  In this article we develop Poincar\'e ideas about a heat balance of ideal gas
considered as a collisionless continuous medium. We obtain the theorems on
diffusion in nondegenerate completely integrable systems. As a corollary we
show that for any initial distribution the gas will be eventually irreversibly
and uniformly distributed over all volume, although every particle during this
process approaches arbitrarily close to the initial position indefinitely many
times. However, such individual returnability is not uniform, which results in
diffusion in a reversible and conservative system. Balancing of pressure and
internal energy of ideal gas is proved, the formulas for limit values of these
quantities are given and the classical law for ideal gas in a heat balance is
deduced. It is shown that the increase of entropy of gas under the adiabatic
extension follows from the law of motion of a collisionless continuous medium.
