Kinetics of collisionless gas: Equalization of temperature, growth of the coarse-grained entropy and the Gibbs paradox

The Poincaré model for dynamics of a collisionless gas in a rectangular parallelepiped with mirror walls is considered. The question on smoothing of the density and the temperature of this gas and conditions for the monotone growth of the coarse-grained entropy are discussed. All these effects provide a new insight of the classical paradox of mixing of gases.