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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Phonon-assisted Boltzmann kinetics of a Bose gas: Generic solution for T⩽ Tc

A. L. Ivanov, Claudia Ell, H. Haug

Published Jun 1, 1997• 29 citations
Publisher

Abstract

The relaxation kinetics of an ideal Bose gas coupled to a phonon bath at temperatures below or equal to the critical temperature ${\mathrm{T}}_{\mathrm{c}}$ is given within a unique scenario. During the first transient stage, which lasts a few characteristic scattering times, the initial distribution disappears. The following nonexponential relaxation towards a quantum degenerate equilibrium state with a Bose-Einstein condensate is a slow adiabatic process. For this second kinetic stage we find an analytic generic solution of the Boltzmann equation. The generic solution is independent of the initial distribution and completely defined by the ratio ${\mathrm{T}}_{\mathrm{c}}$/${\mathrm{T}}_{\mathrm{b}}$ of the critical and bath temperatures.

DOI
10.1103/physreve.55.6363

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The relaxation kinetics of an ideal Bose gas coupled to a phonon bath at temperatures below or equal to the critical temperature Tc{\mathrm{T}}_{\mathrm{c}}Tc​ is given within a unique scenario. During the first transient stage, which lasts a few characteristic scattering times, the initial distribution disappears. The following nonexponential relaxation towards a quantum degenerate equilibrium state with a Bose-Einstein condensate is a slow adiabatic process. For this second kinetic stage we find an analytic generic solution of the Boltzmann equation. The generic solution is independent of the initial distribution and completely defined by the ratio Tc{\mathrm{T}}_{\mathrm{c}}Tc​/Tb{\mathrm{T}}_{\mathrm{b}}Tb​ of the critical and bath temperatures.
The relaxation kinetics of an ideal Bose gas coupled to a phonon thermostat at temperature Tb below or equal to the critical temperature Tc consists of an initial transient and the following adiabatic stage. The transient depends on the initial distribution of bosons at t = 0 and lasts a few characteristic scattering times. The following nonexponential relaxation towards a quantum degenerate equilibrium state with a Bose-Einstein (BE) condensate is a slow adiabatic process. We find a generic solution of the phonon-assisted Boltzmann kinetics at the adiabatic stage. This solution is completely defined by the ratio Tc/Tb. Here, one has a unique example of the fundamental solution of Boltzmann kinetics that is insensitive to the initial distribution. For Tb → Tc from above, we develop a theory of the thermodynamic critical slowing-down which also leads to the nonexponential thermalization of bosons. We apply our results to the relaxation kinetics of para-excitons in Cu2O and reinterpret the recent reports on the observation of BE condensation of para-excitons.
We study, in the framework of the Boltzmann-Bloch-Nordheim equation (BBN), the kinetic properties of a boson gas above the Bose-Einstein transition temperature Tc.{T}_{c}.Tc​. The BBN equation is solved numerically within an algorithm that we test with exact analytical results for the collision rate of a homogeneous system in thermal equilibrium. In the classical regime $(T\ensuremath{\gg}{T}_{c}),$ the relaxation time of a quadrupolar deformation in momentum space is proportional to the mean free collision time ${\ensuremath{\tau}}_{\mathrm{relax}}\ensuremath{\propto}{\ensuremath{\tau}}_{\mathrm{coll}}\ensuremath{\propto}{T}^{\ensuremath{-}1/2}.$ Approaching the critical temperature, quantum statistic effects become dominant, and the collision rate increases dramatically. Nevertheless, this does not affect the relaxation properties of the gas that depend only on the spontaneous collision term in BBN. The relaxation time becomes ${\ensuremath{\tau}}_{\mathrm{relax}}\ensuremath{\propto}(T\ensuremath{-}{T}_{c}{)}^{\ensuremath{-}1/2},$ exhibiting a critical slowing down. These phenomena can be experimentally confirmed looking at the damping properties of collective motions induced on trapped atoms. The possibility to observe a transition from collisionless (zero-sound) to hydrodynamic (such as first and second-sound) is finally discussed.
The process of condensation in the system of scalar bosons with weak $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ interaction is considered. The Boltzmann kinetic equation is solved numerically. We consider two kinetic stages. At the first stage condensate is still absent but there is nonzero inflow of particles towards p=0\mathbf{p}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0p=0 and the distribution function at p=0\mathbf{p}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0p=0 grows from finite values to infinity. At the second stage there are two components, condensate and particles, reaching their equilibrium values. We show that the evolution in both stages proceeds in a self-similar way and find the time needed for condensation.
The mathematical contributions by X.G. Lu (J. Statist. Phys. 98 (5/6) (2000) 1335–1394) and by M. Escobedo et al. (Electronic J. Differential Equations, Monograph 4 (2003)) presented in this Note constitute the first stage in the understanding of the superfluid dynamics, especially of the Bose–Einstein condensation, by means of kinetic models. The Boltzmann–Nordheim equation, which is physically relevant to describe dilute quantum Bose gases, sets important mathematical problems. Nevertheless, under an unphysical truncation of the collision cross-section at low energies, it has been proved that the spatially homogeneous Cauchy problem is well-posed. Furthermore, relaxation towards equilibrium holds in a weak sense, with the appearance of a singularity in infinite time if the initial mass is supercritical, which corresponds to the formation of a Bose–Einstein condensate.
We study the bosonic Boltzmann-Nordheim kinetic equation, which describes the kinetic regime of weakly interacting bosons with s-wave scattering only. We consider a spatially homogeneous fluid with an isotropic momentum distribution. The issue of the dynamical formation of a Bose-Einstein condensate has been studied extensively. We supply here the completed equations of motion for the coupled system, the energy density distribution of the normal fluid and the density of the condensate. With this information the post-nucleation self-similar solution is investigated in more detail than before.