Modern ergodic theory

FEB 01, 1973

There is much more to the mathematical study of Gibbs ensembles than the question of whether or not time averages and ensemble averages are equal

DOI: 10.1063/1.3127948

Joel L. Lebowitz

Oliver Penrose

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).
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Ergoden Theone, Springer, Berlin (1937).
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J. W. Gibbs, reference 14, page 139.
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V. I. Arnold, A. Avez, reference 1, page 8.

More about the Authors

Joel L. Lebowitz. Yeshiva University, New York.

Oliver Penrose. Mathematics Faculty, Open University, Walton, UK.