Editing H-theorem (section)

=== Poincaré recurrence ===

In 1896, [[Ernst Zermelo]] noted a further problem with the ''H'' theorem, which was that if the system's ''H'' is at any time not a minimum, then by [[Poincaré recurrence]], the non-minimal ''H'' must recur (though after some extremely long time). Boltzmann admitted that these recurring rises in ''H'' technically would occur, but pointed out that, over long times, the system spends only a tiny fraction of its time in one of these recurring states.

The [[second law of thermodynamics]] states that the entropy of an [[isolated system]] always increases to a maximum equilibrium value. This is strictly true only in the thermodynamic limit of an infinite number of particles. For a finite number of particles, there will always be entropy fluctuations. For example, in the fixed volume of the isolated system, the maximum entropy is obtained when half the particles are in one half of the volume, half in the other, but sometimes there will be temporarily a few more particles on one side than the other, and this will constitute a very small reduction in entropy. These entropy fluctuations are such that the longer one waits, the larger an entropy fluctuation one will probably see during that time, and the time one must wait for a given entropy fluctuation is always finite, even for a fluctuation to its minimum possible value. For example, one might have an extremely low entropy condition of all particles being in one half of the container. The gas will quickly attain its equilibrium value of entropy, but given enough time, this same situation will happen again. For practical systems, e.g. a gas in a 1-liter container at room temperature and atmospheric pressure, this time is truly enormous, many multiples of the age of the universe, and, practically speaking, one can ignore the possibility.