Hamiltonian Dynamics of a Particle Interacting with a Wave Field

Abstract

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.

Emission of Cherenkov radiation as a mechanism for Hamiltonian friction

“…If the initial speed of the tracer particle is well below the speed of sound in the gas one expects that the motion of the particle approaches a uniform (inertial) motion at large times. A result in this direction has recently been established in a certain limiting regime (the "mean-field-Bogolubov limit") of the Bose gas in [4]. In the present paper, we prove results complementary to those in [4] for the same model: Assuming that the initial speed of the tracer particle is larger than the speed of sound in the Bose gas, we show that this particle decelerates by emission of Cherenkov radiation of sound waves into the gas until its speed is equal to (or smaller than) the speed of sound.…”

“…A result in this direction has recently been established in a certain limiting regime (the "mean-field-Bogolubov limit") of the Bose gas in [4]. In the present paper, we prove results complementary to those in [4] for the same model: Assuming that the initial speed of the tracer particle is larger than the speed of sound in the Bose gas, we show that this particle decelerates by emission of Cherenkov radiation of sound waves into the gas until its speed is equal to (or smaller than) the speed of sound. For some earlier results on related models, see also [15,12].…”

“…It is easy to see that, under rather weak assumptions on the potentials W and φ, Eqs. (1.7) and (1.8) have static solutions, and that if the external force acting on the tracer particle vanishes (Φ ≡ 0) they have "traveling wave solutions", provided the speed of the particle is smaller than or equal to the speed of sound in the Bose gas; see [8,4]. These solutions correspond to an inertial motion of the tracer particle at a constant velocity, with the particle accompanied by a "splash" in the Bose gas.…”

“…In the present paper we consider the supersonic regime, namely the initial speed | P 0 M | of the tracer particle is larger than the speed of sound, v s . For the subsonic regime we refer to our previous papers [4,6]. In contrast to sonic and subsonic particle motions, inertial supersonic particle motions do not exist, i.e., the equations of motion do not have traveling wave solutions propagating at supersonic speeds, as shown in [4].…”

“…For the subsonic regime we refer to our previous papers [4,6]. In contrast to sonic and subsonic particle motions, inertial supersonic particle motions do not exist, i.e., the equations of motion do not have traveling wave solutions propagating at supersonic speeds, as shown in [4]. We propose to construct solutions of the equations of motion corresponding to initial conditions at time t = 0 with the properties that β 0 is "small" in a suitable sense, i.e., the state of the Bose gas is close (or equal) to the ground state β 0 = 0, and the initial speed of the tracer particle is above the speed of sound, v s = λ 2m .…”