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CRC Press eBooks

Kinetics of Bose–Einstein Condensation

Published Apr 15, 2015• 1 citation
Publisher

Abstract

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.We use throughout dimensionless variables in units which minimize the number of prefactors.The physically correct dimensions are supplied, e.g., in [1].Instead of denoting the energy by ε it will be more convenient to use the neutral x, y, z.The distribution function is denoted by f (x), f ≥ 0, with volume element √ xdx, x ≥ 0,

DOI
10.1201/b18346-18

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Cambridge University Press eBooks•Apr 6, 1995•13 citations
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.We use throughout dimensionless variables in units which minimize the number of prefactors.The physically correct dimensions are supplied, e.g., in [1].Instead of denoting the energy by ε it will be more convenient to use the neutral x, y, z.The distribution function is denoted by f (x), f ≥ 0, with volume element √ xdx, x ≥ 0,
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.
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.
In this article we discuss the role played by kinetic theory in describing the non-equilibrium dynamics of dilute systems of weakly interacting bosons. We illustrate how a simple kinetic equation for the time evolution of the spectral particle density can be derived from the spatially homogeneous Gross–Pitaevskii equation. This kinetic equation agrees with the usual Boltzmann–Nordheim equation of quantum kinetic theory in the long wavelength limit where the occupation numbers are expected to be large. The stationary solutions of the Gross–Pitaevskii kinetic equation are described. These include both thermodynamic equilibrium spectra and finite flux Kolmogorov–Zakharov spectra. These latter spectra are intrinsically nonequilibrium states and are expected to be relevant in the transfer of particles to low momenta in the initial stage of the condensation process. This is illustrated by some computations of a solution of the kinetic equation beginning with initial conditions far from equilibrium. The solution generates a flux of particles from large to small momenta which results in a singularity at zero momentum within finite time. We interpret this singularity as incipient condensate formation. We then present some numerical results on the post-singularity dynamics and the approach to equilibrium. Contrary to our original expectations we do not observe the Kolmogorov–Zakharov spectrum during the period of condensate growth. In the closing sections we address the issue of the connection between the Gross–Pitaevskii and Boltzmann–Nordheim kinetic equations. We argue that the two equations have differing regimes of applicability in momentum space, matching in an intermediate range. We make some suggestions of how this matching can be modeled in practice.
We study the kinetic regime of the Bose condensation of scalar particles. The Boltzmann equation is solved numerically. We consider two kinetic stages. At the first stage the condensate is absent but there is a nonzero inflow of particles towards p=0\mathbf{p}=0p=0 and the distribution function at p=0\mathbf{p}=0p=0 grows from finite values to infinity in a finite time. We observe a profound similarity between Bose condensation and Kolmogorov turbulence. At the second stage there are two components, the 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. We do not consider a phase transition from the first stage to the second. Condensation of self-interacting bosons is compared to the condensation driven by interaction with a cold gas of fermions; the latter turns out to be self-similar too. Exploiting the self-similarity we obtain a number of analytical results in all cases.
We apply a Boltzmann approach to the kinetic regime of a relativistic Bose-Einstein condensate of scalar bosons by decomposing the one-particle distribution function in a condensate part and a nonzero momentum part of excited modes, leading to a coupled set of evolution equations which are then solved efficiently with an adaptive higher order Runge-Kutta scheme. We compare our results to the partonic cascade Monte Carlo simulation BAMPS for a critical but far from equilibrium case of massless bosons. Motivated by the color glass condensate initial conditions in QCD with a strongly overpopulated initial glasma state, we also discuss the time evolution starting from an overpopulated initial distribution function of massive scalar bosons. In this system a self-similar evolution of the particle cascade with a nonrelativistic turbulent scaling in the infrared sector is observed as well as a relativistic exponent for the direct energy cascade, confirming a weak wave turbulence in the ultraviolet region.
The kinetics of the formation of coherent correlation properties associated with Bose condensation is studied in detail. The evolution of a nonequilibrium state with no “condensate-seed” is related to a hierarchy of relaxation times. At the first stage, a particle flux in energy space toward low energies sets in. The evolution in this case is described by a nonlinear Boltzmann equation, with a characteristic time given by interparticle collisions. When the particles which will later form the condensate have a kinetic energy which is less than the potential energy, a quasicondensate starts to form. In this stage, fluctuations of the density (but not of the phase) are suppressed and short-range coherent correlation properties are governed by the equation of motion for a quasiclassical complex field. The next stage is connected with the formation of the long-range order. The time for forming topological order and therefore genuine superfluidity proves to be dependent on the system size. The off-diagonal long-range order, arising after the attenuation of long-wave phase fluctuations, has a size-dependent relaxation time as well. Introduction The problem of Bose–Einstein Condensation (BEC) kinetics, being interesting in itself, has acquired a special significance in connection with experimental efforts to observe this condensation in a number of systems with particles with a finite lifetime. Such systems include spin-polarized atomic hydrogen [1], excitons [2] and biexcitons [3] in semiconductors and, more recently, laser-cooled atomic systems [4].