Another argument against the statistical interpretation of irreversibility, the recurrence objection (or Wiederkehreinwand), was raised much later by Ernst Friedrich Zermelo, a collaborator of Max Planck at a time when the latter still opposed atomism, and instead supported the 'energeticists', who attempted to understand energy and entropy as fundamental 'substances'. This argument is based on a mathematical theorem due to Henri Poincaré, which states that every bounded mechanical system will return as close as one wishes to its initial state within a sufficiently large time. The entropy of a closed system would therefore have to return to its former value, provided only the function F(z) is continuous. This is a special case of the quasiergodic theorem which asserts that every system will corne arbitrarily close to any point on the hypersurface of fixed energy (and possibly with other fixed analytical constants of the motion) within finite time.

While all these theorems are mathematically correct, the recurrence objection fails to apply to reality for quantitative reasons. The age of our Universe is much smaller than the Poincaré recurrence times even for a gas consisting of no more than a few tens of particles. Their recurrence to the vicinity of their initial states (or their coming close to any other similarly specific state) can therefore be excluded in practice. Nonetheless, some .'foundations' of irreversible thermodynamics in the literature rely on formal idealizations that would lead to strictly infinite Poincaré recurrence times (for example the 'thermodynamical limit' of infinite particle number). Such assumptions are not required in our Universe of finite age, and they would not invalidate the reversibility objection (or the equilibrium expectation, mentioned above). However, all foundations of irreversible behavior have to presume some very improbable initial conditions...

(The Physical Basis for the Direction of Time. 5th ed., 2010 p.42)

In order to reverse the thermodynamical arrow of time in a bounded system, it would not therefore suffice to "go ahead and reverse all momenta" in the system itself, as ironically suggested by Boltzmann as an answer to Loschmidt. In an interacting Laplacean universe, the Poincaré cycles of its subsystems could in general only be those of the whole Universe, since their exact Hamiltonians must always depend on their time-dependent environment.

(The Physical Basis for the Direction of Time. 5th ed., 2010 pp.56-57)