Ergodic vs. non-ergodic trajectories (Wikipedia) |

...In the physics literature "ergodicity" is taken to mean that a system, including a macroscopic one, visits all microscopic states in a relatively short time...[link]Visiting all microscopic states is not a pre-condition for ergodicity from statistical physics stand point. This form of the theory is the manifestation of strong ergodic hypothesis because of the Birkhoff theorem and may not reflect the physical meaning of ergodicity. However, the originator of ergodic hypothesis, Boltzmann, had a different thing in mind in explaining how a system approaches to thermodynamic equilibrium. One of the best explanations are given in the book of J. R. Dorfman, titled

*An introduction to Chaos and Nonequilibrium Statistical Mechanics*[link], in section 1.3, Dorfman explains what Boltzmann had in mind:

...Boltzmann then made the hypothesis that a mechanical system's trajectory in phase-space will spend equal times in regions of equal phase-space measure. If this is true, then any dynamical system will spend most of its time in phase-space region where the values of the interesting macroscopic properties are extremely close to the equilibrium values...[link]Saying this, Boltzmann did not suggest that a system should visit ALL microscopic states. His argument only suggests that only states which are close the equilibrium has more likelihood to be visited.

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