Statistical Mechanics and the Foundations of Thermodynamics (Lecture Notes in Physics)

Lecture Notes in Physics Edited by J. Ehlers, Munchen, K. Hepp, Zurich R. Kippenhahn, Munchen, H. A Weidenmuller, Heidel...

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Lecture Notes in Physics Edited by J. Ehlers, Munchen, K. Hepp, Zurich R. Kippenhahn, Munchen, H. A Weidenmuller, Heidelberg and J. Zitlartz, Koln Managing Editor: W. Beiglbock, Heidelberg

101 A. Martin-Lof

Statistical Mechanics and the Foundations of Thermodynamics

Springer-Verlag Berlin Heidelberg New York 1979

Author Anders Martin-Lot Institutet for Forsakringsmatematik och Matl'lmatisk Statistik Stockholms Universitet Hagagatan 23, Box 6701 11385 Stockholm Schweden

ISBN 3-540-09255-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-09255-2 Springer-Verlag New York Heidelberg Berlin Ubrary of Congress Cataloging in Publication Data Martin-Lof, Anders, 1940- Statistical mechanics and the foundations of thermodynamics. (Lecture notes in physics; 101) Bibliography: p. Includes index. 1. Statistical thermodynamics. I. Title. II. Series. 0C311.5.M28 536'.7 79-15289

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in da~ banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.

© by Springer-Verlag Berlin Heidelberg 1979 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.

Preface These lectures present an introduction to classical statistical mechanics and its relation to thermodynamics. They are intended to bridge the gap between the treatment of the subject in the physics text books and in the modern presentations of mathematically rigorous results. (So it tries to supply many of the facts that are to be found between the lines both Landau-Lifshitz

Statistical Physics and Ruelles

in

Statistical Mechanics).

We have put some emphasis on getting a detailed and logical presentation of the foundations of thermodynamics based on the maximum entropy principles which govern the values taken by macroscopic. variables according to the laws of large numbers. These can be given a satisfactory formulation using the limits of the basic thermodynamic functions established in the modern work on rigorous results. The treatment is reasonably self contained both concerning the physics and mathematics needed. No knowledge of quantum mechanics is presupposed. Since we present mathematical proofs of many technical facts about the thermodynamic functions perhaps the treatment is most digestive for the mathematically inclined reader who wants to understand the physics of the subject, but it is hoped that the treatment of the basis of thermodynamics is also clarifying to physically inclined readers.

Contents Preface Introduction 1.

Statistical description of systems in classical mechanics

2.

Study of equilibrium distributions

3.

3 14 14

2.1.

The microcanonical distribution

2.2.

The canonical distribution

21

2.3.

The grand canonical distribution

28

The law of large numbers for macroscopic variables and the foundations of thermodynamics

39

3.1.

General study of the probability laws of macrovariables

39

3.2.

Derivation of the basic laws of thermodynamics

58

3.2.1.

The rules for thermodynamic equilibrium

58

3.2.2.

The rules for energy changes; work, heat and

3.2.3.

The thermodynamical description of first order

67

their relation to entropy phase transitions 3.3.

3.4.

Some other uses of the concept of entropy 3.3.1.

Information theory

3.3.2.

Statistical models

Proof of the fundamental asymptotic properties of the structure measure in the thermodynamic limit 3.4."

Properties of the entropy

3.4.2.

The existence of

3.4.3.

A system in a slowly varying external field,

s(e,n)

s(e,n)

87 94

when the interaction

has infinite range the barometric formula 3.5.

76 80 80 84

105 112

The central limit theorem for macroscopic variables, thermodynamic fluctuation theory

References

115 120

Introduction Statistical Mechanics is the branch of theoretical physics which investigates and tries to relate macroscopic properties of systems of many interacting microscopic subsystems (atoms, molecules etc.) to what is known about the laws of the interactions of these atomic parts. The macroscopic description uses a few (<< 1023 ) variables which in general do not indicate the states of individual atoms, and one looks for relations between these variables and tries to describe their evolution with a few (differential) equations only containing these variables (macroscopic causality). These relations are found by considering the state variables of the individual subsystems as random variables whose distributions contain a few parameters. The average values of the macroscopic variables then only depend on these few parameters, and the fact that these mean values well indicate the actual values of the variables depend on the fact that "laws of large numbers" hold for them with overwhelming precision. An example of such a macroscopic theory is thermodynamics which e.g. deals with the equations of state for gases, liquids, chamicai systems in equilibrium, where typical macroscopic state variables are: pressure, densities, temperature, heat content etc. An important problem in this area is to explain phase transitions between gas-liquid liquid-solid, etc. and to understand if such phenomena can he explained only from rather general assumptions about the atomic interactions. Areas where the time evolution of macroscopic systems is studied are hydrodynamics, diffusion, heat conduction, kinetic theory of gases. Here one needs many more state variables (density- and velocity distributions in space e.g.) but one still has a drastic reduction from the microscopic description, and one considers the evolution on a time scale which is very long compared to that of the atomic motions. A fundamental problem is to understand how the equations of motion for the macroscopic variables, which can often be written down intuitively, can be derived from the molecular

dynamics and why macroscopic

causality holds. These dynamic problems are

much more difficult than the equilibrium problems.

In the following we

will consider mostly the latter type of problems. The theory not only gives relations between the average values of the macroscopic variables but also describes their fluctuations

(which are small).

For these in general central limit theorems are valid, because one has sums of weakly dependent variables, Examples:

and one obtains Gaussian distributions.

the theory of Brownian motion and density fluctuations

in a gas

(which gives rise to the blue color of the sky). In these lectures we shall first introduce the probability "ensembles",

appropriate

for describing

distributions,

systems in equilibrium and consider

some of their basic physical applications.

We also discuss the

approach to equilibrium and the problem of how irreversibility dynamics

in the context of the so called Ehrenfest urn modes

problem of comes into the

, which is one

of the simplest systems where these questions can be studied quite explicitlyWe then give a detailed description variables

of how the law of large numbers for macro-

in equilibrium is derived from the fact that entropy is an extenN limit" N,E,V § - with ~ = n = particle

sire quantity in the "thermodynamic E

density, V = e = energy density. fundamental

The discussion

is based very much on the

convexity properties of the entropy and other thermodynamic

functions and shows how the values of the macrovariables appropriate variational principles, dynamical equilibrium.

are determined by

which give rise to the rules for thermo-

In this context we show that one can naturally

how to split the energy changes in a thermodynamical

heat and show how the first and second laws of thermodynamics from the rules for thermodynamical tion of thermodynamics

equilibrium.

see

process into work and are derived

We have elaborated the deriva-

in detail because we feel that it is often insufficient-

ly treated in the physical text books and taken for granted in the more modern treatments

of "rigorous results".

We feel it is satisfactory that the estab-

lishment of the limit of the thermodynamic development

of the mathematical

functions achieved in the modern

aspects of statistical mechanics allows a

more general and logically clear presentation The proofs presented of the thermodynamic sions of those in the litterature.

of the bases of thermodynamics.

limit are somewhat elaborated ver-

Finally we present the basic facts about

I.

Statistical description of systems in classical mechanics

The prototype of such a systems is a gas (liquid) of

N

identical atoms

described as mass points moving according to Newton's equations under the influence of pairwise interactions between the particles and collissions between the particles and the walls of the container: The positions are

(ql,q2,...qN)

qi 6 A C R 3

A = the container. Newtons

equations are: mqi'" = - j#iZ grad V(qi - qj)

for

qi6int

+ the law of reflextion

for

qi ~ BA .

V(q)

A

is the potential of the interaction between a pair whose positions

differ by

q E R 3, V(-q) = V(q).

Typically

V

is repulsive at short

distances and attractive at large ones:

These equations can be written in Hamilton's canonical form like for any mechanical system, and it will be very useful as we will see: Introduce the momenta of the particles a function of

Pi = mqi

and the total energy as

qi' Pi :

mliil2 H(p,q) = Z---~---+ i

Ipil2 Z V(qi-q j) = Z 2--~---+ ~ V(qi-q j) i<j i i<j

Then we have: %H

Pi

BH = ~ grad V(q i ~qi i<j =

Z

-

qj) -

Z grad V(qj - qi ) j
grad V(qi - qj) = - Pi'

j#i which is the cs.nonical form

qi = SH(p,q) SP i

Pi = -

BH(p,q) ~qi

The mechanical theory of such a system consists in solving the system of differential equations with given initial values of course in general impossible for

N ~I023,

(p,q).

This is of

and even if it were possible

one could not hope to measure all initial values needed. Moreover one is not interested in such a detailed description in order to determine the average values of a few quantities over times long compared to the atomic ones. Therefore it is natural to try to treat all variables of whose initial values we have no precise information as random, i.e. to introduce a probability measure in the "phase space" F = {x = (pl,q I .... ,pN,qN )} ~ R 6N (Boltzmann, Gibbs, Maxwell) This is often expressed by saying that we have an "ensemble" of infinitely many identical independent copies of the system whose initial values are distributed according to the given measure having a density hope is then that one can find a

p(x)

p(x)dx.

The

such that averages with respect

to this measure of interesting functions will give their observed values at least if their variances are small.

If

p(x)

is to describe a system

in macroscopic equilibrium it ought to be chosen so that stationary stochastic processr transformations p

x ~ ~ x t = Tt(x o)

defines a stationary measure:

integrable

f(x),

xt

becomes a

The equations of motion define a family of of

F § F,

and stationarity means that

E(f(xt)) = E(f(Xo))

for all

t

and all

which means that

IdT_t(Y)i / f(x)p(x)dx = / f(Tt(x))p(x)dx = f f(y)p(T_t(y)) ~

IdTt~Y )

is the Jacobian of the transformation

dy.

y § Tt(Y)-

The remarkable fact is now that for canonical systems it is easy to find stationary measures: Theorem I (Liouville's theorem) The Lebesque measure

dx = dPldql ... dqN

in

F

is stationary, and more

generally,

if

pendent of t,

p(x)

is a constant of the motion, i.e.

then

p(x)dx

if

p(x t)

is inde-

defines a stationary measure.

Proof: From the above formula we see that it is enough to show that

Jt(x) =

~

= I

dJt(x) - = 0 dt

that

for all

for all

dTt+h(X)

t.

Clearly

Jo(X) = I I I

= I,

so we show

t.

dTh(Tt(x))

Jt+h(X) = ~

= t

dx

I = Jh(Xt)'Jt(x),

because the Jacobian

of a composed transformation is the product of the Jacobian of the factors. For

h

small we have

[

where

x = I

aHrp,q~!

[ Hence

x h = Jh(X) ~

if

x + hx

=

from the equations of motion

B

3p

Jh(X) ~

II + hKl,

~q~p

where

K

is given by

~--~q

K= ~2 H

$2 H

SPSP

SpSq~

We then have - -~2 H

Jh(X) = I + hspK + o(h 2) = I + h i

+

~2H

+ o(h 2)

~Pi~qi

~qi~Pi

= I + o(h2),

so we see that Jt+h - Jt

dJt(x) § o

h

as

In particular, since

h + o,

H(x)

and

-

-

=

dt

0.

is a constant of the motion, we see that

defines a stationary measure if it depends only on

O(x)

=

o(H(x)) .

H(x):

p

For an isolated system like the one we have considered of the motion, so if its value is known, moves on the "energy shell" choice of

p

~

@&~

(The normal to

rE

H(x)

is a constant

we know that the system

defined by this equation. The "simplest"

in this case is to choose the measure

Lebesque measure in

U

induced on

FE

by

P:

Put

du = const, ds____~s dE ds = element of area on

at

simal shell is given by ds(x)

d~(x) = a'(E,N)l~ad

Qi (E,N)

FE

H(x) = E,

x

is

grad H(x),

dE = Igrad HI'ds

H(x)IN:

'

FE

and the thickness of an infiniteWe have:

with

= ~FE lgraddSHiN: = dd_E H(p,q)<__E~ dpdqN! -;

dnA(E,N) dE

q6A N This is called the "microcanonical" measure (Gibbs).

(The factor

N'

will

turn out to be natural later. ) If there are other known constants of the motion it could be natural to let the microcanonical measure be defined on the lower dimensional manifold defined by fixing their values, but in general

H(x)

is the only constant

known (besides the total momentum and angular momentum, but they will automatically have the value zero in a large system, and any other fixed value of these quantite s can be reduced to zero by a suitable choice of the coordinate system). The choice of

u

as a stationary measure was suggested by Boltzmann, who

saw it as a device for calculating time averages of interesting observables. Such averages he considered to be the natural quantities to be compared to observed values.

In order to know that

e.g. for all integrable

f

=

lim ~ f(xt)dt T*~ o

a.s.

one needs to know that the system is ergodic

(Birkhoff~s theorem). It is however only for a few very simplified systems

that it has been possible tc establish ergodicity (Billiards, Sinai so this motivation for the choice of

~

]966)

is not yet available in general.

It is not necessary that the equality is valid for all such

f.

The physi-

cally interesting functions are often of a special kind, namely sums of small contributions from the individual particles (or pairs etc.) which are very many. For such

f

it is often true that

<(f _)2>

is

very small and then as pointed Out by Khinchine in ref. 7 it is also true that I

T f f(xt)dt~

when

_ r

N

is large.

2 + 0

the time average converges to

Also,

if

sufficiently rapidly it is also true that ,

so one should probably make more use

of the structure of the interesting observables in establishing the equality of the two averages. Even if it would be possible to show that a system is not ergodic, i.e. that

FE

can be decomposed into two invariant parts

with positive measure, it is quite possible that all interesting observables are insensitive to this decomposition, and therefore ergodicity is not the single criterion for whether the choice of measure is the appropriate one. The fundamental problem of explaining why macroscopic systems not in equilibrium often quickly move towards equilibrium (described by equ. statistical mechanics) is not solved by showing that the system is ergodic with as stationary measure. Here it is a question about the dynamics of macroscopic variables on a time scale long compared to that of the atoms starting from an initial state, which has extremely small

~-probability.

This problem occupied Bcltzmann very much, and he invented his famous equation to describe the evolution of the state of a gas of the type we have described at least for low densities. In Boltzmanns description one does not give an equation for the whole microscopic state

X t ~ F,

but

the state variable is a much more "coarse grained" quantity, namely the density

f(p,q)dpdq,

(p,q)E R 6 = y 9

f(p,q)

in the state space of a single particle be given on a microscopically long scale, points in

F

whose coordinates

is the particle density

(y-space),

and it is considered to

N.f(p,q)dpdq

(p,q) ~dpdq.

Here

dpdq

pically infinitesimal", but "microscopically infinite", N-f-dp.dq >> I,

and the fluctuations in

f

is the number of are "macroscoso

are supposed to be small.

B's equation is a deterministic differential equation for the evolution of

f,

and for it he could show his

starts from a ft § f~

as

fo(p,q) t ~ ~

H-theorem, which says that if one

different from the equilibrium density f

can be computed from

~

f

described above.

then

This theorem and the question if equilibrium can be established gave rise to controversies

and contradictions.

Zermelo's

built on the observation that the microscopic reversible,

"Wiederkehrseinwand"

if time runs backwards one again gets a possible evolution.

B's equation however is not, if one starts from back to

was

equations of motion are

fo

after a long time, but

f|

f|

one does not come

stays constant,

so it appears

paradoxical that the system has suddenly lost its reversibility.

Poincar~

also showed his recurrence theorem, which says that for every subset with

w(A) > o

xt~A

it is true for almost all

for some

t > o .

xEA

AC F

that if

x = x then o ft' ft § f~ and does

But this is not true for

not come back near its initial value. Since the B-equation was not derive completely of Boltzmann's

from the dynamics in

F

but written down with the aid

intuition it was not clear how the paradoxes could be

explained.

The connection between the microscopic problem how irreversibility

and macroscopic

dynamics and the

can sneak into the equations can be illustrated

in a much simpler system where one can see what happens. This system is the so called urnmodel which was introduced by Ehrenfest Consider a "gas" of two urns

Uo,U I .

x = (x I .... ,XN)

N

particles labeled

The microscopic with

xi = x

state

1,2,...,N, x ET

for this purpose:

distributed between

is given by a vector

iff particle no.

i6Ux,

x = 0,1.

evolution takes place because once every time unit a particle, chosen at random and moved to the other urn, hence forms a Markov chain with the uniform distribution is reversible under

2N

etc.

n,

is

The system

states, and it is easy to show that

~(x) = 2 -N

~: ~(x)P(x,y)

x n § I - Xn,

The time

no.

is stationary,

= ~(y)P(y,x).

and that the process

It is also recurrent,

from every state one sooner or later comes to every other state. This system corresponds

to the mechanical

system with the microscopic

description

and the microcanonical

description

can now e.g. consist in giving only the value of

state

stationary measure. A macroscopic

state

X t = Z xi(t) = no. of particles in UI, a drastic reduction in precision. i The induced distribution for X is easy to see.

Pn = P ( x = n )

=2

-N (~)

For such a macroscopic

.

quantity the law of large numbers is hence valid

with great precision: P(

-

> r

!c~

§ 0

as

N § |

(Here

h(p)

is the usual entropy function h(p) = - p log p - (l-p) log(l-p).) X I are also normal: ~ (~ - ~) has an approximatively normal

The fluctuations distribution

N(0, 89

The macroscopic

as

N ~ |

dynamics can also be studied exactly.

Xt

is also a Markov

chain with transition probabilities

IPn,n+ 1 ;

_ nN

1

n = 0,1,...,N

n Pn,n-I =

The forwards equation for

Pn(t) = P(X t = n) _

Pn(t+1) = Pn_1(t)

(3

n~___~1)+ Pn+1(t )

and it is easy to see that the Under these probabilities

Xt

Pn

is hence

Cn+1 ) "-'-N-'-" ,

above are the stationary probabilities.

is also reversible and recurrent, because

any state can be reached from any other state, and as is easily checked. on a long time scale

N

vary completely deterministically time evolution of step

I

A~ = ~

+

Pn Pn,m = Pm Pm,n'

If we now consider X t on a macroscopic XN" T t = N.T: = f(T) we shall see that

when

= f1(T)

N ~ ~.

and

scale, and f(T)

will

To see this consider the

~--~f2(T)> = f2(T),

For one time

we have: 1

)IfCT) = f )

I I (I= (I - f)Cf + ~) + fCf - ~) = f + N

2f)

1 2 1 2 = (I - f)(f + ~) + f(f - ~) =

= f2 + 2f - hf 2

+ o(!-)

N

So, taking averages of

N2

f(T)

we see that

f1(T),

f2(T)

satisfy the

equations : fl(T + ~) = f1(T) + g(1 - 2f1(T))

f2(T + ~)

= f2(T) +

~(fl(T)

2f2(T)) + o(J-) N2

and it is easy to show that as of the differential equations:

N ~ |

fl and f2

'

converge to the solutions

10

i

df1(r) dr = I - 2f1(r)

[

df2(x) dr

= 2(fl(r)

The variance

dv(r) dT

- 2f2(~)) is hence determined by

v(r) = f2(r) - f~(r)

= 2f 1 - h f 2 - 2f1(1 - 2 f 1) = - h v ( r )

so the solution is:

f f1(~) = ~I + c'e -2T

v(o) e-~r.

v(r)

We hence see that if we start e.g. from an initial state where Pn

=

fn(1-f)N-n

(~)

then

f(o)

is very close to

so it will remain very small, and any time

r

if

f1(o) = f.

f(x)

and

v(o) = 0(~),

will be very close to

f1(r)

Hence we see that with high precision

o,rolves completely deterministically df d-V = I - 2f

f

according to the

at

f(r)

"B-equation"

whose solution rapidly approaches the equilibrium value

I f = ~ 9

In order to get a feeling for how rapidly the irreversible behaviour appears as

N + |

it is instructive to compare the average recurrence time for I I states near equilibrium f = ~ and for states further away from f = ~ . -I It is well known that tf = average recurrence time = PNf = 2N N -I I = (Nf) Hence near f = ~ the local central limit theorem for I

x = /N (f - ~)

with variance

tl/2+x/~[~

~-

I

K

tells us that

e2X 2

~5 t

and at the scale

r = -- 9 N " I

2x 2

r112+xl Jg However when

f - I/2

Tf formula that

N(h( 89 - h(f)) ~ e

ri12

is fixed and

N § ~

it is easy to see using Stirlings

11

States inside the regime of "normal" fluctuations hence have a very short recurrence time, whereas astronomical

states at finite distance from

recurrence time. Take for example then

Tf ~ 9 1 0 - 1 2

to the age of the earth the "cosmic scale"

~5"10

101 ~

N = 1023 ,

017

e

years

have a super-

f = I/2 + 0.001,

1017 TI/2 = 10 -12 sec.,

I/2

,

1016 sec.

(Borel's ech~lle cosmique)

which can be compared So we see that even at

the recurrence times are

"infinite". We will see that it is a general fact that the probability of a macroscopic fluctuation ~

e-N~entropy" difference)

After a long time Boltzmanns

description

in a large system.

tells us that

f = I/2,

but if

one studies the system with a higher resolution one finds that small fluctuations around this value occur all the time. Their evolution in the normal regime can be studied by a suitable scaling. Then for the Markov process I + ~) - X(T)IX(T)

<X(T

12 + ~')

we have:

/N

= x>

= N <(f(T

1 2 - ~.) + f 2

+ f(f

I + ~1 - f ( ~ ) ) 2 1 f ( T )

- 2f(f

+ 1(1

- 2f))]

These infinitesimal moments are those which characterize dx(~) = - 2x(r)dT + dw(T) that

x(r)

,

f(T) = I/2 + X(T)

/N 2x : --~ (I -- 2f) = -- --N

= X>

1 - x(t))21x(T) < ( x ( ~ + ~) = N[~I - f ) ( f

x(T)

Put

= f>

=

1 = ~" 9

a diffusion

and in this case it is not difficult to show

converges to an Ornstein-Uhlenbeck

diffusion with forward

equat ion

=

~

2

~x 2 -2x 2

This is a Gaussian diffusion with stationary distribution

e

2~( 89 " It is also reversible The process differential

(like all one dimensional

stationary diffusions).

x(r)

can be described as the solution to the stochastic dx dw dw equation ~-~= - 2x + d 7 ' where ~ is white noise. This

equation is the B-equation for

f - I/2

with white noise added as a

driving term. We hence see that we have the following description of the dynamics when is large:

12

If we start with

X(o) = f(o)

fixed # I/2

we see with high accuracy

N

a deterministic

motion towards equilibrium

to the normal regime Their dynamics

X(t) N

-

I = x(~) -2

X(t) N = ~I .

When we have come

we can see fluctuations

x(T) = 0(I).

is well described by a diffusion process which is reversible

and is defined by the B-equation near equilibrium with white noise added in the right hand side. All the time we consider the system on the time scale t = N.T . The relation between the stochastic

and deterministic

evolution

is illustrated on the following diagram, which shows three outcomes of simulations of the Ehrenfest process with sponding solutions of the B-equation. is

N = 1000

particles and the corre-

The standard deviation in equilibrium

a'~16.

This picture is believed to be found in general even for complicated realistic systems even if one can not prove it completely, reversible

and irreversible

References:

and we have seen how

behaviour are related.

The Ehrenfest model and its approach to equilibrium is discussed

by Kac in ref. 6. Limit theorems for Markov jump processes converging to the solution of a differential equation are proved by Kurtz in: Kurtz, T.G. Solutions of ordinary differential jump Markov processes.

equations as limits of pure

J. Appl. Prob. 7, h9-58 (1970)

and

Limit theorems for sequences of jump Markov processes approximating differential

processes.

J. Appl. Prob. 8, 3~h-356 (1971).

ordinary

0 o

!

o

t

o g,

o

o

0

N ,t-I

1

0

o

0 qob

0 o

2.

Study of equilibrium distributions

2.~.

The mierocanonical

distribution

In general it is difficult to compute averages in the m.can, distribution. One exception is an ideal gas. Such a gas is characterised by the fact that one neglects completely the contribution calculation of all integrals.

from

V(q)

to

H(p,q)

in the

We hence have:

Ipi12 H(p,q) = Z I

2m

,

and if

[A] = V - volume: 3N

zI Ipila 5 ~ Rn T[ n12 = volume of unit sphere in = (n12): " When we w~nt to compute n we can do it as follows: the average of some function f(x)

with

C

~fE

d

xf) f(x) d__~

FE

if

f(x)

is defined also outside

FE.

similar particles as indistinguishable, f(x)

We shall always in the following regard i.e. we shall only consider functions

which are symmetric when such particles

tion of introducing an

N~

are permuted.

Then the conven-

will turn out to be natural.

Let us study the description of a large system in the limit E,N,V,A ~ E N ~ -~ e, V § n with e,n finite. This is called the "thermodynamic

such that limit". volume

The distribution A%

A = A Is

of the state of the particles in any finite sub-

can then be computed by the following argument: 2,

and

A 2 -~ ~

too.

I If there are + H2(x(2)),

NI

particles

where

is any symmetric

x (I)

{1,2 .... ,N} 9-

and

AI

the total energy is split into

x (2)

are the states in

function of the state of the particles

can be written as follows: set of

in

For any

for which

x = (x I ,x2 .... ,xN)

x i E A I. .

(2)

x

AI

and

in

AI

let

A 2. If f1(x(1),N1 ) then its average

S(x)

be the sub-

can then be split into (2)),

where

the

H~ (x (I)) +

x (1)(x) =

x (i)

15 placed in the same order as in x. (E.g. if x = (x I .....x 5) S(x) = {2,3,5} then x (I) = (x2,x3,x5), x (2) = (Xl,Xh).) Then:

fl(x(ll(x)'

H(x)<_E

dx NI(X)) ~/' = SE H( ~)<Ef1(x (1)(x), N1(x)) ~, = s(x)~s

= E

fl (x(1) , Isl)

f

dx(1)dx (2) Nz

s HI(X(1))+H2(x(2))<_E = E ~N~

I

fl (x(1) ' N1 )c'l-x ( I)

dx(2) = J" H2(x(2))<__E_HI(x(I))

NI\" I/

and = E f f1(x (I), NI)flA2(E-HI(X(1)), N-N I) ----;-~(I) N1

NI"

ax (~) E,N,A = ~(E.N)I. NIE f f1(x(1),N1 ) ~'A2 (E-HI(X(1)) , N-N I) NI .,

We hence see that the probability of having N I particles in A I with state x I has probability density ~'A2 (E-HI(Xl), N-N I) dx I ~(E,N) NI! " In this argument we only used the fact that

H(x) = Hi(x(1) ) + H^(x (2)) but not that H I and H2 correspond to an ideal gas. Putting f1(x~1),N1 ) = I iff Hi(x(1)) ~ El, N1(x) = N I we see that P(HI(X(1)) ~ E I, NI(X) = N I) = I ~(E,N)

~2(E_HI(X(I )),N_NI ) dx(1) NI 2 -

II HI(X

))<__E I

EI I

AI

(HI,NI)~2(E-HI,N-N2)dH I

so the simultaneous density of E I and N I is

18

~I(EI,N I) ~2(E-EI,N-N I) dE 1 9

~(E,~) From this result we see that

~(E,N)

is composed by convolution, because

the total integral is one:

~(E,N) : NIZ ~ ~'AI(EI,N I) ~2(E-EI,N-NI)CI-EI

9

Let us sum up these results: Lemma I.

Consider a system in

Then if

A = AIUA 2

particles in

AI

and

A

with a m.can, distribution defined by

AlflA2 = r

with A2

NI

then

and the state in

AI

has probability

density ~A2(E-HI(Xl)'N-NI )

dx I

~(E,N)

NI!

and hence

EI,N I

have density

n'AI(EI,N 1 ) n~2(E-EI,N-N I )

"

~(E,N)

dE1

and n~(~,~)

:

f ~'AI (EI'NI) ~2(E-EI,N-NI ) dE I 9 NI

Returning now to the case of an ideal gas we have in this case

3_~N a~(E,N)

~

3N

2

3N C~)

: 73~

(-~1 .'

so in the limit

N ~ |

3N

2

(~) E

F'~,v§

E

2 1

V.,'N ~--T '

N

EI,V1,N I = const. aA2(E2'N2)

dx I

n~(E,~)

~i ~

dx 1

~

NI:

3N I --~(~)"

(~) 3N2 (--5--) .'

3N2 1 --E2 2

3--~t E

2

E,N.

and if there is no interaction between

N2 V2

N:

~

~:

17

3___N 2

N1

N2 ( N I

--~1

1

E22

-3NI

(2m',')

2

3N 2

B_AN

3N 2 E

V1

3N I dx I

(nV1)NI

N1 V1

3NI

e -nV I

(2m~) - - 2-

2

3N

3__NN

3NI

Q~e3)

NI!

The last factors tend to: -3__~N 2 I -

3N I

~

3N

-~-]

and

e

_3HI

---.

e

3N 1 --

-3N I

so the probability density for NI

NI

-3NI

(nV I )

-nV I

and

2e

dx I NI VI

e

N I' NI

converges to

-3H I (x I )

2

e

xI

-3NI

-3

=

NI

-nV e

~7 ~

e

I

N I'

dp dq NI VI

We hence see that the probabilistic description of an ideal gas in equilibrium is the following: The positions in space form a Poisson process with density

n

and all the momenta are independent and have a normal (Max2mE - 7 , so the average kinetic energy of a

wellian) distribution with variance 2 particle is

< ~ > =

E 9

In this case we see that the following law holds:

The distribution of the

18 state of a small subsystem in a large system with finite densities of particles and energy has a probability

density proportional to

-~(x)+~ ~x e

where

N' ' 8

and

8u

are parameters.

vered by Boltzmann.

This is a very general law which was disco-

(The exponential

factor is often called the B-factor. ) It

is easy to verify also in the situation where we have an arbitrary system in with a fixed no. of particles

in contact with an ideal gas in

A2,

AI

a heat

bath. The two parts can exchange energy, but the interaction energy can be neglected.

The particles in

AI

and

A2

dx I

O'fi(E'NI'N2) = Ht(xll+~H2(x2)<_E

NI!

need not to be of the same type. Then dx 2

dx I

N2'

-

f -NI! nfi,2(E-H, ~ (x~),N 2)

=

= f ~'A1 (El'N1) ~A2(E-E1)d'E1 (The particles

should not be permuted between

We see as in lemma I

a~2(E-~l(x~),~ 2)

that the probability

AI

and

density for

A2.)

xI

is

dx I NI ! 9

Since 3N 2 ---I

a~2(E2,N2) = K(N2IE 2 2

,

to

3N___2_1

3N2 (E_HI(Xl))

we see that the density is proportional

2

= (const)

~

~I(xI)~ E 2

-

-3H I (x I ) --~ (const) e

2~

-S1~(x t )

This distribution

and it hence converges to

dxl

e

[ ~AI(E1,N1 )

,

3 B--2~

NI.' e -8EI dE I

is called the canonical and is more important in practice than

the m. can. one since very often one is interested in precisely this situation where the system is in contact with a "heath bath". We shall see that in general

19

HI(x I ) NI

has a distribution

very sharply concentrated

around its average when

is large, so one gets the same results concerning most averages as in the

m. can. "ensemble" where H1(x I ) distribution

of

xI

is fixed to this value, because the conditional

given that

H1(x I) = E I

B-factor is constant on each energy shell.) if two systems do not interact:

is the m.can, distribution. The exponential

H(Xl,X 2) = H1(x I) + H2(x 2)

(The

form means that then they are

also independent:

-SH(Xl,X 2) e

-BBI(X 1) -SH2(x 2) = e

e

,

and this is a considerable

simplifica-

tion. (Compare this to coin tossing: xi =

Put

x = (x I ..... x N)

H(x) = ~ x. . I l

~'(E,N) = (no. of

x

s.t.

Then

H(x)=

E)=(~I"

The distribution

of

n

x (I) = (x 1,...,xn),

with

HI(X (I)) = ~ xi I

n~(E-H1(x(1))) P(x(1)) = ~'(E) HI(N_E)n-HI

nl) -H

HI

E

--~

e

is given by

(N-n)! E: (N-E): N! (E-HI)! (N-n-E+HI)!

n-H I (I-~)

=

Bernoulli distribution

as

N +

Nn E N

--

-~

~

~

This distribution

A disression dimensional

is much simpler to handle than the m.can, one.)

concernin~ the definition of the uniform measure on an infinite sphere

The argument above has shown that the uniform measure on the sphere N SN: E x 2i = N I

in

RN

converges to the infinite product measure of independent in the sense that for each I n

2

(2"~) 2 e 2 ZI

n

xi

n

the density of

can

on

R

converges to

This measure should hence in some sense be considered as

the uniform measure on the infinite dimensional This measure

Gaussians

(x I ,... ,xn)

also

be

identified

with

sphere

S

Wienermeasure

"~ x.2 = ~". l I on

20

C(0,=)

via the= following map: Take an orthonormal basis

and put

xn = ~ r v and (x n}

= 6n, m

(Ito integral.)

{r }

Then we have

for

L2(0,~)

E(Xn,X m) = (r162

are independent Gaussian, i.e. have the "spherical" distri-

bution. This point of view is fruitful because it shows that several concepts defined for

SN

can be generalized to

S

also:

The measure is invariant with respect to orthogonal transformations Xn § m~ O n mxm . For L2(SN ) there is a natural orthogonal base system, the so called spherical polynomials. They are eigenfunctions of the Laplacian, and can be defined by a

Pm(X) = Dmlxl a-N =

mI

am2

mI ax I

m2 ax 2 N Iml = ~ mi)

(m = (m I .... ,mN) = multiindex, These have simple limits as

ixl2-N

. . .

1

N -~ ~:

N lim N~-I P (x) = (-1)[mlHm(X) N-~o m Hm(X) = hm1(X I) 9 hm2(X 2) ..with

hn(X) = Hermitepolynomials defined by X

2

h (x) = (-I) n e 2

2

X

mn(e 2)

n

Proof: Choose

M

so that

m. = 0

for

i > M.

N T X.2 = N

Since

1 1

I

Pm (x) = N}-I D m

+ ... +

+ ...

we then have

1-g = N

[r

". §

"---§ N

~

Xl2 + ... + ~__ I-~ N -

=

1 2 § j(x1+'"

.+X~)

N El Dm

1, 2+

X2 + ... + XM~I- { + I"N + 0 [N7

.+X~)

D m e-R~Xl ..

=

It is now true that also in the limit for

L2(S ) = L 2 (Wienerspace)

riables

H

(-1)ImlHm(X) N ~ | {Hm}

forms an orthogonal basis

in the following way: Define the random va-

by m

H m = hml (~ ~1(t)W(dt)) 9 h 2 ( I ~2(t)W(dt) ...

=

2~

It is then true that every random variable

XEL2(Wiener)

can be expanded

in this basis:

X = Z H 9 E(X'H m) . This basis was introduced by Wiener, m m and he called {H } "polynomial chaoses" of orders Iml . One can also define m the limit of the Laplace operator A N on S N , and it becomes 22 A=Z--_xi

~"

~"1

i ~x.2

1

for nice functions of

A

on

S .

r = r

1,...,xn).

This construction

of

As on S

SN

the

Hm

are eigenfunctions

is of interest in the constructive

theory of quantum fields. All these relations

are explalned in detail in the

interesting

Space" by H.P. McKean, Ann.

article "Geometry of Differential

Prob.

I (1973),

197-

2.2.

The canonical distribution

We shall now study a system with a can. probability

law:

-~H(x)

dx e dx f(x) ~7., = ZA(S,N ) NI ,

-~

z^(8,~) -- f e -s~(x)

N: " 8

and give. the physie~l meaning of

and the "equation of state" of the system.

For an ideal gas we hence have -8

g

ipil2

f(x) = e

for

qi s A,

Z

3! so

ZA(g'N) =

(2~12

-7~-' N B

For a gas with interaction we have

Ipi12 -S Z - I 2m

8 Z V(qi-q j) i<j

f(x) = e

e Z

so we see that still all {qi}.

The contribution to

(pi } Z

are independent from the

like for an ideal gas, so we have

3N 2 ZA(8,N) = ( ~ )

QA(8,N)

with

p

gaussian and independent of

integration

can easily be computed

22

-8. Z .V( qi-qj )

QA(S,N) = f e

~<J

Q

is often called the "configuration integral". For an arbitrary gas it is 3m still true that = ~-- , or 2m = <Ekin,i> = ~ " The average ~3

kinetic energy of a particle is

. This is still true if the masses are

different. It is also true in the more general case when

H

is of the form

f I Z Pi Pj aij(q) + V(q) , because then H(p,q) = ~ ij=1 Pi ; Pi @pjS-H-H e-SH dp = ; ---BS-~j (e-SH)dp =

=I~

Hence

Pi e-SH

I

+~f6j

< Pi

<Ekin>

B

.. e-SH

'

i S ; e-SH dp

dp=

B

sothat

< 8 9 ~jPi Pj aiJiq)~=< 89 i

28 "

This is the so called equipartition law: Lemma ~. If the kinetic energy is a quadratic form in <Ein>k = of freedom", then f .

Pi

with

f

"degrees

Example. Consider a system of harmonic occillators:

i N Ipil2 H(p,q) = ~ Z ~

IN T + ~ Z qi Vqi

I

I

This is a model for small occillations of the molecules of a solid. qi = the displacement of molecule

i

from its equilibrium position,

and the potential energy is a quadratic function of the partition law for both

p

and

q

qi " The equi-

then says that

This law describes the total average energy as a function of (temperature)

and says what is the specific heat. (Dulong-Petit's law)

23 I The equation of state and the identification 6 = k-~ The equ~ion of state of a system gives the relation between p,V,N,T, pressure, volume, no. of particles and temperature in equilibrium. To determine it we must calculate the pressure and the temperature. The pressure p is the force per unit area of the container which one has to supply to balance the effect of the collisions of the particles with the wall:

In a collision the momentum of the particle is changed from (px,Py,pz) to (-px,Py,pz). The change is hence Ap = 2Px in the direction of the n o d a l ~ i n t i n g into A. The average of ZAp for all collisions with SA during a short time interval At gives the momentum which has to be supplied to the particles from SA, ~ d this is (total force) xAt. Hence p is given by p.l~Al'At = < ~ A ~ in the l i m t At § 0. collisions to BA in At To calculate

consider a particle very near to 8A at distance x from it. It will see SA as locally flat. Consider a

IA

short At so small that we can neglect the probability that the particle will collide with another particle before hitting BA. The particle X

will hit SA in the next interval At i f - v x _> At' and then it will have Ap = 2Px -- 2mlVxl . Hence the average of Ap for such a particle is _6P 2

8

| 2pdp= mx At

2

mx 2

8m(x e- ~,~" T .

- ~(Z-T) 2/2~m 28 m e

= 2 ~

8

Hence the total average is:

e

A

dq

24

where n(q) is the particle density at q and x(q) the distance from q to ~A. Now, as At § 0 the integrand will have a sharp peak at %A, so if n(q) is nice only its value near ~A will be f{Itered out: Use coordinates as indicated: dq = dxds) o2 the variance of the gaussian is AtBm" Hence as At § 0 we have

~

_i 2 w

S ~A ~At)t as

I

2~ 2

0

nCq) B '

~A smd we finally conclude that

P=

S

ds

~A T h i s e x p r e s s i o n f o r p can be d i r e c t l y

related

t o l o g Z. Consider namely

a small change in A and the corresponding change in log Z:

all qi in AUAA = (N!Q)-I ~

all qi in A

dql ~ e-BV(q) dq2...dqN + O(AV)~ AA

AN-I

(The rema/naer is tl~integral with more than one particle in AA~ But

t -sv(q) dq2...dq N Q AN_I

N!

is the probability density for finding a particle at ql = n(ql)"

25

Hence

log z = ~ n(ql)dq I + O(AV) 2 = hA

= Av. D ~

I n(q)dq § o(~vl2, ~A

so we see that A log ZA(8,N)

I p = lim -Av+O ~

AV

For a large system it will turn out that log ZA(B,N) is very insensitive to the sha~e

of A, so one usually writes log ZA(B,N) ~V

I P = B

for a large system. An alternative derivation of the formula for the pressure: Consider the system together with a small "manometer" consisting of a movable piston with a spring in the wall of the container.The position and momentum of the piston are also included I

1 A~

~

I

as extra coordinatez, and

the total volume

depends on z. Let the energy of the spring be F(z). Then the total B-factor is

.2 -8H(p,q) - 8F(z) - 8M~p e

qi ~ Az

(M = the mass of the piston.) The average of the force on the piston is measured by the manometer as p.A and can be expressed as follows:

~ Z ISdz F,(z)

--

z-1 I dz F'(z) e-~F(z) Z(z) =

=

I ~ e-SF(Z)dz _z-1 I z(z)(- ~)

I e-HCP'q) d ?q

28

= 8-Iz-I i ~ lOgBzZ(Z)Z(z)e-BF(Z)dz

= 8 - I < 3 log Z(z)>~z

(We have neglected the effect of M~ 2 which is completely cancelled.) 2 For a macroscopical!y large manometer the fluctuations of z ought to be small, and then we have with high accuracy

I

A

= 8--A

P =

~logz @z

I

~ 1o~Z

= -8--

3V

'

the same formula as before.

N

For the ideal gas we have log Z = const + N log V, and hence p =-{V ' or

N pV = ~ . From physics in school we remember that the ideal

gas law (Boyle)

is pV = kNT, where T is temperature (on the absolute scale). We have hence I derived this law if we identify 8 = ~ 9 This relation ought then to be valid in general for an arbitrary system, because the canonical distribution describes the state of the system in thermal equilibrium with a heat bath consistning of an ideal gas e.g. If the whole system is described by a caI nonical distribution then 8 as we just said is ~-~, where T is the temperature of the bath. But then it ought to be also the temperature of the system because two systems are in thermal equilibrium iff their temperatures are equal. (We will elaborate this argument later.) For a non ideal gas the equation of state is hence defined by

p =

I

@ log ZA(8,N)

S

~V

I ~ log QA(B,N) = ~

~V

B=m I kT " Boyle~s law

C8~%

be written ~VmV = N, and then it says that for ideal gases

with the same p,V,T

N is also the same. This is Avogadro~s law. Boyle~s

law can also easily be extended to a gas located in an external field with potential U(q), e.g. the atmosphere in the field of gravity. The canonical probability density is then

N ~p: 12 -SZ~ I zm f(p,q) = ZAI(B,N) e

ZA(B,N) (~) =

N -BE U(q i) I I N'-~

3N2(IAe-BU(q)dq~ N ~:

27

The probability density n(q)dq of finding a particle at q 6 A is then given

by

N

-zu(q i)

Ne -SU(q) I e 2 A

n(q) =

dq 2 .... dq N N~

(I e-SU(q)dq>N A N!

Ne -SU(q) n(q) = (e_SU(q)dq

, so the pressure is p(q) = n(q) . 8

A Ex~_~_~A is a vertical cylinder in the field of gravity: U(z) =

m.g.z

~e-SU(q)dq = A~e-Smgzdz = A

A

0

n(q) = 8mg~ e -Smgz

p(q)

A gmg

= m~-Smg

q = (x,y,z)

z

This is the so called barometric formula. Fl___uctuations of the e n e r ~

in the canonical distribution

From the formula for ZA:

ZA(S,N)

= ~

e-~H(X)ax

we see that

N:

~kZA

~

and hence

= I ( - H(x))ke-

SH(X)dx

~-Y

k = .I,2,...

28

_

>

~Z A

~ log

= - z^ I ~ - =

= ZA- I

Var(H(x))

~2Z A ~S 2

ZA

~

~ log Z A 2 S2i~ ZA (___f6__) = _ _ BB2 = -

~S

< H> is the total average energy and ~ < H > I B is called the ~kT = 82 ~8 specific heat C (at constant volume). Hence we have Var(H) = B2C . v v For a large system it is true under quite general assumptions < H>

and C v are extensive,

i.e. they grow essentially proportionally

N

as A,V,N ~ ~ ~ ~ n, as we will see later. ry day experience,

that log Z A, to V

(This is consistent with our eve-

it takes twice as much energy to heat 2 1 of water one

degree than to heat I 1 e.g.). For such a system with a finite energy and specific heat per unit volume I Cv I H I\H 9 H we hence h a v e ~ v 2 finite and Var(v) = V(V-) = 0(V), so V has a very sharply concentrated

distribution

as V + - , and one ought to be able to con-

clude that the canonical law gives asymptotically microcanonical

2.3.

one with E = < H >

the same results as a

9 We will verify this in the following.

The grand canonical distribution

We have seen that the can. distribution

describes

a system in equilibrium

which can exchange energy but not particles with its surroundings. situation which is often encountered

Another

is that of an open subsystem, which

can also exchange particles with the surroundings.

For example when one

considers the state of a small volume of a gas or more complicated chemical system a n d ~ n t s etc. varies.

to study how the density, pressure,

In a homogenous

such variables

composition

system one expects to get the same values of

as in the whole system, and as for the can. ensemble one can

hope that it is simpler to relax the constraints

as much as possible.

Let us hence as in Lemma I consider a subsystem A I , and for simplicity let us asume that the interaction is of finite range R, so that H(Xl,X 2) = = H(x I ) + H(x 2) if x I and x 2 are two configurations

a distance > R apart.

In lemma I we neglected the interaction between A I and the surroundings

A2,

but now let us consider the distribution of the state x I in A 1 given that the state is fixed to x A in a boundary layer A of width R surrounding

29

AI, A 2 = A\(A,~A):

~

A~ I

Then if x 2 is a configuration in A 2 there is only interaction between x I and XA, x 2 and XA, but not between x I and x2, so H(Xl,X 2) = H1(x I) + H2(x2), where the interaction energy with x A is included in H I and H 2 respectively (although x A is not explicitly shown in the notation.) Then the argument of Lemma I shows that the conditional density of x I and N I given x A in the m. can. ensemble is given by: fA1 (xl'NllxA) --~ NI = d Ix

n'A2(En,A(E,N)H1(Xl), N - N I)

N1!dx I

with

~'A(E,N) = NIE ~ ~'AI(EI,NI)a'A2(E - EI,N - N I) dE I Now, let us assume that the following limit exists as A 2 § | ~'

A2

E V§

N e,v § n:

(E - HI, N - N I)

lim

~'

(E,N)

= ~(HI,NI), and that

A2 ~,A(E,N) lim n' (E,N) A2

= Z NI

~ n'

= lim Z ~ n' (E1,N I) NI AI

~'A2(E - EI,N - N 1) n' (E,N) dE1 = A2

(E I,N I) m(E I,N I) dE I AI

Then it is easy to see that ~(HI,N I ) actually has to be of the exponential form encountered for an ideal gas: -BE I + 8~N I ~(EI,N I ) = e for some constants -8,BW. In fact A2(E - E' - E'',N - N' - N'') ~(E' + E''

~'A2(E

= lim

N' + N'') = lim

- E' - E ' ' , N

n'

A2

- ~' - ~ ' ' ) n ' A 2 ( E

(E,N)

- E',~

- ~')

30

= w(E'',N'').~(E',N'), because ~ E- E' + e, N - N' § n also, and V V any Baire function satisfying this equation has to be an exponential. The limiting density is hence dx I fA I(xI'NIIxA) N 'I.

-SH1(x1) + 8~N I = e

GA I(8'ulxA)

-SHI(X1)+8uN I GAI(8,WIXA) = NIZ ~ e

dx 1 -NI!

with

dx I SuN I N1 ! = NIZ e ZAI(B,NIIXA)

This density is called the grand canonical density, and ~ is usually called the chemical potential, and 8 is the inverse temperature. The density of N I is hence

BwN I e

EAt (8,NIIXA)

,

GAI(8,UlX A) and the conditional density of x I given N I and x A is the canonical one with x A fixed:

e-SH1(Xl )

dx I .

ZAI(S,NIIx ~) N1i" Actually this derivation only shows that to each A I there is a 8 and a ~ defining the g.can, density, but a priori different regions could have different 8 and ~ . This is however not the case, because if we have two regions A I A 2 with AIU A I C A 2 and g.can, densities defined by 81,~1,82,~ 2 respectively then the g.can, density in A I has to be equal to the one induced by the g.can, density in A2, and the latter is given by the following calculation:

(8 = 82 , ~ = ~2' A = A I N = N I + N A + N3)

$I

dx I dx A dx 3 fA (Xl 'NI 'XA'NA'X3'N31XA ) N' 2 2 -8(HI(X I) + HA(x A) + HB(X3)) + 8P(N 1 + N A + N 3) =e

dx I dx A dx 3 N~

GA 2

(H I and H 3 include the interaction energy with XA1 and XA2 respectivelM) The conditional density of x1,N I given XAI and XA2 is hence

N' 1 dx3 ' ~ fA 2(''') N-~ N 3 NI' NA.J B 3, NIN3

NI" NA! N3"

dXl

"') dXIN'dx3

In this expression the factors -I GA2

Z

~ e -8(HA(xA) + H3(x3)) + 8~(NA + N3)

N3

dx3 N A .' N 3 '

are cancelled in numerator and denominator, so the conditional density is e-SH1(Xl) + BuN I

dx I ,

i.e. equal to the g.ean, one with the parameters 6 = 82 , U = U2 belonging to the larger region. This argument shows that any two regions have the same 8 and ~ since the values have to be equal to 8 and u of a sufficiently large A containing both of them. Let us now collect the formulas for the g.can, density of a system in A given the configuration y outside A: Lemma 3.

If %he probability density for x,N in A in a large surrounding

system has a limit in the thermodynamic limit of the surrounding system, then it has to he a g.can, density:

f ~ l y N ~ , = e-aH(xly) § S~N GA(~,Uly)

dx

N-~

with

$2

GA(S,~Jy) = Z ~ e-BH(xjy) + BUN N~ N

= Z ~ e -BE + 8~N ~(E,NJy) dE = E e SuN ZA(B,NJy). N N The density of E,N is hence given by

e-BE + BuN ~(E,NJy) dE

GA(S,uly)

e SuN ZA(8,NJy) , and that of N by

GA(S,~Iy)

The average values of E,N are given by log GA(8,UJy)

(Derivation with 8~ = const.)

~8

log GA(B,U Jy)

:•

8

Bu

The conditional density of x given E,N is the re.can, one, and given only N is the can. one defined by 8,N. We have in the derivation assumed that the interaction is of finite range,but the above formulas can make sense also for interactions of infinite range.

33

Spme applications

I.

of the grand canonical distribution

The equation of state of an arbitrar_,y gas in a slowly varying external field

Let us consider a gas influenced also by an external field of the form U(~)

giving a contribution

Z U(~ -~i) to the total energy. Here U(x) i x 6 R 3 , and we shall consider the situation when

is a nice function of L ~ ~,

so that

U

varies very slowly on a microscopic

scale. We shall

however consider the density, pressure etc. on a scale

q = L-x ,

which there is a non trivial variation of these macroscopic Consider now the state in a macroseopieally side

L-Ax

value is

centered at

U(x)

N-U(x).

L-x .

infinitesimal

G

is extensive as

g(8,~) = lim gA(8,wIy) A effect coming from y.

of

gA

for the cell is heoce

since

gA

exists,

and is independent

are convex in 8,~

converge to those of

g ~

= S -I

~g(8,~-U(x) ) BU

This is the general barometric

of the boundary

it follows that the derivatives

S,~.)-

converges to

as

L ~ ~ with

Ax

fixed.

formula for a non ideal gas. To see how g(8,~)

let us independently

of our previous formulas define it by the macroscopic

~Ap

Hence we can

x

the pressure can be expressed in terms of

AX

=

when these exist (which happens

conclude that the average density at nn(X) = 8 -I BgA (B'U-U(x)Iy) SU

gA(~,uly)

A + ~, so that

except at most for denumer~bly many va~iues of

n(x)

with constant

-

log GA(~,uly)

Moreover,

A

in the cell, so its contribution to the energy in the cell

Now, as we shall prove later, it is a basic fact that also = ~

cell

The potential has the esseDtially

The partition function

GA(B,u-U(xlly)

1

on

quantities.

balance equation

34

The increase in

p

in the direction of the force

- grad U

balances

it:

(Ax) 2 (p(x) - p(x + Ax)) = n(x) (Ax)B'grad U(x). Nence

p(x)

is defined by the equation

grad p(x) =

_

n(x) grad U(x)

= B-I grad x g(8,u-U(x)), p(x) = 8 -I g(S,~-U(x))

=

8-I

-

'(8,u-U(x)) grad U(x) =

gu

which when integrated gives if

p = 0

when

g = O.

This is the case if we have a situation when rapidly when p(~) = 0 ,

Ixl § -

so that

g

and

U(x) § ~

n + O

when

sufficiently

Ixl + -

and

i.e. the gas in enclosed in a "potential well".

To see the relation between this and our previous definition of p, P = B- l @ log ZA(B,N) ~V

remember that

GA(B,U) = Z e 8~N ZA(8,N) N

so that at least formally

~GA(8,~) -

-

=

~V

= ~

Z

@ZA(8,N)

eB~N

BV

N

@ log ZA(B,N)

eB~NZA(B'N)

~V

= GA(8,~)B
N

"'" if

N

B log GA(8,U)

~v

= 8 ~ Bp(S, <~>)

has a sharp distribution when

log GA(8,~) = V.g(S,~) + o{V) differentiated

we have

g = BP

as

A § ~

A ~ ~ ,

in the limit

We have so if this relation can be A + -.

(It is not very

simple to complete this argument to a rigorous proof.) The equations

Bp = g(S,~-u(x)) 8n = g~'(S,~-U(x)) define the equation of state of the system in parametric functions of obtained.

B,~.

if

~

form, p,n

is eliminated the relation between

p,n,B

as is

35

The parametric situation when

g

form is however very convenient to use. Consider e.g. the U(x) = mgx I ,

is a convex increasing

a gas in a constant gravitational

function of

~,

field:

and is hence differentiable

except when the derivative possibly jumps.

X!

In an interval where functions of Wc

xI ,

g~

XI

exists

and if

p

and

g~(8,~)

n

are decreasing continuous

makes a jump at a critical value

then the density suddenly drops when

~ - gxl = ~c

but the

pressure is continuous at this point. This indicates that a phase transition takes place, the system consists of a heavy phase (liquid) and above it a light phase (gas) in equilibrium with each other. They are separated by the gravitational at

~ - gxl = ~c"

gate when such phase transitions

2.

field and have a sharp interphase

An important and interesting problem is to investican take place.

The equilibrium rule for a binary chemical reaction

Let us consider such a reaction

A + B~AB

where the molecules

do not interact except when they are formed. one B-molecule required.

form an

AB-molecule

In the reaction § one

and an amount of energy

U

A- and is

Suppose that the state of a single molecule of the respective

types can be described by some coordinates vibrations, NI,N2, N 3

A,B,AB

rotations,

positions

particles respectively

etc.).

x,y,z

resp.

(e.g. describing

The state of a gas consisting

is then described by fixing

of

36

(Xl,...XN1,Yl,...YN2,Zl,...ZN3)

modulo permutations of identical parNI

ticles. The energy of such a state is

N2

N3

Z

H1(x i) + Z

H2(Y i) + Z

I

I

I

H3(z i) +

+ U.N 3 = H(x,y.z) + U.N 3 . HI(X I) etc. is the energy connected with the inner degrees of freedom of the molecules,

and there is no interaction among these. The car.. partition

function for all states with NI,N2,N3 NI

N2

-BE HI(X i) I Z(8,NI,N2,N 3) = S e

NI Zj , N I9

N2 Z2

N3 Z3

N2 '

N3!

given is hence: N3

-BE H2(Y i) I e

-SZ H3(zi)-SUN 3 I dx d~ dz e N]! N2! N3! =

e-SUN3 where

ZI

etc.

are the partition functions

of a single molecule

-SHI(x) ZI = f e

dx

etc.

The can. partition function for a gas with

MI

and

M2

A- and B-molecules

respectively is hence M IAM2 Z(M I,M 2) =

(To form

Z N=0

1 AB

I A

Z(MI-N,M2-N,N)

and

I B

The probability density for

p(N 3 = N) =

is needed.) N3

z (M I-N ,M2-N ,~) Z(MI ,M2) ,

is~hence

and is not so easy to handle.

If we now consider a small subvolume of the system and ask for the equilibrium concentrations of where

MIM 2

~I'~2

and get:

A,B,AB,

are also stochastic.

we can use the g. can. distribution In this case we must use two potentials

37 G(8,~I,~2) = MIM7. 2 e 8~1M1+8~2M2

Z(MI,M 2 )

8wI(N1+N3)+8w2(N2+N3 )

7.

=

N1 ZI

N2 Z2

N3 Z3

NI!

N2!

N3!

e

-SUN 3 e

N I ,N2,N 3 so we get the simple formula 8U 1 log G = Z I e

Pl

and

~2

8~ 2 + Z2 e

are

8(~I+~2-U) + Z3 e

determined by the equations

8~ I MI = 8-I

~ lo~ G = Zl e ~I

M2 = 8-I

~ log G = Z2 e Bu 2

8(~I+~2-U) + Z3 e

8U I

From the formula for

f

G

8(~I+~2-U) + Z3 e

we then also get:

=Z2e B~1

q~

B(~I+g2-U)

=Z3e

so the following relation is valid

Z1(8)Z2(8) =

independently of

Z3(S) ~I,W2.

e sU ~ k(8)

This is the famous law of mass action, which

shows that the equilibrium concentrations

{

are determined by the above

relation and

"I

if

=

MI,M 2

+

are given. We also see that the equilibrium constant

k(8)

38

can be computed from the knowledge of the inner structure of the molecules, and that it has a simple dependence on

U,

the dissociation energy. In

this case it is easy to verify directly that the variances of proportional to ~ N i ~

,

N. are 1 so the law of large numbers holds with great

precision.

3.

The pressure in a.mixture of independent particles~ partial pressure and osmotic pressure

If as in the previous example we have a system of two types of particles A

and

B

we see like in I.

that locally we have for the pressure

grad p(x) = - hi(X) grad U1(x) - n2(x) grad U2(x)

"."

~g1(S,u1-U1) ~I

8 grad p =

= grad (g1+g2)

8p(x) = g1(B,~-U1(x)) p

,

grad U1(x)

~g2(S,~2-U 2) grad Ue(x) = ~U2

and hence

+ g2(8,w-U2(x))

= 8P1(X) + 8P2(X).

is hence the sum of the "partial pressures" of the components.

An example of this is the occurrence of "osmotic pressure".

Consider

a system consisting of a vessel with two parts separated by a semipermeable wall preventing

The pressure to the left is Pr = B-1(gl+g2 )"

B-

but not

A-molecules

Pl = 8-Ig I

and to the right

(The A-particles do not feel the wall, so

computed by integrating over the whole volume, and there, whereas

from passing it:

g2

gl

is

is constant

is obtained by integrating over the right half.)

There is hence a pressure difference called the osmotic pressure. If the hence determined by

~1

Ap = nB.kT,

Ap = 8-Ig2

at the wall. It is

B-particles form an ideal gas it is

the same formula as for an ideal gas.

38

3.

The law of large numbers for macroscopic

variables and the foundations

of thermodynamics 3.1.

General study of the probability

laws of macrovariables

We are now going to give a general argument

showing that macroscopic

variables like the total energy, number of particles sharp distributions

etc. have very

around their average values in the thermodynamic

limit. In this process we will also see to what extent the various "ensembles"

introduced give the same results.

We are also going to see

that the rules for how these averages are related and vary when external conditions

are changed are those prescribed by the laws of thermodynamics.

Let us again state our basic microscopic container

AcR d .

description of a system in a

To begin with we will consider only the part of the

system depending on the position variables part depending on the

Pi

qi

for simplicity.

and afterwards

We also restrict

tion to systems of one type of identical particles. the no. of particles in states

A

with all

qi6A ,

our atten-

For any value

we have the state space

q = (ql,...,qN)CR dN

add the

AN

N

of

of possible

and the basic measure

dql...dq N mN(dq) =

(If the particles are rigid spheres of radius NI

r/2

we should instead take

to the region

c AN

where

mN(d q)

as the restriction

lqi - qjl > r

for all

space for the system with a variable no. of particles with

m(dq)

is

~N(dq)

l

and take

defined on for all

A point of

N.

FA

is hence variable

region

The state

F A = U AN , o by saying that its restriction to A N

FA (For

N = o

we let

A~

is then

consist of a single point

~{l}) = I. )

cal macroscopic v(q) =

of the above measure

i # j.)

Z

1<_i<j<j A'cA

q = (ql,...,qN) (observable)

v(qi - qj) , Z XA,(qi )

if if

q6A N , qEA N

with

N

arbitrary S 0 .

A typi-

is e.g. the total potential energy the total no. of particles

in a

etc.

I

More generally, we can define

for any symmetric function of U(q) =

variables

Z u(qil,...,q i ) (il,...,im)C{1 .... ,N} m

This will be a symmetric function of tionally invariant,

m

(ql,...,qN),

so that it depends only on

and if

u(ql,...,q m) for

u

q E R dN

is transla-

(q2-ql,...,qm-ql) ,

then it

40

will also be invariant when the qi+q

q E R d.

If moreover

U(ql ..... qm ) = 0

qi

as soon as

!

,!

for all

i,j

q

then

We will consider observables

is of finite range

lqi-qj I > R

will have the property that if lqi-qj I > R

are all shifted by the same amount to

u(ql,...,qm)

for some

i # j

can be split into

R, so that then

(q',q")

U(q)

with

U(q) = U(q') + U(q").

U(q)

defined for finite configurations

q

having these properties: a)

U

is symmetric when particles are permuted.

b)

U

is invariant under translations of the configurations in

c)

U

is of finite range

R ~ r,

R d"

as defined above.

This last assumption can be relaxed, but it simplifies the arguments considerably. We will consider a situation where we are interested in a finite no. of such observables

UI,...,UM

which describe our system macroscopieally.

A general m. can. ensemble can be defined by considering all having fixed values of

U.(q)

q E FA

as equally probable (according to

m(dq)),

l

and a general canonical ensemble by saying that

q

has a probability law

M

proportional to

e

-EaiUi(q) I

(Inverse temperature etc.)

~(dq),

where

(a I ..... aM)

are some parameters.

(We use vector notation M

U(q) = (UI(q) ..... UM(q))

a.U = ~ qiUi I

etc.)

These probability laws are expressed in terms of the "structure measure": flA(A) = , , , ( ~ E A

flA(A) = N=OZ ~

, q E F A)

for

AcR M ,

mN(d q) .

,;&N We also introduce the corresponding canonical measure: e-a'U(q) ~(dq) = S nA(A'a) = U(qf_ ~.~," A A

Ia[qcr A (nA(A,o) = flA(A))

e-IAl(a'u)nn(du)

41

We will use a slightly different concept of m. can. measure than before and say that

~

~ 6

where

A,

shell").

is not exactly fixed to a value A

is

small neighbourhood of

a

u6 R M, u

s M

but require that (a thick "energy

We are then going to consider the limit of the m. can. distribu-

tion when first

A § ~

and then

A § u .

(This is technically much

simpler than considering the thin energy shell and physically very reasonable, since it is difficult to keep a macroscopic variable constant with microscopic precision.) The m. cam probabilityTT~l distribution of an observable the restriction

~ 6 A

Uo(q) ~

can then be expressed in terms of

defined by 2A

as follows:

U~ Include for

U~

among

UI,...,U M

and define

~A(AxA) = ~(--~--6A, U 0 ~

A c E I . The probability distribution of Uo(q) , PA ( - E A E A ) I A I

aA(A•

=

aA(AXA)

a^(R~•

(We use the same letter

2A

AxA,A

for both measures and let the difference be etc.)

The can. probability distribution of EAIa) :

is then:

: - aA(~)

clear from the argument

PA(~

6A)

~A (A'a) 2A(RM,a)

~

is expressed by:

AcE M

Uo(q) If we want to consider another observable U~ PA(-~--6A,

~ 6

Bla) :

2A(AXB'(~ RM+I

~A ( =

2A(A• ~A(RM,a)

'

~

AcE I B cR M

as well as above we get:

=

,(o,a)) with the same abuse of notation.

(U(q) will be an extensive quantity, and therefore we consider all the time as

~ A + ~.)

etc. whose distribution will obey the law of large numbers As before we put

GA(a) : ex~IAlgA(a) :

2~RM,a)

42

The most important

special

case is when

U1(q) = the total potential

U2(q) = N(q) = the total no. of particles. analog of ~A(E,N) N u2 = ~ 9

defined

Now, the fundamental

asymptotic

quence of the fact that Theorem

2.

As

The values

_+~

For

we write

s(A,a)

proper$ies

log ~A(A,a)

= s(A,a)

= -|

iff

A -~ ~

so b) c)

s(A,a)

s(A) <__ I,

,

where

C

boxes all of whose

s(A,a)

if and

2

A cB A. i

iff

~A(A) = o

A I + A2 u I + u2 ( 2 = {u = 2

closure

cA.

infinite.

and see how it depends

properties:

on

(A = open convex c~RM)

, A .

. (but not necessarily

is also defined,

>_.

A.

At this stage let us assume

sides become

and

A)

2

with

are open convex, n

s(A,a) = max s(Ai,a) _= V s(Ai,a) i I

s(A 1,a) + s(A2,a)

, a)

for all

property:

s(A,a)

s(A,a) <_ I + sup(-a.u) uEA is bounded on any bounded

s(A,a) <_ s(B,a) n If A -- U A. , I i

s(

of R M

2 later when we have seen its consequences.

has the following

A I + A2

d)

is a conse-

is any open convex subset

is open convex with compact

and

s(A,a)

then

A

which happens

We shall now study the set function

a)

if

has to be made precise.

are rectangular

Lemma h.

defined by

of the probabilities

regularity

We will give the proof of Theorem

A

A

is extensive:

exists,

s(A) = -| ,

.s(A,a) = sup s(C,a) CcA

that

with

log hA(A)

will have the following

The statement

~A(A)

energy,

and the E uI ~

are not excluded.

s(A) = lim ~ Am

s(A,a)

is

a I = 8, a 2 = -~U,

A +

lim ~i,,l ~ log ~A(A,a) Am

a = o

earlier

Then

u I CAI,

u2CA2})

.

A .

@

48

Proof: a)

aA(A) ~

Z f N

N!

= Z

o qEA

.

e

=

,

o

so s(A) ~ I. ~A (A,a) Z ~A (A) exPiAl sup (-a-u) , which gives the bound for s(A,a), uEA b)

c)

is clear. If

A = A I UA2,

with

s(A1,a) ~ s(A2,a)

e.g. we have for any

~ > o

~A(AI ,a) ~ ~A(A, a) ~ ~A(AI 'a) + ~A(A2 'a) 5 2 exp[Al(s(A1,a) + c) if A is big enough. 9"

s(A1,a) ~ s(A,a) ~ s(A1,a) + E

for any

~ > o,

and

s(A,a) = s(A1,a) 9

n

For

A = U A.

d)

For any

A

of width

the

proof

goes

by

induction.

1

1

let

R

A'

consist of two translates of

A

and a corridor,

separating them:

I ^,

A, A"

For any pair of configurations U(q I ) U(q 2) --~--6B I , -~T-EB2 configuration =

q

ql 'q2

AI

and

A2

with

, and no. of particles

NI

and

N2

is allowed in

A'

U(ql) + U(q2) ~ IA,I 6

in

the combined

and has

(B I + Bg~! = B'

and

N = N I + N2

-a.u(q) ~ence

f

e-a'U(q) dq ~_

~B' qEA 'N

f ~-~

e

dq

B'

qE(A '~C)N

N

-a-U(~)

-a'U(ql)

>

e

N I+N~2=N (NI) U(~I ) e -~-- E B I qlEA~ I

dql

U(~2 ) s B2 N2

q2EA2

dq 2

C,

44

(There are

(~i)

ways of chosing

Remembering that

~N(dq) = ~

E f N ~6B'

N I particles to

A I .)

, we get

e-a'U(q) ~N(dq) >_

-a.U(%)

-a.U(q I )

z {ql) NI,N 2 U

e

e

~--6B

~N1(dql)

I

U{q2 ) ~--6B

NI q16A1

~N2(dq2)

2

N2 q26A2

I.e. 2A,(B ,a) ~ 2A(BI,a) nA(B2,a) NOW, as

§ ~I , and

A ~ |

B'

is nearly

BI + B 2 , but an extra 2

approximation argument is needed: Theorem 2 tells us that for any

e > o

we can find

B. cA. i

pact

c-Ai

and

s(Bi,a) > s(Ai,a) - e.

B~ + Be AI + A2 = ~ c ~ U'

in

=

~ u'.

u

with

Hence

us

d(B',B) + o,

B.

u'E B'

we have

d(u',B) ! (I - 2~AA,~) max lul -- o uniformly u6B A I + A2 and B'c----~--- if A is big enough, because

being compact has a strictly positive distance to the complement of which is closed:

com-

i

Then

is also bounded, and for any point so

with

i

A I + A2 2

45

We see that

A2

AI +

nA,(-----~'--, a) ~ ~A(BI,a) ~A(BR,a) when

A

is big, and hence A I + A2

2s(~,

a) ~ S(Bl,a) + s(B2,a)

for all

~ > o ,

Property

c) suggests

centered

at

u.l

that if

then

approximatively A. 9u.. 1 1

which proves

s(A1,a)

+ s(A2,a)

- 2E

d). A

is partitioned into small cells A. n n 1 = ~ s(Ai,a) ~ y s(ui,a) , if s(Ai,a)

s(A,a)

equal to a constant

value

s(ui,a)

is

for a small cell

This is indeed true:

Lemma 5. convex,

s(A,a) with

can be expressed

s(u,a)

s(u,a)

= inf s(A,a). Agu

of

take values

as

s(A,a) = sup s(u,a), us

defined by (This definition

makes

for

A = open

sense also if some components

--M u

axis a)

b)

c)

{-~} U R U s(~,a)

{+~}

-- s(u,0)

s(u)

is

_+~, i.e. when .) -

s(u,a) a.u

-= s(u)

uER

,

where

has the following -

real

properties:

domain

< I, upper semicontinuous (u.s.e.) (also on the extended _M-R ) and concave (possibly = -~ for some u) .

s(u,a)

is uniquely

determined

in the following

= sup ~(u,a) for all open convex u6A is u.s.c, at u .

D = {u; u E R M, s(u) > -~} = U (essential A

Proof:

Since

sequence +|

is the extended

a-~

s(A,a)

d)

R

in

R

range of ~

s(A,a)

in

If

For any u E A

s(u,a)

with

< s(u,a)

s(A u ,a) < r

.

+ c,

s(A,a)

and if

Hence these

> -~

A

of

if

is given by

Clearly

s(u,a), e > o

so if

arbitrary

is an open

s(u,a) = -~ Au

D

= s(u,a)

s(u,a) = lim s(A,a) for any A. § i u. (An open neighbourhood of etc. )

take

finite there

If

~(u,a)

q s s A)

of open convex A. shrinking to i e.g. is an interval A = (a,~) from the definition

sense:

then

and its closure

when

is decreasing

s(A,a) ~ sup s(u,a) us there is equality.

S(Au,a)

is convex,

A,

Au hu

there is an

form an open covering

s(A,a) and

< s(A,a)

- c.

such that

AugU of

r

= -|

A.

such that Take

a

C

with

46

compact

cA

such that

s(C,a)

can be covered by a finite

> s(A,a)

- e. Since n {Au.} I . Hence 1

subcovering

n N e' < s(C,a) ~ s(U Au ,a) = V s(A u.,a) , I 1 I l n and we see that V has to be attained for some

u.

I

is compact

with

it

s(u i,a) > -~

1

and hence

S(Au.,a) 1

a)

< s(ui,a)

+ e _< s u p s ( u , a )

s(A,a)

< s(C,a)

s(A,a)

= sup s(u,a) u~A

If

is a neighbourhood

A

aA(A,a)

so

+ c,

u~g. + e o

and

.

= f e -IAl(a'u) A

of

u

aA(dU)

aA(A) e - l A l ( ( a ' u ) + E l a l )

of diameter

< e

then since

we have

2 flA(A, a) ! hA(A) e - l h l ( ( a ' u ) - e l a l )

and hence

s(A) - (a.u) - clal ! s(A,a)

b)

and

a)

s(u)

 o

or

s(u')

is open i.e. if

s > s(u)

(s-~,s+c)•

< s-c

u' s A .

if

then there

;

is an

Agu

s ~ s(u)}

then there

(s,u)

is also

This property

s(u') ! s(A) ~ s-e

and

for

is clear, ~ > o

u's

.

s(u I) + s(u 2) ~

2

follows

shrink to

ul,u 2.

It then follows that

s(u I) + (I - l) s(u 2)

c)

= {(s,u)E ~M+I

such that

s ( ~ )

and for

u .

s(A) s(u) = inf s(A) Agu s(A) ~ s-E, so that

u I + u2

shrink to

means that the "Epigraph"

and

in the complement, if

A

i.e. its complement

is an open A g u

because

! s(A) - (a.u) § ~la I

~

arbitrary

If

s(A,a)

so

s(u,a) >_~(u,a)

if

0 < ~ __~(u,a) > ~(u,a),

and

~

if

Agu,

u.s.c,

at

U

47

then there w o u l d be a n e i g h b o u r h o o d ~(u',a) ! s(u,a) - e

in

A

,

s(A,a) = sup ~(u,a) < s(u,a) us

d)

A 9 u

and

e > o

such that

but t h e n we w o u l d have ,

contradicting

s(u,a) = inf s(A,a) ABu

.

The p r o o f is given after the p r o o f of T h e o r e m 2 on p. 93.

W e shall now see how the f u n c t i o n

s(u)

can b e used to study the asymp-

totic form of the m. can. and can. d i s t r i b u t i o n s for t y p i c a l m a c r o s c o p i c variables p. hl.

~

.

Consider first the m. can. d i s t r i b u t i o n d e f i n e d on

W e have

uo(q)

§ s(A•

If w e now let L e m m a 6.

If

A

- s(RI•

= sup S(Uo,U) - SUp S(Uo,U) u 6A u o o u E A us

shrink to

s(u) > - ~

u

we filter out the value of

then

Uo(q) 1 l i m lim ,",]qTlog P A ( - - ~ - - 6 A I A )

sup S(Uo,U)

:

u

if t h e s e are u.s.c, at

in case

If

s(Aiu) = -~

U

points

•

Proof:

s(A•

< K

u ,

A, if

and t h e n ,

s(Rlu)

s(Rlu) = s(u). with

s(Alu) = s(Alu)

This happens if

A = [a',a"] w i l l h o l d if

s(a',u) = s(a",u) = -|

o

-

o

s(R[u) = s(Riu)

For a finite interval

=

sup S(Uo,U) -- s(A[u)

-

u6A o

s(Alu) = s(Alu),

S(Uo,U):

if

A = (a',a") s(Aiu ) > -~,

etc. and

also.

•

u n i f o r m l y then the c o n d i t i o n w i l l h o l d at infinite end

.

= sup S(Uo,U) u CA o u s A

= sup sup s(u ,u) = sup s(Alu) uEA u 6 A o us o

.

As a function

48

of

A

with

A

fixed

s(AxA)

satisfies the regularity condition of

Theorem 2, and hence by the uniqueness in Lemma 5 c) if

s(AIu)

= s(u)

.is u.s.c, at

u .

Similarly,

s(AIu) = inf s(AxA) Agu

s(Rlu) = inf s(R• Agu

= inf s(A) = Agu

in this case.

To see that

s(A{u)

is u.s.c, if

s(A{u) = s(A{u)

s > s(A{u) = s(A{u) = sup_ S(Uo,U),

so that

let

S(Uo,U) < s-~

for all

Uo(A Uo~A

and some

S(Uo,U) that

c > 0 .

in ~M+I

For any

Uo~A

,

by the semicontinuity of

there is then a neighbourhood

S(Uo,U') < s-E

for all

(Uo,U')~AUoXdUo

AUoXAUo9 (Uo,U) also.

such

{AUo; Uo~X} forms

an open cover of A, which is compact in R. Hence a finite subfamily n n (AUo,i) I already covers A, so if A = (]I Au~ = open neighbourhood of u

then

= sup

~ceA

S(Uo,U')

< s-e

for all

S(Uo,U') ! s - ~

If e.g.

a'

for

Uos

u'EA,

u'~A ,

so

s(AIu)

is a finite endpoint, and

s(a' ,u) <__ s(Alu)

S(Uo,U) > s(AIu) - E, a'

+

then

2

u.

then

Uo~A

o ~ A

too

is such that

and hence

I

,ul

+ 5 sCAlu) - 5c "

is u.s.c, at

u

U

s( lul >_ s(--r

s(Alu') =

s(AIu) > --~

by the convexity, because if a t +

and

I

s(a',u)+5 S(Uo'U) >-7 s(a',u) +

so that

s(a' ,u) <_ s(Alu) + c

for any

c > o,

and hence

s(a' ,u) 5_ s(Alu)

9

^ - u(q) If

--~--

AC(k,-),

<__k so

then

s(A•

S(Uo,U) = inf

= --

for all

s(A•

= --

A

if

for all

AC(-~,k) u

if

or

k < lUo[ <__|

~-~uo

Agu Lemma 6 leads to the following important rule for how the probability distribution of

Uo(q) ~

the concave function picture is found:

behaves in the thermodynamic S(Uo,U)

for

u

limit:

Consider

fixed. Typically the following

49

s(u) = Sup S(Uo,U) is attained in an interval M = Eu~,u~, which u o can reduce to a single point, finite or infinite. If A is an interval such that s(Alu)

lim A~

is disjoint from

= s(Al~)

< s(~)

,

M

then

so

~ log PA(A[A) = s(AxA) - s(A) = ~ < o l"l

A~)u

if

A

is small enough,

PA(AI~)_<e -IAI~

i.e.

as ^ + ~ ,

and the probability mass in

A

goes to zero very fast.

all probability mass will be concentrated to

M

This means that

in the limit. This is

the basic rule in thermodynamic probability theory:

The probability mass

Uo(q) of a macroscopic variable the restriction that

S(Uo,U)

~

~

~ u

is maximal.

in the m. can. distribution defined by

will be concentrated to values of In particular,

if

S(Uo,U)

U'o giving maximum then the law of large numbers holds: to

u' o

in probability.

We are going to see that

s(u)

is the (Boltzmann)-entropy of the macroscopic state

~

u~

such

has a unique

Uo(q) ~

converges

(or rather k.s(u)) f~u,

and the

above argument makes precise the famous rules of thermodynamics that the probability of a macroscopic state

Uo(q)

-T~T- ~

u~

is proportional to

50

IAlS(Uo 'u) , and that only states having maximal exp (the entropy) k = e entropy are seen in macroscopic systems. In the case where

M

1' o' U'~

is an interval

the above argument does not

tell us how the probability mass is distributed outside

M

in

M, only that the mass

goes to zero.

The corresponding

result for the generalized

canonical distribution

defined

on p. hl follows analogously: 2A(A,a)

PA(U-~6,.,AI~) = aA(RM'a) lim Am

I

.

Hence

log PA(AIa) = s(A,a) - s(RM,a)

(At least if both terms are not +| .) Let us put

s(RM,a) = g(a)

s(A,a) = sup (s(u) - a.u) u(A g(a) = sup (s(u) - a-u) u If

M

is the set

(~M

. ,

and especially

.

where

(S(u) - a.u)

is maximal it follows by

the same argument as above that all the probability mass will be concentrated to

M

in the limit

A § |

Especially if

point the law of large numbers holds for a simple geometric interpretation: plane of slope

a~R M

M

~ .

M

is the set of

touches the curve

consists of a single

The definition of u

M

has

where a tangent

s = s(u):

&

st.

I

This means that

M

is always

a convex set.

M

g(a)

is a convex function of a being the limit of the convex functions

gA(a) 9

(gA(a)=.log f e-IA1(a'u)2A(dU)

is convex,

because in any direction b

51

its second derivative

is

> O:

d2g(a + Ab) > 0 .)

--

s(A,a)

Since

is finite

then the probability escapes

This indicates

in convexity

Lemma 7:

If

g(a)

we see that if

g(a) = +=

will go to zero, i.e. the mass

that the system behaves

in a "cata-

the density will be infinite

with

system.

s(u)

is a well known conjugate

relation

theory:

s(u)

is concave

g(a) = sup (s(u) - a-u) < u

and

A

A

perhaps

close to one in a large

The relation between

g(a)

in any such

way; some quantities,

probability

--

for any bounded

mass

to infinity.

strophic"

d12

for some

a

and u.s.c,

then its conjugate

is convex and lower semicontinuous the relation

is reciprocal

s(u) = inf (g(a) + a.u), with a similar picture a

function (l.s.c.).

If

in the sense that

for the construction

of s:

S~

Two points

in the sup construction, i.e. u g(a) = s(u) - (a.u) if and only if they are related in the inf construca tion, i.e. s(u) = g(a) + (a-u). There is a unique u corresponding to a

(u,a)

are related

if and only if

Proof: then

Consider g(a) = --

g

first the degenerate for all

fined above is the functions

is differentiable

a,

a,

case:

and then

s(u) = -~

u = - g'(a). for all

u,

and

s(u) = inf (g(a) + a.u). g(a) as dea of the family of linear (and hence convex, l.s.c.)

sup u gu(a) = s(u) - a.u.

served under arbitrary

at

sup u

Convexity

operations.

and lower Hence

semicontinuity

g(a)

is pre-

has these properties.

52

Consider now the problem of finding all linear functions and hence

lying above

s~

Epis:

3

J~

Such a function for all u,

u ~ a.u + g

i.e.

iff.

lies above

g ~ s(u) - a.u

Epia

iff.

for all

u,

a.u + g ~ s ( u ) i.e.

iff.

g ~ sup s(u) - a.u = g(a). For a given a the "critical" g is hence u g(a), when one tries to "push down" the function a-u + g as far as possible

towards

Epis,

a.u + g

touches

Epis

correspond to all

u,a

and if at

s > s(u).

g(a) = s(u) - a.u

for some

The possible values of

Epig = {(g,a); g ~ g ( a ) } .

follows that

g(a) + a.u ~ s(u)

inf (g(a) + a-u) ~ s(u). a

Epis

u.

From

u

(g,a)

then hence

g(a) ~ s(u) - a-u

for all

u,a ,

for

and hence

To see that there is indeed equality take any

Then it is always possible to separate the point

(s,u)

from

by a non vertical plane:

S

I.e. a.u

there is + g < s

(g~a)

such that

(Le,~a 8 below).

s > a.u § g(a) ~ i n f a

a-u + g ~ s(u)

Hence

(g(a) + a.u),

S t a r t i n g from the convex function we find that the linear function

g ~g(a), s o in fact g(a)

for all

u

but

and s(u) = inf (g(a) + a.u). a

and applying the above argument

a § s - a.u

lies below

Epig

iff.

53

s - a.u ~ g(a)

for all

a

iff.

i.e.

s < inf (g(a) + a.u) = s(u)

,

a

i.e.

iff.

(s,u)~Epis:

9

$

J

Hence the "critical"

s

for a given

tries to "push up" the function We have hence seen that and

a.u + g(a) = s(u)

s = s(u), If

u

and

u § a.u + g ;

and that

to

a

g(b) > s(u) - b.u

for all so that

g(b) - g(a) > - u.(b - a)

Hence it follows by letting g'(a)

(b - a) , so that

g+'

+ ~b)

# - g'(a,-b)

and

since gl

at all both

a

b

and

not a unique

at

tend to a

u

iff. Epig

g = g(a), at

a

iff.

b

a.) along any direction that if

(b - a).g'(a)

Conversely,

if

= - (b - a).u

g'(a)

such that the directional

- g(a)

are different

~ gl .

(These directional is convex in

for all

for all

is not defined, derivatives

to the right and left:

define supporting lines along

u•

~;

derivatives and

are

g'(a,b) ~ - g'(a,-b).)

b:

~.

w h i c h hence satisfy

and which coincide with u+

at

touches

theorem then tells us that there are two supporting planes

defined by c

Epis

and

g

g(a + ~b)

g(a + ~b) - g(a) ~ ~g~ Hahn-Banach#s

b

to

u = - g'(a).

lim g(a

always defined, Both

touches a + s - a.u

is defined then

then there is a direction

gl ~ g'(a,b)

w h e n one

for all b.

(-u defines a supporting plane

=

s = s(u),

as far as possible towards Epig.

then

g(a) = s(u) - a.u ,

g'(a,b)

is giveh by

s(u) - a-u = g(a). Hence these events are equivalent.

corresponds

the gradient

u

s - a.u

u u

g~

correspond to corresponding

to

along

a

g(a + c) - g(a) ~ c.u• b,

i.e.

b-u• = g~

and they are not equal, a.

.

for Hence

so there is

54

For completeness

we also give a proof of the separation property used

above: L e m m a 8. Let s(u) ! a . u exist

s(u)

+ g

(g,a)

for all

be concave u.s.c, with

for all

u

separating

u,

and

(s,u)

of the coordinates, a.u + g

with

g(a) < |

(g,a).

from Epis,

Then if

for some

a ,

s > s(u)

i.e. such that

i.e.

there

s(u) < a.u + g

s > a.u + g .

Proof. We can assume that

parabolas

and some

(s,u) = (0,0)

so we assume that

g < 0 .

and want to find

The m e t h o d of the p r o o f is to consider the upward

u § alul 2 + ~

to push them towards

b y suitably moving the origin

s(O) < 0 ,

~ > 0,

Epis

y arbitrary

by minimizing

lying above Epis

and try

7 :

S

&m

ml~4L+y

f

If the critical with

Epis

7

is

< 0

then the tangent plane at the point of contact

will be a separating plane. This will happen if

a

is big

enough. The p a r a b o l a lies above critical value of

y

Epis

iff

is given by

u ,

so the

7 = 7(u) = sup (s(u) - ~lui 2) .

s(u) ~ mlul 2 + 7

for all

To see

U

that of

y(a) < 0 s(u)

at

small enough. ~(~)

< max

< max (- 6,

if u = 0

u

is large enough we remember that the semicontinuity means that

s(u) i -

Hence ( sup , sup ) < iuI<_~ lui>~ sup

-

iul>~

<m~(-6, -

sup lul>~

(s(u)

- ~IuI))

< -

Igi + ( l a l - ~ ) l u l ) =

6 < 0

if

lul ! z

and

a

is

55

if

a

is large enough.

Consider all

u ,

plane.

Pick such a value of

y(a) = sup (s(u) - aluI2), u and we can take a = 0, g

If it is

for some u.

> --

for

R

arbitrary

< 0

and an set.

-=

when

6

si~)-~lul 2Lsi~)-~IGI 2

lul § = ,

be the maximal u,

for

to get the separating sup u

some

u,

all

so that

or

s(u) - s(u) < 2m~-(u - u)

u = & + v

we had

is attained

so the

s(u) - s(u) i mlul 2 - alul 2 = 2mu-(u - u) + mlu - ~I 2 that a6tually

for

Sup = sup u IuI<_R function always assumes a maximal

u.s.c,

Let

y = y(a).

s(u) = -=

s(u) - ~]ul 2 ! Igl + laIlul - ~lui 2,

This follows because

big enough,

and take

it is actually finite and the

which is bounded and goes to

value on a compact

m

if it is = -= ,

for all

It then follows

u ,

because

s(u + v) - s(~) > 2mu.v + ~, ~ > O,

if for

then by the

^

concavity of

s

near

u

when

u = & + cv = (I - r

+ r

+ v) we w o u l d

have

s(u) 5 (I - E) s(~) + r

+ v) h s(G) + (2~G).(cv) + r

contradicting

s(u) i s(G) + (2~6).(~v) * ~r when

r

is small enough,

so that

~r

au + g ! s(u)

for all

= y(m) - lul 2 < 0 ,

so it furnishes

The convex function

g(a)

average values

< r

a.u + g ~ 2~6(u

we take the t a n g e n t p l a n e as properties:

2

u ,

We hence see that if

- &) + s(&)

and

g =

it has d e s i r e d

s(u) - 2al&l 2 =

a solution to the problem.

also allows us to study the limits of the

< ~ a , A

in the canonical

ensemble:

=

from the convexity that

g~(a) * g'(a)

whenever the latter exists accor-

ding to the following lemma.

In this case as we have seen

is the value to which

converges

Lena

9. If

then

g(a)

gA(a)

~

are convex,

is convex,

and

Proof. The convexity of finite in a neighbourhood

g(a) of

g A ( a + b) - gA(a) ~ b.g~(a)

in probability as

differentiable

g'(a) = lim g~(a) A is clear. a.

If

A + =

and

g(a) = lim gA(a), A whenever g'(a) exists.

g'(a) is defined then

The convexity of

for all b.

u = - g'(a)

gA(a)

g

implies that

A p p l y i n g this for sufficiently

is

56

small b = •

i = I,...M, c fixed, such that

we find that

I~.ei-g~(a) l ! const as

Any limit point equality

g'

of the sequence

g(a + b) - g(a) ~ b-g'

means that

g' = g'(a)

gA(a + b)

{g~(a)}

for all

is finite

so that Ig~(a)l !c~

^ +-,

will then satisfy the in-

b .

But if

g'(a)

exists this

as we have seen, so all limit points are equal,

which means that the limit exists and is equal to

g'(a).

The relation between the m. can. and can. limiting values of typical observ-

Uo(q) ables

~

can now also be studied:

ensemble defined by

u

was that maximizing

is only one such value. Uo(q)

among

U(q)

Uo(q) PA ( - I ~ - ~ A l a )

The m. can. value of S(Uo,U) ,

a ~ = 0.

in the

supposing e.g. there

In the can. ensemble defined by

and put

u~

a

we include

Then

~A(AxRM,(o,a)) =

,

so as before

GA(RM,a)

lim ~

log P^ = s(A•

- g(a) = sup

(S(Uo,U) - a.u) - g(a) .

Uo~ A u~R M

Hence the probability mass will be concentrated

to values of

u

such that O

sup

(S(Uo,U) - a-u)

is attained.

U O ,U

Supposing that u = - g'(a),

u

has a sharp value,

we know that

sup

i.e. that

g'(a)

exists and

(S(Uo,U) - a.u) = sup (s(u) - a-u) =

U O ,U

U

= s(u) - a-u = Sup S(Uo,U) - a.u = S(Uo,U) - a.u ,

where

u~

is the

U O

m. can. value. Hence if the set

M

Especially

u

has a sharp value in the can. ensemble,

of maximal values of if

u

O

u

then

in the two ensembles are the same.

o has a sharp value it is the same in the two ensembles.

Let us now collect the results about the law of large numbers in the different ensembles: Theorem 3. Suppose that

s(u) > -|

and the conditions of Lemma 6 are valid.

U Cq) Then in the m. can. ensemble defined by

~

where

u~

M~

Especially

is the interval of values of

u

PA (--~I--6MolA) having maximal entropy

§ I S(Uo,U).

if there is a unique maximizing value the law of large numbers

57

Uo(q) holds for

then

~

In the can. ensemble defined by

PA(~MIa)

s(u) - (a-u) of slope g'(a)

~ I , is maximal,

a

touches

exists,

In this case

M

Ig(a) I < |

where

s = g(a) + (a.u)

This set reduces to a unique point

u

iff

M~

for

u = - g'(a).

r

also converges

Uo(q) ~

are related by

if

is the convex set in R M

i.e. where the tsngentp!ane

Epis.

and then

any other variable a

where

a,

to

u,

and the set

is the same for the two ensembles

if

u

and

u = - g'(a).

W e have called

s(u)

d e f i n e d by

It can also be expressed as the entropy of the correspon-

u.

the m. can. entropy of the macroscopic

ding can. density as follows:

state

The can. density is

-a.U(q) f(q) = e GA(a )

hA(a) = ~

,

so if we define its (Gibbs )-entropy per unit volume b y

f f(q) log f(q) m(dq)

Hence we see that if corresponding

to

a

g'(a)

,

we see that it is equal to

is defined,

so there is a unique

u = - g'(a)

then

h(a) = lim h (a) = g(a) + (a.u) Am A with

u = - g'(a)

,

i.e.

h(a) = inf (g(a) + a.u) = s(u) u Hence

s(u)

ding to u.

can also be computed as the can. Gibb's entropy for a corresponLet us now return to the important

in the following, and

U2

namely when

M = 2

and

is the total no. of particles.

will be

a I = 8,

region of e.

.

D g

a 2 = -8~,

and differentiable

UI

special case to be t r e a t e d is equal to the total energy

Then the parameters

and we will show that there,

s(e,n)

aI

> -~

and that it is an increasing

and

H,

a2

in an open function

is given by

g(8,~) = sup (s(e,n) - 8e + 8wn), e,n and will be shown to be finite when

8 > 0 ,

~ arbitrary.

Since

s(e,n)

is

58

differentiable the values of

e,n

the equations

8~ = -s~(e,n)

8 = s~(e,n) ,

g(8,U) = s(e,n) - 8e + ~ n the relation

.

corresponding to

Conversely,

, if

s(e,n) = inf (g(8,W) + 8e - SBn)

B,~

will satisfy

and then g(S,B )

is differentiable

imply that

e = - g~(B,~)

8~ (derivation with

8W = const.),

corresponding to

8,~

3.2. 3.2.1.

n = 8-I g~'(8,B)

for the values of

e,n

Derivation of the basic laws of thermodynamics The rules for thermodynamic equilibrium

In thermodynamics one considers systems consisting of some parts in AI,A2, etc. each of which when isolated can be described by e.g. a m. can. law with a few macrovariables

(el,nl) , (e2,n 2)

etc.

The equilibrium

values of interesting variables are then determined by maximizing the entropies of the isolated subsystems, subject to the constraints stipulated for each of them. When these constraints are changed the values of the macroscopic variables are changed to new equilibrium values, and one is interested in the general rules for the directions of these changes. Especially one wants to study how the energies are changed, what fraction goes into heat, mechanical work etc. Typical such changes: heat flows from a hot body to a cold one when an isolating wall is taken away, a gas in a cylinder expands when a piston is allowed to move, the concentrations of various substances are changed when they are allowed to mix, chemical reactions take place when the amounts of different constituents are varied etc. To start with, let us illustrate the arguments involved by considering three basic situations: thermal, pressure and concentration equilibrium between two systems, and at the same time we will define the basic intensive quantities temperature, pressure and chemical potential. Thermal equilibrium: scribed by

(e~,n I )

and

(e2,n 2)

Consider two systems in

AI,A 2

de-

which together form an isolated system,

and are separated by a wall, which can be changed from isolating to conducting with respect to exchange of energy:

A, I A.

S9

(In the following we usually denote the volume by IA21 = V 2 ,

V = VI + V2 ,

V ,

IAII = V I , vI v2 v i v I = ~-- , v 2 = ~ - .)

and volume fractions by

The total energy of a configuration

x = (Xl,X 2)

is

H1(x I ) + H2(x 2) ,

if the interaction with and across the wall can be neglected. (This interaction if present ought to be a surface effect of the magnitude

V 2/3 ,

and therefore small compared to the total energy of the magnitude V + - .)

This means that for the total system in

(el,n I, e2,n 2)

A = A I U A2

V

when

described by

we have

nA(AIXA2 ) = ~AI(A I) ~A2(A 2)

AI,A 2 C_R 2

(No permutation of particles between

AI

and

A2) ,

and hence the entro-

pies are additive: I

I

log n A = v I 9~

and in the limit

I

log ~AI + v 2 9 V~2 log hA2 ,

AI,A 2 ~ ~

vl,v 2 = const, eI

,

and

HI NI ((V-' V - ) ~ A I

e

H2 N 2 (V-' V-)~A2.

,n

l

we get

s(AI• 2) = VlS1(A I) +

nI

:

e

n2

+

are the restrictions defining

flA(AI•

If the wall permits the exchange of energy then the m. can. law for the system is defined by the restrictions H I + H2 NI nI N2 V ~ e = e I + e2 V I ~ Vl

n2 v2

so if

defined by these restrictions the distribution of e.g.

AC__Rh HI ~-

iS

the set

is asympto-

tically given by: H1

PA (V"-~AIA) =

sup s(el,n I, e2,n 2) - sup s(el,nl, e2,n2) , ADIeI~A} A

and in the limit when

A

are those determined by

e I + e2 = e ,

shrinks we see that the values taken by sup s(el,nl, e2,n 2) eI

i.e. by e

e1

-

eI

n2

when

HI V-

60

eI

The equilibrium values of

~sl(el,n 1 )

hence have to satisfy the equation

~s2(e2,n 2)

3e I

,

~e 2

81 = 82

i.e.

This makes it natural to identify

8 =

~s(e,n) De

as the (inverse) tempe-

rature of a system, because then the equilibrium condition for thermal contact is that the temperatures of the subsystems are equal, and for an ideal gas it coincides with our ordinary scale of temperature. From the convexity of Sl,S 2

8 = ~s(e,n) De

follows the important fact that

creasing function of

e.

is a de-

This means that that system which gives away

energy to the other is the one which increases its 8, i.e. the one I which decreases its temperature (= ~). We hence see that the convexity of

s

is intimately related with our most elementary physical experiences.

The fact that be ~ 0.

s(e,n)

is increasing in

(If a system had

8

e

means that

8

always has to

negative it would always give away energy in

thermal contact with another system however hot, a very weird property. It would he hotter than any other system with Pressure equilibrium:

8 > 0.)

Consider now the situation when the wall is conduc-

ting and also free to move as a piston not allowing exchange of particles.

IA.

A ~

:y

I Include its position where

A

HI v-~el,

I y

dVl

or equivalently

is the area of the piston). H2 v-~e2,

NI ~-~n

I 9

~A

N2 ~-~n

vI

as a state variable

for a state defined by

2 , y~fixed

is hence: de I

dn I

de 2

v~ ) dy ,

eI

s(el,nl,e2,n2,Y) 1

1

nI

= VlSl(~1 , ~11) V

(V log Ay = V log ~ Av I ~ 0

as

so

e2 n 2 + v2s2(~2' ~ 2 )

V § ~

even if

A,~V2/3.)

A

(~--= V

9

61 Hence b y the same argument as above the e q u i l i b r i u m values of valently

v 1,v 2

and

e I ,e 2

are such that

s

is maximal,

y

or equi-

i.e. the con-

dition is:

sup VlS 1 (ve--~11,-~-~) + v2s2 (v~, ~2) when

v I + v2 = I ,

e I + e 2 = e = fixed,

The equ. condition for and for

Vl,V 2 e_! I

el,e 2

we get:

~s I

n l_ 8s I

when

dv I + dv 2 = 0 .

I.e.

with

~s 6 = ~e '

eI

But w h e n

fixed. ~s I as 2 6 1 = ~e I = 82 = - ~e 2 '

gives as before

(Sl - v 1 De---?- v 1 8nl)dVl + (s2

~s 2

n2

3e 2

v 2 ~--~2)dv2 = 0

~s 2

e2

n__22

~11) = (S 2 - 62 ~2 + 82~2 v2) "

81~I

~s 8 =~e

e2

v2

8s 8~ = - ~-~

nI

(S 1 - 61 ~1 +

n I, n 2

~s ~n

8W =

g(8,~) = s(e,n) - 8e + 8wn ,

we have

so the e q u i l i b r i u m condition is 81 = 62 '

gl = g2 ' or

81 = 82 ,

8~Igi = B21g2

E a r l i e r we saw that b r i u m conditions

8-Ig = p = the pressure, so we arrive at the e q u i l i -

81 = 82 " Pl = P2

The i d e n t i f i c a t i o n of A2

p

by a constant force

H I + pAy ,

pA

for thermal and p r e s s u r e equilibrium.

on the piston.

so the equ. v a l u e s of

sup VlS I (v~, v~)

'

can also be seen directly if we replace the system

el,v I

The energy of

AI

is t h e n

are d e t e r m i n e d b y

when

e I + p a y = e = const

,

v I +v2=

I

a=

A v

Hence t h e equ. conditions are g l d V l + 8de I = O, de I + p a d y = 0 , Bp = g

as before.

when dv I = ady ,

i.e.

(gl - B1Pl)dVl = O,

and w e see that

6"2

Also in this case the convexity has a very basic physical interpretation: For

8

fixed

g(8,u)

also increasing in ted N

8P = g

is convex and increasing in

~ ,

but

~-~= Bn ,

n = -- ,

we see that

hence

so we see that if

is an increasing function of

constant, so

U,

p

n.

Hence if

u

V

decreases as

v

~u

is

is elimina-

is varied with increases, as

V

we expect. Concentration equilibrium:

If in the above situation

also free to vary subject to

~s 1 ~s 2 ~n-~ = ~n--~

for

the

equ.

n I + n 2 = n = const,

values,

(The above definitions of

6

i.e. and

81W 1 = 82~ 2 ,

p

s(e,n)

is linear with slope

~s(e,n) De

interval gives

= 8 9

and

n2

are

or

#1 = ~2 "

have the advantage that they do not

suppose that there is a unique value of e.g. If not

nI

then we must also have

8

e

corresponding to a given

on an interval and any

e

in this

This will happen at phase transitions.)

The methods illustrated above show the structure of the rules of thermodynamics which determine the values of macrovariables in equilibrium: A closed system consists of a number of weakly interacting parts

U I ,U 2,. ..

each described by a few macrovariables tions

s1(ul),S2(U2),...

defined by

UI ~-~Ul,

S(Ul,U2...)

= VlS(~1)

The entropy of the state of the whole system

U2 V--~u2 uI

AI,A2,...

having entropy func-

, ...

is then

u2

+ V2S2(~-'~2) + . . .

V. --A1 = v i

is a macroscopic volume, not necessarily = ? V. i z U I U2 The probability distribution of V ' V ' "'" in a m. can. ensemble V

'

where

V

defined by a number of additive restrictions, which can be written in general as Z D.u. + d.v. = u i zl zl with given matrices

D.

and vectors

l

the v a r i a b l e s

u.

1

and

d.

defining the couplings between

l

v.

respectively,

1

i s a s y m p t o t i c a l l y g i v e n by

U.

exp V ~

visi(~.1 ) - s ( u ~ z i

restrictions.

U.

,

where

s(u) = sup ~ visi(~.l) u i ,v i z l

Hence the equ. values of

are those which give the

sup

in

s(u).

ui,v i

under the above

which get probability one

Hence when new restrictions are U.

imposed the system will move to a state such that

Z. v i s i ( ~ ) l l

becomes

8.

6S

maximal under these new restrictions; any change from one to another equ. position that takes place under given restrictions will be such that the U.

total entropy

E visi( ~).. increases to its maximal value i

s(u).

"Die

I

Energie der Weld bleibt konstant, und die Entropie strebt einem Maximum zu." The equ. values of

ui,v i

satisfy equ. conditions which mean that various

intensive variables balance each other. The general structure of these equtions can be seen as follows: Consider also the ensemble where the restrictions are relaxed, and a parameter (row-) vector

a

is introduced corresponding to

u.

Then

g(a) is

given by: g(a)

=

sup

(s(u)

- a-u)

=

U U.

=

U. i

sup E vi(si(~,l) - a'D i ~ ) ui,v i i l i

- a'd i) =

= sup Z vi(sup(si(u i) - a'Diu i) - a.d i) = V. i

i

U. i

= sup E vi(gi(a'D i) - a'd i) 9 V. 1

1

Now, it is easy to see that

s(u)

is concave in

u

(Lemma 10 below). Then

we can use the basic facts about duality in Lemma 7

to

s(u)

and

g(a),

and we find: If we have

a. = a.D. i

in fact

such that

i

E vi(gi(a i) - a-d i) = 0

gi(ai) = a.d i

then

for all v..z Also, if

g(a)

is finite = O,

gi(ai) =

1 U.

U.

U.

l , = Si(~1 ) -- (a i 9 ~?.) I

i.e.

if

a i = s~(~!. I) ,

i

and if

ui,v i

satisfy the

l

restrictions then they give the U.

sup because u

U.

g(a) = Z. vi(si(~,l) - a.D i _!v. - a'di) = i

i

i

U.

= E. visi (''-~)v.- a.u , I

and hence

1

U.

E. v i s i ( ~ ) i

= s(u) ,

because

g(a) ~ s(u) - a-u

always.

i

Conversely, if we have

ui,v i

such that they satisfy the restrictions and

64

U.

Z. visi(~.) = s(u), i

then if there exists a nonvertical tangent plane

i

to

s(u)

at

u

then there is

a

such that

g(a) = s(u) - (a.u).

Then (unless some

all

a. = a.D.

i,

and putting

i

g(a)

is finite and

v i = O)

g(a-D i) = a-d i

for

i

0 = g(a) - s(u) + (a.u) : U.

Uo 1

= ~ vi(gi(a i) - (a.di) - si(~.I) + (a i ~.) + (a.di)) = l

l U.

=

vi(gi(ai)

- si(

) § (a i

l

l

l

U.

But then

U.

v. gi(ai) = si( ~ .) - ai -!I 1 U.

1

U.

for all

i ,

because

1 u. 1

U.

gi(ai ) ~ si v.

i v.

1

1

always.

This means that

and

a. l

-v. 1

are

U.

related by

We also note that if

ai = s~(~).

s(u)

is differentiable

i

at

u

then

Let us collect these results:

a = s'(u).

Theorem h. Consider a system composed of weakly interacting parts aSu. described above. Suppose that

s(u)

defined by

s(u) = sup Z. vis i ( ~ )

when

i

Z.D.u. + d.v. = u , i ,i 11

v. > 0 ---

is u.s.c, and that it has a non vertical tangent plane at {ui,v i} values,

satisfying the restrictions and having i.e. give

i

sup in

s(u)

vi > 0

u.

Then

are equilibrium

if and only if the corresponding

inten-

U U.

sive parameters

ai = s~(~)

satisfy the balance equations:

i

a i = a-D.l ' for some value at

u

then

equilibrium,

gi(ai ) = a'di a

of the parameter vector.

s'(u) = a . and

s(mu) = ms(u)

gi(ai) = a.d i for

If

s(u)

is differentiable

The equations

a. = a.D. express thermal etc. I i pressure equilibrium, s(u) is homogenous:

m > 0 . U.

We also give the proof of the convexity of

vi.s(~.l) l

and of

s(u).

65

Lemma

10.

If

s(u)

same properties

is any concave

as a function

of

u.s.c,

(u,v).

function

Also

then

s(u)

v.s(~)

has the

defined above

is

concave. Proof.

If

(u,v) = A(u',v')

+ (I - k)(u",v")

Uv = (_~_~v')~u' + ((I - v~)V") ~u" , u > ~v' u' s(V) - v s(~) xv's(~) u'

~s(~) If s

To see that s(u)

u" s(7)

or

' .

then this is still true if

Hence it is also true if both

u',u"

c > 0

u

u

s(u) ~ Is(u') + (I - l)s(u")

defined above take for any

ding to

so

+ (I - ~)v"s(~) u"

s - c > vs(~) is u.s.c.

(I - ~)v" ~

+

then

varies

and

v

slightly

because

vary slightly.

if

u = lu' + (I - k)u"

values

(u i,v i) (u?l" v"i )

with correspon-

such that

u! Z. vlsi(~.,) > s(u') - ~ i

etc.

1

Then

(ui,v i) = k(u~,v~)

restrictions

correspond

+ (I - k) (u[,v[)

are linear,

to

u

because

the

and

U.

s(u) ~ Z. v i s i ( ~ ) 1

1

u!

u?

_> ~z. v[si(~) + (I - ~1 z. v"sii (-~v''.)-" 1

1

1

1

h Xs(u') + (~ - x)s(u") - z . Since this is true for any Remark:

r > 0

s(u)

If there is only one system the above U1

having the restrictions s(u,v)

is convex.

= v-s1(~).

~-~u

,

-V - =

v

This is the convenient

one wants to consider

situation

corresponds

to

V1

and

V § =

Then

way of scaling the variables

a system whose volume

VI

as well as

UI

can be

varied. If moreover

u = (e,n),

a = s'(u,v)

are

so

s(u,v) = vs(e, n)

a I = s' = 8 , e

~

a 2 = s' = - B'~ n

then the components and

of

when

66

a3 = S . .e .s, . V e ds(e,n,v)

n s, = g = 8.p ' v n =

8de

-

The equation for

8udn

a3

+

so in an infinitesimal

change we have

8pdv.

can also be written

s(e,n,v) = 8e - 8wn + 8pv , which together with the previous is homogenous

one expresses

the fact that

s(e,n,v)

(Euler's theorem).

We can now prove a version of the statement that the probability a small subsystem

AI

in contact with a big system

A2

law for

(heat bath)

is

given b y the can. law: The equ. values of sup

VlS 1 ( v ~ ) +

uI

and

v2s2 (v~)

u2

are determined by

when

u I ,u 2 u I + u 2 = u = fixed Vl,V 2 = fixed

.

'he equ. equations uI

are hence

u2

If we n o w consider the situation when u2 uI vq = ~ - v~ ~ ~

we see that

sider any limiting value of equ. equation

Sn(U)

If

if it is differentiable.

continuous. determined by uI

Especially, it as

remains bounded.

s(u)

if

s! 1

v2 § ~

Hence if we con-

are continuous.

This is n o w

is concave and differentiable

Hence b y Lemma 9 ,

u___+ ~ = fixed, v2

we see that it has to satisfy the

~ s(u + u n) § s(u) ~ s(u + u),

s'(u + u n) § s'(u + u)

value of

uI

u1' Ul + Ul

s~ (~V) = s~ ([) ~ a

the case by Lemma 9: then

when

V 2 § ~, V I fixed,

and especially

because

s(u)

s~(u) § s'(u) s'(u n) § s'(u)

and

un ~

is continuous , ,

i.e. i.e.

s'(u)

is

In Theorem 3 we saw that the values of u I in the can. ensemble Ul a were those where s I' (~i) = a. Hence we see that any limit is an equ. value in the can. ensemble determined b y if there is a unique such value we see that :

uI

a = s~(u).

converges

to

67

Theorem 5. Any equ. value of with a large system for

A2

A I determined by

v 2 + ~,

u_ u . v2 +

uI

in a small subsystem

a = s~(u)

Especially,

in the limit

if

a

will converge to this unique value as

3.2.2.

AI

in contact

will he an equ. value in the can. ensemble

determines

u I + u 2 = u, uI

v~

fixed~

uniquely then

uI

v2 §

The rules for enersy changest work~ heat and their relation to entropy

With the help of the general rules derived in the previous section we can now study how the energies of the subsystems change when the equilibrium values of the state variables change. In this process we will see how to distinguish between that part of the energy which is called heat and that part which is called (useful) work. We will see how the celebrated rules for the operation of machines converting heat and work into each other come out of the laws for the equilibria. Consider first the definition of work. A typical system where energy can be stored and be made useful at a later time is a suspended weight:

~2

Its energy is a function of y:

E = Mgy + (internal energy),

and by raising

it potential energy is stored, which can be retrieved when needed by lowering it. Its entropy depends only on the internal energy and is hence constant when

E = Mgy + E (Neglecting energy lost by friction in the wheel etc.) o This is the property that makes it useful for storing energy, because when coupled to a similar system

F7

e.g.:

F7

88 the equilibrium positions of

y

in the total system are such that the

total entropy is maximal under the condition that the total energy is fixed. Since the entropy is independent the maximization,

of

and by an infinitesimal

y

it is not determined by

perturbation

(kick)

it can

be moved a large amount back and forth. Then the other system can be disconnected

and work has been taken in or out essentially without loss.

Similarly,

if it is coupled to the system with the piston considered before:

A

A~:

J then as we saw the equ. position is determined by v - ay

=

v

o

sup vs (~, ~ ~) v

when

const.

=

because there is no contribution to the entropy from the weight, and we assume that

A

and the weight are enclosed by isolating walls. If we

design the machine carefully by making ever make

vs(

, ~)

independent

of

M

y.

a function of

y

we can how-

The condition for this to happen

is that dM

s'de + gdv = O, or de = - pdv. If M is changed by an amount e which is chopped off and left at the height where it is located then

no work is done in this process, the mass chopped off is constant, varied so that be independent infinitesimal

Mg = P a of

v,

and the energy of the whole system including i.e.

de + Mgdy = O.

all the time when

v

is

Hence,

vai~gdthen

if

M

is

vs (~, ~)

will

and as before it can be varied back and forth by an

effort, so that work can be taken in or out of

A

and be

stored in the weight. In the above discussion we have considered macrovariables (variable)

infinitesimal

changes where the

all the time take their equilibrium values determined by the

restrictions.

Such changes are called quasistatic,

if the rate of change of the restrictions

and take place

is slow compared to the time it

takes for the variables to reach their equilibrium values with constant restrictions. constant

Such a change in which the total entropy of the system is

is called reversible.

Any change in a closed system with given

69

restrictions

which can also go backwards must be reversible

sense, because -

As > 0

in both the entropy change is ~ 0;

then

As = 0.

In general a reversible

to run hack or forth b y an infinitesimal

if

in this

As ~ 0

change can be made

extra effort as above. A gene-

ral system of the type considered before whose thermodynamic determined by a (vector-) variable

u

and has entropy

called a work sourte for some set of possible of

u

AI

change of its state is called work, because a work source

A2

state is

s(u)

quasistatic

if its entropy is constant under these variations.

ly, the change in the energy of a system

and

can be

variations Corresponding-

which occurs in a reversible if the system is coupled to

so that

2 Z D.u. + d.v. = u = const. ll ll I and energy is exchanged between the systems then the work can be transferred to

A2

in a reversible

change of the total system with an infinite-

simal effort as in the example.

In an infinitesimal

part of a quasistatic

change of a system we have: ds = a.du

since

s'(u) = a .

Hence if the components called

of

(8,-8m2,.. .-Sm L)

u

are

(e,u2,...uL),

then for a reversible

and those of

change we have

a

are

ds = 0,

L or 8de - 8 Z aldu I = 0 . Hence the infinitesimal work put into the sys2 L tem is given by ~w = Z aldu z, and the total work put in is the integral 2 of this expression u = (e,n,v),

along the quasistatic

6w = ~dn - pdv

(p.

66)

.

path describing the change.

When

We now come to the definition

of

heat. A typical heKt source is a gas in a container with constant volume, which only changes its state by exchange of energy quasistatically

through

thermal interaction with other systems. The energy put into it in such a change is called heat. sidered we have

If the system is of the general type we have con-

ds = 8de

if only

e

and not

u 2 ,...u L change.

the heat added in an infinitesimal

quasistatic

given by

du i = O, i = 2,...L.

~q = 8-1ds = kTds

when

Hence

change of a heat source is If an arbitrary

sys-

t e m A I is coupled to a heat source and only the energies are allowed to vary subject to

e I + e2 = e

heat.

In an infinitesimal

only

e4,e

change,

then the energy change in the system will be called quasistatic

change we also have

so also for an arbitrary

6q = 8--ds ~ 2 when only the energy changes,

ds I = 81de I

system

and not the other variables.

when

70

If we consider an arbitrary

infinitesimal

L

9

with

ds = a-du = 8de - 8 Z ~idul then 2 L de = ~w + ~q with ~w = Z aldu I and 2 6q = 8-1ds

change in a system

de

can be split into

.

Correspondingly change when

quasistatic

9

the change can be achieved in two steps: First a reversible

ds

stays constant,

then an irreversible

change of

e

with

du2,...du L = 0.

The first change can be thought of as brought

about by the interaction with

a suitable work source, and the second one by the interaction with a heat source,

and in this process

~w

and

~q

are the work and heat added to

the system. Hence we have arrived at the two fundamental

laws for quasistatic

changes

in thermodynamics de = 6w + 8q

(first law)

~q = (kT)ds

(second law)

The total change in u'

and

u"

e

and

s

along a quasistatic

path between two states

is hence given by U"

e" - e' = f 6w + 6q , uI

and

U"

s,,_s,=i~ u' kT Here the integrals

of

u'

e" - e',

to

since

u" , e

and

This property

but s

~w

and

6q

and

are functions

depend on w h i c h path is chosen from s" - s'

of

are independent

u = (e,u 2 .... u L)

is the important one in the treatment

of the path

. of thermodynamic

problems. In a given change

6w,~q

and

kT

can in principle be measured empirically,

and then the second law shows h o w (Hence thermodynamics w i t h thermometers

s

can be found empirically.

is the science where probabilities

and calorimeters.)

can be m e a s u r e d

71

Let us now see how these concepts

can be used to analyse the possibilities

of converting heat into work and vice versa~ a steam engine of some sort)

Consider a system in

A

(e.g.

which is coupled to two heat sources and

a work source:

,%

i

,%

The combined

system is isolated,

and for simplicity

sources are so big that their temperatures

assume that the heat

do not change m u c h when heat

is taken in or out. Suppose that the machine can run in a cycle thereby taking an amount of heat

ql

and delivering part of it as work

the work source and part of it as lost heat the maximal

w

that can be obtained.

Since the state of

A

q2"

returns to its initial value its energy and entropy its energy b y

w~

entropy does not change. The changes in the heat baths are '

Ae2 = q2' As2 = 82%2

from one equ. state to another.

Ae = 0

and its Ae] = - q1"

respectively.

(Assuming that the changes in them are quasistatic.) system is closed we must have

A

to

Consider one cycle of the operation.

does not change. The work source increases

ASl = - 81qi

w

We want to know what is

and

As > O,

Since the combined since the motion is

(We can imagine an isolating wall enclosing

opened at the beginning of the cycle and closed just when

its initial state, this state being an equ.

state of

A

A

returns to

when it is isolated.)

Hence we must have ql - q2 - w = O,

and

82q 2 - 81q I ~ 0 . This means that if

w > 0

then

ql > %2 '

and

82q 2 > 6 1 % 2 , i.e.

82 > B I.

72

(We can not have

ql > % = 0

q2 = 0

because then

81qi <__ 0

contradicting

.)

Hence we arrive at the fundamental

conclusion that we must have

T I > T2

if the machine is to deliver work. This is the famous statement that a perpetum mobile of the second kind is impossible. deliver a positive

w

with

81 81 <-- ql - ~ ql = ql (I - ~2)

T I <_ T 2.

Such a machine would

We also see that

with equality iff

As = O,

w = ql - q2 <-i.e. if the whole

system moves reversibly during the cycle. This important conclusion means T2 ~ I <-" (I - ~ )

that the efficiency

,

and that the maximal efficiency

achieved if the system works reversibly. see that

w

>_ (I - - -

.

Also, if

w < 0, ql < 0 ,

is we

This means that if heat is to be transported

from a cold to a hot reservoir it is necessary to spend work, and the

- -

T2 T1

The interest in these conclusions

is that they assume very little about

the structure of the systems and that the efficiencies tions of the temperatures

Similarly,

are universal

func-

and do not depend on the systems used.

we can consider a system in

A

whose state

u = (e,u 2

..u L)

changes by Au

from a given initial to a given final state thereby deli-

vering a work

w

vironment by

-(Ae + w,Au2,...AuL).

and we ass~mle that the environment changes by

to a work source, and changing the state of the enAgain, the whole system is closed

a = (8,-8a2,. 9 .-8~ L) ,

the intensive parameters

do not change much in the process, L 9 9 -S(Ae + W) + 8 Z ~ I A u l 9 2

If the process

so its entropy

is a possible motion then we have: L As - 8(Ae + w) + B Z ~IAul ~ 0 , with equality if the process is 2 reversible. Hence

of

73

L

.

.

w < 8-1As - Ae + Z a i A u I = w , and again we see that the m a x i m a l -rev 2 w o r k is obtained in a reversible change. We also see that if the change is cyclic so that

Au = As = 0 ,

then

w < 0 ,

so no cyclic

m a c h i n e taking in energy from one f i x e d environment and d e l i v e r i n g it L as w o r k can be constructed. (The function -B-Is + e - Z aiui = 2 =-8-I(s - a.u) m e a s u r e s the u s e f u l w o r k that can be obtained from the system in state

u

in an environment w i t h parameters

a .

It has b e e n

called exergy and is nowadays m u c h u s e d in discussions of e n e r g y resources. It takes its m i n i m a l v a l u e when

u

is the value w h i c h is in

e q u i l i b r i u m with the environment. When

u = (e,n,v)

, a = (8,-SU,BP)

e.g. we have

_8-I sl _ a.u) = -8 -Ivs.e ~ , ~n) + e - ~n + pv v

)

V

An example of a cyclic m a c h i n e that can in principle t r a n s f o r m heat into w o r k and conversely is the Carnot machine.

It consists of a c y l i n d e r w i t h

a gas and a moveable piston. It can be e x p a n d e d or c o m p r e s s e d either in heat contact with the heat sources at constant t e m p e r a t u r e t h e r e b y t a k i n g in or delivering heat, or in isolation t h e r e b y d e l i v e r i n g or r e c e i v i n g w o r k from a work source ~ n d l o w e r i n g or raising its temperature. quasistatically,

It runs

and its cycle of operations is the following:

T

6

d

a + b:

%$

The gas is in heat contact w i t h the hot reservoir at

TI

and w i t h

a w o r k source, and it expands at constant t e m p e r a t u r e taking u p an amount of heat

ql

and d e l i v e r i n g an amount of w o r k

e n t r o p y increases b y process is reversible.

B1q I

wI

to the w o r k source.

Its

just the amount lost by the hot source, so the

74

b § c:

At

b

the cylinder is isolated,

so no heat can be exchanged,

and

then it is allowed to expand reversibly in contact with the work source delivering to

T 2.

w2

to it. This goes on until its temperature

Since

c § d:

6q = 0

Similar to

reservoir at T 2. w3

has been lowered

all the time the entropy remains constant.

a ~ b,

compression

in heat contact with the cold

T2 q2 = TVI ql

The amount of heat

is delivered to it,

and

is taken in. Again the total entropy of the system remains constant.

d ~ a: Similar to rises to The net in at

TI

b § c, compression in isolation until the temperature

again,

wh

is taken in, and the entropy remains constant.

effect of the operation of a cycle is hence that TI

and

w = w I + w 2 - w 3 - w~,

work source and the cold reservoir at

and TI

q2

ql

is taken

are delivered to the

respectively,

in a reversible

way. We see that in order to get a high efficiency one ought to try to make the operation of a heat engine as reversible temperature

ratio of the heat taken in and out as high as possible.

When e.g. a Carnot machine is run backwards E.g. we get arefrigerator

it can be used as a heat pump.

if the cold reservoir is the volume we want to

keep cold. and a heating device for a house a house,

as possible and the

e.g. if the hot reservoir

is

and the cold one the exterior of it (at least in wintertime).

The usual approach to thermodynamics

is to take as basic postulates the

first law, conservation of energy and the second law, e.g. the impossibility of a cyclic perpetum mobile of the second kind. Then one can introduee the entropy and the temperature

from the equation

ds = ~ T

for

quasistatie

changes, because one shows that for any cyclic such change u J ~ = 0, so the equation s(u) - s(u O) = ; ~ defines uniquely a funcu o tion of the state u. Then one can go on and show that this entropy has the extremal properties we have found directly from its probabilistic definition. We see that the probabilistic basic laws of thermodynamics probability

approach gives a unified derivation of the from one basic assumption,

the microcanonical

law, and that in this way the mystical concept of entropy gets

a natural interpretation

and is related to the microscopic

description

of

75

the systems. An example of the relation between increase Consider from

in entropy and loss of work:

a gas in a container with a piston which is allowed to expand

vI

to

v2

in heat isolation:

If we expand it without work the total entropy with

e,n

const. This increase

away the possibility

e

~n , ~)

increases

in entropy means that we have thrown

of getting some work, which can be obtained if

instead we let the gas expand reversibly decreasing

v-s(

in contact with a work source

and having the total entropy unchanged.

Let us compute

the two cases for an ideal gas. Its entropy can easily be obtained from

g(8,~):

~lPI 2 GA(B,U) = E e 8~N 0

-- f e

Pm

dp

=

3N = Z e 8~N

--

=

0

3_ 2 =

exp e Bu v ( ~ ) _3 2

Hence

g(B,~)

is

g(8,U) = e 8W (~)

(c = 2mw)

,

and

e.n

are o b t a i n e d

from 3 e = - g~ = 2--8 " g 9 (Derivation with n=8

Hence

-I

8~ = const.)

g v =g

3n 3n s(e,n) = g + 8e - 8~n = n + ~ - - n log n (~ec)

5n 5n 3n 3n 2c = ~ - - ~-- log n + ~ - log e + ~ - log ~-,

vs(~, ~) n = 5n 2 We see that if

v

3/2 =

and hence

5n log n + ~3n 3n 2c - l o g e + n l o g v + ~-- l o g ~ - - .

2

increases with

e,n const,

the entropy increases b y

76

v2 n log~

2

whereas

,

if it is constant we must have

so that

log e + log v = const.,

el 2 v2 log--=e2 ~ log v~1 '

and

The same analysis applies

w = e I - e2 > 0 .

if we have a dilute solution in the container

and liquid outside the piston.

A

If we make a hole in the piston and pull it out without concentrations

will equalize and the entropy increase,

make the piston semipermeable a reversible

doing work the whereas

if we

and let the osmotic pressure do work in

expansion we can get work in the equalization

(This could be a method of extracting work by mixing

process.

sea- and river-

water reversibly.)

3.2.3.

The thermodznamic

Consider

u

and having entropy

a small but macroscopic A)

.

of first order phase transitions

a closed system in a large container

law defined by

of

description

AI

subvolume

AI

A

S(Ul).

In t h e o r e m 5 we saw that the possible values of a = s'(u)

systems goes to infinity. the tangentplane

(Let us put uI

of

v I = I.)

are those given by

in the limit when the size ratio of the

I.e. these values are the convex set

s = a.u + g(a)

M

uI

(not too close to the boundary

also has entropy function

the can. law w i t h

described by a m. can.

s(u). Consider the state

touches

Epi s:

M

where

77

M

reduces to a single point

otherwise

s(u)

corresponding i.e.

A

iff

g'(a)

varies linearly on

to a

then

uI

is very homogenous,

M.

exists,

and then

M = {- g'(a)}

When there is a unique u-value

has this value in any small subsystem

AI,

and we do not find that its local properties

vary much in space. If on the other hand there is a phase transition, macroscopic

regions where the local properties,

so

A

is filled with

density etc. are drastig

cally different and can take a finite no. of values

u" [ I ) ,~

%

u [f)

charac-

teristic of the different phases of the system (ice, water, vapour etc.), then at least if

AI

is chosen at random in

A

the value of

uI

should

be f v(i)u(i) '

=

u1 where

1

v (i)

is the

with probability

fraction

v li)"'

A1

of

IAI

o c c u p i e d by p h a s e n o .

will

fall

into

the region

i.

of phase

Because i,

where

!

uI

takes the value

u ~IJ,

and with negligible probability

lap with more than one such region. f v(i)u(i) u = Z ,

because

U(q)

All

u <'i)g M

since

it will over-

u I~ M.

Also

is essentially additive

I

f = ~.(Z

U(q (i)) + boundary terms)

where

q(i)

is the part of

q

I

located in the region of phase

u((i)) If now v (i)

u

i ,

and

.)u(i)

completely determines the probabilit~ law in

are uniquely determined by

varied in M. This means that

M

u

from

u = Z v(i)u (i) I ought to be a simplex in

that any point in it has a unique representation

U=

A

then also the when R M,

u

is

i.e. such

f (i) (i) Z V U ,

with

1

v (i) > 0

fZ v(i) = I 1

(Simplies in R2:

~

;

1tl

If this is the case we must have

f < M + I.

phase rule, which e.g. says that if bility law then

f ! 3.

(e,n)

) This is the famous Gibbs uniquely determine the proba-

(There are at most three coexisting phases in

an one component system, e.g. ice, water and vapour.)

,

78

If this situation occurs then it is easy to see that the average of any U o ~

other quantity u~

= f v(i)u(i) Z o " I

will also vary linearly with

u ~ M,

so that

as should be expected if the above picture with regions

with coexisting phases is correct. Proof: We have shown in lemma 6 f.f. sup S(Uo,U ) = s(u) u o is linear in u E M ,

is attained. because if

s(u') = s(u~,u'), S(Uo,U)

u = ~u' + (I - l)u"

s(u") = s(u" u") o" ~

is concave

that the value of

u

0

S(Uo,U) A Is(u~,u')

= XS(U') + (I - X)S(U") = S(U) 9

u

is that where

o

If this is unique for all

u6 M

e.g. and

= ~u' + (I - ~)u" 0 0 + (I - l)S(U~,U")

(Remember that

Hence

u is the maximizing value corresponding o summarize these results:

then it

s(u) to

then because :

is linear on

u,

M.)

q.e.d. Let us

If for a certain value of the intensive parameters

a

ding

then this set ought to

u-values

M

consists of more than one point

the set of correspon-

be a simplex if there are only finitely many phases and if determines the m. can. probability law. Any point in represented

as a mixture

extreme points phases, s(u) in

u (i)

and the

in

v (i)

f v(i)u(i) u = Z I M.

v (i)

completely

can be uniquely

f v(i) Z = I I

These represent the values of

u

of the in the pure

are the volume fractions of the different phases.

and the values of any other quantities Mj

~ O,

M

u

U o ~[

vary linearly with

u

which reflects the fact that the system is filled with a mixture

of pure phases in the proportions point iff.

g'(a)

directional

derivatives

is not defined, i.e.

" g'(a,b) # - g'(a,-b). transition.

v (i)

g'(a,b)

M iff.

consists of more than one in some direction

b

the

are different to the right and left:

(This is the origin of the name first order phase

The symptom is a jump in the first order derivatives

of

g(a).)

It is a difficult problem to completely verify that the above picture with a mixture of different phases really occurs from first principles.

This

has been done with great effort by Minlos & Sinai for the Ising model. Example

I. The Clausius-Clapeyron

one component

system.

formula for the vapour pressure in a

79

Let us consider

a system described

for some range of

(8,U)-

0 < I < I,

with

of

The tangent

(8,~).

precisely

for

I.e.

by

M

(e,n)~M,

s(e,n) = g(8,U)

for

u = lu (I) + (I - l)u (2)

for the pure phases being

s = g(8,~)

i.e.

which has two phases

is the segment

u (i) = (e(i),n (i)) plane

u = (e,n)

+ 8e - Sun

(e,n)~ M

+ 8e - Sun ~ 8(p(8,~)

is tangent

we have both

+ e - un)

and

=

In particular

the first equation

holds

for the pure phases:

" (kT)s(e(1),n ( i) ) - e(i) + ~n (I") = p(8,~) If we vary

(~,~)

slightly we have:

kdT.s (i) + (kT)ds (i) - de (i) + ~dn (i) + d~.n (i) = dp but from the second equations ds (i) = 8(de (i) - ~dn (i))

above we have ,

kTds (i) = de (i) - ~dn (i), ks(i)dT

+ n(i)d~ = dp

We can eliminate

d~ =

d~ ,

or

so

for

i = 1,2 .

and get

ks (I) _ ks (2) n(2) _ n(1) dT ,

dp = ks(1)dT + n (I)

ks(1) - ks(2) n (2) _ n (I)

nC2)s (I) _ n(1)s(2)

dp=k(

dp = k

~

n(1)

s(1) ~(2) n~1) n(2) I

1

n(1)

n (2)

dT .

) d~=

dT

functions to

Epi s

80

s (i) -~

Here

is the entropy per particle,

and

v(i) _ n ~I

the specific

n

volume per particle. The latent heat per particle in the transition from one ,s (1)

phase to the other is

_ s (2)

q = kT [n--V~

n-T~) ,

so we arrive at the famous

formula d_l~_ q dT - T(v (I) _ v(2)')' which expresses how the pressure changes in terms of the measurable quantities

3.3.

q,T,v

(i)

Some other uses of the concept of entropy

3.3.1.

Information theor~r

Let us consider the basic asymptotic problem in coding theory treated by Shannon, that of estimating how many messages have effectively to be considered when coding the output of an information source. For simplicity let us consider a markovian source (as Shannon did). The messages are long sequences alphabet for pairs

X.

x = (Xl,X2,...XN)

Let us make the m. can. markovian probability assumption

x~.

I.e. take as basic macrovariables

i,j

of elements from a finite

~ong

x I ....XN,

matively given values of

Uij(x) = no. of consecutive

and consider all sequences having approxi-

Uij(x)

as equally likely:

let

N

frequencies, and let

8

all sequences

having

xd~

u..

be the

~J

be a neighbourhood of UN--~ A

u = (uij). Then consider

equally likely with probability

~ ( A ) -I , ~ ( A ) = the no. of such sequences. The basic interest in coding the sequences is to estimate

~(A)

no. of sequences instead of IXI N bility assumption is valid.

when

N

is large, because only that

need to have code words if the proba-

Let us see that the estimation of

can directly be done using the methods developed before. think of

x~P

A = ~ ,N~,

~N(A)

(We can actually

as the states of a one dimensional "crystal" in

each point

n~ A

one of finitely many states

being occupied by an atom which can be in Xn6 X .)

81

As

N § ~

we have

I lim ~ log ~N(A) = s(A) = sup s(u) , N-~ u~A so the b a s i c problem i s t o compute

s(u)

.

This can be done by first con-

sidering the corresponding can. law and calculating

d e f i n e d by p a r a m e t e r s

a = (aij)

corresponding to

PN(Xla) = GNI(a) e x p - ~. aij Uij(x)

g(a).

This law is

U = (Uij) ,

and

xEX N .

1J -a..

Putting

A.. ij = e

zO

this can be written as

P~(xla) = a~l(a) ~1"x2

ax2,x 3 . . . AN_I,xN

ON(a) =

A

[

A

X l , . . . x N Xl'X2

...

A

x2,x 3

,

with

=

XN_I,x N

Xl,X N

(This is actually a markovian law, because for any

(AN-I)xl,x N

I < n < N

we have

A ...A ...A XlX 2 XnXn+ I XN_1,x N "'" XNIXl .... Xn) = ~ A ...A ...A

PN(Xn+1 ,

Xn+1...xNXlX2

A

...

XnXn+1

XN_1,x N

A

XnXn+ I

XN_iX N

=

,

with

GN_n(Xn,a)

GN_n(Xn,a) =

[ A Xn+1...x N XnXn+1

... A

XN_1,x N

= [ (A N-n" xN )Xn'XN '

and we see that this conditional probability depends only on on

Xl,...Xn_ I .

The transition probabilities

Xn,

and not

are

Axn,Xn+ I GN-n-1(Xn+1'a) PN(Xn+1]x n) =

GN_n(Xn,a) and the initial probabilities

P~(x I ) =

GN_1(xl,a) [ GN_1(xl,a) xI

Q~_1(xl,a) =

so we see that the markov chain

82

is not stationary when

N

is finite).

From the expression above for g(a): A = (Aij)

GN(a)

we can immediately find

is a matrix with positive coefficients. The Perron-

Frobenius theorem then tells us that is has a largest eigenvalue

l(a)

which is positive and simple, and that the corresponding left and right eigenvectors 1 so that

and

l.r = 1.

GN(a) =

r

have positive components. They can be normalized

From this follows that

IN-1(a)

~

r

1

+ O( 12 N-I )

with

12 < l(a)

so that

XlXN xl ~N l g(a) = lim ~ log GN(a) = log l(a)

From the fact that I,'i and r l(a)

l(a)

is simple it is not too hard to show that

are differentiab!e in all Ai~ , and that the derivatives of

can be obtained by the formal perturbation calculation:

I

t=

Xr

-----7

A + dA)(r + dr) = (I + dk)(r + dr)

Adr + dAr that ~k

~A.. ij

~

Idr + dl-r , multiply by

1.A = l.l: = 1.r..

i

1.dAr~

1

dl(1.r) = dl .

from the left, and remember Hence

dl = [

.. 1j

From this follows that

=

1

1j

j

so

j

g(a)

is differentiable, and

dg = d log I =

= k-ldl = I-I ~j i i dAij rj = - I-I i~ ~ li Aij daij r.j ,

~

1. dA..r. ,

-

I -I

8a..

i.

1

ij

A..

Ij

r.

so that

.

j

The correspondence between the can. law is centered at

u

and u

a

which says that

u = - g'(a)

if

hence gives

I-I i. A.. r. = u . . . i ij 0 ij Let us introduce We then have

Pij and Pi

Pi~ > 0 ,

by

Pi.i = I-1 r-11 A..Ij r.0 '

[. Pi~ = I , j

[. Pi = I , i

and

Pi = i. r.. l l

[. Pi Pi~ = Pj ' I

so

these quantities define a stationary markov chain with the right pair

8S frequencies: uij = pi.Pij

9

(Actually,

the non stationary

finite

converge to

N

GN_n(Xn,a)

P..

transition as

= AN-n r x "I i

GN-n-1(Xn+1'a) GN_n(Xn,a)

-I ~

PN(Xn+iiXn)

§

-I

1

r

~

when first

probabilities

r

rxn

Xn+1

,

so that

and

I -I r -I A r = P x n Xn,Xn+ I Xn+ I XnXn+ I

PN(Xl)

obtained above for

This is easy to see, because

+ O(k~ -n)

For the initial probabilities

FN(Xl )

N + ~.

as

N ~

similarly we have

x~

~ xl

N ~ ~

#

Px]

,

but

PN(Xn)

§

Px

xI

n

and then

We can now calculate

s(u)

n + ~

.)

from

S(U) = inf (g(a) + a-u) = g(a) + a-u

when

u = - g'(a)

.

I.e.

a A..

s(u) = log I - [. uij . log . .Aij = - [ uij log I-~1=I ij ij = - [- Pi Pij log r i P~j rj I = - ~. Pi Pij log Pij + 13 IJ + ~. Pi Pi~ (log r i - log rj) . 13

The last term is

~ Pi log r i - [. p~ log rj = 0 , l j

s(u) = h(P) = - [ Pi Pij log Pij , ij Let us sum up our basic results: can. laws defined by

Uij(x)

.

so we see that

which is Shannons famous formula.

We have considered the m. can. and For finite

stationary markov chain whose probabilities stationary markov chain defined above by

N

the latter is a non

converge to those of the Pi' Pij

having

Pi Pij = uij

"

84

Since

g(a)

is differentiable

our basic result about equivalence

ensembles, Theorem 3, tels us that for any neighbourhood PN ( U N - ~ A I a )

+ I

if

u = - g'(a)

and the number of such sequences in the sense that

~ log ~ ( A )

of

we have

,

QN(A)

N's(u)

increases as

: sup s(u) u&A

I lim lim ~ log RN(A ) : s(u) : h(P) a+u N-~

A 9u

,

e

=

eN-h(P)

so

.

This is Mc. Millan's theorem for this particular markovian case. If we consider the interpretation "crystal" the "energy" a sum of contrihutions is differentiahle transition.

I

of the above system as a one dimensional

aij Uij(x) = axlx2 @ ~x2x3 + "'" § a

from nearest neighbours.

is

The basic result that

g(a)

then means that such a system does not exhibit any phase

This result is easily generalized to any finite range inter-

action using similar arguments as above.

3.3.2.

Statistical models

As an example let us consider the so called Wilson model for predicting traffic flows between a given set of origins and destinations

in an urban

area. In this field it is very fashionable to use entropy arguments to derive several formulas. Consider the daily traffic from a given set of origins

i : 1,...n

a given set of destinations

persons each making

j = 1,...m.

There are

N

such a trip, and one is interested in predicting the total flows between all

O-d

x = {(in,~n)

pairs.

A microscopic

n = I,...N}

For a given set of

U.. 90

of the o-d with

U.

N'

J~uj

= ~

ij

U_.'

ij-

z

uij9

N

9

1O

U..

state description is a list pairs of'each individual person.

E U.. : N ij 10

having these flows is

to

the total no. of microstates

85

N N N' ~ , (e) ,

By Stirling's formula,

this means that

I s(u) = N~lim~ log ~N(U) : - ijZ u.1~ log uij

as N -~ ~

u.. = zj

U~ . i~ = const. N

One common m. can. probability assumption is to assume that all

x

having given values of

and

of

T(x) = Z

Oi(x) : Z Uij(x) j

and

Dj(x) = Z Uij(x) i

li~ Ui~(x) = the total travelled distance are equally

ij likely. The corresponding then obtained from

~N(O,d,t)

and entropy

~N(O,d,t) = . jZ u .ij

Z =o.

~

s(o,d,t)

is then

(u) ,

I

Zu..:d. i ij j Z i,. u.. = t ij z,] zJ and the principle that

"only the largest term contributes"

s(O,d,t) = sup - [. uij log uij u

gives

when

iJ

jE U.. ij

=

O.

I

Eu..=d. i

zj

Z

i.. u.. = t

ij

zj

J

zj

U .lJ. >-0 The argument used extensively before then says that the probability density of

limit

N § =

ui~(x) N

aN(u)

in this ensemble is

all probability mass of

values giving the s~D s(u) u

above.

~N(O,d,t)

U(x)

~

=

so that in the

will be concentrated to the

Hence this maximization principle

gives the values of N ~

'

u.. iJ predicted by the m. can. law in the limit The corresponding can. law is defined by parameters ai, bj, c

corresponding to

0~, D:, T J

and has density

88

PN(Xla,b,c) = GNI exp - Z

(a i + b. + clij) Uij(x)

GN(a,b,c) = Z exp - Z. (ai + bj + clij) Uij(x) = x ij N' : Z U

-(ai+bj+clij)Uij = (Z

II U . . '

H.. e

99 zj

zj

zj

The can. density of

-(a.+b.+cl..) N e

i

j

IS

)

ij U..

is then multinomial:

zJ

U.

N'

PN (Uja'b'c) = H U..' ij zj with

P.. = ij E

H ij

~

Pij IJ

e-(ai+bj+clij) -(a.+b.+cl..) " z j zj e

ij We see that

g(a,b,c) = log (Z

e

-(a.+b.+cl..) 1 J iJ ) which is differentiable,

ij so our Theorem 3 tells us that if the can. law is centered correctly, i.e. if -

' ga.

=

o~

=

dj

=

t

1

-

~.

J --

gc!

U.

then the averages of

9

which are Pij(a,b,c)

N

m. can. averages defined by

o,d,t.

will be the same as the

The centering equations are:

-(ai+bj+clij+g) ~e

=o.

j

-(ai+bj+clij+g) e

= d. J

i

-(ai+bj+clij+g) Z

i.. e

= t

These non linear equations have to be solved by some iterative procedure to give

a,b,c and then

Pij(a,b,c).

Reference: A.G. Wilson. Entropy in Urban and Regional Modelling. Pion, London (1970).

87

3.h.

Proof of the fundamental asymptotic properties of the structure measure in the thermodynamic limit

Let us now return to the proof of Theorem 2, which establishes the basic asymptotic property of the structure measure: lim

I

log fiA(A,a) = s(A,a) = sup (s(u) - a.u) , A

IAI

^-~

being an open

u~A

convex subset of RM. We first recall the basic definitions and assumptions in section 3.1. preceding the formulation of Theorem 2. The above limit is usually established by an argument based on superadditivity as follows. For any bounded region R

A

to

let

A c,

A'C A

be the set of points in

and define

~(A,a)

as

A

tion that all particles are restricted to be in

s (A,a)

with distance at least

flAiA,a) but with the extra restricA':

-a'u(q)

u•e

=

e(dq) = f l A , ( ~ A , a )

.

A

q6s A ,

(~ = 0

Then If

if

A'

is empty).

log fl~ (A,a) A = AIUA 2

is superadditive in

with

AI,A 2

A:

disjoint then

2~(A,a) > n'

(A,a)-fl' (A,a), AI A2 fl' we only consider configurations where all particles are A --

because if in in

A~UA~

because

,

ql

so and

q = (ql,q2) q2

I^11 U(q 1)

with

qiEA!1

then

are at least a distance

IA21 U(q2)

R

U(q) = U(ql) + U(q2) apart and

U(qi)

I

so just as in the proof of d) in Lemma h ' (A,a)

~A

>

I

9~ ~

~

~ (NI) ; NI+N2=N ~ A

--N

e

-a.u(%)

q16(A'1 )N1 = ~~

(A,a)

AI

9 ~'

(A,a),

A2

U(~ e dql IA21 -A

-a.U(~2)dq2

q.26(A'2 )N2

=

88

First we let

i^41=

I s 4 =-~--[ of

A

be a cube

IAoI2 s

2

log ~'A (A,a)

translates of

ss I ~ ss

As

with side

L-2 s

and volume

s = 0,1 ..... Then from the superadditivity follows that increases with

As

The limit of

so

4,

~'As

because

A4+ I

is the union and hence

_> (~'As

thus exists, call it

ss

s(A,a) = lim ss = sup ss 4~ 4>0 s(A,a)

will have the regularity required:

s(A,a) = sup s(C,a), CCA

where

C

is open convex with compact closure CA. This is easily seen first if s(A,a) = - | ~ lI o g

If

s(A,a) > - ~,

n'As ( A , a )

~( A , a ) . is a

> s(A,a)

-

for any

take

A4

such that

s

being a Borel measure on

C

e > 0

RM

has the above regularity, so there

such that

I

log n'A4 (c,a) > ~ ~

log a'A4 (A,a) - 3

and hence

s(C,a) 5 ~ l o g

n~(c,a) > s(A,a) - c 9

We can now directly prove that the properties described in Lemma 4 a)-d) are valid for A = A IUA 2

s(A,a):

with

~4(A1,a) ~ ~s so if

~

s(A1,a) = - ~

s(A1,a) > - ~ 1 ~log

nAs

a) and b) are clear.

s(A1,a) ~ s(A2,a)

e.g.

(A1,a) + fl~s

To prove c) suppose that

Then ~ 2 exp IAs s(A1,a) ,

then all terms are zero and

s(A,a) . . . .

then

1,a) § s(A 1,a)

and hence I

~log

nAs

§ s(A1,a) too,

i.e.

s(A,a) = s(A1,a) .

If

89

To prove d), for any > s(Ai,a) - e AI

and

A2

for

of the form

A(I ),...A(2d)

or

=

A2

,As

A6

Decompose

As I

~

61

into

log ~s

and 2d

62

>

needed for

translates of

As

As I

q(i)~ F (i)'' and with A respectively for half of the boxes. Then with

~d-1 ~ )

~ 7

such that

Consider only configurations in

q = (q(~) ,...q (2d)

)6 A I

~

take

i = 1,2. (The larger of

will do.)

and call them

~

e > 0

2zd U( (i))\

~

+ 2d_1+i~-~;

so we get a lower bound to

A I + A2 G' ( - a) A6+ I 2 '

AI + A2 ~

2

'

by considering only such

configurations. Hence A I + A2

s ( ~2

,

a) > ~ l Io g _

n' As I

A I + A2 ( 2 '

a) ->-

L

I

2 - - ~ l~

I

n'A 6 (At,a) + ~ 1 o g

I

>--3 s(A1'a) + ~ s(A2'a) - e,

(If

s(Al,a}

or

with

s(A2,a) = - ~

n~6(A2,a) t

E

arbitrary.

d) is trivially true.)

On the basis of Lemma h we see that Lemma 5 is valid for

s(A,a),

i.e.

s(A,a) = sup (s(u) - a.u) , uEA s(u)

having the properties described in Lemma 5.

In order to establish the existence of the limit for more general regions A ~ ~

we have to introduce a condition on

fects can be neglected"

as

Definition: A sequence

{A i}

van Hove if

and if

IAil § -,

A

saying that "boundary ef-

A § is said to tend to infinity in the sense of

l^i(d)l

- ~ - §

the set of points with distance at most

0 d

for any

d,

where

from the boundary of

Ai(d) A.. 1

is

90 Lemma 11.

If

lim ~

A ~ ~

in the sense of van Hove then

log fiA(A,a) ~ lim ~

Proof: We try to approximate converges to As

E.g.

log fii(A,a) ~ s(A,a) .

~i(A,a)

s(A,a). To this end fill

consider

Rd

of

A'

are contained in

Let

Ns

~A' !

~I^~[ ~

and

Take any

with many translates of

A(D+R)

L-2 s

if

D

Those which cover the boundary

is larger than the diameter of

be the number of those which are inside Ns

be their union, covering

A'

as the union of all such translates with the

vertices being integer multiples of

As

from below by something which

A ~ = [J I Ai

I^(D+R)l,

Then and

IA[

A',

and let

A o

IA'] - IAol ~ the volume of those

- IA'I ~ IA(s)l,

so

I

i^ol --1~+

0 ,

§ I .

C~A

as in the definition of regularity above. Then for con-

figurations of particles

qi 6 FA!

the combined configuration

q

is

1

EFA,,

=

if

A

u(q i)

and if

T ~ C

then

Z U(qi)dZ i i

is big enough,

C =-~---

so that

~

CCA

is close to I.

Hence integrating only over such configurations

q 6 FA

we get the lower bound

Ns fi~(A,a) ~ (fi~s

I log and as

Q~(A,a)

when

>

A

is big enough.

This gives

IA~II(log~(C,a))-~Ng"IAg"I

A § ~ 9

>~ logn~ (C,a} A~-lim

for all

s .

91

Hence for all

lim > s(C,a)

CCA

,

and finally !am > s(A,a) = sup s(C,a) A~ CCA In order to get an estimate from above we make another regularity assumption about

A,

which is somewhat restrictive but adequate for most com-

mon shapes. Definition:

{A i}

tending to infinity in the sense of van Hove is said

to be approximable by cubes if one can find such

A. ~ l

A. 1

with

l^il lim--

= inf d:o = e > 0 .

then the condition is true, and thickness The side of

~

(E.g. if

but if

A.

A. 1

i

is a spherical shell of radius

i

it is not. )

Ai' Li'

can always be taken of the form

because we can always take

s

= [l~

.L. i > 2-2 = ~I , -L.

c

to at most

Ai

is a sphere of radius

decreasing

we can always take

IAil

I

< ~

+ I

e.g.

s Li = L.2 1

if necessary,

and

c .4-d. By enlarging if necessary

and then

A . \ A. § ~

in the sense

of van Hove. Lemma 12. If

A ~ ~

in the sense of van Hove and is approximable by cubes

then lim L,~IT~log ~A(A,a) < s(A,a) Am or if

s(A,a) = - |

D = {u; s(u) > - ~}

and

A

if

s(A,a) > -

has positive distance to the domain of

In the latter case

Proof: Consider first the case

s(A,a) > - |

as in the definition above such that

A~ ~ A ,

~A(A,a) = 0

for

and take cubes and put

all

s(u) A .

A~A

A2 = A 1 k A .

If we consider only configurations having particles only in get a lower bound for

O'

(A,a)

9 O'

--

hI 1

A UA~

9 flh(A,a)

(A,a)

,

and

A2

log

,^,,

{A,a)

+

log

(A,a)

AI As

A ~ ~

we

g' (A,a): AI

.

A2

the left side tends to

s(A,a)

so using Lemma 11 for

'

~' A2

we get s(A,a) L c lim

c > 0

If

s(A,a)

s(A,a)

~

log

> i~

.

= - ~

flh(A,a) = 0

then

~a(A,a) +

~

(A,a)

(I - c)s(A,a)

= 0,

unless we knolw that

but

~' (A,a) 9 O. A2

argument by considering configurations

u(q2) --~A2,

~

where

we know that

A2

(q,q2)

and since

not

conclude

s(A2,a) 9 -

|

in

AUA~

Ac .

D,

U

so

we have

A 2 ~__ A c , is defined as

A c = {Au + (1-1)u2;

with

Hence the previous bound is replaced by

~2[A2,a)

> 0

if

~

is sufficiently large since

A2 ~ "

in the sense of van Hove and

s(A2,a) > - |

choose

A2

also we see that

such that

S(Ac,a) = - ~

Hence if we can ~' (Ac,a) = 0 AI

~A(A,a) = 0 , and

log flA(A,a) . . . .

To find a good choice of

A2

let

d = d(A,D) 9 O,

cd d(u,A) ~ d + ~-- e.g. and let

Then we claim that hence

in

~2(A2, a) 9 ~A(A,a).

Lemma 11 implies that

that

u

~

u E A , u2~ A2, A ~ c }

lim~ Am

U(q) ~--6A,

with

Then we know that for the total

A +

where the convex set

and hence

that

We therefore modify the

is a small neighbourhood of a point

U(q) + U(q2) ~ [All ~

GXI(Ac,a)

we c a n

,

S(Ac,a)

= sup u~A

d(Ac,D) ~ (s(U) C

A2

> O, -

a.u)

and take

be a sphere of radius so

= - -

Ac

uED

such

cd ~ - around

is disjoint from

D

and

u.

93

In fact if

vx = Av + (1-X)v 2

ux = Xv + (1-A)u we D

then

with

v~A,

v 2 ~ A 2, ~ ~ c

,

and

Ivx - uAl _< (1-1)Iv 2 - u[ _< 3--'cd and for any

we have

lw-u~i§ lu~-vl&lw-v[~d

and

lw- v~l + Iv~ - u~I & iw- u~I. Since

cd cd ~ (1-c)(d + ~-) ~ (1-c)d + ~--

lu I - v[ = (1-~)(u-v)

lw-v~l ,d- lu~-vl - Iv~-u~l 9 --

we see that

cd=c__d

--

3

3

3

as claimed above. Lemma

11 and 12 together

Theorem

2.!f

A

approximable s(A,a)

9-

establish

tends to infinity

in the sense of van Hove and is

by cubes then

~,

or if

lim i,~iT~log RA(A,a) = s(A,a) exists if A~ s(A,a) = - = and d(A,D) 9 0 . (s(A,a) = + =

possibly. ) s(A,a)

is inner regular:

s(A,a)

= sup s(C,a), CCA

where

C

is open convex with compact

and it has the properties

described

in Le-~a 4.

d(A,D)

= 0

A

90

then

~A(A,a)

for

If

sufficiently

s(A,a)

following

does not influence

the use of

s(u)

= R ,

If

u~5. If

> 0

when

of

D

in Lemma 5 d):

u~R

~A(dU)}

then the same is true for some open

for all

A ,

and by L ~ a

11

Agu,

s(A) = - |

so

so that ACD c

and

Hence ~ .

u~R

then for some

QA(A)

90

such

A

because

Hence

d(A,D)

in the discussion

where

R = ~J{the support of A Proof:

and

Le~mma 6. )

At this stage we can prove the characterization

GA(A) = 0

= - |

cA,

large.

(The fact that we have to make the extra assumption s(A) = - ~

closure

for all open either

otherwise R__CD,

A

u

is in the support

A~u.

s(A) 9 - |

~A(dU),

i.e.

From Le na 12 then follows that for all or

for sc:e open

and t h e r e f o r e

of

d(A,D) A~u

R~D .

= 0. d(A,D)

This implies that = d(A,D)

9 0,

and

u~D, s(A) . . . .

S4

3.4.1.

Properties of the entropy s(e,n)

We now consider the most important special case when a = (8,-8~)

g(8,~) = sup (s(e,n) - 8e + 8~n) e,n We will also prove that in

(e,n).

Since

creasing in

U = (H,N)

and

and we will give conditions to ensure that

e

s(e,n)

8 = s~(e,n)

is finite.

is increasing in

e

as remarked in 3.2.1

is equivalent to

8

being positive,

and differentiable s(e,n)

being in-

which is needed to

have a "physical" behaviour of the system. Consider first the situation when U2(q) = N,

U1(q) = total potential energy and

and call the correponding functions

natural condition to ensure that

g(B,~)

s(e,u), g(8,~).

is finite for

8 SO,

The ~

arbi-

trary is the following: Definition: U1(q) and some

is called stable if

K ~ O.

U1(q) ~ -

Stability implies that

g(8,B)

K.N

for all

q ~ R Nd

is finite, because

we have the bound: -SU1(q)+8wN

dq

<

N>_O q6A N

<

IAI N!

E N>O

= exp JAJesK+B~

so that g(8,~) = l i m ~ A We also have

log GA(8,V) ! e 8K+Sv

g(8,~) ~ 0

because

GA(S,V) ~ I .

(In Ruelle: Statistical Mechanics criteria are given for a pair interaction

UI(q) =

Z

i<j

u(qi-qj)

to be stable, and it is shown that if

u(q)

95

is u.s.c, then stability is also necessary in order to have finite for any bounded

Ipi12

N

energy

GA(B,~)

A.) We now want to include also the kinetic

Uo(P) = Z I

2m

in the energy and replace

U (q) by I

H(p,q) = Uo(P) + U1(q) . In order to do this include also the momenta p = (pl,...pN)E RNd

among the coordinates, so the state is described by

(p,q)~ ~] RNdxA N with the basic measure dP~N(dq) , and add U0(P) N>0 the othe~ observables and define the extended structure measure for

to A ~ R3:

-SU0(P)-SU1(q)+8uN ~A(A,a) =

Z f e N>_0 IAI-I(u0,uI,U2)EA

dPmN(dq) ~

(p,q)~RNdxAN (a =

(S,-S~)

,

Also, define log ~i(A,a)

U 2 = N.)

~(A,a)

as before with the restriction that

will be superadditive just as before, because there is no

coupling between different

Pi

or between Pi and qj,

still additive for configurations at least = s(A,a),

R.

(p,q)

All the arguments of

with

s(A,a)

3.h. showing that

can then be applied to the extended

~A(A,a),

we get following bound: Suppose that

~A(A,a) <

-

Z

so

I

N>o u0(p <~IAI

e

-SU0(p) dp

U(p,q)

is

having the q's separated by lim

having the properties describe

Lemma 4, a) is not valid, because the

<__ N>o

q E FA, . Then

Pi

AN

log ~A(A,a) =

except that the bound of

vary over all of R d.

u0 ~ c

q~

I

Lemma 4 and 5

when

Instead

(Uo,Ul,U2)~ A,

e(SK+8~)N

then

dq

N! i

dpe("K§ iPl2<__~elAi p~R Nd

Nd

z

(2"rrmcJAI) 2

N>_0

Nd , (-~1.

Nd

i^1 N e(SK+8~) N N!

(,2',,mc I^! ) 2 -expT At e 6K+6~ 0

(-{),

X

But

x.' _> (e)

for

eb x (~--) = eb

sup ~ sup N~O

x _> 0

SO

$

for

b =

2mclA{,

and

x>O

~A(A,a) ~exp{A{(2rmc + e 8K+BW) .

Hence the bound of Lemma h a) can be replaced by s(A,a) ~ (2wm) sup u 0 + e ~K+SW,

and

monA S(U) --< (2wm)u 0 + I , so s(u) < + | s(u0,ul,u 2)

always. can be expressed in terms of the "potential entropy"

S(Ul,U 2) as follows: Let

A = A0xAI•

be a small neighbourhood of

= z

;

dp ui(~)

]~TF.A2]L~L2 2mA0 - - [ ~ A

s

(u0,ul,u2).

Then

=~(dq) 1

q~A~ The p-integral is Nd

Nd

(2rmlAlu~) 2

- (2rm}A{%)

(Nd ~)

2

if

A0

=

(u'

"~

. 0,u0J

,

:

2rm{A{u3

so with sufficient accuracy it is equal to

(

Nd 2e

is approximatively equal to

~u2{A{ s

so when

~

4rmeUO (d-~--2 )

{A} ~ |

2

and then

s

) ,

A § (u0,ul,u 2)

we get

Nd 2 )

, and

hA(A)

97

du 2 s(u0,ul,u 2) = 7 1 o g

4~meu 0 _ (---~--u2) + S(Ul,U 2) 9

Let us call the first part The entropy

s(e,n)

of the total system described by

H(p,q) = U0(p) + U1(q) , s(A) =

sup

~(u0,u2), the "kinetic entropy".

s(e,n)

and

for

N = U2 A C R 2,

can now be identified from Indeed,

(e,n)EA

corresponds to

(e,n)6A (u 0 + Ul,U2)~A,

s(A) =

so

sup s(u0,ul,u 2) = sup (sup s(u0,e-u0,n)) (u0+ul,u2)~A (e,n)~A u 0

,

and hence s(e,n) = sup S(Uo,e-Uo,n) = sup ~(UO,n) + s(e-Uo,n) u0

u0

if this is an u.s.c, function of to check that if Note that because

s i ~ s(ei,ni)

s(u,n) = - U1(q)

for

(e,n) and

(si,ei,n i) § (s,e,n)

u < 0 and

is stable, so in the

according to Lemma 5 c). We have

s(e-u,n) = - -

0 < u < e + Kn

attained for some

in this interval because

u,s,c,

Let

~(u,n)

s(ei,ni) = ~(ui,ni) + s(ei-ui,ni) ,

sequence suppose that

u. § u. l

s ~ s(e,n).

e-u < - Kn

sup ~(u,n) + s(e-u,n) only values in u take part. Hence sup is actually u

the compact inverval u

then

for

and

s(e-u,n)

are

and by passing to a sub-

Then

s < lim ~(ui,n i) + l[mn s(ei-ui,n i) !

i

i

! ~(u,n) + s(e-u,n) ! s(e,n) as claimed.

~(u,n)

is increasing and differentiable

perties are inherited by e' > e"

and

s(e,n)

as a function of e.

s(e',n) = ~(e'-u',n) + s(u',n)

etc.

in

u,

and these pro-

Indeed, suppose

Then

s(e',n) i ~ ( e ' - u " , n ) + s(u",n) > ~(e"-u",n) + s(u",n) = s(e",n). From Lemma 7 concerning conjugate functions we recall that a convex function is differentiable check for

s(-,n).

iff its conjugate is strictly convex. This is easy to In fact its conjugate is:

g8

h(8,n)

= sup (s(e,n)

- Be) =

e = sup ( ~ ( u , n ) e~u

+ s(e-u,n)

= sup (~(u,n) u

- 8u) + sup (s(e,n) e

= ~(8,n)

and

+ h(8,n)

~(8,n)

dn= 2u

is strictly

is defined by

dn 8 , u = ~ ,

8u -

8(e-u))

convex in h(8,n)

- Be) =

8

because

is differentiable,

i.e.

8 ,

dn ~ = ~-- log (

Hence we see that it is the presence important

s(.,n)

.

Su(U,n)

~(8,n)

for the physically

=

,

so the same is true for (~(8,n)

-

dn nd 2wm11 ) - ~ - = 2 log (S Y of the kinetic

fact that

s(-,n)

energy which accounts

is increasing

and

8

posi-

tive. We can now also prove that (e,n)

.

plane at each point s~(e,n)

s(e,n)

is differentiable

Lemma 7 tells us that this happens

exists,

directions strictly conjugate

(e,n),

because

and since

as a function

iff there

si(e,n)

is a unique

of

supporting

exists this happens

such a plane is determined

iff

if its slopes in two

are given. We now claim that

convex of

in

BU.

s'(e,n) exists iff g(8,U) is n In fact, as a function of 8U g(8,W) is the

h(8,n):

g(8,~)

= sup (s(e,n) e~n

so Lemma 7 tells us that rentiable.

(Check that

~( 8 ,n) = ~-n d log (

) ,

g(8,') h(8,')

- Be + 8~n) = sup(h(8,n) n is strictly = ~(8,')

convex

+ h(8,')

so it is enough to check

if

+ 8wn),

h(8,-)

is convex~ h(8,n)

=

is diffeu.s.c.,

sup (s(e,n)

- Be).

e>:En The

sup

e § + |

is always attained because

s(e,n)

- 8e ! I - 8e § - |

If

- 8el

and

hi ~ h(8'ni)

= s(ei'ni)

- Kn i _< e i _< 8 -I (I - hi ) , nuity follows

as for

n = kn' + (1-k)n",

s(e,n) h(8,n')

(hi,ni) § (h,n),

so we can suppose that above. = s(e',n')

Convexity - Be'

as

e l. * e

and

is easily checked:

etc.

then if

then

u.s.

conti-

if

e = ke' + (1-k)e"

99

h(8,n) ~ s ( e , n )

- 8e ~ X(s(e',n')

=

-

kh(8,n')

(Remark:

(1

+

- Be') + (I - X)(s(e",n")

- 6e") =

k)h(S,n").)

Also

h(S,n)

= sup (s(e,n)

- Be) =

e

= sup (s(u,n)

- 6u) + sup (s(e-u,n)

U

- 6(e-u))

e-u

is always attained, the first one is for Moreover

h(8,')

(with slope g(8,W)

is differentiable

-6W)

is,

because we have just seen that the last sup nd u = ~-r. This fact will soon be needed.)

for each

= sup (h(6,n)

n.

iff there

-6U

is a unique

supporting

and

line

is such a slope iff

+ 8~n) = h(B,n)

+ 6wn = s(e,n)

- Be + Sun

for some e,

n

i.e.

iff

(8,-8u)

defines

a supporting

a plane is unique as remarked

above

plane to

iff

s

s'(e,n)

at

(e,n),

is defined,

and such and

n

8W

=

-

s~(e,n)

.

Hence it now remains

to prove that

g(6,~)

If it is not there is an interval where

g~(8,~)

~g(B,u)

for

= lim - -

where

convex

in 8U

it is linear,

9

i.e.

that then U2 ~2gA(6,~) 0 = g~(6,~ 2) - g~(8,~ I) = lim f A ~I 3u2

so

From Lemma 9 follows

~' < WI < ~2 9 W" 9

Hence,

if we can show that

of each point Let

(~',~")

exists and is constant.

is strictly

A

(6,~)

before

it follows that

be a big cube,

cubes with side

L. =

~(~)2

> --

~2

As e.g.,

Take e.g. Vat(N)

~

c > 0

g(8,~)

uniformly

in the vicinity

is strictly

which can be partitioned L > hR.

convex into

As we have seen several

in the g. can.

ensemble

in

A

in

6~.

K = 2s times

defined by i-

(8,B),

so we have to show that

Var(N)

> c{A I > O.

Let

{Ai}~~_

be the

K cubes of side

L

making up

A,

and consider

A

as

A0

Ai,

K

A0 = Ak

(J A! . I

l

Any configuration

x = (p,q)

in

A

is correspondingly

d~

100

partitioned into and similarly since

(Xo,X 1,...xK)

K N = X Ni . 0

depending on whether

Given

x 0 {x i}

and

qi~AO

{Ni} I

or

are independent

{qi}~. have no mutual interaction, only interaction with

The general relation for random variables

A!

q~u "

X,Y:

Var(X) = E Var(XIX) + Var E(XIY) ~ E Var(XIY) K Var(N) ~ E Var(Nlx 0) = X E Var(Nilx O) I

then gives

The distribution of tion in of

Ni

given

x0

is determined by the g. can. distribu-

A~I with interaction energy

qo

which is in

Ai~h!z

@

H(qilqo)

between

qi

and that part

Hence

"

-8H(x)-SH(qlqo)+8~n f x~Rnd•

e

dPmn(dq)

Pn(Xo) = p(N i = nlx O) = Z n

If we can bound e,g.

Po(Xo)

then we can also bound I Put

if

and

P1(Xo)

Var(Nilx O)

from below by e.g.

p(x O) > 0

from below:

N. 1

y = I0

d:o

if

Then Var(Nilx O) > E Var(NilXo,Y) > PoPl

I

P(Xo)

P(Y = I) 9 Var(NilXo,Y=1 ) = (Po + Pl ) (Po + Pl )2 To find

p(x O)

in

A.\ A: i

PO- I + p;1 ->

2

suppose that the interaction energy has the bound:

H(qlqo) > - K-N.-M. --

=

I

for some

being the number of particles

i

interacting with

I

a pair interaction

K > 0 , M,

I

A: .

Such a bound is valid if there is only

i

bounded below by

-K

e.g.

We can then bound

Po(Xo):

101

-8H(x)-SH(q]q0)+8~n dpdq I

-

Z

Po(Xo )

#

d - = exp ( )2

if

M. < M 1

n!

n>O (p,q)s

_ ~1~12 2m e z f n>__0 pERnd

<

e

8Kn+SKnMi+6un dp

IAi In n!

e

8K+BKMi+SU e

and

(L - 2R) d ! GM

(8,~)

vary in a neighbourhood of fixed values. Similarly:

--

d

(~)2 I P1(Xo) ~G~M

f (p,q)6Rd•

'

GM

because if we restrict the particle to be in at all with

eSU(L_~R)d

e-SH(x)+Su dpdq =

(A~)'

it does not interact

q0 "

Hence we have a bound

p(x O) ~ 2qM > 0

if

M I !M

and

(8,U)

vary near

fixed values. Let

KM

be the number of

Ai

which have

Mi ~M.

Var(N) ~ E(KM'qM) = qM-E(KM) . The remaining so

(K-KM)'M ~ Z. M i --
Then we see that

K-KN cubes have

M i > M,

, and

l

K~>_K-~,

E(K M) _> K -

We have

=

for any

W' > ~

Hence

~

~IE ( ~ )

.

8gA(B,~) gA(8,U ') - gA(8,B) ~(8~------5--! 8(~' - ~) by the convexity of

is bounded as

A § ~

gA(8,~) .

because

gA(8,~)

g(8,~)

i

K'L d =

so we finally see that Var(N) _> qM(JAJL -d if

M

constM JAJ) = JAJ qM(L-d

is chosen big enough.

const)M = JAJ c > 0

IAI

102

Let us f i n a l l y

study the convex region

show that this happens

in the i n t e r i o r

DCR 2

where

s(e,n)

> - =. W e

of a r e g i o n o f t h e f o l l o w i n g

shape:

o

{(e,n);

e > e . (n) mln

e . (n) > - K n mln In fact:

0 < n < n m a x} = int D

is a c o n c a v e f u n c t i o n

of n.

s(e,n) = sup (~(u,n) + s ( e - u , n ) )

e _> e m i n ( n )

then asUsoon

e - u i emin(n) to s h o w that then take

,

i.e.

sup s(e,n) e

as

e > emin(n)

so that > - |

~(u,n)

A = RI•

with

=

on

E

so if

s(e,n)

+ s(e-u,n)

> - =

u > 0

> - =.

for s o m e

so t h a t

We h e n c e h a v e

0 < n < nmax,

a n d can

for t h e s e n - v a l u e s .

U1(q)

A = (n',n")

,

we can find

in some i n t e r v a l

e . (n) = inf e mln s(e,n)>--

If w e p u t no r e s t r i c t i o n

2A(A)

,

in the d e f i n i t i o n

of

s(A)

,

i.e.

if

then

f mN(dq) q6A N

A

If t h e r e

are no h a r d core r e s t r i c t i o n s

integral

is

> ,~_N INAIN --~ N ~ '

and

between

the particles

1 2A(A) >

IAl(n"-n')

t h e n the

nlAl

inf (n) n~A

so t h a t w e get:

s(A) = sup sup s ( e , n ) n~A e for a n y f i n i t e For a n y

interval

n > 0

> inf (n l o g !) n n&A

A.

w e can h e n c e f i n d

sup s ( e , n l ) , s u p s ( e , n 2) > - | e

If that

n1< n < n 2

by taking

A

with below

o r a b o v e n.

e

n = ~n I + ( I - ~ ) n 2 , sup s(e,n)

s ( e i , n i) > - | ,

n

max

= + |

e = ~e I + ( 1 - A ) e 2

>_ s(e,n) >_ Xs(e 1,n I) + ( 1 - ~ ) s ( e 2 , n 2) > - =.

e

I.e.

> _ |

--

in t h i s case.

t h e n w e see

103

If there are hard core restrictions particles then

n

max spheres. To see that

nma x > 0

in this ease is obtained if side

L

so that

lq i - qjl >_ r > 0

for all

is at most equal to the close packing d e n s i t y of we remark that a lower bound to

A

is a cube w h i c h contains

regularly spaced w i t h spacing

~ ~~

N

~A(A)

cubes of

L + 2r:

Ifweconsider onlyconfigurations wherethere is only one particle in each small cube we see that

S

N(L + 2 r ) d < IAI

if

m N ( d q ) > (Ld)N

,

q~A N

and as before s(A) = inf n log L d > - | n~A The best choice of

L

is

if

L = n

n < (L + 2r) -d .

-1/d

s(A) > inf nd log(n -I/d - 2r) > - ~ n~A sup s(e,n) > - ~

for

- 2r, if

so we have

n" < (2r) -d

0 < n < (2r) -d ,

so

n

and as b e f o r e

> (2r)-d . max --

e

By a similar argument we can see that s(e,O) > - ~ In fact let

also for all A = AIXA 2

s(O,O) = 0 > - ~

be a n e i g h b o u r h o o d of

r e s t r i c t e d configurations above but w i t h have

U1(q) -~-- = 0 E A I ,

s(A) > inf

so that

e > 0 .

r

(0,0)

and consider the

changed to

R.

All of t h e m

so they are allowed, and again we have

nd log (n -I/d - 2R),

and as

A

shrinks to

(0,0): ~(0,0) > O.

ngA 2 An u p p e r bound to

hA(A) i

w h i c h gives

Z

s(A)

is o b t a i n e d b y ignoring the r e s t r z c t l o n on

<

N: --

s(A) <_ sup

z

(e-~NA)N

n log (e)

,

,

and as

A

Let us now collect the properties of

s(e,n)

we have found:

shrinks to

nEA 2

s(0,0)

< o.

(0,0):

U1(q):

104

Theorem 6.

Let

H(p,q) = ~ I

UI(q) is stable,

I~I2 +

- -

U1(q) > K . N

,

that for any two configurations interaction

U1(ql,q2)

the observables

for

(U I,N).

s(e,n)

is

)

;

ql 'q2

s(e,n) =

with

N I ,N 2

in

e > emin(n)

is increasing in

e

R.

,

where

Suppose also

particles the mutual Then the entropy for

sup s(u,n) + s(e-u,n), 0
is the entropy for

s(e,n) > - ~

int D = {(e,n)

and has finite range

- U1(ql ) - U1(q2) >_- K.NIN 2.

(H,N)

s(u,n) = ~nd - log (

U1(q) = U0(P) + U1(q)

D C R 2,

(U0,N)

and

s(e,n)

where

the entropy

where

, 0 < n < nma x} 9

and differentiable

in

(e,n)

,

and bounded,

s(e,n) ~ ~(e + Kn,n) + I 9 The conjugate function

g(8,~) = sup (s(e,n) - 8e + 8Bn) e,n

is bounded:

d 2 0

i g(s,~)

in

8~ 9

<_ (~)

e 8K+Bu

for

B > 0 , U

arbitrary and strictly convex

105

3.h.2.

The existence of s(e,n) when the interaction has infinite ranse

In this section we show how the proof of Theorem 2 can be modified when the interaction is of infinite range. For simplicity we only consider the case U(q) = (UI(q),N), a = (8,-8~), i.e we have only potential energy (and omit the bar used in the notation in the previous section). We have to make some assumption about the decay of the interaction energy between configurations far apart however, and a useful one is the following: Definition: U1(q) is called tempered if for some R>O and 6>d IU1(ql,q 2) - U1(q 11 - U1(q2) l ~ K'N1"N2(d(q1>q2)) -6 for any two configurations ql,q 2 with NI,N 2 particles respectively when their distance d(ql,q 2) [ E. A pair interaction UI(q) =i~u(qi-qj) is tempered if J

[u(x)[ ~

K'Ix[ -6

for Ix[> R

x e R d.

We assume that U1(q) is stable and tempered. As before we first consider the special cubes As with sides L'2 s and define ~s by re! ! stricting all the particles to be in A s At, where As is a cube with side L'2s163

We shall let Rs

, but more slowly than L.2 s

We also

make the restriction U1(q)" ~ ) s

A

more restrictive by shrinking A C R 2 by an amount es +0 in the e-direction. Hence ~e-SUl(q) + 8~N

!

~s

=

~(dq)

~es q~rA[ To start with we shall assume that A is bounded in the n-direction: n ~ c when (e,n)~A, and we shall see that we can take Rs = R~2ps U

es = e02-gs

for suitable R0, e0, e > 0, 0
106

The temperedness implies that if we have several configurations (q1' q2 "') with (N I N2... ) particles and d(q i, qj) ~ D then if q = (q1' q2 "'')

lu1(q) - z u1(qi) I ~ KD-6(Z NiN3) ~ KD-6(Z i

i<j

i

Ni)a. T

In comparing ~s

and ~s

we note that in As I we can pack 2 d

translates of As so that their distances are at least (L.2 ~+I - 2R~+ I) - 2(L-2 s - 2Rs R'2 ~s

= 4Rs - 2Rs I = 2Ro2Ps

if 2R0(2 - 2p) ~ R.

tt

Ag

^r

Hence i f in A~+1 we consider only configurations q =

(q( 1 ),

with q(i) in these translates of A~ and U1.(q(i) ) + N~ (~ - e~,l~l ) ~ A

then

JUt(q) - Z U1(q(i)) I < K(clAs

-6 g A

and

i

U1(q)

< 2-d

U1(q(i))

+ es

uICq) T~qT-

Z(

> 2-d

e~_

+ r163

U1(q(i)) Z(

r

a

A

so that by the convexity of A

Ul(q)

- 2~)

A

,~)~A.

(N

ZN( i ) ) i

...

q(2d) )

107 Hence if es I _< es -

~

u~(q)

A

N

(A~'~+I I -+ r163 ' ~

we have

) 6 A,

so q is allowed in As I and as before a[+l(A,a) ~ (a~(A,a))2d'e -SA , so for ss = ~

I

log as we have

ss I ~ ss - 8 ~

A

[ ss + 8(es I - es

or

ss I - 8Es I ~ ~s - 8~s So this time ss - 8~s is increasing, and has a limit s(A,a), and since s~

0 ss § s(A,a) also. Just as before it is easy to check that s(A,a)

is inner regular in A and has the properties of Lemma h, so it is given by s(A,a) = sup (s(e,n) - 8e+B~n) with s(e,n) = inf s(A,0). (e,n)6A Ag(e,n) It remains to check that es can be found so that

r163- r163 [ ~ d

= Kc2H06(2L)d2 ~(d-p6)

(es - es

) = e0(1-2-r

-es

Hence we see that since 6 >d we can take p d, and then 0 < e ~ (P6-d), and finally e 0 so that

r

[ K~2R06(2L) d.

In order to treat more arbitrary A § = we have to slightly strengthen the van Hove condition: Definition: {A.} is said to tend to infinity in the strong van Hove sense I

if IAil§174~nd if IAi(R)I

Rd f f (~)

, where f(x) is a continuons decreasing function

108

for x > 0 with f(O) = O. The proof of Lemma 11 then goes roughly as before: Lemma 11' If A ~ | in the strong van Hove sense and if A ' ~ A

is de-

fined by deleting a corridor of width R A along the boundary of A then if

R~

A

r~

+

0 and if ~(A,a) is defined by confining the particles to

!

A

I li___~m ~ l o g

I ~log

2A(A,a) > lim

~(A,a) > s(A,a)

if A is bounded in the n-direction. Proof: As before we fill A' with Ns translates of As N if AO = IU~Ai

{Ai}Ns

that

SO

l^'l -IAol < IA(D+R)I , I^1 -I^'l <_ IA(~)I. Take C ~ A

and consider only configurations q = (q1' q2 '''') with

qi~A~, A' being centered in A. with side L.2s163 l 1 by the temperedness JU(q) - ~ U(qi) I <_ Kc2(Ns163 1

Rs as before. Then

-~ --

Ns

- --~

Ns

^

I

T~

U(o.~ ) ~.

N~

Hence If T~ T § 0, ~ + I we see that if ( ~ - IAs i then ( ~ , ~) ~= A if A is big enough so tha~t q

e

d for all in A',

and as before we have

' ' nA(A,a) >_ (2s

NA -SA e

!

I

!

log 2A(A,a) > Ns163 I

I

log fls

- 8 A

Hence if we can arrange so that A ~ ~,s ~ ~

together so that

Ns163 I -~ I we get lira > s(C,a) for a l l C c A

and hence lim > s(A,a).

~7

A

We need to have D d

r~

~

+ O,

A

§ O, i.e. 2 s

~

§ O in addition to R d

U7

+ 0 in order

109

that ~N~I^~I

§ I as before. ~ ~

~

~C2Ro-61AI2-~

SO we need to have

IAi2-Zp6--~ O, and IAi2-Zd + ~. This is achieved if e.g. s is chosen so that Le~

IAI~2 (pa+d)~/2 as IAI §

~,

12' obtained by requiring that A + ~ in the strong van Hove sense

and that n is bounded in A can then be proved roughly as Lemma 12. t

Proof: AID A and A2 are defined as before, and A2 is obtained from A 2 by deleting a corridor of width E L along aA2, A I having side L'2 s

We

!

take C c-A and consider only configurations ql = (q' q2 ) in A U A 2 with restrictions defined by A and C respectively. Then

Iu1(qI) -

u1(q) - u1(q2)I ~ KC21AII2(R02~) -~ ~

I ~

~ -

1^21

Since T ~ T

u1(q) IA21 u1(m2) < i--rrr-

- -rr7

-

1

>- ~ the sum of the two energies is contained in

{Au2+(1-X)u; u2~ ~ u @ A , X ~ I/2}, and since C is compact C_A this set has a positive distance to A c. Hence A since T ~ § 0 we can conclude that

U1(q I ) t

if ~ is big enough so that, q.i is allowed, in the definition of ~s and as before s163

~ s163

A 2 also tends to infinity in the strong van Hove sense because

1^2(~)1 ~ I A(R) 1 + IAI(R)I ~d 89 (1-c)lAll

~ IA212 89

I^21H)I <

~ IAI ~ clAll,

so +

R)I<

const. T'AT' 'id

Rd Rd I/d Rd I Rd -< f(_~_1-cT ~ ) + const. ( T ~ ) -- f 2 ( T ~ ) § 0 as T ~ T § O. d Rs ~ 2~d(p-1 )§ A Now T ~ T const. O, and T ~ § o, so as before we can apply

110 l

Lemma 11

!

to 9A2 and get

c lim

log aA(A,a) + (1-c)s(C,a) <_ s(A,a)

for all C ~ A, and hence

lim < s(A,a) if s(A,a) 9 -~. A The case s(A,a) = -~ can be treated as before. We finally get rid of the restriction that n is bounded in A. For any open convex A ~ R 2 we define s(A,a) by s(A,a) = sup (s(e,n) (e,n) ~ A If n 9 c

8e + 8wn).

-

in A we have: -SU1+8~N le mN(dq) < NEcIAI qE AN

~^(A,a) < E "

<

e (sK+8~)N IAI N

Z

N!

~IAI

-

(Because ~ x _N hN:

= ~ b

<

(eBK+Sw+1cIAl

9-

e 8K+8~

c

x N bN <

(~)

if c

,)

-

x b ; bN =

(~)

~.'-

0N-T

Hence s(A,a) ~ c(8K+8~+I) - c log c

ex b

if b > x.)

%-)

if n ~ c in A.

If s = s(A,a) 9 -~ take c so large that n(SK+8~+1) - n log n < s-1 e.g. for n -9 c.

Then if A is partitioned into A ' U A ' '

according to whether

!

n < c+I or n 9 c

II

respectively we have s = max(s(A,a),

s(A

,a)) and

I!

s(A

) ~ s-l, SO '

I

'

s = s(A ,a) = lim I ~ [ l o g ~A(A ,a) In particular 9A(A',a) > e IAl(s-1) if A is large,and since 9A(A''

,a)~elAl(s-l)

by the estimate above we have t

t

aA(A ,a) ~ aA(A,a) 5 aA(A ,a) + aA(A lim I

log ~A(A,a) = s

also.

t!

!

,a)<_ 2aA(A ,a), and hence

111 !

If s(A,a)

= -~ and d(A,D)

for A sufficiently

large.

all A. Then ~A(A,a)

t

> 0 then s(A ,a) = -~ for any e, so ~A(A ,a) = 0 '' n sAl I For any s take c so that 2~(A ,a) < e' ' for

~ 2A(A

,a) + ~A(A

,,

<el^Is,,

,a) _

"

-

if A is sufficiently

large, and

lim I A-~~ ~

log 2A(A,a)

The kinetic

< s.

Hence lim . . . . A-~~

energy can now be taken into acccuntjust

proof that s(e,n)

is differentiable

requires

as before,

new methods

but the

to bound IAI

from below. Let us collect the results: Theorem 7. Let U(q) = (UI(q),N), tempered,

a = (8, -B~) where UI(q)

is stable and

and suppose that A -~ ~ in the strong van Hove sense.

Then

I lim ~ [ A-~=

log ~A(A,a)

with s(e,n)

= s(A,a)

= sup (s(e,n) (e,n)6A

- 8e + 8Bn)

= inf s(A,0) A~(e,n)

exists

if s(A,a)

s(e,n)

> -~ has the shape described

> -~ or if s(A,a)

int D = {(e,n);

e > e

= -~ and d(A,D)

(n), 0 < n < n min

> 0. The domain D where

before: }, max

and s(e,n) has the bound

sup s(e,n)

~ n log (~). As before

e

g(8,~)

= s(R2,a)

is bounded:

0 ~ g(8,U) ~ e 8K+Su.

112

3.h.3.

A system in a slowly varying external field, the barometric formula

In chapter 2.3 we considered a system described by a g. can. law influenced by a slowly varying external field giving a contribution V(kq) = Z v(kq i) i so V(kq)

to the total energy,

varies on a scale

k -I

v(x) x 6 R d

is a nice function,

which is long if

~ § 0,

We argued

that one can then regard the system as consisting of macroscopically infinitesimal cells of size

k-lAx

in which

stant. The cells are however microscopically

v(x)

is essentially con-

infinite, so one can hope

that their interaction can be neglected, and the total partition function is approximatively the product of those of the cells: Gk(8,v) ~

ff Gcell(8,-v(x))

.

Then as

k ~ 0

we ought to get:

X

(Ax) d lim k d log Gk(8,v) = lim Z ~ k+O ~+0 x

log Ocell(8,-v(x))

= ; g(8,-v(x))dx

.

Making suitable assumptions about the interaction this argument can now be made precise: Theorem 8: Suppose that with

u(q) >_- K ,

U1(q) =

Z u(qi-q j) i<j

lu(q) l < K I q 1 - 6

when

is a stable pair interaction

lql >_R

for some

and suppose that there are hard core restrictions

so that

always and the number of particles in any region

A

some

c > 0 .

Then if

e -By(x)

~ > d, R > 0, lq i - qjl > r > 0

is at most

is Riemann integrable and

clA 1

-s(u I (q)+v(~q)) Gk(8,v)

= ~ e

~(dq)

is finite,

and

lim kd log Gk(8,v) = f g(8,-v(x))dx < |

Proof: From the stability follows that

%(B,~/_<

z N>__O

=

Z N>O w

e~

f

Gk(8,v) is finite:

e-sv<~) ~ =

q~RNd

N = k-d eBKN N---7 (f e-BV(X)dx) x-Nd exp eBK~

k d log Gk(8,v) ! e 8K f e-SV(X)dx

.

e-BV(X)dx

for

; e-SV(X)dx <

,

and that

113

The same bound is valid for

f g(8,-v(x))dx.

The following bounds hold for the interaction energies since the density is at most

c:

The interaction between one particle at the origin and

those at a distance

I

> D > R

is bounded by

E u(qi) I < const. Z (s lqiI~D -D

- s s

< const, =f s163 -D

< const Dd_6 --

The interaction between one particle at the origin and all others is bounded below by Z u(q i) =

i

Z

u(qi) +

lqil!R

E

u(q i) ~ -

const.(R d + R d-6) = - const.

lqiI~R

The interaction between the particles in a cube

A

with side

L > R

and those outside is bounded below by L

E qi6A

E u(qi-q j) = E E + Z qj~A c d(qi,AC)<_R s

> _ const.(R.id-1 + --

>

s

--

- const.(RL d-1 + L d-1 ~ s163 R

Ak

think of

Rd

having side

E

(L-s d - (L-s s

Now, let

Z s163

be a cube with side

~-

L

const.L 2d-6.

centered at

L.k, k ~

d ,

and

as partitioned into ~ A k . Let A~ be concentic with k -d L - 2D. Finally let A be a big cube with side s

consisting of a certain no. of the

Ak'S.

12 I: r:

Ak

114

First we get a lower bound for with particles only in Ak!

A~

Gk

for

by considering only configurations

Ak~A.

Then the interaction between

and the other particles is bounded above by

Z

Z

u(q i - qj) ! c'Ld'Dd-6 --- a ,

qiEA~ q ~ -Sv~ so if

e

= inf

e-SV(kq)

q~A k then -SZ(U1(qk)-SNkV~-SA) k

n ~(dqk) = H G~(B,-v~) e-B~

e

k

qk~TA~

k

and

xd log G/e,v) L Z (Xn)d L -d log aA,(8,-v ~) - B~dIAIL-d~ . k As

~ ~ 0

where

and

x = iq

s

o

is fixed, so that

ranges over the cells

~dlA I = s l-A k

the sum is a Riemannsum

with side

Ax = A-L.

Hence we

have

lim Ad log

GA(8,v)

L -d log GA,(8,-v(x))dx - 8s

Ix l!~ then as

L, D ~ ~

with

D ~ § 0

d-6

o the integrand converges to

g(8,-v~x)),

so by bounded convergence we have lim >

[

and finally as

g(8,-v(x))dx ,

s § |

lim > f g(8,-v(x))dx . l+O To get an upper bound we note that the previous estimates of the interaction tell us that if

qk ~ F A k

for

Ak~A

and

q~rAC

and

total configuration then

U1(q) - Z U1(qk) - U1(q~) >_- C'L 2d-6" IAI'L -d = A k

,

q

is the

115

so if

e

-By" -Sv(lq) k = sup e q6A k

then

GI(8,v) _< (Hk GAo(8'-v~'))~ GAc(B,-v) e 8A . By the same estimate as in the beginning

I d log GAC(8,v) ! eB~xgl.A c~ as

s

the side of

e -Sv(x) dx = es § 0

h-A, § =

As before we thus have

li--~ I d log Gk(8,v) <

~

L -d log G A (8,-v(x)) dx * aZ + 8CL d-6

-Ixil!~

x~

and as

L + =

and then

o

s +

ii--~! / g(s,-v(x)) dx .

3.5.

The central limit theorem for macroscopic variables~ thermodynamic fluctuation theory

When we are interested in the distribution of the macroscopic variable Ui(q)

in a small but macroscopic region

A

we have seen that it is

given by the can. law with

gA(a) = ~

log

f e -a'U(q) ~(dq) q~T A

.

In Theorem 3 we showed the law of large numbers for u = - g'(a)

uA

=

is defined then

(r

~ §

=-g~(a) §

u

~

:

If

in probability and

as A + -

A,a Since

U(q)

is the sum of many small contributions it is not unlikely U(q) - lAInA that the central limit theorem should hold for ~ X(q)

IA1112

116 The covariance matrix of

X

is

g~(a) ,

because

a~gA(a) A,a = a a"~ a "J ~= (gX(a))iJ

<XiXj) Hence if

g~(a) § g"(a)

we can expect that the distribution of

X

con-

verges to a gaussian with these covarianees. This can also be seen by considering the generating function for X.

~e-b-U /\A , a

= exPtAi(gA(a@b)

~/e - b x

-- e~Pl^lCgA(a+blAI - ~ / 2 )

\ /A,a

, f (l-t)

= exPlAI

- gA ( a ) )

'

so

- gA(a) - IAt -1/2 b ' g ~ ( a ) )

=

d~g^(a+tblA1-1/2) dt =

0

dt 2

I

--

exp

j" ( l - t )

Is-g~Ca+tblAI-t/2)b

dt

0 by Taylor#s formula. We hence see that if (g~(-)}

are equicontindous at

a

g~(a) § g"(a)

~nd if

e.g. then

<e-b'X>A,a+ exp -b'g"(a)b2 and

X

has the gaussian limit distribution with covariance matrix

g"(a).

Its density is usually derived in physics by the following argument: As we have seen in the proof of Theorem 3 ? $

P(!~du)

~

explAl(s(u) - a-u - g(a)),

so if this approximation is good enough

P(X~) Since

~ exPlAl(s(u+xlAj -1/2) - a-(u+xlAI - l / a )

u

and

a

correspond by

we have as in Lemma 7.

P(Xe~)

~

g(a)

expIAI(s(u§

=

u = - g'(a) s(u) - a-u

- g(al)

in the duality relation and

a = s'(u).

Therefore

-I/2) - s(u) - 1^I-I/2x.s,(~))~

exp--, 2

so the limiting gaussian should have this density, and hence the covariance matrix

(-s"(u)) -j.

To see that this is the same as

duality correspondence says that the mappings

g"(a)

u = - g'(a)

we note that and

a = s'(u)

are inverses of each other. Hence their Jacobians are also inverses of

117

each other, i.e.

du

~da~ -I = (s"(u))-~

(~)

=

-

g"(a)

=

"du"

An important feature of this limit theorem is that the limit law is completely determined by derivatives thermodynamic

significance

of

g(a)

which have a direct

and can be experimentally measured.

an important insight of Einstein:

thermodynamics

This was

is also related to

fluctuation phenomena and gives their variances e.g. Brownian motion or density fluctuations the case

u = (e,n)

in a gas. To see this relation let us consider

a = (B,-8~)

dg = g'(a)da = - uda

and

,

du = - cda

g = 8P.

Then if

" 'a) ~ cij gijt

gives

d(Sp) = - edS + nd(S~) ~de = - c11d 8 + c12d(Bu)

I

dn

- c21d8 + c22d(8~)

If we eliminate

d(8~)

.

we get

d(B~) = ~ d8 + I d(Sp) = e+p dS + ~ dp n ~ n

le

=

n

(c12e+--t n p-(c22

e+p n

c11

)dB + c12 ~B dp

c21)d8

+

c22

~

dp

Thus we see that e.g.

c22 and

c22

~n = = (~PP)T the isothermal compressibility is the variance of the density fluctuations

etc.

Let us also consider the derivation of the famous formula for the motion of a small but macroscopic state is determined by

brownian particle moving in a fluid or^gas.

(p,q)

and its energy is only kinetic

I~~-

Its

.

It is small compared to that of the whole system, so the probability

law

~1~12 of p

(p,q)

should be given by the e x p o n e n t i a l law

has a maxwellian distribution,

and

q

c'e

"2m dpdq ,

has a uniform distribution

i.e. in

the container. The average motion of the particle can for moderate velocities and accelerations

be described by the equation

force excerted by the m e d i u m = calculated in hydrodynamics

~ = the average frictional

- mv = - a ~m ~ where the coefficient is (Stoke's formula) . a = 6W~r for a sphere

118

of radius

r ,

fluctuations

where

in

p

n = the viscosity of the medium. The small random

should then be described by the same equation but

with white noise added,

~ = - ~ p + ~

according to the general philo-

sophy described for the Ehrenfest model in ch. I. fluctuations

in

p

should be described by a Gauss-Markov process

t - ~(t-s) S e m ~(s)ds

p(t) =

-= with variances

This means that the

r

=

,

whose equilibrium distribution

7 e -i~t m a2dt

mo2 . = 2~--

is gaussian

This variance has to

0 agree with the maxwellian one to

o

2

2a =--~

m ~ ,

which means that

a2

has to be equal

This is called the fluctuation dissipation relation,

because it says that the coefficients

o

and

a

describing these two

types of forces have to be related in a specific way. The correlation tion of

Pi(t)

is then given by m - ~ltl = ~ 6ij e

p~(t~

t q(t) - q(0) = S p(s) ds m 0

and since

t t

qi(t)-%(o))

. --If

-

2

t

u

0

0

is large.

- ~(u-v)

du dv = m~

2

t

- s

2t

at

m

It is in fact not hard to show that as

approximatively ~P I ~-~= aS A P .

is given by

lu-vl

e

O0

m8

its variance

a Wienerprocess

This is Einsteins

m

m

i small

q(t)

is

governed by the diffusion equation

celebrated result that the position of the brownish

particle is described by a Wiener process with diffusion constant I kT D =5= 6=,r Another example of the fluctuation dissipation relation is the Nyqvist formula for the fluctuating current in an electric circuit. Consider a simple one, e.g. an

R-C circuit:

func-

119

in thermal contact with a heat bath with inverse temperature

8 . The

state of the system is e.g. described by Q, the charge in the capacitance. ~2 Its energy is ~ , so the exponential probability law is _

const e

BQ2 2c

giving

Q

c ~ . The average equations of V e Q ~ = I = - ~--= - R-~ ' so the fluctuations

the variance

motion are given by Ohm's law

should be described by the Gauss-Markov This time the

F-D relation says that

process defined by ~2Rc = ~ , 2 8

i.e.

2

Q = - ~- + a~ Rc 2 = ~-~ .

The

equations of motion can be written RI

=

RQ

=

- ~

c

+

R~

=

-

V

c

+

R~

,

which means that the noise can be thought of as generated by a voltage source

Ra~

in series with R:

It has correlation function with the constant density circuits:

each resistance

(2kTR)~(t-s) , and hence power spectrum dm (2kTR) ~ . This is a general rule for gives rise kTR

with it having power density

References:

and lb. The barometric

in ref. 9

formula is derived in:

Fields.

of Particle Systems in the Pre-

Commun. math. Phys. 27, lh6-15h (1972).

The Central Limit Theorem for thermodynamic R.L. Dobrushin,

in series

limit of entropy etc. is discussed

E. Presutti. Thermodynamics

sence of External Macroscopic

Equivalence

source

w

The thermodynamic

C. Marchioro,

to a white noise

B. Tirozzi.

of Ensembles.

variables is discussed in:

The Central Limit Theorem and the Problem of

Commun. math. Phys.

and in several references given there.

5h, 173-192 (1977),

120

References I. 2.

H.B. Callen. Thermodynamics. Wiley 1960. C. Domb, M.S. Green. Phase Transitions and Critical Phenomena Vol. I. Academic Press 1972.

3.

P. & T. Ehrenfest. Begriffliche Grundlagen der statistischen Auffassung in der Mechanik. Enc. der Math. Wissenschaften bd. h, Teil 32 (1911).

h.

W. Gibbs. Elementary Principles of Statistical Mechanics. Dover 1960.

5.

K. Huang. Statistical Mechanics. Wiley 1963.

6.

M. Kac. Probability and Related Topics in Physical Sciences. Interscience Publ. 1959.

7.

A.I. Khinchin. Mathematical Foundations of Statistical Mechanics. Dover 1949.

8. 9.

L. Landau, E.M. Lifshitz. Statistical Physics. Pergamon Press 1969. O. Lanford. Entropy and Equilibrium States in Classical Statistical Mechanics. In Statistical Mechanics and Mathematical Problems. Springer Lecture Notes in Physics 20. Springer 1973.

10.

R.A. Minlos. Lectures on Statistical Physics. Russian Mathematical Surveys 23 (1968) no. I.

11. 12.

A.B. Pippard. Classical Thermodynamics. Cambridge University Press 1957. C. Preston. Random Fields. Springer Lecture Notes in Mathematics 534. Springer 1976.

13.

R.T. Rockafellar. Convex Analysis. Princeton University Press 1970.

lb.

D. Ruelle. Statistical Mechanics. Benjamin 1969.

15.

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Texts and Monographs in Physics Editors: W. Beiglbock, M. Goldhaber, E. H. Lieb, W. Thirring ABohrn

Quantum Mechanics 1979. 105 figures. Approx. 570 pages. ISBN 3-540-08862-8 O. Bratelli, D. W. Robinson

Operator Algebras and Quantum Statistical Mechanics Volume 1 The Mathematical Theory ofC*-and W*-Algebras

1979. Approx. 500 pages. ISBN 3-540-09187-4 HPilkuhn

Relativistic Particle Physics

1. Kessler

Polarized Electrons 1976.104 figures. IX, 223 pages. ISBN 3-540-07678-6 W.Rindler

Essential Relativity Special, General, and Cosmological Second Edition 1977. 44 figures. XV, 284 pages. ISBN 3-540-o7970-X K Chadan, P. C. Sabatier

Inverse Problems in Quantum Scattering Theory 1977.24 figures. XXII, 344 pages. ISBN 3-540-08092-9 1. M. Jauch, F. Rohrlich

The Theory of Photons and Electrons

RD. Richtmyer

The Relativistic Quantum Field Theory ofCharged Particles with Spin One-Half Second Expanded Edition 1976. 55 figures, 10 tables. XIx, 553 pages ISBN 3-540-07295-0

Principles of Advanced Mathematical Physics I

C. Truesdell, S. Bharatha

1979. Approx. 400 pages. ISBN 3-540-09348-6

1978.45 figures. XV, 422 pages. ISBN 3-540-08873-3 R M. Santilli

Foundations of Theoretical Mechanics I: The Inverse Problem in Newtonian Mechanics 1978. 5 figures. IX, 266 pages.

The Concepts and Logic of Classical Thermodynamics as a Theory of Heat Engines Rigorously Constructed upon the Foundation Laid by S. Carnot and F. Reech 1977.15 figures. XXII, 154 pages. ISBN 3-540-07971-8

ISBN 3-540-08874-1 M. D. Scadron

Advanced Quantum Theory oUts Applications through Feynman Diagrams 1979.78 figures. Approx. 300 pages. ISBN 3-540-09045-2

Springer-Verlag Berlin Heidelberg NewYork

Selected Issues from

Lecture Notes in Mathematics Vol. 561: FunctionTheoretic Methods for Partial Differential Equations. Darmstadt 1976. Proceedings. Edited by V. E. Meister, N. Week and W. L Wendland. XVIII, 520 pages. 1976. Vol. 564: Ordinary and Partial Differential Equations, Dundee 1976. Proceedings. Edited by W. N. Everitt and D. Sleeman. XVIII, 551 pages. 1976.

a

Vol. 565: Turbulence and Navier Stokes Equations. ProceedIngs 1975. Edited by R. Temam.IX, 194 pages. 1976. Vol. 566: Empirical Dislributions and Processes. Oberwolfach 1976. Proceedings. Edited by P. Gaenssler and P. Revesz. VII, 146 pages. 1976. Vol. 570: Differential Geometrical Methods in Mathematical Physics, Bonn 1975. Proceedings. Edited by K. Bleuler and A Reetz. VIII, 576 pages. 1977. Vol. 572: Sparse Matrix Techniques, Copenhagen 1978. Ediled by V. A Barker. V, 184 pages. 1977. Vol. 579: Combinatoire et Representation du Groupe Symetrique, Strasbourg 1976. Proceedings 1976. Edite par D. Foata. IV, 339 pages. 1977. Vol. 587: Non-Commutative Harmonic Analysis. Proceedings 1976. Edited by J. Carmona and M. Vergne.IV, 240 pages. 1977. Vol. 592: D. Voig~ Induzierte Darstellungen in der Theorie der endlichen, algebraischen Gruppen. V, 413 Seiten. 1977. Vol. 594: Singular Perturbalions and Boundary Layer Theory, Lyon 1978. Edited by C. M. Brauner, B. Gay, and J. Mathieu. VIII, 539 pages. 1977. Vol. 596: K. Deimling, Ordinary Differential Equations in Banach Spaces. V~ 137 pages. 1977.

Vol. 648: Nonlinear Partial Differential Equations and Applications, Proceedings, Indiana 1976-1977. Edited by J. M. Chadam. VI, 206 pages. 1978. Vol. 650: C·-Algebras and Applications to Physics. Proceedings 1977. Edited by R. V. Kadison. V. 192 pages. 1978. Vol. 656: Probability Theory on Vector Spaces. Proceedings, 1977. Edited by A Weron. VIII. 274 pages. 1978. Vol. 662: Akin. The Metric Theory of Banach Manifolds. XIX, 306 pages. 1978. Vol. 665: Journees d'Analyse Non Lineaire. Proceedings, 1977. Edite par P. Benilan et J. Robert VIII, 256 pages. 1978. Vol. 667: J. Gilewicl, Approximants de Pade. XIV. 511 pages. 1978. Vol. 668: The Structure of Attractors in Dynamical Systems. Proceedings, 1977. Edited by J. C. Martin, N. G. Markley and W. Perrizo. VI, 264 pages. 1978. Vol. 675: J. Galambos and S. Kotz, Characterizations of Probability Distributions. VIII, 169 pages. 1978. Vol. 676: Differential Geometrical Methods in Mathematical Physics II, Proceedings, 1977. Edited by K. Bleuler, H. R. Petry and A Reetz. VI, 626 pages. 1978. Vol. 678: D. Dacunha-Castelle, H. Heyer et B. Roynette. Ecole d'Ete de Probabilites de Saint-Flour. VII-1977. Edite par P. L Hennequin. IX, 379 pages. 1978. Vol. 679: Numerical Treatment of Differential Equations in Applications, Proceedings, 1977. Edited by R. Ansorge and W. Tornig.IX, 163 pages. 1978.

~1.

Vol. 681: Seminaire de Thilorie du Potentiel Paris, No.3, Directeurs: M. Brelo~ G. Choquet et J. Deny. Redacteurs: F. Hirsch et G. Mokobodzki. VII, 294 pages. 1978.

Vol. 606: Mathematical Aspects of Finite Element Methods. Pro· ceedings 1975. Edited by I. Galligani and E. Magenes. VI, 382 pages. 1977.

Vol. 682: G. D. James, The Representation Theory of the Symmetric Groups. V, 156 pages. 1978.

805: Sario et aI., ClassificationTheory of Riemannian Manifolds. Xx, 498 pages. 1977.

Vol. 607: M. Metivier, Reelle und Vektorwertige Quasimartingale und die Theorie der Stochastischen Integration. X, 310 Seiten. 1977. ~1.

615: Turbulence Seminar, Proceedings 1976/77. Edited by P. Bernard and T. Ratiu. VI, 155 pages. 1977.

Vol. 618: I. I. Hirschman, Jr. and D. E. Hughes, Extreme Eigen Values of Toeplitz Operators. VI, 145 pages. 1977. Vol. 823: I. Erdelyi and R. Lange, Spectral Decompositions on Banach Spaces. VIII, 122 pages. 1977. Vol. 628: H. J. Baues, Obstruction Theory on the Homotopy Classification of Maps. XII, 387 pages. 1977. Vol. 629: W.A Coppel, Dichotomies in Stability Theory. VI, 98 pages. 1978.

Vol. 684: E. E. Rosinger, Distributions and Nonlinear Partial Differential Equations. XI, 146 pages. 1978. Vol. 690: W. J. J. Rey. Robust Statistical Methods. VI. 128 pages. 1978. Vol. 691: G. Vienno~ Algebres de Lie Libres et Mono'ides Libres. 111,124 pages. 1978. Vol. 693: Hilbert Space Operators, Proceedings, 1977. Edited by J. M. Bachar Jr. and D. W. Hadwin. VIII, 184 pages. 1978. Vol. 696: P. J. Feinsilver, Special Functions, Probability Semigroups, and Hamiltonian Flows. VI, 112 pages. 1978. Vol. 702: Yuri N. Bibikov, Local Theory of Nonlinear Analytic Ordinary Differential Equations. IX, 147 pages. 1979.

Vol. 630: Numerical Analysis, Proceedings, Biennial Conference, Dundee 1977. Edited by G. A Watson. XII, 199 pages. 1978.

Vol. 704: Computing Methods in Applied Sciences and Engineering, 1977, I. Proceedings, 1977. Edited by R. Glowinski and J. L. Lions. VI,391 pages. 1979.

Vol. 636: Journees de Statistique des Processus Stochastiques, Grenoble 1977, Proceedings. Edite par Didier Dacunha-Castelle et Bernard Van Cutsem. VII, 202 pages. 1978.

Vol. 710: Seminaire Bourbaki vol. 1977/78, Exposes 507-524. IV, 328 pages. 1979.

Vol. 638: P. Shanahan, The Aiiyah-Singer Index Theorem, An Intro· duction. V, 224 pages. 1978.

Vol. 711: Asymptotic Analysis. Edited by F. Verhulst V, 240 pages. 1979.

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