On the Motion of an Elastic Body in a Free Gas

We study the motion of an elastic body immersed in a three-dimensional perfect gas (Knudsen gas) in the mean-field approximation. The body is a homogeneous cylinder moving along an x-axis perpendicular to its bases, which as a consequence of collisions with the gas particles, and of its elasticity, modifies its length along the x  -axis. The body interacts with the gas particles by means of elastic collisions. We perturb initially the body, and study the approach of the body to equilibrium (rest), proving that, depending on the initial conditions, it can reach equilibrium with an exponential rate e−|α1|te−|α1|t or with a power-law t−4t−4 The exponential approach is characterized by the absence of recollisions between gas particles and body (for kinematic reasons), while the power-law approach is due to the presence of recollisions, which affect the motion by a long memory term.