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 [FSSG10], describing a tracer particle in a dispersive medium is investigated, and decay properties of the solution are proven. This work is an extension of [FGS10]; 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. * danegli@itp.phys.ethz.ch † zhougang@itp.phys.ethz.ch 1 2 −δ dr .(2.1)By numerical simulation we find that there exists a constant δ * ≃ 0.66 such that Ω(δ * ) = 1.Moreover, for any constant δ < δ * , we have Ω(δ) < 1.For the system of equations (1.4) we prove the following main theorem,

Hamiltonian Dynamics of a Particle Interacting with a Wave Field

“…In [12] and in [10] (where a more general result going beyond the weak-coupling limit is proven), some of the authors have analyzed particle motion in an ideal Bose gas, that is, for µ = 0. In an ideal Bose gas, the speed of sound is zero, hence there is only supersonic particle motion.…”

Ballistic motion of a tracer particle coupled to a Bose gas

“…The following models are of interest (see [5,4]): B -Model: κ = 0 (ideal Bose gas), g → 0, see [4]. C -Model: κ = 0, but g = 0; see [3]. E -Model: 2κρ 0 /g 2 := λ =const., with g, κ → 0 ("Bogolubov limit").…”

Emission of Cherenkov radiation as a mechanism for Hamiltonian friction

“…The following models are of interest (see [8,7]): B -Model: κ = 0 (ideal Bose gas), g → 0; see [7]. C -Model: κ = 0, but g = 0; see [5]. E -Model: 2κρ 0 /g 2 := λ =const., with g, κ → 0 ("Bogolubov limit"); this paper.…”