Ruppeiner(1979)

PH

YSICAL

REYIE%

A _VOLUME

20,

_{%UMBER 4}

_OCTOBER

₁₉₇₉

Thermodynamics:

A Riemannian

geometric

model

George Ruppeiner

Department _ofPhysics, Duke Uniuersity, Durham, North Carolina 27706

(Received 22 January 1919)

Byincluding the theory offluctuations in the axioms ofthermodynamics it is shown that thermodynamic systems can be represented by Riemannian manifolds. Ofspecial interest is the curvature ofthese manifolds which, for pure fluids, is associated with eA'ective interparticle interaction strength by means ofa general thermodynamic "interaction hypothesis.

"

This interpretation ofcurvature appears to be consistent with hyperscaling and two-scale-factor universality. The Riemannian geometric model isanew attempt to extract

information from the axioms ofthermodynamics,

I.INTRODUCTION

Geometry has been much used in

thermodynam-ics.

'

'

Previous works, however, lack a meaning-ful metric structure, an expression for distance between equilibrium

states. It is

shown in this paper that ifthe theory offluctuations is included

in the axioms of equilibrium thermodynamics,

there

exists

acorresponding Hiemannian metric which enables us to represent thermodynamic

systems by Riemannian manifolds. Of special

in-terest

is the curvature ofthese manifolds which is associated with interactions. This interpretation is anew attempt to

extract

information from the axioms of thermodynamics.

II. FOUNDATIONS OF THE GEOMETRICAL MODEL

The geometry is based ontwo fundamental axioms

ofequilibrium thermodynamics.

For

simplicity, only single-component fluids in one phase will be

discussed in this short paper.

I.

For

apure fluid, given any thermodynamic system A~ with some fixed scale X,~there exist

equilibrium states which can be represented by

points in atwo-dimensional manifold

e„which

is

differentiable everywhere except at phase

transi-tions and

critical

points.

The above axiom differs from the corresponding

one ofGibbs' since there is no explicit mention of

afundamental equation. The fundamental equation

is not included because we wish to develop a

co-ordinate independent intrinsic geometry as op-posed to the more common extrinsic geometry,

which is the foundation of most previous

geomet-rical

constructions in thermodynamics.

The second axiom deals with systems in contact.

Any model of thermodynamics must take this into account. We deal here only with the simplest and

most important

case

of systems in contact with

ve ry large

rese

rvoirs.

Denote the state ofA~

W(x,

x')dx'=B

exp

S,

'

x, x')

'

dx',

k~

(2.

1)

where x—=

_(x„x,

₎and x'=-(x,

',

x,

')

are

the coordinates of(Pand(P', respectively,

_S,

(x,

x') is

the total

en-tropy ofthe

reservoir

and A,((P'), ks is

Boltz-mann's constant, and

8

is

anormalization

factor.

Each particular

reservoir

has been labeled with

the coordinates of the most probable state of

A,

in contact with

it;

toevery state

A,

((P)there

cor-responds a

reservoir.

_S,

(x,

x')

is assumed to be

adifferentiable function of

x'.

If all of the

reser-voirs have the same

scale,

S,

(x,

x')

can be con-sidered adifferentiable function ofx

also.

S,

(x,

x')

cannot be represented as a

scalar

func-tion on

8,

.

Instead, the information contained

in

_S,

_(x,

x')

will be represented by the second

mo-ments offluctuation which are defined at the

point with coordinates xby

g-'(x) -=ax'a _fx D &xx' w(x x')dx' (&.2)

where 4x„'=x„'

—

_x„.

It is

not difficult to show' that

(2 3) Toa very good approximation,

W(x,x')dx'

=(2v)

'

exp[--,

'

_g,

_,

(x)~x,

'.

_~x,

_{.]v'g(x)dx,dx,}

,

(2.

4) where g(x)—=detg(x). From this expression we see that the line element

~f

=-[

g,

(x)rx~px']'

"

(2.

5)

corresponding to the point

I

on

8,

by A~(g). Place

A,

in contacte with afixed

reservoir

and let a steady-state condition be reached. According to the theory of fluctuations,

'

A~ will fluctuate among states A,((p'), which neighbor the most probable

state A,((P), according to the probability

distribu-tion

THERMODYNAMICS:

A RIEMANNIAN

GEOMETRIC

MODEI 1609

has the appearance of

a

distance between nearby'

states

ifby

"distance"

we have probabilities in mind

—

the

less

probable afluctuation between

states,

the further apart they

are.

It should also

be noted that dA=_gg(x)dx,'dx2 has the form ofan

infinitesimal

area

on a Riemannian manifold with metric

g(x).

The above considerations motivate the following axiom, which

is

-to represent the second law of

thermodynamics:

II.

There

exists

on

8,

a

positive definite

Rie-mannian metric

_g(x),

which is determined at the point

0'ce,

by the condition that the components

of

_{g '(x)}

in a particular coordinate system are the

second moments of fluctuation of the state A~((P).

The second moments offluctuation provide an

ex-cellent approximation for IV(x,

x'),

which

is

usual-ly negligible if higher-order terms in the

expan-sion for

_S,

(x,

x')

make acontribution; such higher- .

order terms cannot be represented geometrically on 8~ since they are generally nontensorial. The well known condition that the intensive parameters

of two systems in equilibrium

are

equal

is

repre-sented a bit indirectly in this model. It appears

since g(x)

is

a tensor

field.

"

Including the theory offluctuations, which

is

usually associated with

statistical

mechanics, in

the axioms ofthermodynamics was

first

done in

1931

by Lewis, who argued that the second law of thermodynamics cannot be properly stated without

it.

"

fallen

also discusses such an extension of

thermodynamics.

'

If the theory of fluctuations

is

not included, the second law can be represented geometrically by the recent construction of Weinhold, who

as-sociates

inner product spaces with equilibrium

thermodynamics.

"

However, it is not possible

to build a meaningful Riemannian geometry on such

a

foundation since

a

concept of distance between

states

is lacking. Also, the inner product in that work

is

ambiguous at least with

respect

to an

ar-bitrary positive definite scalar-multiplying

func-tion.

Therefore,

it does not seem possible to

ex-tract

fundamentally new information from Wein-hold's geometry.

The Nernst principle

is

not considered in this

paper.

III. INTERACTIONS AND CURVATURE

Thermodynamic systems have been represented with curved Riemannian manifolds in quite a reasonable fashion; therefore,

it is

meaningful to inquire into the physical significance oftensors which can be constructed from the

metric,

in

par-ticular,

curvature

tensors.

The present model has the advantage that it allows the construction of

invariants not previously encountered in thermo-dynamics.

Exactly what information curvature tensors

provide

is

not

clear

from the outset. However, the following ideal-gas result suggests a solution.

Define A~to be

a

system with fluctuating energy

and particle number Uand _{N, respectively,} and with fixed volume V. Such systems will be called "open" and

areof

primary

interest.

Let T be the

temperature and p=N/V the density ofA

.

It is

straightforward to show'~ that in

_(T,

p) coordinates the square of the line element defined in

Eq. (2.

5)

ls

(3.

l)

where C~

is

the heat capacity at constant volume

and._K~the isothermal compressibility ofA.

_~.

The ideal gas has the equations ofstate

P

=_{pk~T and}

C~=Nf(T), where

P

is

the pressure and

_f(T)

is

a

positive definite function of

T.

We find

for

the

ideal gas

.

r T)

B

(3.

2)

On making the successive coordinate

transforma-tions

(3.

3)

and

x,= v'2Vp(cos-,

't+

sin-,

't),

x,

=v'2Vp (cos-,

't

—sin-,

't),

(3.

4a)

(3.

4b)

where T0is an arbitrary positive constant, we obtain

(3 5)

Nfl =IX +CfX2q

the line element

for a

plane, which

possesses

zero curvature.

"

To describe the internal behavior of

a

fluid, we should consider open systems since they best

mimic the situation in the interior of a fluid, where both energy and particles

are

free

to move about

with no external

barriers

to impede them in any

way. Therefore, since the ideal gas is

character-ized by the absence of effective interparticle

inter-action strength, the remarkable result that open

systems composed

of

the ideal gas have zero

cur-vature motivates us to

associate

curvature with interactions. To arrive at an exact correspondence

between curvature and interactions, we proceed

by analogy with general relativity, where

1610

GEORGE

RUPPEIXER

20

C(x)=_VK,

_(x),

_(3.

₇₎

which

is

independent of V. Aside from

scalar

functions related by dimensionless multiplicative constants of proportionality, which can be ab-sorbed in w, it appears that this is the only

rea-sonable tensor field constructed solely from curv-ature which we can

associate

with interactions.

"

We have from

Eq.

(3.

1)and a standard

expres-sion

for

the Gaussian

curvature"

that

C(x)=~I

(x),

(3.

8)

where C(x) is a tensor field constructed only from the curvature of

8»,

I(x)

is

a tensor field which

de-scribes

effective interaction strength, and ~

is

a dimensionless constant ofproportionality which the geometry cannot determine. This relation

will be called the interaction hypothesis.

The interaction tensor I(x) is at

first

somewhat mysterious since there is no physical quantity which corresponds to it in thermodynamics as it is generally known. C(x), being amathematical

object constructed only from thermodynamic

quan-tities,

is not

as

enigmatic, so it is best to begin

by examining

it.

Riemannian curvature is de-scribed by the fourth-rank Riemannian curvature

tensor field

R(x).

R(x)has only one independent

component on two-dimensional manifolds,

"

so a

scalar,

the Gaussian curvature, provides all the

information about curvature on such manifolds.

"

The Gaussian curvature function

for

_8»,

K»(x),

de-pends on the volume as 1/V, so it cannot be C(x)

since

a

description of intrinsic interaction strength

should not depend on

V.

Therefore, we take

02

k

A₀

(3.

9)

the last equality following by using

z

=0.

1and

y

=1.

197'

The second term in

Eq.

(3.

8)is more difficult to evaluate; careful linear model

calcula-tions"

on the

critical

isochore yield

ing, relates the exponent v

for

the fluctuation

cor-relation length $(x) to the specific-beat exponent

z

through the equation vd

=2-

_z,

where d is the spatial dimensionality ofthe system,

"

which in our

case is

3.

As was

first

demonstrated by Nidom,

"'

the hyperscaling relation in d

=3

systems follows if the singular part of the

free

energy is afunction

only of

$'(x).

"

The role of

P(x)

as "interaction

tensor"

in hyperscaling plus the fact that I(x)has the dimensions ofvolume leads us to speculate that I(x)

is

in fact the correlation length cubed,

aplausible idea since

((x)

is the only intrinsic parameter with the dimensions oflength related

to fluctuations.

'The preceding assertion can be tested by

com-paring

Eq.

(3.

8)with available experimental

re-sults.

Qn apath with constant

critical

density

p„

the

critical

isochore, sufficiently near the

critical

point, C»/V =A,e and

_Kr

_=Kr,

e

",

where

A, and

Kr,

are constants and e =(T —

T,

)/T„where

T,

is the

critical

temperature. e

is

taken as

pos-itive since we consider the fluid only in the single

phase. Substituting these expressions into the

first

term in

Eq.

(3.

8),

and keeping only the most singular term, yields

2~g 8T

_(~g

BT)

(

2

2)

A

V 8 1 ag_TT

Z60'-2

2'»

vg &p

(3.

10)

where

_g»

——C»/k~T',

g„—

=_{V/k~Tp'Kr, and}

g=

gr~„.

There are two features of C(x) which im-mediately strike us

as

interesting. The

first

is

that C(x) has the dimensions ofvolume and the

second is that C(x)diverges at the liquid-gas

critical

point, as demonstrated below. The latter feature indicates, by the interaction hypothesis, that interactions become dominant near the

critical

point. Therefore, any description of interactions

must provide the relevant information about

a

fluid in the

critical

region; in particular, the

free

ener-gy should be

a

function only ofthe interaction

scalar

I(x)sufficiently near the

critical

point.

That all ofthe exponent relations of scaling can be derived from the statement that the singular part

of the

free

energy

is

afunction of only one diverg-ing parameter near the

critical

point is easy to show.

"

Stronger than scaling is hyperscaling, which, in addition to including the results of

scal-where ~Z~~

0.

01(0.

21ks/A,

).

Since the

correla-tion length has not been measured to better than 10/o, the second term will be neglected and we take

I(p„e)

=0.

21(ks/vA, )e

(3.

11)

near the

critical

point. We see that I(x)has the same

critical

exponent as the hyperscaling

pre-diction for

P(x).

Also of importance is the result of

Eq.

(3.

11)

that

I

p ~E')C»6

0.

21

=universal constant

Vks

(3.

12)

=_{universal constant,}

Vk~

(3.

13)

to better than -1/o. The hypothesis of two-scale-factor universality" relates the correlation length

THERMODYNAMICS:

A RIEMANNIAN

GEOMETRIC

MODEL 1611

which has been verified experimentally to

-10%.

"

Equations

(3.

12)and

(3.

13)provide further support

for the premise that I(x) is

P(x);

the universal constant in

Eq.

(3.

13)has been measured to be

0.

19+

0.

04,

'

so we require w=

1.

0+

0.2.

The above results indicate a remarkable

consis-tency between the interpretation ofthe Riemannian

geometric model and hyperscaling and

two-scale-factor

universality. Further

tests

ofthe geometric

model with experimental data

are

difficult because ofthe limited nature of measurements ofthe

cor-relation length. Perhaps magnetic systems can

provide some additional

results.

IV. PATHS AND PATH LENGTHS More important than further experimental

comparison

is

some understanding based on

sta-tistical

mechanics ofthe relation between inter-actions and curvature. Any attempt tofind

a

microscopic connection between interactions and

curvature must deal with paths and path lengths

since Riemannian geometry

rests

ultimately on

this concept. Toobtain insight into the meaning ofpaths and path lengths we introduce in this

sec-tion athermodynamic

process

which

is

somewhat analogous to Brownian movement. This

process

will be used to demonstrate the uniqueness ofthe present geometrical model.

At time t

=0

consider the system A~ in the state

A, (a,

).

Place it in contact with the

reservoir

with which

_{A, (a,}

)

is

in equilibrium and leave it in

con-tact for a

period oftime &before removing

it.

We

restrict

7 only by requiring that it be much

larger

than any relevant dynamic time constants in the

system; v

is

so

restricted

since otherwise the

fluc-tuation probability distribution is no longer given

by

Eq.

(2.

4)and we would be outside the scope of

the formalism used in this paper.

A,

will have

fluctuated to some new state

A,

(&P,

).

Now place

A,

in contact with the

reservoir

with which

A,

(a,

)

is

in equilibrium and leave it in contact

for a

per-iod oftime v before removing

it.

A,

will have

fluctuated to another

state,

A,

(a,

).

Repeat this

procedure n times;

after

a time t =n7 system

A,

will be found in the state A~(a'„).

The above procedure is a Markovian stochastic

process

somewhat analogous to Brownian

move-ment. Define

p(x,

t)dA =the probability that system A~ is found in

a

state within the infinitesimal

area

dA on

(t,

containing the point with coordinates x at time

t.

We

have"

p(x,

t+

~) =

p(x+x',

t)

exp[-

—,

'

g,

.z(x)xIx~]dA'.

(4.

1)

For

the ideal-gas system considered at the

begin-ning of

Sec.

III,

in the Cartesian coordinates

de-fined by

Eqs.

(3.

4),

we obtain, by expanding P on

the left-hand side of

Eq.

(4.

1)

to

first

order in

r

and on the right-hand side to second order in

x',

the diffusion equation

BP 1 BP BP

Bt 27' BX BX

(4.

2)

It

is

easily verified that the problem of diffusion

out from apoint with

x,

=xyo x2 x20 and t

=0

has the solution

p(x„x„

t)= (2vt/7 )-'

_{exp[-6/2(t/r)],}

_(4.

₃₎

where s =_{(x, —}

_x„)'+

_{(x, —}

_x„)2

_{is the square of}

the geodesic distance from the starting point. This result has the same form

as

the probability

dis-tribution for Brownian movement in two dimen-sions from a point source with Cartesian coordin-ates (d„&,d20),

p(d„d„

t)=(4vDt)

'

exp(—d'/4Dt), where d'

=(d,

—d»)2+(d, —d»)2

is

the square of the straight-line distance from the source.and D

is

the diffusivity constant.

"

We have thus ob-tained the very important result that

for

the

ideal-gas system considered, the relevant quantity, in addition to time,

for

the diffusion of states

process

is

the geodesic distance on the corresponding

Rie-mannian manifold C~; this distance

is

quite

anal-ogous to distance

as

it

is

used in Brownian move-ment. These considerations establish that the Riemannian metric defined by

Eq. (2.

3)gives a meaningful definition of distance

for

open

ideal-gas systems globally as well

as

locally. More-over, this metric

is

unique up to an arbitrary

dimensionless constant ofproportionality. This

constant ofproportionality, which we take to be unity,

is

ofno

real

concern since it can be ab-sorbed in ~ in

Eq.

(3.

6)when we consider the in-teraction hypothesis.

Curved Riemannian manifolds

are

locally flat

so

the diffusion ofstates

process

for non-ideal-gas open systems has,

for

sufficiently small time, the solution given by

Eq. (4.

3).

Again, it is

dis-tance as defined by the Riemannian geometric

mod-el which is relevant.

Therefore,

the metric

is

unique also

for

non-ideal

gases.

The

process

described above, which

is

a well-defined feature ofthe thermodynamic formalism, resembles perhaps most closely the foundation

of the renormalization group theory.

Renormal-ization group

is

based on the same type of idea

behind the diffusion ofstates

process

—

fluctuations

within fluctuations within fluctuations

.

.

.

.

"

So

far as

the author

is

aware, an approach to

renor-malization group with path lengths has not been

GEORGE

RUPPEIX

E R

Itmust be realized that to achieve by the

pre-sent methods of

statistical

mechanics a proper

understanding of the interaction hypothesis

is

not an easy

job.

Statistical mechanics

treats

interac-tions with models, such as the Ising

or

lattice gas models, in which interaction strength

is

measured

by the magnitude of couple. ng constants, which are not intrinsically functions of

state.

In contrast, in-teractions appear inthe Riemannian geometric model

through curvature which isindependent ofany

statis-tical

mechanical model, requires

for

input only

ther-modynamic quantities,

is

based onfluctuations, and gives a measure of effective interaction strength

for

each state of athermodynamic system. To reconcile these two differing points ofview

is

difficult but

necessary,

so

far

as the interaction

hypothesis

is

concerned, since any thermodynamic

result must ultimately be derivable from

micro-scopic theory.

"interaction

tensor"

was constructed from the

curvature; it was found to agree with the fluctua-tion correlation length in rank, units, the

ideal-gas limit, and the

critical

exponent. In addition,

consistency with the hypothesis of

two-scale-factor

universality was found. The reasoning used was of

somewhat the same type as that which has proved

successful in general relativity. However, such reasoning is no substitute

for

an understanding

based on

statistical

mechanics. It is hoped that

the results presented in this paper may lead to new ideas in

statistical

mechanics, ideas which

could constitute

a

major advance in the theory of

collective phenomena.

Note added in

proof

In .a recent paper,

"

H. Dekker has applied Riemannian geometry to the general problem of Qnsager-Machlup

processes.

The choice ofmetric in the present paper

is

formally equivalent toDekker's.

V. SUMMARY ACKNOWLEDGMENTS .

Ithas been demonstrated in tQis paper that if

the theory of fluctuations

is

included in the axioms

ofthermodynamics, thermodynamic systems can be represented by Riemannian manifolds.

It

was

found that the curvature of

a

manifold representing

an opensystem composed of the ideal gas is

zero.

'This result led us to

associate

curvature with

ef-fective interparticle interaction strength. An

The author

is

indebted to

Professor

R.

B.

Griffiths for helpful correspondence concerning an

earlier

draft of this manuscript. Useful con-versations with

Professor

B.

Buck,

Professor

J.

H. Meyer,

Professor

R.

G.

Palmer, and with

Mr.

Baird 3traughan

are

also gratefully acknow-ledged. This work was supported in part by a

grant from the NSF.

~Laszlo Tisza, Generalized Thermodynamics (MIT, Cambridge, Mass., 1966).

R.

B.

Griffiths and

J.

C.Wheeler, Phys. Rev.A2, 1047 (1970)

.

3F.Weinhold,

J.

Chem. Phys. 63, 2479 (1975); 63, 2484

{1975); 63, 2488 {1975); 63, 2496 (1975);65, 559

(1976);Phys. Today 29, {3),23 (1976).

Afixed scale is necessary, otherwise the system is undefined; seeRef. 1, p. 250. The scale isusuallytaken tobe either the volume or the particle number. H.

B.

Callen, Thermodynamics (Wiley, New York,

1960), Postulate

I,

p. 12.

"Contact" means that all of the parameters ofA_~ are allowed to fluctuate but the fixed scale X. Note that

the coordinates we choose below for 8~ need not be the extensive parameters ofAz but can be any pair of

in-dependent functions ofstate.

L.

D.Landau and

E.

M. Lifshitz, Statistical Physics

(Permagon, New York, 1977), Chap. XII.

Reference 7, p.347, Eq. (113.8)

The criterion forjudging whether states are "nearby" is that the contribution of terms higher than second order in the expansion forS&(x,

x')

be negligibl.

The components ofthe matrix of second partial

deri-vatives ofS&(x,x') transform under achange of coor-dinates

x;=x; (x')

according to

a2st Bxu Bx,' a St Bst 82 h

~x ~xy ~xg ~x ~x'

Since BS&/Bx';~

„„.

=0, the second term on the right-hand side vanishes, and the components of g(x)

trans-form as those of a second-rank tensor. The condition

BS&/Bx ~

„„,

=0is the source ofthe equality between

the intensive parameters ofthe reservoir and A~((P).

G.N.Lewis,

J.

Am. Chem. Soc. 53, 2578 (1931). 12Reference _5, Cha-p.

15.

F.

Weinhold,

J.

Chem. Phys. 63, 2479 {1975). Landau and Lifshitz (Ref. 7)demonstrate the procedure

for making such a calculation in Sec.114,p.348. ~The existence of a transformation to bring a line

ele-ment to the form in Eq. (3.5)isboth necessary and sufficient to prove zero curvature; see

I.

S.Sokolnikoff, TensorAnalysis (Wiley, New York, 1966), p. 96. Zero curvature for the ideal-gas manifold can also be

proved directly from Eq. (3.2) by use,ofthe

expres-sion for the Gaussian curvature in Eq. (3.8).

See,

e.

g., C.W.Misner, K. S.Thorne, and

J.

A. Wheeler, Grat itation (Freeman, San Francisco, 1973),pp. 404—407.

Reference 15, p.167.

The Gaussian curvature E(x)isrelated to the

THERMODYNAMICS:

A RIEMANNIAN

GEOMETRIC

MODEL 1613

by K(x)= —~R(x);see Ref.15,p.168. For an

ac-count ofthe extraordinary properties ofthe Gaussian curvature one cando no better than K.F,Gauss

Genera/ Investigations ofCurved Su/aces (Raven, Hewlett, N._Y., 1965).

~Generally, we must first try a'series form

where Co(x)=VK~(x) contains all the information

about curvature, $0, P~, P2, areuniversal fluid

constants, $0 being chosen tomake p~dimensionless,

and

0

is some real exponent. Since Co(x) has the

dimensions ofvolume, P2has the dimensions ofim

verse volume, p3 has the dimensions of inverse volume

squared, '

'.

However, there is no universal constant

with the dimensions ofvolume either in

thermodyna-mics or statistical mechanics; therefore, P2, P3, must be zero. To use nonuniversal constants, such as

k&T, /P~, where

T,

and

_P,

are critical temperature

and pressure, respectively, is inconsistent with the

interaction hypothesis since itis then no longer the

case that curvature alone isthe measure ofeffective.

interaction strength.

0

must certainly be positive since

9

negative would imply infinite effective interaction strength for the ideal gas. Forsimplicity, we take

0

=1.

The choice of aparticular positive ~does not

ef-fectour interpretation below ofI(x)as the coherence

length to some power. @Op&is taken as dimensionless

since there is no dimensioned universal constant

which seems appropriate. We can take Jog&=1 since dimensionless constants of proportionality can be

ab-sorbed in K.

20Reference 15, p.168.

See David L.Goodstein, States of Matter

(Prentice-Hall, Englewood Cliffs, N.

J.

, 1975),pp. 478—483.

A good reference on hyperscaling in fluids isthe

ar-ticle by

J.

V. Sengers and

J.

M. H. Sengers, in Progress in Liquid Physics, edited by C.A. Croxton

(Wiley, New York, 1978),pp. 164-165. 23B. Widom,

J.

Chem. Phys. 43, 3892 (1965).

B.

Wldom Physlca 73, 107 (1974).

25_For

the latest experimental values ofthese exponents see Ref.22, p.143.

26Theinformation formaking thiscalculation was obtained

from

J.

V.Sengers, W.

L.

Greer, and

J.

M.H.Sengers,

J.

Phys. Chem. Ref.Data 5, 1 (1976). He4, Xe, He3,

steam, and 02 yielded ~Z~/(0.21ks/Ao)~ 0.004, while

CO2 had the anomalously large ratio 0.22;the origin of this discrepancy is not clear. Inobtaining the ex-ponent

e

—2, the scaling relation y+2P+0.=2was used.

2~D. Stauffer, M.Ferer, and M. Wortis, Phys. Bev.

Lett. 29, 345 (1972). See also Ref. 22, p.166.

J.

V.Sengers and M.R.Moldover, Phys. Lett. 66A, 44 (1978).

2~This number was calculated with the experimental data from

He,

Xe, CO2, and He . The specific-heat

data were taken from Sengers, et

al.

(Ref. 26); the correlation length data were taken from Ref. 22, p.

166.

3_{Brownian movement is governed by the same}

type of integral equation; seeA. Einstein, Investigation on

the Theo~ ofthe Brownian Movement (Dover, New York, 1956),p.14.

Reference 30, p.

16.

2See,

e.

g., the article by Leo

P.

Kadanoff, "Scaling,

Universality and Operator Algebras,

"

inPhase

Tran-sitions and Critica/ Phenomena, edited by

C:

Domb and

M.S.Green (Academic, New York, 1976),Vol. 5a, pp. 10—12.