We consider a body immersed in a perfect gas, moving under the action of a constant force E along the x axis. We assume the gas to be described by the mean-field approximation and interacting elastically with the body. Such a dynamic was studied in [S. Caprino, C. Marchioro,. In these studies the asymptotic trend showed no sensitivity whatsoever to the shape of the object moving through the gas. In this work we investigate how a simple concavity in the shape of the body can affect its asymptotic behavior; we thus consider the case of a hollow cylinder in three dimensions or a box-like body in two dimensions. We study the approach of the body velocity V (t) to the limiting velocity V∞ and prove that, under suitable smallness assumptions, the approach to equilibrium is |V∞ − V (t)| ≈ Ct −3 both in two or three dimensions, with C a positive constant. This approach is not exponential, as is typical in friction problems, and even slower than for the simple disk and the convex body in R 2 or R 3 .

We consider a body immersed in a perfect gas, moving under the action of a constant force E along the x axis . We assume the gas to be described by the mean-field approximation and interacting elastically with the body, we study the friction exerted by the gas on the body fixed at constant velocities. The dynamic in this setting was studied in [3], [4] and [5] for object with simple shape, the first study where a simple kind of concavity was considered was in [10], showing new features in the dynamic but not in the friction term. The case of more general shape of the body was left out for further difficulties, we believe indeed that there are actually non trivial issues to be faced for these more general cases.To show this and in the spirit of getting a more realistic perspective in the study of friction problems, in this paper we focused our attention on the friction term itself, studying its behavior on a body with a more general kind of concavity and fixed at constant velocities. We derive the expression of the friction term for constant velocities, we show how it is time dependent and we give its exact estimate in time. Finally we use this result to show the absence of a stationary velocity in the actual dynamic of such a body.

Abstract. We consider the motion of a particle along the geodesic lines of the Poincaré half-plane. The particle is specularly reflected when it hits randomlydistributed obstacles that are assumed to be motionless. This is the hyperbolic version of the well-known Lorentz Process studied by Gallavotti in the Euclidean context. We analyse the limit in which the density of the obstacles increases to infinity and the size of each obstacle vanishes: under a suitable scaling, we prove that our process converges to a Markovian process, namely a random flight on the hyperbolic manifold.