Thermalization of the ideal gas in a one-dimensional box

The time dependence of the phase-space distribution function in a model system of a one-dimensional perfect gas is studied. It is assumed that the velocity of a particle after each collision at the walls takes a random value governed by a given probability distribution function, irrespective of the velocity before the collision. Formulas giving the time evolution of the phase-space distribution function are presented. A necessary and sufficient condition for the presence of the nontrivial steady state solution is given. The asymptotic behavior of the model system at large time is examined. The obtained behaviors are compared with numerical results. It is shown that the system has a steady state, other than the trivial one in which all the particles have zero velocity, if and only if a specified integral converges.