## Two “paradoxes” in Statistical Mechanics

When **Boltzmann** announced the H theorem a century ago, objections were raised against it on the ground that it led to “paradoxes”. These are the so-called “reversal paradox” and “recurrence paradox”, both based on the erroneous statement of the** H theorem** that dH/dt≤0 at all times. The correct statement of the H theorem, is free from such objections.

The “**reversal paradox**” is as follows. “The H theorem singles out a preferred direction of time. It is therefore inconsistent with time reversal invariance”. This is not a paradox, because the statement of the alleged paradox is false. We have seen in the last section that time reversal invariance is consistent with H theorem, because dH/dt need mot be a continuous function of time. In fact, we have made use of time reversal invariance to deduce interesting properties of the curve of H.

The “**recurrence paradox**” is based on the following true theorem.

**Poincaré ’s Theorem**: A system having a finite energy and confined to a finite volume will, after a sufficiently long time, return to an arbitrarily small neighborhood of almost any given initial state.

By “almost any state” is meant any state of the system, except for a set of measure zero (i.e., a set that has no volume, e.g. a discrete point set). A neighborhood of a state has an obvious definition in terms of Γ-space of the system.

This theorem implies that H is an almost periodic function of time. The “reccurence paradox” arises in an obvious way, if we take the statement of the H theorem to be dH/dt≤0 at all times. Since this is not the statement of the H theorem, there is no paradox. In fact, Poincaré ’s theorem furnishes further information concerning the curve of H.

Most of the time H lies in the noise range repeat themselves. This is only to be expected.

For the rare spontaneous fluctuations above the noise range, Poincaré ’s theorem implies that the small fluctuations in the noise range repeat themselves. This is only to be expected.

For the rare spontaneous fluctuations above the noise range, Poincaré ’s theorem requires that if one such fluctuation occurs another one must occur after a sufficiently long time. The time interval between two large fluctuations is called a Poincaré cycle. A crude estimate shows that a Poincaré cycle is of the order of e^{N}, where N is the total number of molecules in the system. Since N~10^{23}, a Poincaré cycle is enormously long. In fact, it is essentially the same number, be it

10^{10^23} sec or 10^{10^23} ages of the universe,

(the age of the universe being a mere 10^{10} years.). Thus it has nothing to do with physics…..

“*Statistical Mechanics*”, **Kerson Huang**

## Leave a Reply