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arXiv:hep-ph/9409202 · Physical Review Letters

Kinetics of Bose Condensation

D. V. SEMIKOZ, Igor I. Tkachev

Published Apr 17, 1995• 198 citations
arXivHTMLPDF

Abstract

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 $\mathbf{p}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ and the distribution function at $\mathbf{p}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ 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.

DOI
10.1103/physrevlett.74.3093

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The process of condensation in the system of scalar bosons with weak \ensuremathλ\ensuremathφ4\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}\ensuremathλ\ensuremathφ4 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.
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 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 evolution equation of a weakly interacting boson system is derived. The model includes the interaction between the atoms in the condensate and the surrounding gas of noncondensed particles. The Bogoliubov transformation is introduced in a full quantum context and the scattering kernel between dressed particles and the condensate phase is obtained. The final system is expressed by the Boltzmann evolution equation for noncondensed particles coupled to the Gross-Pitaevskii equation for the condensate. We consider an out-of-equilibrium situation that induces a fast production of condensed particles. We apply our model to study the condensation dynamics of positronium atoms by evaporation.
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].
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.