will be shown below. Given the wave length X and the modulus k, eq. (3,7) gives the frequency a. From eq. (3.4), (3.6) and (3,8) we have U

M - ( - i

))-#])•

(3.12)

If the modulus k is very small, we have jniiSsinH ,

1

E

k j '

1

k Z(H)S?—sin 2u ,

(2.13)

Therefore if it
oik' . I 2rtn\ • sin ( ait H I ,

oiSz2V~r~ sin-y- ,

j—ab.

(3.14) (3.15)

Thus if the amplitude of the wave is small, our solution (3.6) reduces to that of the linear case with the force constant j. For sufficiently small * or small amplitude of the wave, we have the wave, for the original chain,

100

Morikazu

434

(Vol.22,

TODA

By adjusting appropriately the end conditions r for the 0-th and f(t) for the A'-th particles, cyclic boundary condition can be realized to yield the fi "normal modes" in the sense that they reduce to the N normal modes in the limit of small amplitude. We consider the ensemble of the chains. If each normal mode is excited in each chain, it will exhibit expansion because of the anharmonicity. If the amplitude is small we may expand the righthand side or eq, (3.4) as 0

Since the time average of s* vanishes, the time average of r„ is, to the first approximation,

F,=-^7-77.

(3-18)

in which we may use the zero-th approximation for s„. In the zero-th approximation in turn we have r =—j»/afi and the chain reduces to a linear chain with force constant j=ab. Therefore in this approximation the equipartition of energy gives (£ = Boltzmann const.) B

B

Z.—:

r^rr.^Z"

(3.19)

Thus it is shown that r7=^-k T.

(3.20)

B

On the other hand if we expand ip'(r) for small r we have

Therefore eq,

(3,20)

can be written as T.^kBT

(3.22)

This is in accordance with the result which is given by the conventional method of statistical mechanics. Appendix I. The following relation can be verified easily: sn\u + v - J J ! ' ( K - I > ) -

I--

———

—

.

(Al -1)

dv l—k'sn'usn'v where * denotes the modulus of these Jacobian elliptic functions. Since !

!

dn'u= l - * i n u ,

(AI-2)

if we define the function ,du l(H)=j <SAn

(AI-3)

and use its derivatives l

s'(it)=dn u ,

E"0)= —2k'snucnudnu ,

(AI-4)

101

1967)

Vibration of a Chain with Nonlinear Interaction

4

we have

By simple transformations we easily arrive at eq. (3.6) in the text. The same conclusion can be drawn by using the relation to the ^-function (tf,=i!,)

and the relations

3s*

B

'

fl.(0)

L *<(<>) J U w J

i

rf,(»+w)^t»-iv)[^tO)] =[*t{i')'J-t&MftMt"

.

(Al-8)

Appendix II. We shall here treat the wave with the shortest wave length. This wave exhibits the character of stationary wave as well. Consider the mode rtn=J+2i,

rpki.t=4—2x ,

(All-1)

in an infinite chain. Here A denotes the average value of r„ and ± x represents the displacement of each particle from their average positions. The equation of motion for a particle is seen to be of the form (mass m = l) M

J

x= {e- +">-e-'" -'"} .

(AH-2)

a

or *=-a'sinh 2fct,

(All-3)

with 11

a'—ae- .

(All-4)

The solution of the above equation is e

±i».

= e

dnKValTct.k)

(All-5)

where k is the modulus, and 1 c —V I - * '

(AIL 6)

1

If there is no external force, J represents the natural expansion of the chain as it vibrates. In this case J is given by the condition that the average value of j„ is zero, or ^= - V^=0, dr o

(AD.TJ

o

However e-»-=cl

dn'(Va'bct)dt j - ^ ^

= c

c

^

(AH-8)

where K and E are the.complete elliptic integrals of thefirstand the second kind of modulus k. We have therefore e-o'c— = Q . K

(AlI-9)

102

436

Morikazu Ton*

(Vol. 22,

The coefficient of r in (All-5) is therefore JWbc^^cTb^.

(AIL 10}

This wave has the wave length J=2. On the other hand eq. (3.7} gives, for 1 — 2, the coefficient of (in eq, (3.12) 2A-,= V ^ / / J .

CAIM1)

51

Further, since we have the relation

l

•^l_ dn (2Ku!- K) = V{^k'/dn'(2Kv!) , ks

r

(AH-12}

C we see that (he wave treated here is in agreement with the special case 2 = 2 of the wave given by eq. (3.12). 113. References 5) H.B.S. Jeffereys: Mathematical Physics (Cam1) J. Ford: J. math. Phys. I (1961) 387. bridge University Press, 1956). 3rd. ed., Chapter. 2) J. Ford and J. Waters: J. math. Phys. 4 (1963) 25. E. T. Whittaker and G. N. Watson: Modern 1293. 3) M. Toda: J. Phys. Soc. Japan 20 (1955) 2095. Analysis (Cambridge University Press, 1927), 4 th 4| M. Toda: Progr. theor. Phys. Suppl. 36 (1966) ed.. Chapter 20-22.

103

J O U R N A L OF T H E P H Y S I C A L S O C I E T Y O F J A P A N , Vol.

23, No.

3 , S E P T E M B E R , 1967,

Wave Propagation in Anharmonic Lattices Morikazu

TODA

Department of Physics, Faculty of Science, Tokyo University of Education, Tokyo (Received May 31. 1967) Analytic solutions to the equation of molion in anharmonic one-dimensional lattice are given. Wave-trains and solitary-waves which propagalo in the lattice arc studied with reference to the limiting cases of the svstem of hard spheres and to the continnm limit. g 1. Introduction In a preceding paper" the author gave exact solutions to the equation of motion in a lattice of particles interacting with nonlinear force. The adopted interaction potential between adjacent particles is of the form.

solutions are stable with respect to small perturbations, the system is non-ergodic. A slep to approach this problem was given by the numerical calculations by Zabusky and Kruskal.' They studied the so-called Korleweg-de Vries equation, or the long-time asymplolic behaviour of long waves in anharmonic lattices, and observed that ^R^/fe-w+aJt+consl. (A, a. 6>0) (1.1) solitary waves or "solitons" pass through one where R is the distance between the particles, and another without losing their identity. Thus we see A, b and a are constants. If natural distance that the solitary-wave solutions are stable. On between the particles is denoted by D, (i(R) is the oiher hand Ford and Waters" had shown that rewritten as the standing normal modes are stable. It seems likely that nonlinear interaclions do not guarantee jK*M-4- e-'f-^ + ofR-D)-const. (1.2) the ergodicity of the system in general. b 1

and if we expand d-(ff) in power scries of R — D § 2. Equations of Motion we get Let usfirstconsider a uniform chain composed N)connected by p(R)=const.-|- — ( R - D ) - — (ff-D) -! . or jV-f-1 particles(n=0.1, 2, ;V springs which exert nonlinear force. We as2 6 (1.3) sume that the panicle n = D isfixedat the origin, Therefore iT we keep a*=finiLe and take the limit .V.. -!). ir the natural length of a spring is denoted by D, the equilibrium position of the .i-th particle b->Q, a—'co, we obtain the harmonic case. On the other hand if we take 6-»:o we get the limit is nD when there is no external force. Now the of hard sphere interaction. Thus the adopted kinetic energy K and the potential energy U of the potential have wide applicability. The aim of this system arc given by paper is to extend the treatment of the preceding paper, to the case where uniform strain is present (2.1) and to the solitary-wave solution. In addition, the hard sphere limit and the continuum limit will be given. Ford and others' showed by numerical cal- where m stands for the mass of a particle, X the culation and perturbation method that the equipar- position, X„ the velocity of then-th particle, #(K„) tition of energy does not take place among oscil. the potential energy of a spring as a function of lators coupled by nonlinear forces, and further the distance that there exist "normal modes", which means the R^X.-X,-, , (2.2) non-ergodicity of the system. Recently Saito between adjacent particles, and F the external showed that some correlation functions do not vanish after a long lapse of time, which indicate force acting on the /V-th particle. The generalized momentum j-„ which is canonthat the system does not reach the state of thermal ically conjugate to R„, is defined by equilibrium. 1

1

11

1

n

11

If the "normal modes" or the exact analytic SOI

s„ = dLISR» ,

(2.3)

104

Morikazu

502

(Vol. 23.

TOD*

We can therefore remove the restriction that the where L denotes the Lagrangean, L — K—U. system befiniteand use eqs. (2.11) to (2.13) also Since" fljjr.=J»—J.+i , (2.4) for an infinite chain. Actually we can derive eqs. (2.12) and (2.13) directly for this case. Since the Hamiltonian of the system is given by the equation of molion for an infinite chain can be written as (2.5) with

(2.14) subtracting the same form of equation for mX*-, from eq. (2.14), we obtain

the equations of motion,

,

m

mR,=a{2e,-*t''*-'» —e-"*.-i-*>—e-*u «+i- ) . (2.15)

R*=3HI3s, and j.= - 3 / / / S « , , are

If we introduce D' and a' related by mJ?. = 2 i , — , *„=-$•(«„)+*•.

i

(2.6)

J

As in the preceding paper we assume the potential of the form (2.7)

b which yields

(2.16) a eq. (2.15) turns into eq. (2.13). Alternately, if we introduce a' and j/„ by - ,it„-i > 5l±Z- _ a eq. (2.15) yields e

6

l

=

(2.17)

i

-(&'(«)="{e-""'- "-11 (2.8) ~ - T ^ - = - — C 2 > . - 7 w - J - « . ) . P.") dt a +y« m In the presence of the constant external force f%' of each spring which is seen to be equivalent to eq. (2.12) after putting is given by the equation i»=0, or e

-l|D'-B)_

y. = d$.ldi.

(2.9)

_

(2.19)

§ 3. Wave Train and Solitary Wave As in the preceding paper," we obtain the wave a'^o-i^O). (2.10) train, the solution to eqs. (2.12) and (2.13), of the We note a'—a means the uniform stress exterted form on the chain and D'—D the uniform strain given thereby. btm \ Now. inserting eqs. (2.8) and (2. ID) into eq. (2.6) -Hir.-D'i we rewrite the equations of motion as a'blm\_ \ \ Al\ mR„ = 2j,—J.-i—J.+i, 1 (3.2) e-« -- ''=(o'+j )M. J ' where we have defined a' by

e

K

D

t

2

U

=

)

n

where JC and E are the complete elliptic integrals of thefirstand the second kind respectively.

We may eliminate /(„ to obtain —

[2fc—fc-t—J.+i) , in or. alternately, eliminate j . to obtain

(2.12)

-J.

,,

D

/7i/t.=(i'(2e-*<.- ''—e-""

'—e-* u i » + i - * ' i [ .

So far, we have been considering the case of a finite chain to clarify the nature of the forces and especially the strain given to the system. In the case of a finite chain the uniform strain is clearly caused by the external force F, which, however, is only implicitly included in eqs. (2.12) and (2.13) through a' and D'.

' . . .

E

dn'udw—— ,

(3,3)

and f is the frequency given by » ~ v f { - v - . i r <»> A as a function of the wave length A expressed in terms of the number of particles. The constants a' and D', mutually related byeq. (2.16), are the measure of the uniform part of the stress and the strain in the system. The wave form and the +

105

Wave Propagation in Anharmonic Lattices

1967)

503

amplitude depend on these parameters and the the maxima get sharper and minimaflattertill the modulus it of the elliptic integralsand the Jacobian wave turns into an equidistant succession of elliptic functions. In Fig. 1 we show the function 3-iunctions. f(x)=dn'(2K.c)-E/K for A ' = 0 . 9 9 2 . For small The behaviour of the function f(x) for large k indicates the solitary wave in the limit k-tl, /(x)=l^-cos2x. For large k (,k-*l). zf—>co. Keeping a=2K[A constant, we take the

Fig. 1. /[j) = dn!{2Kj)-£/Kasafunctionori, limit A—>l. and get the solitary wave

for #=0.992 where it is the modulus.

we obtain the hard sphere repulsive core. In this limit, D gives the diameter of the core cr '.he (3.5) shortest distance of approach. Keeping D' or

St — — '— tanh (an—,5r) , b -nn„-j>'. _ t j.sinh n-Sech (an-,5r) , (3.6) where j} is a constant given by 1

1

e

joTb . . p= Y —• sinh a .

(3.7)

b

6

<J'

(4.1)

finite, we take the limit 6 - > ™ u'->0. We have also to take the limit k-'\ to keep motion in the chain. By eq. (3.4) we have

The fact that eqs. (3.5) and (3.6) satisfy eqs. (2.12) 2X/A S\-V and (2.13) or the eqs. (2.11) can be verified directly. (4.2) V m \ sn 2 The wave form of the solitary wave is illustrated in Fig. 2 by giving sech' (on) as a function of an. Since in the limit k-*l, we have sn x-»tanh x, cn jc^sech .v, we obtain sech (a/j) !

2

10/

*-i 2A v

(4.3)

2

m

Thus, to keep u=const, we have to put 4K

b= AA

(4.4)

and further

\ _ •—i

1 1

i

0

r

1 2

z

i

i

3

4

Fig. 2. sech (nn) as a function of an.

2

a=tnA (uA) b .

an

sin 2JT'X

The velocity of the solitary wave is given by a V a which depends on the amplitude through a and on the strain through a and D .

(4.5)

On the other hand, we have the well-k-rtoww trigonometric expansion of the Z-function (4.6)

where

m

q tr-*K'l* , =

K=K(k) , X'sXfVT^F) .

For the limit k-tl, we have § 4. Hard Sphere Limit ==l-e , E = T /2K<1 . The range orthe repulsion force of our potential is characterized by Ijb. Taking the limit b-too, Therefore for the hard sphere limit !

9

(4.7)

(4 8)

106

J04

Morikazu T O D A

(Vol. !3,

is shown in Fig. J, in which K, is also shown. In Fig. 4 the diagram of the motion in the chain is shown in the case or A—*. The motion in , (4.9) general is such that the n-th particle is moving to where /(x) is a periodic function of x with period the right with the velocity v{A— 1)J. while the n+1 to (n+/i —l)-th particles arc moving to the I, and left with the velocity — >A. When collision takes r i - 2 * (0<x<\) (4.10) place the velocities of the colliding particles are ' 12|*|-1 (-l<Jt<0) interchanged. The velocity (see eq. (2.4)) The motion of the solitary wave in the case j.=f>.-<„+i)/m , (4.11) of hard spheres is very simple. In this case «->0, ZmuAd "

*h = -

IT mvAA

,2

,

ST* " H9

B

3K

-v

A

B C Fig. 3. Hard sphere limit, A=4. a) The velocity X. of the n-lh particle as a funciion of I. b) The dislance Jt. between adjacent particles as a function of f. -9+1

X -nO n

Fig. 4. (X, i) diagram in the hard sphere limit, showing the time evolution of the system, for A=i.

107

Wave Propagation, in Anharmonic Lattices

1967)

505

A-*oo. A panicle is moving to the right while X(x,t) = -^x~-?-te h£-{x-a). (5.8) others are standing still. This is the motion of D o D coins when they are placed equidislantly on a with smooth table and a coin is fillipped from the left. It should be noted that the motion in the hard c* = c>*(\-b4+^ . (5.9) sphere Jimit is very simple. For instance if two solitary waves collide, they go through each other where a and J are arbitrary constants. AID without any disturbance. Any (X, r) diagram stands for the strain BXjdx at x= + co. We have also the exact solution to eq. (5.6) composed of straight lines represents the feature of the possible motion. In this sense, a kind of R(x, 0 = 0' +— scch — (i-ci) , (5.10) "superposition of motion" holds in the hard sphere b D limit. where n

!

§5.

Continuum Limit

D' = D+A

When the wave length A is very large we may anticipate the continuum limit. In this limit, the density p , the sound velocity c , the coordinate x specifying the position of the n-th particle and the wave length 2 are given by 0

Equation (5.2) is the special case Of the equation X -c,\l+ X.)X„ tl

= i*X.„,

l

, (|f|
a

p,—mlD, c-DVablm,

x-nD,

i = AD. (5.1)

It is easy to get the continuum limit. Denoting the displacement by X(x, t), we obtain from eq. (2.14)

which appears in manyfieldsto describe phenomena in hydrodynamics, shallow water waves and plasma waves. By the asymptotic transformation proposed by Gardner and Marikawa. (5.12) 'A'=2C;(£,T) + 0 ( O ,

and pulling X„ - e.'(l - DbX,)X =c * ^ IX

0

X„

(5.2)

"(£. r)=— . 0? (5.13) The fourth derivative X can not be ignored. o = s'l2c,' . If this term is ignored, the so-called overtaking phenomena takes place. An approximate solution we get the so-called Korteweg-de Vries equation to eq. (5.2) can be obtained in the form Ki-rWHj-f oH;;;=0 . . (5.14) IIXI

!

!

(5.3) where q is a constant and ;=x—ct , D* 12* \-\

(5.4) (5.5)

Zabusky and Kruskal" studied this equation using computer. They showed that the solitary waves or solitons are stable; that is, when solitons collide, they pass through one another unaffected in size, shape and speed. Soliton, or the exact solitary wave solution to eq. (5.14) is given by

H ( l i t - «•) sech' (.{£—rr)/r] , (5.15) Using this solution we see that the nonlinear term with DbXJC,, on the left hand side of eq. (5.3) is of the same order of magnitude as one of the terms £ = n>,-f (itj—n„)/3 , (5.16) included in D'X,„, on the right hand side. f = ;v'12/(n -n .) Similarly in the continuum limit we get We shall show that the eq. (5.15) can be derived R„-c*[R„-b{R,'+iR-D)R„}]=c>~ff„„ . directly from eq. (5.8). Since the solitary-wave solution of eq. (5.11) corresponding to eq. (5.8), (5.6) written in the form of eq. (5.12), is D' — ]Sit—CD J„=C '— I „ « . (5.7) X"=r<x+/itanh (x—ct) , D

^ 5

}

2

1)

Now, for the solitary wave, we have the exact solution to eq. (5.2), which takes the simple form

(5.17)

108

506

Morikazu

wilh constants r and and *=B*{\2,

(5.18)

we have

(5.19)

TODA

(Vol. 23,

the waves in connection with the non-ergodic character of nonlinear systems, in tbe harmonic limit, while sinusoidal wave trains are stable, we have no stable solitary waves in the lattice. On the other hand, in the hard sphere limit, while solitary waves are stable, we have no stable wave trains. Between these limits it seems that the waves of special wave forms can be stabilised in virtue of the nonlinearity of the interaction forces.

Therefore we have to put Acknowledgements The author wishes to express his sincere thanks Since we are assuming |<|-<1, we get fromeq. (5.9) to Professors E. Teramoto and H. Matsuda for stimulating discussions and comments. Thanks are also due to Professors T. Tatsumi, T. Kakutani, and N, Yajima for inspiring conversations =(•{«..+("»-«-);?} (5.:i) on hydrodynamical aspect of the problem. The Therefore most part of the present work was done during the author's stay at The University of Kyoto from Oct. 1966 to March 1967, being financiated by the (S.22) Japan Society for the Promotion of Science. Thus we see that eq. (5.19) coincides with References eq.(5.15). 1) M. Toda: J. Phys. Soc. Japan 21 (1957) -131. 2) J. Ford: J. math. Phys. J (1961) 387; J, Ford § 6. Concluding Remarks and J. Waters: J. math. Phys. 4 (1963) 1293. After Zabusky and Kruskal" solitons are stable. E. A. Jackson: J. math. Phys. 4 (1963) 551. 686. This fact, in turn, indicates the stability of the 3) N.Saitoand H. Hirooka: J. Phys. Soc. Japan solitary waves given in S3, which propagate in M (1967) 167. the nonlinear chain. It is important to examine, 4) N.J. Zabusky and M. D. Kruskal: Phys. Rev. by computers or by theories, the stability of Letters 15 (1965) 240. •f="~-

5

20

(- >

109

JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN

VOL.

26, SUPPLEMENT, 1969 1968

PROCEEDINGS OF THE INTERNATIONAL CONFERENCE ON STATISTICAL MECHANICS

Mechanics and Statistical Mechanics of Nonlinear Chains Morikazu TODA Faculty of Science, Tokyo University of Education, Tokyo, Japan

I would like to point out certain facts concerning the motion in nonlinear one dimensional lattice. Especially I want to discuss the recurrence phenomena observed by Fermi-Pasta-Ulam (FPU) in computer experiments. 2 ) On the other hand, the asymptotic behavior of nonlinear lattice is described by the Korteweg-de Vries equation (KdV), which has been more thoroughly studied by synergetic approach than the nonlinear lattice. ) ' The ccncept of soliton and the explanation of the recurrence phenomena of KdV in terms of solitons will be ex ended to FPU case. Schematically, I intend the following: nonlinear lattice (FPU)-----continuum (KdV)

"'"recurrence phenomena/ 1. As for the nonlinear lattice it has been found convenient to use the equation of motion such as l ) Sn= ;

fJ,

fJ= lab 2 sinh L

Ym

with (2 )

represents a solitary-wave pulse, which we may call the (lattice-) soliton for the reason to be shown below. The solution of the form

e±b"-I=~.!!:....S a dt2 n, with (3 )

rand BIA are functions of It and p; there are two cases: r= lab 2 sinh.!.... cosh L m 2 2

cosh.!.... , 2

2

Y

.

BIA= cosh (1t/2)/cosh (pI2) .

This gives the state in which two solitons are running in opposite directions. at the lattice boundary is included in it. (ii)

2

2

BIA= sinh (1t/2)1 sinh (p/2) .

}

(4)

The reflection of a soliton

2 sinh..E.. cosh ~ r= Ylab m 2 2

fJ= lab 2 sinh.!.... cosh L .

Ym

( 1)

e±b,· -1 = sinh2 It sech 2 (ltn - fJt) ,

log [A cosh (Itn-fJt)+Bcosh (pn-rt)] ,

represents a two lattice-soliton state. (i)

mrn= ±a(e±b'n-l+e±b'n+1-2e±b'n) ,

where Tn is the mutual displacement r'=Y'-Y'_I, Y. being the displacement of the nth mass. a and b are positive constants, K=ab being the linear spring constant, a= ± bl2 the nonlinearity constant used by FPU.2l The solution of the form

•

}

(5 )

This is the state two solitons are running in the same direction . For small It and p. it reduces to the two soliton state of KdV. 2. For the wave travelling to the right we transform into a frame of reference moving with the velocity co=v'ablm h, where h is the natural length of the spring, and write

i;=x-cot.

(x=nh) •

11:

T'=

2' feo(ay/au)t •

u= ± -

1

(aulay)Y••

11:

235

I

(6 )

110

236

Morikazu TODA

where y(jr ,l)=y,{t), y. = dyldx; a„ and a, are con-recurrence can be expected if the solitons move as stants. Assuming f to be small, we get the KdV independent entities. If we use the periodic conequation ' dition 0
K

,

with

2

itH ft/a r«= Ac —= ,2froVa„ , ,— • (16) ^=(A /12^i[a./a,! . (7) The coefficients in the above transformation have Zabusky and Kruskal found, for 3 = 0.0222, been so chosen as f = x at r=0 and the initial con- = 1, Tji — iOAIir, while eq. (16) gives rj, = 40/jr. The descrepancy seems to be due to the fact that when dition y,\,=D=a sinjrJ:, [ 8 ) solitons interact they accelerate. 3. In the case of nonlinear lattice, if we assume of a progressive wave travelling to the right is the progressive wave with the initial condition eq. transformed into the initial condition for u: (8), the recurrence time in the above approximau,=o=a. cos i £ . I 9 ) tion, is given by eq. (6), or t =TiiHirl2)ec<,ia,jau). It is convenient to normalize the length in such a After some replacement we get way that N masses are included in the unit length: . _ _3 N<* x = nlN. £=I/»&p , /OI. (10] It has been shown useful to consider the eigen- where fz=2/©> is the so-called linear period. FPU used the fixed boundary condition: ' value equation" i

f

K

1

1, 1

y,\,=a= sin xx , y,\, —0 , yo=yn=0 . 6tfV«'-|r/-A)r ' =<>, (t/=-a) . (11) TheeiBenvalues2are independent of time, though In the linear case, a standing wave is a superposiu evolves with time and splits into solitons, each tion of two progressive waves, the amplitude of each being the half. Therefore, for small nonof which has the form" u,=u+A secti'{(Z-c,T}ir } . (12) linearity we may put a,= l/2 for FPU case. Then we get The speed of the /th soliton is given by =B

(

l

l

s

,

l jtL=06N - la" , which is to be compared with the empirical result In calculating h we may use U at r = 0, for which due to Zabusky:" we assume eq. (9), and for largest solitons the exttln= lA2N'- 'la" (/v-=16~64) . pansion ci=u+A,!3 ,

A,= -2{h+a) .

K

(13)

3

(/|,> =- |,=p=-a. + ^ - V 0

References 1) cf. M. Toda: J. Phys. Soc. Japan22 [1967) 411; 23 (1963) 501. 2) E. Fermi, J. R. Pasta and S. M. Ulam: Collected Papers of E. Fermi (1965) Vol II p. 977.

114)

U

to obtain

3) ef. N.J. Zabusky: Proc. Symp. Nonlinear Partial

[1=0, 1,2, ••-) •

(15) Differential-EgaotiimjlAcadernicPress. N.Y. 1967). Since the differences Ac of the speeds of successive 4) C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura: Phys. Rev. Letters 19 (1967) 1095. solitons are a constant in this approximation, the DISCUSSION The result given earlier by me (Proc. Conference on Mathematical Models in the Physical Sciences, ed. S. Drobot, Prentice-Hall, 1963, pp. 99-133) showed /j (r = I.42/V -™/(««) - . The N dependence was estimated from three runs [N—Xd, 32, and 64) and it was suggested (p. 112, above reference) that iV and not fF was the dependence on N. Here a is the maximum amplitude of the standing wave at r=0, y\ —asin-x, y,| =0. If we assume the N"Haa} dependence and use the data for /V=32 and N—64, (ora = ]/4) in the table on p. 112 we see that the constant becomes: ,

i

l,s

,

i

t

m

at

0

0

2

!

! !

!

^=82:145/[32}" [1/4)"=0,401 jV=64:341/(64)'(l/4]"=0.333

Ill

Mechanics and Statistical Mechanics of Nonlinear Chains and nol 1.42 as indicated in the above paper. These new results compare favorably with the result (given in my paper at this conference for progressive wave initial conditions a ] namely l^lL^6" "- 'AN IH a )->l =0.31 « " K | - ' " = 0.44Af ' («o)-" , Ci4=0.7! is the constant in !«lt = AI3.) The value, 0.44, for the fixed boundary lattice is smaller than Prof. Toda's result of 0.6. Note, this scaling law also applies to strong excitations; that is, where solitons may be 4 or 5 lattice intervals wide and the continuum approximation is questionable. The importance of Prof. Toda's resull should be emphasized, for it provides us with a procedure for studying the interaction of lattice solitons, that is exact stationary solutions of the coupled nonlinear differential equations of an exponential lattice. In the representation given in an earlier paper his equation (exponential lattice) can be written as: Sn — ll+tTS,j6, S, and the "cubic" equation can be written as S„= [l + iS,)'' i, 5„. Thus, for small amplitudes the results should be comparable and, since the KdV was shown by me to be applicable to the latter, then (in the same order expansion of square root) it is applicable to the former. Also, for people engaged in numerical analysis and numerical simulation of physical systems, these exact solutions should provide a wonderful test for numerical integration algorithms. p

!

,l

1

1

a

r

!

J

3

J

e

!

:

J

A. S C O T T I : If one can construct solutions which correspond to different initial conditions (corresponding to a set of positive measure on the energy surface) of the type just discussed by Prof. Toda, one could settle the problem of ergodicity. Is this the case 1 M. T O D A : I think yes, for the mode! I described, I found family of solutions which differ by the value of some continuous parameters. The solutions therefore cover the domain of positive measure on the energy surface.

112

174

Supplement of the Progress of Theoretical Physics, No. 45, 1970

Waves in Nonlinear Lattice

Morikazu

T O D A *

1

Department of Applied Physics, Faculty of Science Tokyo University of Education, Tokyo In this article waves in nonlinear lattice or in nonlinear medium are studied. One of the aims is to seek for the point of view to deal with the great majority of phenomena related to nonlinear waves in general. For one dimensional nonlinear lattice analytic and computer-experimental treatments have been developed. It has been found that a certain kind of pulse-like naves (solitons) is the fundamental motion in nonlinear lattice vibration. If two or more solitons collide, they interact nonlinearly, pass through one another and, when they separate, return to their original forms. Thus solitons are conserved and behave like particles.

§1.

Introduction

The problem of wave motion in nonlinear media is interesting not only as purely mechanical problem, but also i n connection with many physical phenomena such as shallow water-waves, plasma waves and heat conduction in crystal lattices. Since nonlinear phenomena have infinite variety compared with linear cases, it seems quite important to find the way to extend our pattern of thinking as to make it possible to understand the essence of nonlinear world. Vibration of a system of particles joined by harmonic springs can be described by superposition of normal modes which are mutually independent. For instance, if we excite a normal mode, its energy is not transferred to Other normal modes. The system of harmonic oscillations never reaches the state of thermal equilibrium, and is non-ergodic Although the harmonic oscillator models give good results for the problems such as the specific heat and other equilibrium properties of crystals, the nonlinear terms ignored in these models have been considered by many people to play an essential role in the problem of approach to thermal equilibrium. However, since the nonlinear terms make calculation insurmountably complex, it is usually assumed that the nonlinear terms guarantee the ergodictty of the system and its approach to thermal equilibrium. Fermi, Pasta and U l a m ( F P U ) " intended to verify this expectation by computer experiments. But contrary to their expectation the one-dimensional nonlinear lattice showed recurrence phenomena. Ford, ' Jackson ' and others further examined the same problem, and it was clarified that one-dimensional nonlinear lattices marvellously sustain the characters of linear lattices. 1

3

*> Present address: Institute for Optical Research, Tokyo University of Education.

113

Waves in Nonlinear

Lattice

175

Now, one-dimensional nonlinear lattice can be considered as the model of nonlinear continuous medium, which can be obtained as a limit of long wave length or of small distance between particles of the lattice. If the deformation of the springs is small, the Hooke's law or the linear term w i l l suffice. So for a little larger deformation, we may write the force of the spring as [linear term] + [nonlinear term]. When the nonlinear term is absent, i n virtue of the superposition principle, we can analyze the general motion of the system in terms of suitable fundamental motion. A s such, we usually use normal modes represented by sinusoidal waves. I n the case of continuous harmonic medium, where dispersion phenomenon is absent, since arbitrary waves are propagated with the same speed keeping their initial wave forms, we can also take, for example, a suitable set of pulse-like waves as the fundamental motion, and express the general wave as the superposition of such pulses. If the nonlinear term is regarded as perturbation, the expansion i n terms of the normal modes yields the so-called secular terms which increases indefinitely with time. If we try to avoid such a kind of fault, as i n the theory of nonlinear oscillations, we are led to unreasonably complicated analysis which lacks any sound mathematical basis. In place of such perturbational methods, some Other ways must be sought for in order to develop new pattern of thinking essential to nonlinear problems. Although any concept cannot, of course, be almighty as to cover all the nonliner phenomena, the concept of soliton,*' which w i l l be described below, seems to have very wide applicability at least i n the case of one-dimension a I waves. A soliton is a pulse-like wave which travels through nonlinear media without changing its wave form. Its speed depends in general on the height of the pulse. When two solitons approach, they interact, pass through each Other, and return to their initial forms. Thus soliton behave like stable particles and seems to be the fundamental motion i n nonlinear wave propagation. §2. 2-1

Finite

Equation of motion of nonlinear lattice

lattice

First we consider a finite uniform one-dimensional nonlinear lattice, which consists of N particles of mass m connected by nonlinear springs. We assume that the lattice is fixed at the left end, label the particles as M==1, 2, N from left to right and assume pressure p applied on the right end particle (n=Nj. If the potential energy of a spring is denoted by '(.y.-y.-ij my- =-4-'(ys-y»-C>-p, N

+'(y,n-y.), («=i,2. - , N - i )

.

(2-1)

114

176

M . Toda

where y, denotes the displacement of the n-th particle measured from the equilibrium position for p—0. Naturally y„=0, and the mutual displacement of adjacent particle, or the elongation of the n-th spring, is given by

r,=y.-y.-i,

(2-2)

or conversely

y.=n+r,+—+r,.

(2-3)

The momentum 5, conjugate to r , is, by definition,"

where T is the kinetic energy

T=±-^-yl

(2-5)

with

y.=h+h+

—+r,.

(2-6)

Thus it is seen that

my, — s, -

,

=0.

(2-7)

Since the pressure p can be considered to be due to the potential energy py«, the hamiltonian of the system can be written as H=

S ( s . - O ' + S t > ( r . ) +^>r.}.

(2-8)

Therefore the canonical equations of motion are given by

r, = ^-(2s,-s.- -s,^'), 1

s.= -{
(2-9) (2-90

If we eliminate s„ we get mr'.=4'(r.- ) + # ' 0 , ) - 2 # ' ( r . ) , I

+1

#(>> i>=—# +

(2-10) (2-100

which are equivalent to E q . (2-1). On the other hand, if E q . (2-9') is solved for r . to yield r

"

=

" i "

x

(

i

"

)

eliminating r , from Eqs. (2-9) and (2-11), we get

(2-11)

115

Waves in Nonlinear

Lattice

177

-§-ay&) = i _ , + i - , - 2 s . . ar

(2-12)

This equation can be interpreted as the equation of motion of a linear chain of particles with "displacements" s. and "momenta" z(s'.). In effect, we have interchanged the roles of r . and s„ and E q . (2-12) describes the motion of the "dual lattice"." If we further introduce S. = ^sjt,

(2-13)

E q . (2-12) can be written i n a more convenient form: XGS.) - S.-> + S

M

- 25..

(2 • 14)

Though the above treatment assumes finite length of the lattice, if we write n—N/2 i n place of n and take the limit N-*°°, we get an infinite lattice for which all the above equations are applicable. 2-2

Exponential

lattice

In what follows we shall adopt the potential function which has been found to be quite appropriate and convenient. This is of the form' 1

$(r)=-f-(e--"-l)-r-flr, b

( » 0 )

(2-15)

where the arbitrary additive constant has been chosen in such a way that 0(0) =0. Since r = 0 represents the natural state of the spring, ^ ' ( 0 ) = 0 . For small \b\ we may expand the right-hand side of E q . (3-1) to yield *(r) = ^ - | W - . - ) .

(2-16)

Thus, for small displacement the spring obeys the Hooke's law with the force constant K=ab. If we keep K=ab finite and take the limit b^O we get harmonic system. If we put r—R—D and take limit b^>°° we will get the system of hard rods of diameter D where the distance between the centers of adjacent rods is represented by R. If the one-dimensional lattice is subject to a constant pressure p, its effect is equivalent to that of the additional term pr in the potential energy (cf. Eq. ( 2 - 8 ) ) . That is, we have only to replace <j>(/) by >r

*('") +pr=-je~ + = 4- ->" e

where

(a+p)r + t I

+ const

V-l-const,

(2-17)

116

178

M- Toda /

a =a+p,

(2-18)

and -

d

= r > -

r

= j l

0

g

J -

. )

(2

19

represents the contraction of the nonlinear spring due to the pressure. Therefore, rewriting r and a' as r and a we can include the effect of the constant pressure in our formalism. I n what follows, we w i l l not explicitly deal with the pressure. In this and the following chapters we want to simplify our notations by replacing r, and t by dimensionless quantities: br. -* r., •J^-t—t. » m

(2-20)

T h e n the equation of motion (2-10) reduces to f. = 2(T'' - e-'-' -
(2-21)

W e may call this lattice the exponential lattice. Now, let us introduce v- by * r " - l = ?„

(2-22)

/ - „ = - I o g ( l + ?.).

(2-23)

or by Equation (2-21) yields + . -2 ..

- C - log(l+ .) -

V +1

5

V

(2 • 24)

Further if we introduce s„ = Jij„*fr, we have r „ = - I o g ( l + s".)

(2-25)

and we get the following equation, which corresponds to E q . (2 -12):

jUog(l + i „ )

=5„_

T

+ s*, - 2 s . .

(2 • 26)

In the right hand side of the above equation, the integral constants have been set equal to zero, which is possible for an appropriate definition of s„. Alternately if we define S* = \sJ&, we have

r , = -log(l + X )

(2-27)

e-'—l=S..

(2.28)

which implies that

117

Waves in Nonlinear Lattice

170

Then E q . (2-26) yields ;

log(l + i .) -

+

- 25.

(2-29)

and from E q . (2-28) we get r. = 2 5 . - 5 _ , - 5 . „ . n

(2-30)

In the above equations the integration constant has been chosen appropriately, so as to give the simplest expressions for the relation between r , and S,. N o w , we shall describe some properties and particular solutions of our nonlinear lattice.

2-3 Expansion due to vibration It can be shown that in general motion in the lattice gives rise to expansion (if £ \ > 0 , and contraction if £ < X J ) . W e have to show that the average of r, is positive. The force on the n-th spring is /. = -

(r.) =

-1=5?.-

(2-31)

The average of the force must vanish if the lattice as a whole is at rest. So, using E q . (2-22) we have

^=7^—1=0.

(2-310

However the same equation yields r . = - l o g ( l + *.) —

(v.-W+"0.

(2-32)

Therefore we see that for small oscillation r~.=U

(2-33)

which indicates that the lattice expands as it vibrates. If we assume thermal equilibrium, the law of equipartition of energy can be used to estimate the effect of thermal motion i n the weakly nonlinear lattice. Since in the usual units, the 'force constant of the spring is K=ab and the elongation is r./b, the equipartition of energy can be written as (2-34)

where k is the Boltzman constant. Using E q . ( 2 - 3 2 ) or r . = —JJ,, we find that Tjl^ib/ajkeT, and therefore that a

£-£*7\

(235)

118

180

M- Toda

This gives the thermal expansion of the system and is in accordance the result given hy the conventional statistical mechanics.

§3. 3-1

with

P a r t i c u l a r solutions

Soliton

In this section we describe particular solutions of Eqs. (2-21), (2-26) and (2-29) for r., s. and S.. First we note the solution which represents a soliton, e~'— 1 = fi'sech' (an +

fit),

(3 1)

s.= - £ t a n h ( < m + / K ) ,

(3-2)

5, = log cosh(art + 0r) +const,

(3-3)

tf=sinh*.

(3-4)

where

The velocity, 0/a, 3-2

Collision

increases with the height of the pulse. of two

solitons"

The state with two solitons can be represented by 5 . = l o g { c o s h ( « 7 i - ^ ) +B c o s h G t f i - r J + 3 ) } +const,

(3-5)

where £, r and B are functions of * and M, and 8 is an arbitrary constant. While E q . (3-5) gives the state with a soliton when it represents twosoliton state when K^TP. There are two cases. One of them is the case two solitons are running in the opposite directions, and the other is the case they are running in the same direction. (i) The solution representing the state with two solitons running in the opposite directions is given by 0 = 2sinh-|cosh-|-, r=2sinb-|-cosh-| , B=cosh(*/2)/cosh(/'/2).

(3-6)

For t—* — °°, the asymptotic form of the wave is 1

e-' - 1 = & sech (can and for t-*°°,

it is

ft*+A),

i = 1,2,

(3 - 7)

119

Waves in Nonlinear

Lattice

181

5

e-'- - " 1 = $ sech v%n - $,t -h),

(3-8)

where 2 E

•

2 (3-9)

-'=T/£^.

In the ahove asymptotic forms » = 1 and 2 represent two solitons. (ii) The state in which two solitons are ranning in the same direction is described by /3=2sinh-|-cosh-^-, r—2 sinh-^-cosh-^-. B=sinhCA:/2)/sinhCV2).

(3-10)

Asymptotic forms for /—>±°o are also given by the same equations as Eqs. (3-7) to ( 3 - 9 ) .

3-3

Wave train

There is another type of particular solutions which represent periodic wave or wave train. This can be written as 51

!

1

tr'--l ^=( 2 ^ v ) [ d n { 2 ( v r + ^ - ) x ) - ^ ] ,

(3-11)

where the frequency and the wave length A satisfy the dispersion relation

^

, \ K

~

1

' ' K )

(3-12)

•

In the above formula sn and dn are the Jacobian elliptic functions, and K and E are the complete elliptic integrals of the first and the second kind respectively. These are a l l of the same modulus which we shall denote by k(0
sm:=sn(x, !

E=E(k),

k),

dnx =Vl

The function d n ( 2 r . K ) — E/K 1, and its Fourier expansion is

where

— k'sifx=dn(x,

k).

is a periodic function of x with the period

120

M

182

-

T o d a

K'=K(k'j,

k'= VT^F.

(3-14)

W e see from the above formula that the average over a period of the right hand side of E q . (3-13) or of E q . (3-11) vanishes. This implies, as easilyseen from the form of the function e~ ', that the lattice expands as a whole: r „ > 0 . This is in accordance with the argument in § 2 - 3 that the lattice expands in general when it vibrates. 1

If the modulus k is very small, K' is very large and we may use approximation for that

the

expi-xK'/K)—£716,

snx—sin.r.

(3-15)

Then Eqs. (3-11) and (3-12) reduce to

r

.^-J^lcos( ,r+^-), 0

a>=^2sln~. A

(3-16) (3-17)

Therefore, for small modulus k, the wave train reduces to sinusoidal wave whose amplitude is proportional to k . W e see that the solution (3-11) contains a l l the normal modes of the linear lattice vibration as the limit of small amplitude. In the same limit, we have 1

S P ^ t t ^ M ^

(3-18)

M

,.=i.-^cos( ,r+^-). 0

3-4

(3-19)

Relation between the wave train and solitons

If the modulus k approaches to 1, the amplitude of the wave train or the height of the spikes gets larger, and the wave train takes the form of a sequence of pulse-like waves. This can be understood if we note the identity

d

" '

(

2

^ - f ^ ( - w ) * J -

h

1 ^ ^ - o | - w

(

3

'

2

0

)

so that the wave train can be expressed as < T ' - - 1 = S f sech

!

AO)-2$v

(3-21)

121

Waves in Nonlinear

Lattice

183

with K'A

0 = -#v.

'

"

(3-22)

K'

The right-hand side of E q . ( 3 - 2 1 ) represents a sequence of infinite pulses at equal intervals of A, and with a downward shift of 2j3v. Each pulse is a sech -wave and, as is seen in the next section, it is indeed a soliton: The wave train is a sequence of solitons progressing equidistantiy. These solitons are mutually interacting, not independent of each other, and their speed is given by the dispersion relation of the wave train (Eq. ( 3 - 1 2 ) ) . 2

If we take the limit
§4. 4-1

in E q . ( 3 - 2 1 ) , the wave

W a v e t r a i n and e l l i p t i c ^ - f u n c t i o n s

Wave train in terms of t?n

The elliptic t?-function, &n(,x), has the period of 1 and satisfies the relation"

where

* W G 0 =

Comparing E q . ( 4 - 1 )

,

f

(

2

f f i

r

•

C4-3)

with E q . ( 2 - 2 9 ) , we find a special solution of the

form ^ l o g ^ r - - ^ ) / ^ ' where y = l/A,

1

] ,

(4-4)

or

)I=

1+

^ [^i?r- l|'

m

This solution represents the same wave train we have been discussing i n the preceding section. W e can show this by using the relation between the ^-function and the Jacobian 2-function, or zn-function:

122

184

M . Toda

4-1)

, = v

S.=5.

~v-

7

where Z ( w ) is related to dn by Z(tt) - H d n * « -

.

(4-8)

Therefore, - - 1 = 5 . = ( 2 & 0 "[dn

e

- - § - W - J|]

2

(4 • 9)

coincides with E q . (3-11), i.e. the wave train described in the preceding section. N o w , to see the amount of elongation due to vibration we take a part of the lattice which contains JV particles such that TV is a multiple of the wave length A. In other words, we think A to be, or very close to, a rational number and N includes an integral number of the wave length A. Then we have r +i — r-,; that is, the lattice is cyclic. The length of this part of the lattice is H

r + r + - + r =(2S -S ,-S ) 1

z

N

1

<

+

2

( 5 * r — 5nr+i)

( 2 5 , - 5 , - 5 , )

(5D

+

••• + ( 2 S „ - 5

W

_ , - 5

W

+

1

)

Si)

= log

= JVlogC. We have used the fact that N/A with the period of 1, so that

(4-10) is an integer and i?o is a periodic function

It is therefore shown that the average length of the spring is given by ?=logC.

(4-11)

For small amplitude ( ^ ^ 1 ) , expanding the right hand side of E q . (4-5) i n powers of k (cf. Appendix I ) , we get 2

1

logC^-g-sin ^. io A

(4-12)

123

Waves in Nonlinear

Lattice

185

Thus we have verified the fact that the lattice expands when it vibrates. Equation (4-12) is in accordance with the result using Eqs. (2-33), (3-17) and (3-19). 4-2

Decomposition

of a wave train

£ ( a 0 is defined in product form as 0

2

* ( x ) =tf.(x, q)

1

1

fi (l-2q '- CQs27ix+q"- ),

(4-13)

where *G7) = i i ( W ) .

(4-i4)

The parameter q is related to the modulus k by the relation ,

q=e** >*>

(4-15)

N o w , since ( 1 - 2 ? , cos 2iez + ? , ) ( l - 2 g , c o s 2 » ( x ± i ) ± ? ? ) = l - 2 ? i C O s

Anx+qi, (4-16)

J

i?o(2x,
(4-17)

V

where tpitf) is a functional of q. This identity gives the following decomposition of a wave train into two wave trains: S. [vt —

=

log t? (vi - - - , qj + const 0

or, in terms of e~'*—1, !

e

!

~ : - 1 = {2K(k)v} [dn

+

fdn

jalrf--^jjKik),

2

- ^

i) - - g g - ]

- 1 ) X W , *} -

where the moduli k and i are related by

,

(4 • 19)

124

186

M . Toda

Equation (4-19) shows that the wave train with the wave length A can be decomposed into two wave trains, component waves, each with the wave length 2A and progressing at the distance A. This means, conversely, that wave trains of the same form, with the wave length 2A and set at a distance A, can be superposed to yield a wave train. The speed of the wave train is determined by the dispersion relation of the resultant wave and not by that of the component waves. In general, the speed of a wave is controled by the presence of other waves. This is one of the characters of nonlinear decomposition and superposition of nonlinear waves. Each of the component wave with the wave length 2A can be further decomposed into waves each with the wave length AA, and so forth. Or, more generally, we can decompose a wave train into three, or four, or into any integral number of wave trains. This can be verified by a extension of E q . (4-17), that is, by the relation 9$ (Ix, cf") = - ^ ^ p - n

(4-21)

Using this relation we get

£(*) K(K)

(4-22) with K(JT> _ K(x') lK(k) K{*) '

(4-23)

This decomposition means that the wave train with the wave length A can he considered as a superposition of I wave trains each with the wave length lA and set equidistantly at the distance A. If we take the limit l^-oo we see that *—»-l and E q . (4-22) reduces to E q . (3-21) as it should. A wave train can be considered as a superposition of pulse like waves, which, as will be seen in what follows, are actually solitons. 4-3

Solitons and wave trains in a ring {Cyclic boundary

condition)

Though we have treated a wave train as if it is i n an infinite lattice ( — o o < « < ™ ) , we can consider such a periodic wave as a wave in a ring ( 0 < n < J V ) whose length is an integral multiple of the wave length, i.e. N/A — integer. Therefore, we can superpose or decompose a wave train under

125

Waves in Nonlinear

Lattice

187

periodic boundary condition in a similar manner as we have done in an infinite lattice. W e have only to renew the labels appropriately; that is, in place of the labels ( - , - 1 , 0,1, - , N N+l, •••), we have (»;N-1,N,1, •••,;v,i,-0.

W e consider a ring of N and r i = n , or r .=r.. w t

particles n = l,2, —,N.

n = 0 means

n—N,

Ht

The solution representing a wave with only a spike in the ring may be called a soliton under the cyclic boundary condition. Such a soiiton is given by & = i o g [ i f c ( , . ( - - j ) j {ca/N)

}•">],

«T—1 = l2K(Kj,r[dn^2(,,t-j^K(Kj,Jt

— f § - ] ,

(4-24)

where the modulus is written as ie and the frequency as v„ which is given by (4-25) t

s

n

N

If we set two such solitons so that they are separated from each other by the distance N/2 apart, a wave train with two spikes in the ring is obtained. The modulus k of the wave train is related to the modulus of the solitons by E q . (4-20), and the frequency or the speed of the wave train is given by E q . (4-6) with A^N/2. In general, from the solutions for an infinite lattice, for instance E q . (4-9), we can make the wave trains in the ring. In the limit of small amplitude, they reduce to the normal modes of the ring in the harmonic limit. Thus the wave train with, say m spikes, may be called the m-th "normal mode" of the nonlinear ring. However, the nonlinear normal mode has the peculiar character that it can be decomposed into other normal modes as stated above: If the number of spikes is a multiple of some integer /, it can be decomposed into / wave trains, and it can be decomposed into as many solitons i n the ring as the number of spikes present. Thus it is seen that the nonlinear "normal modes" are not the fundamental motion, but they are superposition of solitons. Therefore the solitons are to be considered as the fundamental motion of the nonlinear lattice. A soliton i n the ring has a spike where r,<S) (lattice contracts) and a tail where r „ > 0 (lattice elongates). It is given by E q . (4-24). However, we shall study it for a large ring ( J V ^ l ) . It is convenient to use E q . (3-21) with A — N for such a soliton. If we want to keep the width aT'^K'N/nK finite for N^l, we have to assume the case K/K^l or the case where the modulus k=I. Thus for N^l,

126

188

M . Toda

the soliton can be approximated by ,

e

-'.-].=g'aech C^-ttn)-2ft.,

( - ^ < « ^ ^ )

(4-26)

where @=2Kv. However for the limit k-*l, since s n x ^ t a n h x , the dispersion relation (3-12) yields (a=2K/N~) fj=2&=sinh«,

E/K—Q,

(4-27)

which is the dispersion relation for the soliton, its speed being 0/a—slnha/a. The first term of E q . (4-26) represents the spike where the lattice is contracted and the second term represents the tail where it is elongated. The total change of the length of the ring can be evaluated using E q . (4-10) or (4-11). In the limit stated above,

and therefrom ^ | A ^ ( M ) _ « | . The change i n the length of the ring is ^ S r . = -Wlog C=

( 4

.

2 9 )

(a=2K/N)

2a\(^^-J

-l}

(4 - 30)

which is always positive. It is therefore shown that, although the spike of the soliton represents compression of the lattice, the trough gives small (—1/.ZV) but non-negligible contribution to elongation, and as a whole, the soliton gives elongation of the lattice. In the case of infinite lattice the infinitesimal elongation i n the infinitely long trough gives rise to a finite elongation of the lattice given by E q . (4-30). So we must note the fact that soliton expressed, for instance, by Eqs. (3-1) to (3-4) implies no elongation formally, it should be complemented by the total elongation R given by this equation. §5.

Continuum l i m i t

(Korteweg-de Vries equation) If the wave form varies slowly compared with the distance between particles, the continuum limit is generally valid. W e may apply the operation rule =/C«+D

to r(n,t) = r.(t)

and rewrite E q . (2-21) as

(5-D

127

Waves in Nonlinear

189

Lattice

If we neglect higher order derivatives, and higher powers of r, we can rewrite the above equation as

V 9(

2 8n

2 9n ^ \ dt

2 &n

2

dn)

The operators here are to operate on all the terms on the the wave which advance to the right, we have 3

\dt

right.

+ 2sinhi--?__^J-WO 2 dn 2 dn/

Thus for

(5-3')

K

or, if we expand sinh, 9 , 9 V9r dn

1 d" -\r4-\=Q. 12 9K* 2 3n/

(5-4)

(For the wave advancing to the left we have to change n by — n.) If we recover the usual unit to see the effect of potential parameters, looking back E q . (2-20), we get /m_dr_ dr_ V ab dt dn +

1 &r __ ft _Sn 12 dn 2 dn

=

(c. c\ ^

0

3

J

Now, we introduce

3

where r> & W

3 1 1 0

du , du , d u ^ ^ are constants. Then E q . (5-5) reduces to + s2

"ft

»8e

=°

f

c-

( 5

'

7 )

(

'

8

with ~~2j^N'

'

"

W

5

)

9

Equation (5-7) is the Korteweg-de Vries equation ' which is used to describe shallow water wave, plasma wave and nonlinear lattice waves. K d V equation has been discussed using various scales, which correspond to suitable choice of the constants a, (3, M and N. Let us use the units for which
u = w . + A sech a (f - cr),

(5-9)

128

190

M . Toda

where

This can be derived quite easily as a hmit of the lattice soliton given by E q . (3-1) using the transformation described by E q . (5-6), and some modification to include « „ , the uniform strain at infinity. In a similar manner, we can derive the state with two solitons of K d V equation from E q . (3-5). Further, corresponding to the wave train in the lattice we have the wave train of the K d V equation: CS-11)

t

u=u„+Aci\ a(£-c-i), where (_k is the modulus of the cn-function)

+

^ * -

i (

2

- i r ) '

8

This is called a cnoidal wave. ' Equation (5-11) approaches to a soliton, E q . (5-9), as the modulus k approaches to 1. It has been shown very useful to consider (Schrodinger type equation)" 6 6 ^ ^ - ( I 7 - W '

J

the eigen-value equation

=0

(5.13)

for the K d V equation (<»=1), where (5-14)

f / = - M ,

since the eigen-value X, are independent of time, though u evolves with time according to the K d V equation:" - ^ - = 0. at

(5-15)

Such a invariant property of the eigen-values may be used to predict the number and the height of solitons which w i l l emerge out of the initial wave form. Z

If the initial wave t/(f, 0) is very large, or if 8 is very small, there will be many eigen-values, and these may be evaluated by means of the W K B method, or by the "quantum condition" where i A 2 5 * replaces K: /

J - | ^ = ( 7 i + 5)27r lzF V

O = 0 , 1 , 2 , -••)

(5-16)

129

Waves in Nonlinear Lattice

191

in which the momentum p is given, in terms of energy, by

p=V2(E-U).

(5-17)

Therefore the number of eigen-values between E and E + dE proximated by

dn . dE -dE= f f . dE 2nV\2tf J p

can be ap-

(5-18)

J r

For instance, if {/(£, 0) is maximum at £ = 0 and symmetric with respect to 5= 0, such that I7=-C7«#Cf),

(5-19)

where E7 is the maximum value at £ = 0 . Putting 0

*(£)=*•

—jf-=*(*>

C5-20)

we have

Since the height of the soliton with the eigen-value .i^cT is equal to 2\E\, we put

and write the number of solitons between i} and ij + di} by

= 0.

(,^2)

A=

f(ji)dii. W e get

(5-23) 10

This is i n accordance with the result given by Karpman. ' For example, if the initial wave is !

Jj(f,0)= -i7„sech AS we get 2

48V ' A n d the total number of solitons is

Whereas the exact eigen-values in this case are

(5-24)

130

192

M . Toda

,

*.= - 6 a « ( - | — « ) * .

(5-27)

U^efpip+n).

(5-28)

where we have put

If Ut is very large, the total number of the eigen-values —V Uo/Gtf/a, in accordance with the above result.

§6.

is

N—p/a

Recurrence phenomena

Studying the K d V equation by computer-experiment, Zabusky and K r u s k a l " found recurrence to initial state. This phenomenon can be interpreted as follows. 1

Consider the initial wave of the form K| _o=Acosjif. T

(0<£<2)

(6-1)

W e take the cyclic boundary-condition « ( f ) = « ( £ + 2). Assuming A much smaller than V 12j*, we expand the potential U= — w| -o as

to be

T

U=-Acosi£^-A+^-F

(6-2)

to obtain the approximate eigen-values 2. - - A + in + }) V123" An*.

(6-3)

The solitons associated to these eigen-values travel, interact as they collide and pass through one another. If we neglect the acceleration during interaction, solitons w i l l travel independently. The velocities of solitons in the above approximation form an arithmetical series with the common difference Je-tyl&At.

(6-4)

When these solitons move i n the region of length 2, there comes again the state with the same mutual situation as the initial state after the time interval _

r r

K

2 _

_ 0.364

~ ^ ~ ^ n V A T ~ ^ M

,v •

C 6

"

5 )

This is the recurrence time in our approximation. If we put A=l, £=0.0222 we get the recurrence time T =4X)/n, which is to be compared with the experimental value r = 30.4/jr observed under the same condition. The discrepancy w i l l be due to the change in speed of solitons during interaction. In actuality, solitons were rather close to each other and therefore not mutually independent i n this experiment. B

s

131

Waves in Nonlinear

Lattice

193 11

Fermi, Pasta and U l a m " found the recurrence of the nonlinear lattice. " In this case computer experiment was done under fixed-end condition. The relation between the K d V continuum of length 2 to the lattice of length 2N, can be obtained if we refer to E q . ( 5 - 6 ) : v

=-b,

!

!

p ^ - i , A - l , 5 =1/12JV .

(6-6)

Denoting by y, the displacement of the n-th particle i n the lattice, we assume the initial condition ;y„Uo=-Bsin^-

(6-7)

with fixed ends. The stationary wave thus generated may be approximated by the superposition of two progressive waves travelling in opposite directions, v.|,= =-|-sin^-.

(6-8)

0

which corresponds to &y,

nbB

.

,~ ~^

umr-*-~-^m**.

(6-9)

Therefore if we put

A=--^-

(6-10)

E q . (6 -7) takes the same form as E q . (6-1). Therefore the recurrence time of the lattice is, referring to Eqs. ( 5 - 6 ) , (6-6) and ( 6 - 5 ) , given by 3

2N yab/m

3iV " , x b *y/B m

f R

ll

where t —2N/Vabjm is the time necessary for a wave of extremely long wave length to travel the lattice of length 2N. Equation (6-11) is to be compared with experimental results, which reveals that ' L

13

3

0.4JV "

r f i

t J

«

where a=bh/2 is the nonlinearity constant used F P U , h represents the natural length of a spring, B/h is the initial amplitude measured i n units of h. W e see that theoretical expectation (6-11) is very close to experimental results (6-12). Appendix Formulas of elliptic

I functions

Calculations involving elliptic functions are sometimes annoying because

132

194

M . Toda

of the lack of suitable tables of formulas. Here we summarize those formulas, which seem appropriate to be listed in connection with the present calculation, including some integrals.

a)

Complete elliptic integrals: V'l-J&'sin'fl

dx £>'(l-x Xl-A .r ) ' I

£=£(*)

,

=

1

Vl-esm'edd

^(^)->log .

4

,

(*-i)

vl-k*

K(kT) =K', b)

E(_k!)^E',

k'=V\-P.

Jacobian elliptic functions:

sn

=

u sin tp — sn (u, A),

cn w=cosy = cn(w, A), l

s

d n « = Vl — k sn w = dn(w, A), sn(w, 0) = s i n u , c n ( « , 0) = cos« , s n ( « , 1) =tanb «, cn(w, 1) = d n ( « , 1) =sechw ,

d sn u — cn u dn u , du d du c n w = — sn u d n ( i , d

du

dn

1

M =

—

9_

k sn « cn u ,

~

no+ifz

Waves in Nonlinear • d

n

dn d,=

(

^

rc\ , 2ir A M

=

^

+

q"

-—^—-z ^

)

-

T

f=

c

x . ? . T + ^

u

Lattice

o

I mzX \ s

l i 2 ? r j «

( ^-j,

rCOS

( ^ ) ; S s e c h ^

r

(

„ - / )

2KK'

z « ) = 5 ; ( d n ^ - f ) ^ (

2rc ^

q"

. ( mat \

K -^i sin(nim/K) K . - i sinhCTiTrX'/K) '

k}

1

1 +

f

+

k

l k

i

+

" ) '

^ 1

Z(u + v)+Z(.u-v)-2Z(u)

2

= - 2k

1

1

sn u cn u dn u sn u Z>(w,

1

dn (a+w) + dn ( H - v) - 2dn u = 3

W)

D(u,v),

2

{ d n ( « - r - f ) — dn (u —v)}dudv=log D(u, v), 2

!

t

l

D(u, v)—l—k sn vsn u Elliptic

2

2

= cn' v+sn vdn u

.

^-functions:

0 , 0 + 1 ) = 0 . 0 ) ,

* C * | r ) = / ^ ' ' * * ( f I "7") '

!

= l + 2SC-l)V cos27r«x •=1

= VA t f i S * S ^ " ^ ' c o s h - ^ - ( 2 n + l ) x , 2

* A.

»-«

A

134

196

M . Toda

!

=#(
Mx)

(X-^Mm

9

I-l

= * (?) II (1 - 2 ^ ' " '

E

cos 2^r - "- ) 9

** » /"--ir \ - cosh-^-(/-x)cosh-^(/+x) ^.(ov^'cosh ^ r ) n ^ _ * *(s) = n ( l - 0 .

^io

£

g

l

?.(-^)=d *«-A, n

p.(Q)Tpi(*)T

b ^ j ^ J P

{Mx)Y

- [* (0)]

2

0

o

r

b

0

J

L t f l C

^ l*C^"J J J

2

=c|l+v ^-logtf„(x)(,

!

/=v ( ')=sn (2A^)MO)/C2^) , a

3

3

D 0 ) = l-(l-J-)sn (2A' -) 2

I

i

= cn (2Ay) + -|-sn (2itx). 2

D(u, v) = l

—

2

1

!

k sn u

K

2

2

!

r

r iog/j>(»,^)^=4A io 3

5

sn u = cn v + sn i;cln M , g

'

•

Appendix Another

II

expression for wave train

Rewriting E q . (3-11) for a wave train, we can easily verify that

135

Waves in Nonlinear

AU)

Lattice

197

'

where 2

!

2

D(ic,«) =cn ic + s n j r d n M , 2

2

D,Or) = cn je + - ^ - s n * ,

u ^ t - ^ J K . If we put r=logC with

then we have Z>(*, a). However, since we have (cf. Appendix 1)

W e see that the time average of r» is just f: \

(_r.-r~)dt = 0.

J period

Appendix Momentum a)

III

and energy of a soliton

Momentum T h e momentum of a particle in the lattice is given by my,~~s„-s. .

( A III -1)

+1

For the soliton in a ring as given by E q . (4-24), s, is cyclic,

-A-,,

4HT)

•*H — " J * — * V 7

Y

136

M

198

-

T o d a

^ K v . Z ^ . t - ^ K }

,

(AIII-2)

and therefore the total momentum vanishes: S / « i , = 0.

(Ain-3)

• -1

This can also be shown by the time average, y.dt~^s,dt

-

=6 •

For a large ring, we can use the approximation (4-26) form, s.=^e tanh (fit - an) - 2v, (fit -an)

( A TTI • 4) and its integrated ( A III • 5)

which yields S

my.~B

7

^ ~ ,

"

>

( A III • 6)

!

sinh \Bt - an — | j + codf— and the time average vanishes again:

J e

J

!

- s i n h ^ r + cosh f

(AIII-7)

The fact that the soliton we have been discussing has no total momentum comes from the choice of the form of s,. Without violating the equation of motion (2-26) we may add to s. a term which is linear with respect to n: s.-*s. + cn + S,

( A III • 8)

where c and 3 are arbitrary constants. Then we have an extra momentum — c for each particle, which means a translation of the whole ring along itself. This kind of motion is also seen in linear case, where we can superpose the motion given by the lowest mode t» = 0 to any other normal mode. The above mentioned term cn+S gives the same kind of superposition to our nonlinear modes. W e shall disregard this extra momentum in the following. b)

Energy The average kinetic energy of a particle in soliton is given as

Isinh'iSc + c o s h ' y l = 2^(acoth«-l).

(AIII-9)

137

Waves in Nonlinear

Lattice

Since there are N particles in the ring, v,N=()/a see that the total kinetic energy of a soliton is

199

(speed), and |9 = s i n h « , we

m h

T~2sinhffl(cosh«-^ -^-) .

(A HMO)

Now, as for the potential energy we have $(r„) = Or'- - 1 )

+7-..

( A III -11)

Since the average of the first term vanishes because of Eq. (2-31), we are left with the total potential energy ( A III-12)

•-1

which is equal to the amount of elongation in our units. therefore, from E q . (4-30)

CAHM3)

U^2a[(^^y~l]-

Appendix Partition

For a soliton,

IV

function

It is interesting to note that the potential energy l

«(r)=|(«- '-D+«r

(AIV-1)

affords analytic expression for the partition function Z of the lattice under the pressure p.

W e have 2m

T

W

2=( ^ ) V

(AIV-2)

in usual notations and Q=J~

-M->+><)i"dr.

e

(A IV'3)

In our case after simple transformations we get

where r is the r-function. W e see readily the relation between the averages of the potential energy and the elongation:

138

200

M . Toda

(A IV-5)

KUkT)

hgQ

(A IV-6) where iff stands for the di-gamma function. If the pressure is absent, we see that ar=W)

(AIV-7)

which is in accordance with E q . ( A III-12). References

1) E. Fermi, J. R. Pasta and S. M . Ulam, Collected Papers of E. Fermi (Univ. of Chicago Press, 1965), Vol. II, p. 977. 2) J. Ford, J. Math. Phys. 2 (1961), 387. J. Ford and J. Waters, ibid. 4 (1963), 1293. 3) E. A . Jackson, J. Math. Phys. 4 (1963), 551, 686. 4) N. Zabusky, Proceedings of the Symposium on Nonlinear Partial Differential Equations (Academic Press, N. Y„ 1967). 5) M. Toda, J. Phys. Soc. Japan 22 (1967), 431; 23 (1967), 501. 6) M. Toda, Proceedings of the International Conference on Statistical Mechanics, Kyoto 1963 [Suppl. to J. Phys. Soc. Japan 26 (1969), 235]. 7) cf. Appendix I, and A . G. Greenhill, The Application of Elliptic Functions (1892), Dover edition 1959. 8) D. J. Korteweg and G. de Viies, Phil. Mag. 39 (1895), 442. 9) C. S. Gardner, J. M. Greene, M. D. Kruskal and M. Miura, Phys. Rev. Letters 19 (1967), 1095. R. M. Miura, C. S. Gardner and M. D. Kruskal, J. Math. Phys. 9 (1968), 1204. 10) V. I. Karpman, Phys. Letters 25A (1967), 708. 11) N. Zabusky and M. D. Kruskal, Phys. Rev. Letters 15 (1965). 240. 12) R. Hirota and K. Suzuki showed recurrence phenomena using a nonlinear L C circuit: J. Phys. Soc. Japan 28 (1970), in press. 13) cf. reference 6), the discussion by N. Zabusky, on p. 236-237.

139

The Criterion for the Existence of a Gap in the Optical B a n d of a Disordered M i x e d Crystal

Morikazu Toda

Institute for Optical Research, Tokyo University of Education, Tokyo (1970)

M a k i n g use of the Rayleigh's theorem for an ocsillator-system the exact criterion of the existence and non-existence of a gap i n the optical band is given for an isotopically disordered mixed crystal of the type AB C\- . X

X

The

Saxon-Hutnei type theorem thus derived gives a possible classification of the optical spectra of mixed crystals into two types (persistence and amalgamation types).

51.

Introduction

It is known that the optical spectra of substitutional mixed crystals of the type AB C\X

X

can be classified into two types. The infrared reflection or

Raman spectrum of a certain kind of mixed crystals, including Na CI + K CI, contains only one maximum, whose frequency changes as x changes. O n the other hand the reflrction spectrum of some other mixed crystals, including Ga As^Pi-x,

has two maxima whose intensities change gradually as x changes.

We have thus two types of mixed crystals. Hence Matsuda and M i y a t a ( M - M ) called the former the amalgamation type (a-type) since the different

140

constituents seem to produce amalgamation effect, and called the latter the persistence type (p-type) because the two frequency-maxima, characteristic ot 1

the two pure crystals AB and AC, seem to persist i n the mixed c r y s t a l ' . It looks quite natural to hypothesize with M - M that i f there is no gap in the optical band of the mixed crystal it will be of the a-type and i f there is a gap it will be of the p-type. Although strict criterion will require detailed calculation of the shape of the frequency spectrum and the optical density of the optical band, the above hypothesis seems to work quite nicely. A direct examination of the optical bands of pure and mixed crystals by neutron scattering or other methods is necessary i n this connection. M - M investigated the criterion for the existence of the gap by making use of the Hermitian characer of the dynamical matrix for the lattice vibration. They proved thus the so-called Saxon-Hutner type theorem in the case of the substitutional isotopic mixed crystal. It can be stated as follows. The frequency range i n which the pure AB and the pure AC crystals have no eigen-frequency remains to have no frequency i n the mixed crystal AB G\_ , X

X

provided that the

force between A and B is the same as the force between A and C (including the forces beyond the nearest neighbors). The above is the sufficient condition for the existence of the relevant band gap for a mixed crystal. B u t it is also the necessary conditin for a random mixed crystal, because there is a non-zero probability for large domains to have pure AC

OT AB arrangements, so that i f the range has some eigen-frequencies of

either pure AB or AC crystal it has also some eigenfrequencies i n the mixed crystal. The above theorem was extensively studied by Hori by the so-called evap2

oration m e t h o d ' . The aim of this paper is to show a simpler method of proving the Saxon3

Hutner type theorem by making use of the Rayleigh's theorem ' for a system of interacting harmonic ocsillators.

§2. F r e q u e n c y B a n d s o f M i x e d C r y s t a l s First, let us consider the case where the optical bands of pure AB and pure

141

AC crystals overlap as shown in F i g . 1. Of course the number of eigenfrequencies of both optical and acoustic bands are the same and are of the number of atoms in the crystal. B u t , for simplicity, the bands are shown in the figure as i f there were only five levels (eigenfrequencies) in each band, which means that the number of atoms of each component were of the order of five or so. In actual we are thinking of a real crystal consisting of a large number of atoms. We can, however, use the scheme as shown in F i g . 1 with only a few levels in the following consideration. In F i g . 1. it is assumed that the mass Mc of the C-atom is smaller than the mass MB of the S-atom, so that the bands of a pure AC crystal are higher than the corresponding bands of a pure AB crystal: MB > Mc3

Now, the Rayleigh's theorem ' (for a system of interacting harmonic oscillators) is that, i f we replace one of the masses of the oscillators by a lighter (heavy) one, keeping the forces unchanged, then each of the eigenfrequencies shifts upwards (downwards) or is kept still, but in any case it cannot move beyond the original next higher (lower) eigenfrequency. The theorem includes the similar case where we change one of the force constants. B u t we are now interested only i n the case of changing the mass.

Opt.

Opt .

AB

Fig. h

AC

Frequency bands of the pure crystals AB and AC {MB > Mc)the optical bands overlap.

When

142

Starting from a pure AB crystal we replace a 5-atom on an arbitrary lattice point by a C-atom. Then, since we are assuming that MB > Mo, by the Rayleigh's theorem, all the levels are shifted upwards (some may be kept still), but cannot exceed the original next higher levels. So we get the level scheme as shown by the second column from left i n F i g . 2. T h e n we replace another fi-atom by C-atom, and so on. The shift of levels will be as shown in F i g . 2. W h e n all the fi-atoms are replaced by C-atoms we get the pure AC crystal. We could have started from the pure AC and proceed reversely from right to left i n F i g . 2. The dotted lines i n F i g . 2. indicate the fact that the optical band in this scheme is divided into two triangular parts by the dotted hues which step down to the right in this case from the upper edge of the optical band of the pure heavier constituent AB to the lower edge of the optical band of the pure lighter constituent

AC.

In the triangular parts of F i g . 2. we have no gap. The detailed level structure of the mixed crystal in this scheme depends on the process we substitute 5-atoms by C-atoms. If we made some ordered arrangement of B and C , say at 50 - 50 % concentration, we would have a gap or gaps between the dotted lines. We see also the fact that the gap of the ordered mixed crystal remains

opt.

opt.

pure AB

F i g . 2.

mixed crystal

pure AC

Level scheme for successive substitution of the atoms B by C. Overlapping case (a-type).

143

even i f the ordering is disturbed by replacing a considerable amount of B or C-atoms by C or fl-atoms in the ordered arrangement. This circumstance is shown in F i g . 3.

pure

Fig. 3.

ftB

ordered mixed c r y s t a l

W h e n ordered mixed crystal is formed we may have a gap.

However, for a random mixed crystal we have no possibility of a gap i n the optical band i f the optical bands of the pure AB and pure AC crystals overlap as i n F i g . 1. and Fig. 2., since there is non-zero probability for a large domain to have pure AB or AC arrangements. We have already mentioned about this case i n § 1. Thus we conclude that i f the optical bands of the pure AB and AC crystals overlap, the random mixed crystal has only one optical band and will be of a-type. Now we come to the case where the optical bands of the pure AB and AC crystals do not overlap. We can apply the similar argument as before. It will be seen that the level scheme i n this case necessarily has to take the form shown in F i g . 4. In order to fulfill the requirements of the Rayleigh's theorem and the condition that the optical bands of pure crystals do not overlap, the levels have to shift sometimes highly enough and result in a gap of the optical band of the mixed crystals. This level scheme can be clearly understood when we make an arrangement of match-sticks or the like to express the level scheme of F i g . 2. and raise the right end (pure AC) untill the overlapping of the optical bands of the pure crystals is taken off, keeping the triangular parts unbroken. The result of such a match-stick game will be the scheme of Fig. 4.

144

ODt.

common gap

opt.

pure AB

F i g . 4.

pure AC mixed crystal

Level scheme for successive substitution of the atoms B by C. Nonoverlapping case (p-type).

In F i g . 4. we have a common gap i n the optical band of the mixed crystals between the dotted lines indicated. Thus, we conclude that i f the optical bands of the pure AB and AC crystals do not overlap, the mixed crystal has a gap in the optical band and will be of p-type. Similar statements can be made with respect to the existance or nonexistance of a gap between the acoustic and optical bands. But this will not be so interesting as far as only the optical properties are concerned.

§ 3. C o n c l u s i v e R e m a r k s

Applicability of the above results is restricted to isotopic substitutional mixed crystals, since we have assumed that the forces between atoms are the same for AB and AC and that the crystal structures of the pure AB and AC

145

crystals are identical. But there has been no further assumption such as the near est-neighbor interaction. If we have sufficient data of the neutron-scattering or other experiments which give information about the band structure of lattice vibration, we can have direct test of the above-mentioned assertion on the mixed crystal. A n d , i f we have more detailed theory on the intensity of infrared absorption spectrum or of the reflectivity of mixed crystals, we may test the criterion for the demarcation of a and p-types of these crystals. B u t , at present, we have to rely on approximation, as M - M did, to compare the above theory with experiments. The above mentioned result can be expressed as follows: i f the optical bands of the pure crystals AB and AC do not overlap, the common band gap in the pure crystals remains as a band gap in the mixed crystal AB C\X

Z

(p-

type), and i f this condition fails there can be no relevant band gap (a-type). This is one of the cases of the Saxon-Hutner-type proposition.

References 1.

H . Matsuda and T . Miyata: Prog. Theor. Phys. Suppl. E x t r a Number (1968) 450.

2.

J . H o r i : P r o c . Phys. Soc. 92 (1967) 977.

3.

A . A . Maradudin, E . W . Montroll and G . H . Weiss:

"Theory of Lattice

Vibration i n the Harmonic Approximation" - Solid State Physics Suppl., 3 (1963) (Academic Press, New Y o r k ) .

146

Interaction of Solitons with Electromagnetic Waves MORIKAZU TODA Institute for Optical Research, Tokyo University of Education, Japan

Toda. M. Interaction of Solitons with Electromagnetic Waves. Physica Norvegica, Vol. 5, Nos. 3-4,1971. Analytic treatment is given for a one-dimensional lattice with special reference to the continuum limit. An electromagnetic wave incident upon such a medium will be scattered by elastic waves, and especially by solitons or the pulse-like fundamental excitations in the nonlinear medium. At some special frequencies, the electromagnetic wave will be transmitted without reflection if certain conditions for the occurrence of transparent solitons are satisfied. Morikazu Toda. Institute for Optical Research, Tokyo University of Education, Kyoiku University, Tokyo, Japan

INTRODUCTION The time-dependent behavior or irregular lattices and of nonlinear lattices has received much attention in recent years. The study of the effect of impurities implies the problem of scattering of waves by in homogeneity. On the other hand, in the study of nonlinear lattice vibrations, we have been led to the concept of a stable pulselike motion called soliton. When two solitons approach, they interact, pass through each other, and recover their initial forms. In this sense solitons can be said to be mutually transparent.

and has been attacked computer-experimentally (Ooyama & Saito 1970) as well as analytically (R. Hirota, private communication). However, we have here another very interesting problem of interaction between solitons and other wave-fields. If the latter are electromagnetic waves, they may provide the means to detect solitons by direct observation. In addition, this problem is closely related to the mathematical method of solving the famous nonlinear equation, the Korteweg-de Vries equation, which can be conveniently written as 8u

du

s

8u

+ dx = 0.

(1.1)

Now, we may ask about the scattering of waves by the inhomogenuity induced by solitons. The waves can be small disturbances coexisting with the solitons in the same medium. This is an interesting problem in connection with the stability of solitons against perturbations.

Gardner et al. (1967) found that if J. is the eigenvalue of the Schrodinger type equation,

The present work is dedicated to Professor Harald Wergeland on the occasion of his sixtieth birthday.

J. is independent of time even though u evolves with time according to the KdV equation. The

1

d'v + {l-u)tii = 0, dx 1

d-2)

147

204

Morikazu Toda

8h *r 13 1 3* \* wave v has been introduced for formulating the 8t eigenvalue problem and there can be other ways of mathematical manipulation to state the same thing. They pointed out, however, that Kay & Moses (1956) had developed the method of solving the with the abbreviation r = r(j,t) = r,(t). If we inverse scattering problem, that is, the method of neglect higher order derivatives and higher solving the above Schrddinger type equation for powers of r, the above equation can be written as u, using what is known about the asymptotic J_J M b^ 8_ 1 m 8 _(3_ behavior of the scattered wave ip. From this, 24 3j>) 2 8j \ ab ~S~iU Kruskal et al. reduced the KdV equation to a linear integral equation. \-il~m~ $ 18 1 8 \ +

+

r

s

"\y^b-87

The aim of this paper is to apply the abovementioned method to a more realistic problem, where is formally replaced by electromagnetic

SOLITONS IN NONLINEAR MEDIA As a model of a nonlinear lattice we think of the system of particles of mass m connected by anharmonic springs (Toda 1967CT, b, 1970):

~ 2

t ,

w

1

(2.1)

[-87

+

U W) (2.6)

r

Here the derivatives are to operate on all the terms on the right. Thus we have a class of motion expressed by .. 8r 3r •ab 8 i - 3 j +

dbj = a(2£- J-e- J-i-e-"^+'), d!

+

+

1 B'r b Br 24 W~^ W = °' r

(2

_ -

7)

We shall use the frame of reference moving to the right with the speed

where r, denotes the amount of elongation of the jth spring, and a and b are constants character(2.8) i n g the force exerted by the spring, a, b > 0, since the lattice should have the stable state where D stands for the natural length of a spring. r, = 0. By the Galilei transformation (n,r) -f <£,r), where We have obtained exact particular solutions of (2.9) the above equation (Toda 1967a, b, 1970). Among these we have a family of solutions each (2.9') r = cl , of which represents a soliton: we get -"->- 1 = sinh a • sech (aj-Bi), ( 2 . 2 ) 0

J

!

e

with ab

- 1

3x sinh a ,

2 ' 84

+

24 a?

1

(2.10)

(2.3) which is nothing but the KdV equation

where a is an arbitrary constant. Now, the above equation of motion can be rewritten as m^f+a

[ 2 sinhi -|r] V"J = 0. (2.4)

Further, for small disturbances, we may use the approximations, e 3

_tr

= x+^x , to yield

= \-br+ ^ r'and sinh x

In this continuum limit the soliton solution reduces to 4= -&.*mt U - ^ , ) . e u „ D b This is a compressional pulse, which will result in a change in the refractive index of the medium

148

Interaction of Solitons with Electromagnetic Waves where

and thereby scalier the electromagnetic wave incident upon the soliton.

-

g

~

g

q

(3.7)

Qe

denotes the change in the density e of the medium. At this point, we may use the LorentzLorentz formula.

SCATTERING OF ELECTROMAGNETIC WAVES

If we denote by y{x,t) the field of the electromagnetic wave, we have the following equation for the wave propagating in the ^-direction: to get iPv

I n'-l = const. 6

(3.8)

W-l)( ;+2) 30o n

(3.1)

where k denotes the wave number of light in vacuum, and n = n(r) is the local refractive index which may deviate from the constant value n , because of the soliton motion in the medium. Since the velocity of each soliton is very small compared with that of light, we can ignore the motion of solitons and assume that they are standing still while they scatter light incident upon them.

(3.9)

Thus, using Eq. (2.11) and replacing £ by x, since the soliton moves only slowly, we get

0

V(x) = - lixa} sech* OJC ,

(3.10)

with («i-l){«;+2) D

(3.11)

If we refer to the Lennard-Jones type interacFor the scattering problem the wave equation tion potential is equivalent to the integral equation (A,, — kn ) a

iy(x) = e"" *(*--OF(*>(*')dx'

3 2

<'>

where e""* represents the incident wave and the second term on the right band side is the (3.12) scattered, reflected and transmitted waves. V(x) stands for the scattering potential, which is so as to compare with our potential V(_x) = k'(n'(x)-rtl) (3.3) a *(r) = -r (e + 6r) 4-const. in our case, g(x) is the Green's function defined b by ab , ab' S const.+ - x - r 7(3.13) L o ( " £ +*s)iH*-AT m 6(x-x'), (3..4) we can express the parameters a and b in terms of and is, in the one-dimensional case, #o and D. Thus we see that _h-

1

irW = e'i*.'o/2,* .

abil = 360JD",

(3.5)

0

When there is a soliton given by Eq. (2.11), we will have a disturbance which will be proportional to the compression —r/D:

b = 21/D,

(3.14)

and In' f = -^-W-1)W+2)(|J , !

3rf\ _r_ de ja D

1

(3.6)

(3.15)

where we have introduced the wave length X of

149

206

Morikazu Toda

the electromagnetic wave by k = 27[1J...

1

(3.16)

Eqs. (3.10) and (3.15) give the scattering potential induced by the soliton. The result can be generalized for any disturbance including the case of many solitons. But we shall investigate the possibility of perfect transmission of the electromagnetic wave incident upon a soliton in the next section. POSSIBILITY OF A TRANSPARENT SOLITON

(4.1)

where a is an arbitrary constant (Kay & Moses 1956). In our case this can be attained when /1 =

1

=

V

:3

7[

V(n~- J) (1I~+2)

(4.4)

It is easy to show directly that this satisfies, with

Eq. (4.1), the integral equation (3.2) for the scattering problem. The incident wave as given by (4.4) is IfI(X) = elkox (x --+ - 00) ,

and the transmitted wave is

tan fJ

= 2akl(k' - a') .

DISTURBANCE TRANSPARENT AT ALL FREQUENCIES Now, following Kay & Moses, we can proceed a little more in seeking the disturbance which is transparent to all frequencies. Starting with Maxwell's equations for a plane wave, propagation in x-direction through an inhomogeneous medium (E and B are the electric and magnetic field intensities respectively),

(4.2)

In view of Eq. (3.15) this condition implies that the wave length should be chosen in such a way that

~

•

with the phase shift fJ given by

In the preceding section we have obtained the formula for the scattering of electromagnetic wave by the disturbance of the medium by the presence of a soliton. It will turn out that the Born approximation gives only a poor approximation for this problem. However, it is known that the incident wave shows reflectionless transmission if the scattering potential has the form Vex) = - 2a' sech' ax ,

lfI(x) = --=-k (iko-a tanh ax)e'k OX I o+a

aE ax =

iwB,

aB

.

(5.1)

n'(x)

ax = ,w e ' E

(5.2)

we apply the transformation (x --+ ,,), where (4.3)

in order to have a transparent soliton. Since the right hand side of this equation is of the order of unity for some realistic values of the refractive index no, the condition will be satisfied if the wave length of the electromagnetic wave is of the order of the atomic spacing D in the crystal. For example, if no ~ 1.3, the appropriate wave length is A. ~ 0.8D. And if no ~ 2.4, A. ~ 3D ~ 10 A for the usual atomic spacing. In the above transparent soliton, the incident wave will suffer a phase shift when passing through the soliton. Actually the perfectly transmitted wave is in this case given by

d" = dxl
(5 .3)


(5.4)

E(x) =
(5.5)

with

and

to get d'