  We study the deterministic dynamics of non-interacting classical gas
particles confined to a one-dimensional box as a pedagogical toy model for the
relaxation of the Boltzmann distribution towards equilibrium. Hard container
walls alone induce a uniform distribution of the gas particles at large times.
For the relaxation of the velocity distribution we model the dynamical walls by
independent scatterers. The Markov property guarantees a stationary but not
necessarily thermal velocity distribution for the gas particles at large times.
We identify the conditions for physical walls where the stationary velocity
distribution is the Maxwell distribution. For our numerical simulation we
represent the wall particles by independent harmonic oscillators. The
corresponding dynamical map for oscillators with a fixed phase (Fermi--Ulam
accelerator) is chaotic for mesoscopic box dimensions.
