Is ergodicity a reasonable hypothesis for macroscopic systems?

In the physics literature “ergodicity” is sometimes taken to mean that a system, including a macroscopic one, visits all microscopic states in a relatively short time. However, many authors have realized that this is impossible and we provide a rigorous bound demonstrating this fact. A related concept is the “thermal distribution.” This enters in an understanding of dissipation, comparing the thermal state (the Boltzmann or Gibbs distribution) to its time evolute using relative entropy. The thermal distribution is based on the microcanonical ensemble, whose equal probability assumption is another phrasing of ergodicity in a macroscopic physical context. The puzzle then is why the results of these assumptions are in agreement with experience. We suggest (as others also have) reasons for this limited agreement, but note that the foundations of statistical mechanics make much stronger assumptions, assumptions that do not have the support of either reason or experience. This article is supplemented with comments by P. Gaspard, Y. Pomeau and H. Qian and a final reply by the authors.