Axiomatics of Classical Statistical Mechanics
By Rudolf Kurth
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Axiomatics of Classical Statistical Mechanics - Rudolf Kurth
Mechanics
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
t is the time variable; dots denote differentiation with respect to t, the xi’ 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
(x¹, x², ..., xn), and the Xi(x, t)’s are continuous functions of (x, t) defined for all values of (x, tof Γ and at each moment t, there is a uniquely determined solution
of the system of differential equations (*) which satisfies the initial conditions
Then the principal problem of mechanics reads: for a given force
X(x,t) and a given initial condition 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 imposes certain restrictions on the functions Xi(x, t) which are not assumed in the general theory of differential equations. (Cf. §§ 6 and 10.)
If the number n 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 provides 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 discussed in Chapter II.
CHAPTER II
MATHEMATICAL TOOLS
§ 2. Sets
2.1. "A set is a collection of different objects, real or intellectual, into a whole. (
Eine Menge ist die Zusammenfassung verschiedener Objekte unserer Anschauung oder unseres Denkens zu einem Ganzen"—CANTOR.) 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 s ∈ S, and the sentence "the set S consists of the elements s1s2, ..." by the formula S = {s1s2, ...}
It is formally useful to admit sets consisting of only one element (though there is nothing like collection
or Zusammenfassung
) and even to admit a set containing no element at all. The latter set is called an empty set.
2.2. DEFINITIONS. A set S1 is called a subset of a set S if each element of S1 is contained in S. For this we write S1 ⊆ S or S ⊇ S1. If there is at least one element of S which does not belong to S1 the set is called a proper subset of the set S. In this case, we write S1 ⊂ S or S ⊃ S1. If for two sets S1 and S2 the relations S1 ⊆ S2 and S2 ⊆ S1 are valid at the same time, both sets are called equal and we write S1 = S2. The empty set is regarded as a subset of every set.
A set is called finite if it consists of a finite number of elements. Otherwise it is called infinite. A set is called enumerable 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 {S1S2 ...} be a set (or, as we prefer to say for linguistic reasons, an aggregate) of sets S1S2, ... ; then the sum (S1 + S2 + ...)of the sets S1S2, ... is defined as the set of all the elements contained in at least one of the sets S1, S2, ... . If the aggregate {S1,S2, ...} is finite or enumerable we denote the sum (S1 + S
The intersection S1S2 ... or S1.S2 ... of the sets S1S2, ... is defined as the set of all the elements contained in each of the sets S1S2, ... . If the aggregate {S1S
The (Cartesian) product S1 × S2 × ... of the sets of a finite or enumerable aggregate of sets S1S2, ... is defined as the aggregate of all the sequences {s1,s2 ...} where is any element of Sx y 1 of the (x,yx y 1.)
Let S1 be a subset of S. Then the set of all the elements of S which are not contained in S1 is called the difference, S – S1 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 S1 be a subset of a space S. Then the difference S – S1 is called the complement of the set S1.
An aggregate A of subsets S1S2, ... of a space S is called an additive class if it satisfies the following conditions:
(i) S is an element of A ;
(ii) if S 1 is an element of A is also an element of A ;
(iii) if each set S 1 , S 2 , ... of a finite or enumerable aggregate { S 1 S 2 , ...} of sets S 1 , S 2 , ... is an element of A , then the sum S 1 + S 2 + ... is also an element of A .
Example. The aggregate of all the subsets of the space S is an additive class.
2.3. THEOREMS. The addition and intersection of the sets of a finite or infinite system of sets S1, S2, ... are associative and commutative. Thus, in particular,
(so that we may write without any brackets S1 + S2 + S3 and S1S2S3 for (S1 + S2) + S3 and (S1 S2)S3) and
For the following pairs of set-operations the distributive law holds: addition and intersection, subtraction and intersection, addition and Cartesian multiplication, subtraction and Cartesian multiplication. Thus,
These statements can be made intuitively evident by figures of the following kind:
2.4. THEOREM. Let {S1S2, ...} be a finite or enumerable aggregate of finite or enumerable sets S1, S2, ... . Then the sum S1 + S2 + ... is a finite or enumerable set.
Proof. Write
Then the sequence {s11, s12, s21, s13, s22, s31, s14, s23, ...} yields an enumeration of S1 + S2 + ... . (The elements having already occurred in the enumeration have to be omitted.)
2.5. THEOREM. The set of all rational numbers is enumerable.
Proof. The set of all rational numbers can be represented as the sum of the following enumerable sets:
Now apply Theorem 2.4.
2.6. THEOREM. Let S1S2, ... be subsets of a space S, and let the complement of a set be denoted by an asterisk. Then
Proof. Let s be an element of (S1 + S2 + ...)*, and denote by the symbol ∉ the negation of the relation ∈. Then
Hence
By inverting the chain of arguments, it follows that
and both inequalities together imply the first statement. The proof of the second one is similar.
2.7. THEOREM. Let {S1S2, ...} be a finite or enumerable aggregate of sets belonging to an additive class A. Then the intersection S1S2 ... is an element of A.
Proof.
(by Theorem 2.6).
are elements of Ais an element of A, too.
§ 3. Mapping
3.1. DEFINITIONS. Let X and Y be two sets (which need not be different), and suppose that to each element x of X there corresponds a unique element y of Y. (The same element of Y may correspond to different elements of X.) Then we write y = f(x) where x ∈ X, and say: y or f(x) is a function of x defined in the set X ; the set X is mapped into the set Y ; and the set of elements f(x) which correspond to the elements x of X is the image of the set X. It is denoted by f(X). If f(X) = Y (so that each element y of Y is the image f(x) of at least one element x of X), then the set X is said to be mapped on the set Y by the function f(x).
Like the word collection
(Zusammenfassung) in 2.1, the word correspondence
denotes an elementary act of our mind which, by its nature, does not admit of a formal definition.
If f(x) is a function defined in a set X such that f(x1) ≠ f(x2) for x1 ≠ x2, then the function (or the mapping) f is said to be bi-uniform. If Y = f(X) is the bi-uniform image of X, then to each element y of Y there corresponds exactly one element x of X such that y = f(x). This correspondence is called the inverse function of the function f and is denoted by f–1.
3.2. THEOREM. Let f(x) be a function defined in a set X. Then, for any subsets X1, X2, ... of X,
Proof. From
it follows that
f(X1) + f(X2) + ... ⊆ f(X1 + X2 + ...).
Conversely, if x ∈ X1 + X2 ..., then x belongs to at least one of the sets X1, X2, and, therefore, f(x) belongs to at least one of the sets f(X1),f(X2), ... .
Hence f(X1 + X2 + ...) ⊆ f(X1) + f(X2) + ... .
This yields, together with the above inequality, the first statement. The proofs of the other statements are similar.
3.3. THEOREM. Let f(x) be a bi-uniform function defined in the set X. Then, for any subsets X1, X2, ... of X,
f(X1X2 ...)