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arXiv:1211.6154 · Communications in Partial Differential Equations

Hamiltonian Dynamics of a Particle Interacting with a Wave Field

Daniel Egli, Jürg Martin Fröhlich, Zhou Gang, Arick Shao, Israel Michael Sigal

Published Jun 24, 2013• 10 citations
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Abstract

We study the Hamiltonian equations of motion of a heavy tracer particle interacting with a dense weakly interacting Bose-Einstein condensate in the classical (mean-field) limit. Solutions describing ballistic subsonic motion of the particle through the condensate are constructed. We establish asymptotic stability of ballistic subsonic motion.

DOI
10.1080/03605302.2013.816857

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We study the Hamiltonian equations of motion of a heavy tracer particle interacting with a dense weakly interacting Bose-Einstein condensate in the classical (mean-field) limit. Solutions describing ballistic subsonic motion of the particle through the condensate are constructed. We establish asymptotic stability of ballistic subsonic motion.
We study the motion of a heavy tracer particle weakly coupled to a dense interacting Bose gas exhibiting Bose-Einstein condensation. In the so-called mean-field limit, the dynamics of this system approaches one determined by nonlinear Hamiltonian evolution equations. We derive the effective dynamics of the tracer particle, which is described by a non-linear integro-differential equation with memory, and prove that if the initial speed of the tracer particle is below the speed of sound in the Bose gas the motion of the particle approaches an inertial motion at constant velocity at large times.
Mathematical results on a model describing the motion of a tracer particle through a non-interacting Bose-Einstein condensate are described. In the limit of a very dense gas and for a very large particle mass, the dynamics of the coupled system is determined by classical nonlinear Hamiltonian equations of motion. The particle's motion exhibits deceleration corresponding to friction with memory caused by the emission of Cerenkov radiation of gapless modes into the gas. A more general class of models involving interacting Bose gases will be studied in forthcoming papers.
We study a system consisting of a heavy quantum particle, called tracer particle, coupled to an ideal gas of light Bose particles, the ratio of masses of the tracer particle and a gas particle being proportional to the gas density. All particles have non-relativistic kinematics. The tracer particle is driven by an external potential and couples to the gas particles through a pair potential. We compare the quantum dynamics of this system to an effective dynamics given by a Newtonian equation of motion for the tracer particle coupled to a classical wave equation for the Bose gas. We quantify the closeness of these two dynamics as the mean-field limit is approached (gas density →∞\to\infty→∞). Our estimates allow us to interchange the thermodynamic with the mean-field limit.
We study a system consisting of a heavy quantum particle, called tracer particle, coupled to an ideal gas of light Bose particles, the ratio of masses of the tracer particle and a gas particle being proportional to the gas density. All particles have non-relativistic kinematics. The tracer particle is driven by an external potential and couples to the gas particles through a pair potential. We compare the quantum dynamics of this system to an effective dynamics given by a Newtonian equation of motion for the tracer particle coupled to a classical wave equation for the Bose gas. We quantify the closeness of these two dynamics as the mean-field limit is approached (gas density →∞\to\infty→∞). Our estimates allow us to interchange the thermodynamic with the mean-field limit.
We study a system of a heavy quantum particle, from now on called tracer particle, coupled to an ideal gas of NNN light Bose particles with a ratio between the masses of the tracer particle and of a gas particle being proportional to the gas particle density. While all particles obey the non-relativistic dispersion relation, the tracer particle is also driven by an external potential and couples to the gas particles through a pair potential. We compare the corresponding quantum dynamics to effective dynamics, given by a Newtonian equation for the tracer particle and a corresponding effective equation for the Bose gas. We quantify the closeness of these two dynamics depending on the gas particle density and the volume of the support of their wave function. The provided estimates are such that we get convergence in any limit taking the density and the volume to infinity. In particular, they allow to interchange the thermodynamic with the mean field limit.
A paper presents a theoretical investigation of subsonic and supersonic effects in a Bose-Einstein condensate (BEC). The BEC is represented by a time-dependent, nonlinear Schroedinger equation that includes terms for an external confining potential term and a weak interatomic repulsive potential proportional to the number density of atoms. From this model are derived Madelung equations, which relate the quantum phase with the number density, and which are used to represent excitations propagating through the BEC. These equations are shown to be analogous to the classical equations of flow of an inviscid, compressible fluid characterized by a speed of sound (g/Po)1/2, where g is the coefficient of the repulsive potential and Po is the unperturbed mass density of the BEC. The equations are used to study the effects of a region of perturbation moving through the BEC. The excitations created by a perturbation moving at subsonic speed are found to be described by a Laplace equation and to propagate at infinite speed. For a supersonically moving perturbation, the excitations are found to be described by a wave equation and to propagate at finite speed inside a Mach cone.