Ergodicity

H(,) = E

We then have

as there is only one trajectory and this trajectory fills all of the surface H(,) = E.

The charming aspect of ergodicity is that Boltzmanns H-teorem shows that ergodicity leads to the SL-IV form of the second law of thermodynamics.

The less attractive aspect of ergodicity is that it is plainly wrong.

Loschmidts Umkehreinwand attempted to demonstrate that ergodicity is impossible due to the time-reversal symmetry of the dynamics.

Zermelos Wiederkehreinwand attempted to demonstrate that ergodicity is impossible due to Poincare recurrence. Interestingly, Zermelo reached the correct conclusion, i.e ergodicity is impossible, but his argument is incorrect. Recurrence for points in phase-space does not imply recurrence for the distribution of points in phase-space.

Ergodicity is impossible due to a theorem in topology. In the equation