Cedric Villani and the Trend to Equilibrium in the Boltzmann Equation

摘要

If, as an initial state, all the molecules of a gas are put in one part of a container, with vacuum in the rest, the gas will expand throughout the container within a short time, and nobody will expect to ever observe a state close to the initial state again. On the other hand, such recurrence events are predicted to occur infinitely many times by the Poincare recurrence theorem. The catch in this seeming contradiction is a quantitative one. At atmospheric pressures and temperatures, 1 cm~3 of air contains on the order of 10~(20) molecules. Due to this huge number, recurrence times for a typical gas container can be estimated to be much larger than the age of the universe.