The founding fathers of statistical mechanics, Boltzmann, Maxwell, Gibbs and Einstein, invented the concept of ensembles to describe equilibrium and nonequilibrium macroscopic systems. In trying to justify the use of ensembles, and to determine whether the ensembles evolved as expected from nonequilibrium to equilibrium, they introduced further concepts such as “ergodicity” and “coarse graining.” The use of these concepts raised mathematical problems that they could not solve, but like the good physicists they were they assumed that everything was or could be made all right mathematically and went on with the physics.
REFERENCES
1.
V. I. Arnold, A. Avez, Ergodic Problems of Statistical Mechanics, Benjamin, New York (1968);
J. Ford, “The Transition from Analytic Dynamics to Statistical Mechanics,” to be published in Advances in Chemical Physics;
J. L. Lebowitz, “Hamiltonian Flows and Rigorous Results in Non‐equilibrium Statistical Mechanics,” in Proceedings of the IUPAP Conference on Statistical Mechanics, University of Chicago Press (1972);
O. Lanford, “Ergodic Theory and Approach to Equilibrium for Finite and Infinite Systems,” to be published in Proceedings of the 100th Anniversary of the Boltzmann Equation, Vienna, 1972;
Ya. G. Sinai, “Ergodic Theory,” to be published in Proceedings of the 100th Anniversary of the Boltzmann Equation;
A. S. Wightman, in Statistical Mechanics at the Turn of the Decade (E. G. D. Cohen, ed.) Marcel Dekker, New York (1971).
2.
“Ergodicity of Boltzmann's Equations” in Statistical Mechanics, Foundations and Applications (T. A. Bak, ed.) Benjamin, New York (1967);
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O. Penrose, Foundations of Statistical Mechanics, Pergamon, Oxford (1970) chapter III.
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D. ter Haar, Elements of Statistical Mechanics, Holt, Rinehart and Winston, New York (1964), Appendix 1, paragraph 5;
O. Penrose, Foundations of Statistical Mechanics, Pergamon, Oxford (1970), page 40;
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7.
A. N. Kolmogorov, “Address to the 1954 International Congress of Mathematicians,” [translated in R. Abrahams, Foundations of Mechanics, Benjamin, New York (1967), Appendix D].
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Enrico Fermi: Collected Papers, Volume II, University of Chicago Press, Chicago (1965) page 978.
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11.
V. I. Arnold, A. Avez, ref. 1, page 25.
12.
13.
Ergoden Theone, Springer, Berlin (1937).
14.
J. W. Gibbs, Elementary Principles in Statistical Mechanics, Dover, New York (1960) page 144.
15.
Oeuvres 7, 262;
J. W. Gibbs, reference 14, page 139.
16.
T. Erber, B. Schweizer, A. Sklar, to be published in Comm. Math. Phys.
17.
D. Ornstein, B. Weiss, to be published in Israel J. Mathematics.
18.
P. R. Halmos, Lectures on Ergodic Theory, Mathematical Soc. of Japan (1965) page 9;
V. I. Arnold, A. Avez, reference 1, page 8.
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© 1973 American Institute of Physics.
1973
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