Combinatorics of recurrence theorem

Question

I have read about the Poincaré recurrence theorem and I am also aware that as systems become larger, recurrences become extremely unlikely. But I wonder whether combinatorics has been taken into consideration in the proof of this theorem as well? Because if one considers a system of identical and indistinguishable particles then there are many states which are equivalent and the number of equivalent states increases "factorialy". So dp recurrences become more likely if one takes into consideration that particles are indistinguishable?

Answer 1

No, the combinatorics of those equivalent states is not taken into account in the proof of the recurrence theorem, and I think you're right about they making recurrence happen more often. But you're probably better off adding this piece of info to your model (thereby simplifying it significantly) instead of trying to consider it only in the analysis phase.

Say we're talking about ideal gas in a containment vessel. In a microscopic description of the system (with $6N$ degrees of freedom for $N$ particles), my bet is that, if our definition of equivalence is "corresponding to the same macrostate", then considering equivalent states probably reduces the recurrence time — actually to zero, if the system is at equilibrium, or to longer-than-the-age-of-the-universe if very far from it (say, all gas concentrated on half of the volume) — but this calculation would be simply reproducing the usual calculations from statistical physics. Now if equivalence is not defined in terms of pressure, temperature, etc. but based only in the indistinguishability of the particles, then we probably reduce the recurrence time (from mindbogglingly large to absurdly large), but I'm not sure we can gain anything from it.

More generally, the question hints at a connection between dynamical systems and statistical physics. If that's what one's interested in, then it's worth pointing out that, although these are distinct fields (e.g., classical dynamical systems theory concerns itself mostly with deterministic, low-dimensional systems, compared to the statistical physics' typically stochastic systems with degrees of freedom numbered in the Avogadro number range), there are indeed meaningful contact points. A number of examples and references for this connection can be found in the answers to the question Statistical Mechanics & Dynamical Systems.

Answer 2

The Poincare theorem is about classical Hamiltonian systems, made of distinguishable particles, with finite number of degrees of freedom and with finite volume of accessible phase space.

If we introduce indistinguishability of particles into such a system with many identical particles, the recurrence time gets greatly reduced, but is still immense for macroscopic systems, effectively infinite.

E.g. consider 1 mole of gas that is expanded into twice the volume. In order for this process to be reversed, with allowance for indistinguishability, any state obtained by permuting the $N=$6e23 particles could happen. For any microstate, there is 6e23! of similar microstates that can be treated as indistinguishable. Naively, by this factor the recurrence time gets divided. But still, for one mole of particles, we expect that time to be immensely long, mainly because we have never documented such recurrence and also because naive estimates of its value produce times much longer than the estimated age of the universe.

If, with introducing indistinguishability, we also want to change the whole model into a quantum theory model, then there is a similar recurrence theorem for quantum systems:

Bocchieri, Loinger: Quantum Recurrence Theorem, Phys. Rev. 107, 337 (1957)

Isolated quantum system with discrete spectrum only will eventually, after finite time, return arbitrarily close to the initial state. However, for a gas in a box, the space of states seems even bigger than in classical theory, as we have infinite sequence of state variables - complex coefficients $c_k(t)$ for expanding the quantum state into some orthonormal basis.

A common feature of these theorems is they don't say what is the time to recurrence. So it is not easy to compare the values and decide whether the time is greater or smaller in quantum theory.