Some Hamiltonian models of friction II

Abstract

In the present paper we consider the motion of a very heavy tracer particle in a medium of a very dense, non-interacting Bose gas. We prove that, in a certain mean-field limit, the tracer particle will be decelerated and come to rest somewhere in the medium. Friction is caused by emission of Cerenkov radiation of gapless modes into the gas. Mathematically, a system of semilinear integro-differential equations, introduced in Fröhlich et al. [“Some hamiltonian models of friction,” J. Math. Phys. 52(8), 083508 (2011)10.1063/1.3619799], describing a tracer particle in a dispersive medium is investigated, and decay properties of the solution are proven. This work is an extension of Fröhlich et al. [“Friction in a model of hamiltonian dynamics,” Commun. Math. Phys. 315(2), 401–444 (2012)10.1007/s00220-012-1564-2]; it is an extension because no weak coupling limit for the interaction between tracer particle and medium is assumed. The technical methods used are dispersive estimates and a contraction principle.