Modern ergodic theory 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

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… (More)