# Poincare Recurrence Theorem (1890 - 1897)

If you play bridge long enough you will eventually be dealt any grand-slam hand, not once but several times. A similar thing is true for mechanical systems governed by Newton's laws, as the French mathematician Henri Poincare (1854-1912) showed with his recurrence theorem in 1890: if the system has a fixed total energy that restricts its dynamics to bounded subsets of its phase space, the system will eventually return as closely as you like to any given initial set of molecular positions and velocities. If the entropy is determined by these variables, then it must also return to its original value, so if it increases during one period of time it must decrease during another.

This apparent contradiction between the behavior of a deterministic mechanical system of particles and the Second Law of Thermodynamics became known as the "Recurrence Paradox." It was used by the German mathematician Ernst Zermelo in 1896 to attack the mechanistic worldview. He argued that the Second Law is an absolute truth, so any theory that leads to predictions inconsistent with it must be false. This refutation would apply not only to the kinetic theory of gases but to any theory based on the assumption that matter is composed of particles moving in accordance with the laws of mechanics.

Boltzmann had previously denied the possibility of such recurrences and might have continued to deny their certainty by rejecting the determinism postulated in the Poincare-Zermelo argument. Instead, he admitted quite frankly that recurrences are completely consistent with the statistical viewpoint, as the card-game analogy suggests; they are fluctuations, which are almost certain to occur if you wait long enough. So determinism leads to the same qualitative consequence that would be expected from a random sequence of states! In either case the recurrence time is so inconceivably long that our failure to observe it cannot constitute an objection to the theory.

Updated 24 August 2001