Axiomatics of classical statistical mechanics

185 Pages·2014·10.164 MB·English

by  Kurth, Rudolf, Sneddon, I. N., Stark, M., Ulam, S

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AXIOMATICS OF
CLASSICAL STATISTICAL
MECHANICS
RUDOLF KURTH
Department of Mathematics
The Durham Colleges in the University of Durham
PERGAMON PRESS
OXFORD · LONDON · NEW YORK · PARIS
1960
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Copyright
©
1960
PERGAMON PRESS LTD.
Library of Congress Card Number 60-8973
Printed in Great Britain by John Wright ώ Sons Ltd., Bristol
By the same author:
Von den Grenzen des Wissens (Basel, 1953)
Zum Weltbild der Astronomie (with M. Schürer) (Bern, 1954)
Entwurf einer Metaphysik (Bern, 1955)
Christentum und Staat (Bern, 1957)
Introduction to the Mechanics of Stellar Systems (London, 1957)
Introduction to the Mechanics of the Solar System (London, 1959)
"Wer dem Gangé einer höheren Erkenntnis und
Einsicht getreulich folgt, wird zu bemerken habén,
dass Erfahrung und Wissen fortschreiten und sich
bereichern können, dass jedoch das Denken und
die eigentlichste Einsicht keineswegs in gleichem
Maasse vollkommen wird, und zwar aus der ganz
natürlichen Ursache, weil das Wissen unendlich und
jedem neugierig Umherstehenden zugánglich, das
Überlegen, Denken und Verknüpfen aber innerhalb
eines gewissen Kreises der menschlichen Fáhigkeiten
eingeschlossen ist."
GOETHE, Annalen, 1804
PREFACE
THIS book is an attempt to construct classical statistical
mechanics as a deductive system, founded only on the equa­
tions of motion and a few well-known postulates which formally
describe the concept of probability. This is the sense in which
the word "axiomatics" is to be understood. An investigation
of the compatibility, independence and completeness of the
axioms has not been made: their compatibility and inde­
pendence appear obvious, and completeness, in its original
simple sense, is an essential part of the mos geometricus itself
and, in fact, of the scientific method in general: all the assump­
tions of the theory have to be stated explicitly and completely.
(I know there are other interpretations of the word ' 'complete­
ness", but I cannot help feeling them to be artificial.)
The aim adopted excluded some subjects which are usually
dealt with in books on statistical mechanics, in particular the
theory of Boltzmann's equation; it has not been and cannot
be derived from the above postulates, as is shown by the
contradiction between Boltzmann's jET-Theorem and Poincaré's
Recurrence Theorem. By this statement I do not wish to
deny the usefulness of Boltzmann's equation; I wish only to
emphasize that it cannot be a part of a rational system founded
on the above assumptions. Instead, it constitutes a separate
theory based on a set of essentially different hypotheses.
My aim demanded that the propositions of the theory be
formulated more geometrico also, that is, in the form "if...,
then...", which, in my opinion, is the only appropriate form
for scientific propositions. For convenience in derivation or
formulation the assumptions were sometimes made less general
than they could have been.
It was intended to make the book as self-contained as
possible. Therefore the survey of the mathematical tools in
Chapter II was included so that only the elements of calculus
and analytical geometry are supposed to be known by the
reader. In order to confine this auxiliary chapter to suitable
vii
viii PREFACE
proportions, however, full proofs were given only in the
simplest cases, while lengthier or more difficult proofs were
only sketched or even omitted altogether. Thus, in this
respect, I have achieved only a part of my aim.
The set of the references given is the intersection of the set
of writings I know by my own study and the set of writings
which appeared relevant for the present purpose. A part of
the text is based on investigations of my own, not all of which
have yet been published.
It is a pleasure for me to remember gratefully the encourage­
ment I received from Dr. D. ter Haar (now at Oxford University)
and Professor E. Finlay-Freundlich (at St. Andrews University)
when I made the first steps towards the present essay and had
to overcome many unforeseen obstacles. Several colleagues at
Manchester University were so kind as to correct my English.
Particularly, I am indebted to Dr. H. Debrunner (Princeton)
for his many most valuable criticisms of my manuscript. Last
but not least, I gratefully acknowledge that it was A. J.
Khinehin's masterly book which gave me the courage to think
for myself in statistical mechanics, after a long period of doubt
about the conventional theory. To them all I offer my best
thanks.
Cheadle (Cheshire) RUDOLF KUBTH
CHAPTER I
INTRODUCTION
§ 1. Statement of the problem
In this book we shall consider mechanical systems of a
finite number of degrees of freedom of which the equations of
motion read
χ,^Χ^χ,ή, i = 1,2,...,n. (*)
t is the time variable; dots denote differentiation with respect to
t, the a;/s, i =1,2, ...,n, are Cartesian coordinates of the
n-dimensional vector space Rn, which is also called the phase-
space Γ of the system; x is the vector or "phase-point"
(x ,x , ...,x), and the X^x, £)'s are continuous functions of
1 2 n
(x, t) defined for all values of (x, t). Further, we assume that,
for each point x of Γ and at each moment t, there is a uniquely
determined solution
Xi = Xi(xXt) i=l,2,...,n, (**)
f
of the system of differential equations (*) which satisfies the
initial conditions
x = (xJJ), i = 1,2, ...,n.
{ Xi
Then the principal problem of mechanics reads: for a given
"force" X(x,t) and a given initial condition (xj), to calculate or
to characterize qualitatively the solution (**). In this formulation,
the problem of general mechanics appears as a particular case
of the initial value problem of the theory of ordinary differential
equations. It is, in fact, a particular case since mechanics im­
poses certain restrictions on the functions X^x, t) which are not
assumed in the general theory of differential equations. (Cf. §§ 6
and 10.)
If the number n of equations (*) and unknown functions (**)
is very large, two major difficulties arise. First, the initial
phase-point x of a real system can no longer be actually deter­
mined by observation. (It is, for example, impossible to observe
1
2 AXIOMATICS OF STATISTICAL MECHANICS
the initial positions of all the molecules of a gas or of all the
stars of a galaxy.) Secondly, even if the initial point x of such a
system is known, the actual computation of the solution (**) is
no longer practicable, not even approximately by numerical
methods.
But it is just this embarrassingly large number n which pro­
vides a way out, at least under certain conditions which will
be given fully later: it now becomes possible to describe the
average properties of these solutions, and it seems plausible
to apply such "average solutions" in all cases in which, for any
reason, the individual solutions cannot be known.
The average behaviour of mechanical systems is the subject
of statistical mechanics. Its principal problems, therefore,
read: to define suitable concepts of the average properties of the
solutions (**), to derive these average properties from the equations
of motion (*); and to vindicate their application to individual
systems. Before starting this programme in Chapter III, the
principal mathematical tools which are required will be dis­
cussed in Chapter II.
CHAPTER II
MATHEMATICAL TOOLS
§2. Sets
2.1. "A set is a collection of different objects, real or intel­
lectual, into a whole." ("Eine Menge ist die Zusammenfassung
verschiedener Objekte unserer Anschauung oder unseres
Denkens zu einem Ganzén"—CANTOB.) This sentence is not
to be understood as a definition, but rather as the description
of an elementary intellectual act or of the result of this act.
Since it is an elementary act, which cannot be reduced to any
other act or fact in our mind, the description cannot be other
than vague. Nevertheless, everyone knows perfectly what is
meant by, for instance, an expression such as "the set of the
vertices of a triangle".
The objects collected in a set are called its elements and we
say: "the elements form or make the set", "they belong to it",
"the set consists of the elements", "it contains these elements",
etc. The meaning of terms such as "element of", "forms",
"consists of", etc., is supposed to be known. The sentence,
"s is an element of the set S", is abbreviated symbolically by
the formula seS, and the sentence "the set 8 consists of the
elements s,s ,..." by the formula S = {8 s,...}.
x 2 l9 2
It is formally useful to admit sets consisting of only one
element (though there is nothing like "collection" or "Zusam­
menfassung") and even to admit a set containing no element
at all. The latter set is called an empty set.
2.2. DEFINITIONS. A set Sx is called a subset of a set S if each
element of S is contained in S. For this we write S c S or
x x
S^S If there is at least one element of S which does not
V
belong to 8 the set 8 is called a proper subject of the set S.
V 1
In this case, we write S <=: S or S => S If for two sets S and S
1 v x 2
the relations S ^S and 8 ^S are valid at the same time,
1 2 2 1
both sets are called equal and we write $ = S . The empty
χ 2
set is regarded as a subset of every set.
3
4 AXIOMATICS OF STATISTICAL MECHANICS
A set is called finite if it consists of a finite number of
elements. Otherwise it is called infinite. A set is called enumer
able if there is an ordinal number (in the ordinary sense) for
each element and, conversely, an element for each ordinal
number, i.e. if there is a one-to-one correspondence between the
elements of the set and the ordinal numbers. A sequence is
defined as a finite or enumerable ordered set, i.e. a finite or
enumerable set given in a particular enumeration. If any two
elements of a sequence are equal (for example, numerically)
they are still distinguished by the position within the sequence;
thus, as members of the sequence, they are to be considered as
different.
Let {Si S ....} be a set (or, as we prefer to say for linguistic
2
reasons, an aggregate) of sets S S , ·..; then the sum
V 2
(S + S + ···) of the sets SS ,... is defined as the set of all
1 2 v 2
the elements contained in at least one of the sets S S ,....
l9 2
If the aggregate {S S ,...} is finite or enumerable we denote
l9 2
n oo
the sum (á^-h^-t-...) also by 2 & or Σ S.
v v
The intersection S S ... or S S ... of the sets S S ,... is
1 2 lm 2 V 2
defined as the set of all the elements contained in each of the
sets S S ,.... If the aggregate {S S ,...} is finite or enumer-
1} 2 l9 2
n oo
able, the intersection is also denoted by Π 8 or Π $„.
V
The (Cartesian) product S xS x ... of the sets of a finite or
1 2
enumerable aggregate of sets S S ,... is defined as the aggre­
l9 2
gate of all the sequences {s s ,...} where s is any element of
l9 2 x
S etc. (Example: the square 0<a:^l, O^y^l of the (x,y)
v
plane is the Cartesian product of both its sides 0 ^ x ^ 1 and
0<y<l.)
Let Si be a subset of S. Then the set of all the elements of S
which are not contained in S is called the difference, S — S
1 ly
of both sets.
The operations which produce sums, intersections, Cartesian
products or differences of sets will be called (set) addition,
intersection, (Cartesian) multiplication or subtraction.
If all the sets occurring in a theory are subsets of a given
fixed set S, this set S is called a space. Let 8 be a subset of a
λ
space S. Then the difference S — S-L is called the complement Sf
of the set S^