Does Poincare recurrence show that Gibbs entropy is not strictly increasing?

Question

Statistical Mechanics setup: Consider a box of gas with $N$ particles in some initial macro state (fixed temperature, energy, etc.). A configuration of particles giving a fixed macro state is called a micro state corresponding to that macro state, and the set of all micro states is called the phase space, and this is a discrete subset of $\mathbb{R}^{6N}$. The points in phase space are microstates. A macro state can also be seen as a probability distribution on phase space. If $\mu$ is a macro state and $x$ is a microstate with $\mu(\{x\})>0$, then $x$ is a microstate corresponding to the macro state $\mu$. Because a single micro state can only correspond to one macro state, if $\mu_1$ and $\mu_2$ are two macro states (measures on phase space), then $\mu_1$ and $\mu_2$ are disjointly supported. Then entropy of a macro state $\mu$ is given by $E(\mu)=\sum_n \mu(\{x_n\})\log\left(\frac{1}{\mu(\{x_n\})}\right)$. Since this Gibbs entropy is only defined for discrete probability distributions, this is why we assumed phase space is discrete.

Poincare Recurrence: Now, If $T:\mathbb{R}^{6N}\to \mathbb{R}^{6N}$ is the time evolution of phase space, then by the Poincare recurrence theorem, for any set $A\subset\mathbb{R}^{6n}$, the set of points in $A$ that return to $A$ infinitely often has measure $\mu(A)$, where here $\mu$ is the Lebesgue measure on $\mathbb{R}^{6N}$. So if the system is initially in some state, with corresponding microstate $x$, then there is a sequence $(n_k)_k$ of natural numbers such that $d(T^{n_k}x,x)\to 0$. See for example, https://mathoverflow.net/questions/145005/poincare-recurrence-theorem-and-convergence-on-compact-metric-spaces. Now, I've seen it claimed in Huang's Statistical Mechanics book, that the Poincare recurrence theorem contradicts that the Gibbs entropy is strictly increasing in time. Intuitively, one starts with a macro state and a corresponding micro state, eventually that micro state is arbitrarily close the original one, so the entropy should be get closer to the original value, and so it can't be strictly increasing.

My question is, does this actually contradict that Gibbs entropy is strictly increasing? For one, it could be that $T^{n_k}x$ just stays in the same macro state as $x$ for all $k$,in which case, the entropy remains constant. It could also be that $T^nx$ leaves the macro state that $x$ is in, for some $n$, and doesn't actually return to it, even if it approaches the original macro state (the original macro state could just be a dirac measure on $\{x\}$, for example). The other thing is that there is not even a guarantee that there is any $x$ for which $d(T^{n_k}x,x)\to 0$, since phase space is a discrete subset of $\mathbb{R}^{6N}$ and therefore has measure zero. So I'm not sure if I have something wrong with the formalism, or I'm not understanding how to apply the Poincare recurrence theorem in this situation, but it doesn't seem like a contraction.

Answer 1

Your perplexity and observations about Huang's statement are well-founded. Indeed, Huang acritically repeats Zermelo's and Poincaré's arguments against Boltzmann's ideas.

One flaw in using the recurrence theorem as an argument against the Statistical Mechanics approach to entropy is based on the wrong assumption that the statistical mechanics entropy is a function of the microstate. It is not. It is a property of the macrostate, and each macrostate usually corresponds to a huge number of microstates. Failure to understand this point is quite frequent, even today.

The second flaw is ignoring that Thermodynamics can be recovered from Statistical Mechanics only in the so-called Thermodynamic Limit, i.e., for an infinite system. In the same limit, the recurrence time diverges, preempting any use of the recurrence theorem. By the way, this was the original Boltzmann's line of defense against Zermelo's argument.

A final word of caution about your statement

phase space is a discrete subset of ${\mathbb R}^{6N}$ and therefore has measure zero.

The discreetness of the phase space results from identifying all the classical microstates in a small volume of size $h^{3N}$. The result is that any finite phase space volume corresponds to a finite number of microstates. However, this is not equivalent to saying that one is considering subsets of zero measurements.

Answer 2

Intuitively, one starts with a macro state and a corresponding micro state, eventually that micro state is arbitrarily close the original one, so the entropy should be get closer to the original value, and so it can't be strictly increasing.

Let me start with pointing out that there are several distinct quantities called entropy, which are related, but not identical - see Is information entropy the same as thermodynamic entropy?. Notably, the original phenomenological definition of Clausius and Gibbs was intended to describe the experimentally observed fact that the macroscopic processes have a preferred direction in time, despite the reversibility of the underlying microscopic dynamics. This phenomenological definition, framed as the second law of thermodynamics, is just a statement of an experimental fact - there is nothing paradoxical about it.

The paradoxes started to emerge when one tried to explain the thermodynamics laws in microscopic terms, notably in terms of atoms/molecules in a gas. This is when various version of entropy resembling Shannon's information entropy has emerged (although historically Shannon's work came much later.) Loschmidt and Zermelo paradoxes, associated respectively with the reversibility and the recurrence of microscopic motion, are the two most notable. See also Why is the second law of thermodynamics not symmetric with respect to time reversal?

@GiorgioP in their answer points out that in purely mathematical terms there may be solution of the paradoxes due to the thermodynamic limit: $N\rightarrow \infty, V\rightarrow\infty, N/V=\text{const}$. From physics viewpoint, there are several reasons for why the systems either approximate this thermodynamical limit or otherwise bypass the pre-conditions for the supposed paradoxes:

• Isolated system - we never really deal with an isolated system. Indeed, a gas in a container interacts with the container walls, which are themselves a thermodynamical system and which are coupled to the rest of the Universe.
• Long observation times any thermodynamic phenomena observed during rather short times (compared to the recurrence time suggested by the Poincare theorem) - the duration of the experiment, or PhD program, or the experimentalist lifetime. The upper limit is the time of the existence of the Universe. We do know that in some cases this time is too short, as in the case of second order phase transitions - the fixed value of an order parameter, like the magnetization of a ferromagnet, remains stable, even though all directions of the magnetisation are equally energetically favorable. This can be ultimately traced to violation of the core assumptions of the statistical mechnaics - e.g., the ergodicity assumption (which in itself is based on phenomenological grounds.)
• Mechanistic approach and Ideal gas An alternative way of obtaining thermodynamic distributions is by maximizing the information entropy as a measure of our ignorance about the actual microstate of the system - see Microcanonical ensemble through Maximum Entropy method. The increase of entropy then simply amounts to adopting configuration where the microstates are equiprobable, reflecting our ignorance of them. Note that some statistical physics texts prefer to use Ising spins (rather than a gas) as their basic model - completely forgoing the question of mechanics, but assuming (more or less explicitly) the equiprobability of microstates.