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Physical review. B, Solid state

Time evolution of a Bose system passing through the critical point

Evgeny Levich, Victor Yakhot

Published Jan 1, 1977• 57 citations
Publisher

Abstract

Kinetics of Bose condensation of an ideal Bose gas and of the classical Bose gas of hard spheres is considered. Deviations from the equilibrium state are not assumed to be small. An analytic solution for the problem of an ideal Bose gas is found and the role of collisions between the Bose particles themselves is discussed qualitatively. It is shown that Bose condensation is a slowing down process which can be completed at $t\ensuremath{\rightarrow}\ensuremath{\infty}$ while the formation of the high-energy tail of the distribution function is relatively fast. The transition time is finite if the nuclei of new phase are present in the beginning of the cooling process.

DOI
10.1103/physrevb.15.243

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Kinetics of Bose condensation of an ideal Bose gas and of the classical Bose gas of hard spheres is considered. Deviations from the equilibrium state are not assumed to be small. An analytic solution for the problem of an ideal Bose gas is found and the role of collisions between the Bose particles themselves is discussed qualitatively. It is shown that Bose condensation is a slowing down process which can be completed at t\ensuremath→\ensuremath∞t\ensuremath{\rightarrow}\ensuremath{\infty}t\ensuremath→\ensuremath∞ while the formation of the high-energy tail of the distribution function is relatively fast. The transition time is finite if the nuclei of new phase are present in the beginning of the cooling process.
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].
We examine the time evolution of a weakly interacting Bose gas in the course of the Bose-Einstein phase transition and show that, in contrast with previous claims in the literature, the relevant time scale for the appearance of the condensate is finite and, under the conditions we consider, of O(\ensuremath{\Elzxh}/kB{\mathit{k}}_{\mathit{B}}kB​Tc{\mathit{T}}_{\mathit{c}}Tc​), which is very small compared to the characteristic lifetime of the system due to inelastic collisions in the gas.
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.
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.
In this paper, we present a method for simulating the kinetic evolution of a dilute gas of atoms that are cooled below the critical temperature for Bose-Einstein condensation. Our method gives insight into the formulation of physical kinetics by illustrating directly the decomposition of the distribution function into an infinite sum of single-particle trajectories. This approach is valid for the entire range of phase-space densities, although we limit the discussion here to exclude the region where the condensate fraction is close to unity and the effect of the mean field is significant. We present explicit calculations of finite number effects on equilibrium, the dynamic build-up of the ground state, and simulations of evaporative cooling.
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.