Axiomatics of Classical Statistical Mechanics
By Rudolf Kurth
5/5
()
About this ebook
Read more from Rudolf Kurth
Introduction to the Mechanics of the Solar System Rating: 0 out of 5 stars0 ratingsIntroduction to Stellar Statistics: International Series of Monographs in Natural Philosophy Rating: 0 out of 5 stars0 ratingsDimensional Analysis and Group Theory in Astrophysics Rating: 0 out of 5 stars0 ratings
Related to Axiomatics of Classical Statistical Mechanics
Titles in the series (100)
Rational Mechanics: The Classic Notre Dame Course Rating: 5 out of 5 stars5/5Theory of Linear Physical Systems: Theory of physical systems from the viewpoint of classical dynamics, including Fourier methods Rating: 0 out of 5 stars0 ratingsStatistical Fluid Mechanics, Volume II: Mechanics of Turbulence Rating: 0 out of 5 stars0 ratingsEquilibrium Statistical Mechanics Rating: 4 out of 5 stars4/5An Introduction to Acoustics Rating: 1 out of 5 stars1/5Quantum Mechanics with Applications Rating: 2 out of 5 stars2/5The Theory of Heat Radiation Rating: 3 out of 5 stars3/5A First Look at Perturbation Theory Rating: 4 out of 5 stars4/5Treatise on Physiological Optics, Volume III Rating: 0 out of 5 stars0 ratingsThe Philosophy of Space and Time Rating: 5 out of 5 stars5/5Electronic Structure and the Properties of Solids: The Physics of the Chemical Bond Rating: 3 out of 5 stars3/5Readable Relativity Rating: 4 out of 5 stars4/5Problems in Quantum Mechanics: Third Edition Rating: 3 out of 5 stars3/5Mathematics of Relativity Rating: 0 out of 5 stars0 ratingsBrownian Movement and Molecular Reality Rating: 0 out of 5 stars0 ratingsGravitational Curvature: An Introduction to Einstein's Theory Rating: 0 out of 5 stars0 ratingsSymmetry: An Introduction to Group Theory and Its Applications Rating: 4 out of 5 stars4/5An Elementary Survey of Celestial Mechanics Rating: 0 out of 5 stars0 ratingsQuantum Mechanics of One- and Two-Electron Atoms Rating: 0 out of 5 stars0 ratingsDynamic Light Scattering: With Applications to Chemistry, Biology, and Physics Rating: 5 out of 5 stars5/5Light Rating: 4 out of 5 stars4/5Introduction to Electromagnetic Theory Rating: 0 out of 5 stars0 ratingsAn Elementary Treatise on Electricity: Second Edition Rating: 0 out of 5 stars0 ratingsGroup Theory in Quantum Mechanics: An Introduction to Its Present Usage Rating: 0 out of 5 stars0 ratingsGeneral Relativity and Gravitational Waves Rating: 5 out of 5 stars5/5An Elementary Treatise on Theoretical Mechanics Rating: 5 out of 5 stars5/5A History of Mechanics Rating: 4 out of 5 stars4/5Theories of Figures of Celestial Bodies Rating: 0 out of 5 stars0 ratingsGet a Grip on Physics Rating: 3 out of 5 stars3/5Thermoelectricity: An Introduction to the Principles Rating: 4 out of 5 stars4/5
Related ebooks
Nonequilibrium Statistical Thermodynamics Rating: 0 out of 5 stars0 ratingsEquilibrium Statistical Mechanics Rating: 4 out of 5 stars4/5Gravitational Curvature: An Introduction to Einstein's Theory Rating: 0 out of 5 stars0 ratingsMathematical Foundations of Quantum Statistics Rating: 0 out of 5 stars0 ratingsDifferential Forms with Applications to the Physical Sciences Rating: 5 out of 5 stars5/5III: Scattering Theory Rating: 0 out of 5 stars0 ratingsAsymptotic Expansions Rating: 3 out of 5 stars3/5Elementary Principles in Statistical Mechanics Rating: 5 out of 5 stars5/5The Classical Groups: Their Invariants and Representations (PMS-1) Rating: 4 out of 5 stars4/5Foundations of Statistical Mechanics: A Deductive Treatment Rating: 0 out of 5 stars0 ratingsI: Functional Analysis Rating: 4 out of 5 stars4/5The Conceptual Foundations of the Statistical Approach in Mechanics Rating: 3 out of 5 stars3/5Mathematical Foundations of Quantum Mechanics Rating: 4 out of 5 stars4/5Algebraic Methods in Statistical Mechanics and Quantum Field Theory Rating: 0 out of 5 stars0 ratingsLectures on Homotopy Theory Rating: 0 out of 5 stars0 ratingsStatistical Physics: A Probabilistic Approach Rating: 0 out of 5 stars0 ratingsLectures on Ergodic Theory Rating: 0 out of 5 stars0 ratingsFinite Quantum Electrodynamics: The Causal Approach, Third Edition Rating: 0 out of 5 stars0 ratingsPeriodic Differential Equations: An Introduction to Mathieu, Lamé, and Allied Functions Rating: 0 out of 5 stars0 ratingsFoundations of Modern Analysis Rating: 1 out of 5 stars1/5A Mathematical Companion to Quantum Mechanics Rating: 0 out of 5 stars0 ratingsSolved Problems in Classical Electromagnetism Rating: 0 out of 5 stars0 ratingsFunction Theory on Planar Domains: A Second Course in Complex Analysis Rating: 0 out of 5 stars0 ratingsPerfect Form: Variational Principles, Methods, and Applications in Elementary Physics Rating: 0 out of 5 stars0 ratingsMathematical Foundations of Statistical Mechanics Rating: 4 out of 5 stars4/5Topology and Geometry for Physicists Rating: 4 out of 5 stars4/5Mechanics: Classical and Quantum Rating: 0 out of 5 stars0 ratingsRadiative Transfer Rating: 4 out of 5 stars4/5Topological Transformation Groups Rating: 3 out of 5 stars3/5Gauge Theory and Variational Principles Rating: 2 out of 5 stars2/5
Science & Mathematics For You
The Big Book of Hacks: 264 Amazing DIY Tech Projects Rating: 4 out of 5 stars4/5The Joy of Gay Sex: Fully revised and expanded third edition Rating: 4 out of 5 stars4/5How Emotions Are Made: The Secret Life of the Brain Rating: 4 out of 5 stars4/5Ultralearning: Master Hard Skills, Outsmart the Competition, and Accelerate Your Career Rating: 4 out of 5 stars4/5The Gulag Archipelago [Volume 1]: An Experiment in Literary Investigation Rating: 4 out of 5 stars4/5Outsmart Your Brain: Why Learning is Hard and How You Can Make It Easy Rating: 4 out of 5 stars4/5The Psychology of Totalitarianism Rating: 5 out of 5 stars5/5Homo Deus: A Brief History of Tomorrow Rating: 4 out of 5 stars4/5The Systems Thinker: Essential Thinking Skills For Solving Problems, Managing Chaos, Rating: 4 out of 5 stars4/5Activate Your Brain: How Understanding Your Brain Can Improve Your Work - and Your Life Rating: 4 out of 5 stars4/5Becoming Cliterate: Why Orgasm Equality Matters--And How to Get It Rating: 4 out of 5 stars4/5Memory Craft: Improve Your Memory with the Most Powerful Methods in History Rating: 3 out of 5 stars3/5Feeling Good: The New Mood Therapy Rating: 4 out of 5 stars4/5The Way of the Shaman Rating: 4 out of 5 stars4/5Free Will Rating: 4 out of 5 stars4/5On Food and Cooking: The Science and Lore of the Kitchen Rating: 5 out of 5 stars5/5The Gulag Archipelago: The Authorized Abridgement Rating: 4 out of 5 stars4/5A Letter to Liberals: Censorship and COVID: An Attack on Science and American Ideals Rating: 3 out of 5 stars3/5Why People Believe Weird Things: Pseudoscience, Superstition, and Other Confusions of Our Time Rating: 4 out of 5 stars4/5No-Drama Discipline: the bestselling parenting guide to nurturing your child's developing mind Rating: 4 out of 5 stars4/5The Wisdom of Psychopaths: What Saints, Spies, and Serial Killers Can Teach Us About Success Rating: 4 out of 5 stars4/5Lies My Gov't Told Me: And the Better Future Coming Rating: 4 out of 5 stars4/5The Complete Guide to Memory: The Science of Strengthening Your Mind Rating: 5 out of 5 stars5/5Suicidal: Why We Kill Ourselves Rating: 4 out of 5 stars4/52084: Artificial Intelligence and the Future of Humanity Rating: 4 out of 5 stars4/5The Invisible Rainbow: A History of Electricity and Life Rating: 4 out of 5 stars4/5The Big Fat Surprise: Why Butter, Meat and Cheese Belong in a Healthy Diet Rating: 4 out of 5 stars4/5The Trouble With Testosterone: And Other Essays On The Biology Of The Human Predi Rating: 4 out of 5 stars4/5No Stone Unturned: The True Story of the World's Premier Forensic Investigators Rating: 4 out of 5 stars4/5Other Minds: The Octopus, the Sea, and the Deep Origins of Consciousness Rating: 4 out of 5 stars4/5
Reviews for Axiomatics of Classical Statistical Mechanics
1 rating0 reviews
Book preview
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 ...)