On a microscopic model of viscous friction

We consider a body moving along the x-axis under the action of an external force E and immersed in an infinitely extended perfect gas. We assume the gas to be described by the mean-field approximation and interacting elastically with the body. In this setup, we discuss the following statement: "Let V-0 be the initial velocity of the body and V-infinity its asymptotic velocity, then for vertical bar V-0-V-infinity vertical bar small enough it results vertical bar V(t)-V infinity vertical bar approximate to Ct(-d-2) for t large, where V (t) is the velocity of the body at time t, d the dimension of the space and C is a positive constant depending on the medium and on the shape of the body". The reason for the power law approach to the stationary state instead of the exponential one (usually assumed in viscous friction problems), is due to the long memory of the dynamical system. In a recent paper by Caprino, Marchioro and Pulvirenti,(3) the case of E constant and positive, with 0 < V-0 < V-infinity, for a disk orthogonal to the x-axis has been discussed. Here we complete the analysis in the cases E > 0 with V-0 > V-infinity and E = 0. We also approach the problem of an x-dependent external force, by choosing E of harmonic type. In this case we obtain the power-like asymptotic time behavior for the body position X(t). The investigation is done in detail for a disk orthogonal to the x-axis and then, by a sketched proof, extended to a body with a general convex shape.