Kinetics of a collisionless continuous medium

The ideas of Poincaré on thermal equilibrium of an ideal gas regarded as a collisionless continuous medium are developed. A study is made of the evolution of the particle distribution function with respect to space and with respect to velocities. It is proved that the solutions of Liouville's equation have a weak limit as the time goes to infinity. As a corollary it is shown that, regardless of the initial distribution, a gas in a box with mirror reflection irreversibly distributes itself uniformly over the whole volume in the course of time. The existence of a weak limit of the probability distribution density leads to a new interpretation of the second law of thermodynamics on the growth of entropy.