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arXiv:cond-mat/0311622 · Journal of Physics B Atomic Molecular and Optical Physics

Temperature-dependent Bogoliubov approximation in the classical field approach to weakly interacting Bose gases

Mirosław Brewczyk, Peter Borowski, Mariusz Gajda, Kazimierz Rza̧żewski

Published Jun 22, 2004• 30 citations
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Abstract

A classical field approximation to the finite temperature microcanonical thermodynamics of weakly interacting Bose gases is applied to the idealized case of atoms confined in a box with periodic boundary conditions. The whole isolated system is described by a set of classical amplitudes. We show that if the system is in equilibrium the classical amplitudes follow the equipartition theorem which allows us to assign a unique temperature to each given energy state. We also analyse the spectral properties of classical amplitudes and obtain nonperturbative results for the chemical potential as well as temperature-dependent Bogoliubov frequencies and their damping rates. Where available, we make comparison with the analytical estimates of these quantities.

DOI
10.1088/0953-4075/37/13/010

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A classical field approximation to the finite temperature microcanonical thermodynamics of weakly interacting Bose gases is applied to the idealized case of atoms confined in a box with periodic boundary conditions. The whole isolated system is described by a set of classical amplitudes. We show that if the system is in equilibrium the classical amplitudes follow the equipartition theorem which allows us to assign a unique temperature to each given energy state. We also analyse the spectral properties of classical amplitudes and obtain nonperturbative results for the chemical potential as well as temperature-dependent Bogoliubov frequencies and their damping rates. Where available, we make comparison with the analytical estimates of these quantities.
The thermodynamic properties and finite-size scaling of weakly interacting Bose gases with a finite number of particles confined in a cubic box of volume L3{L}^{3}L3 with periodic boundary conditions are investigated theoretically based on an exact recursion relation for the canonical-ensemble partition function. In a similar manner to the case of an ideal Bose gas, the recursive scheme of the interacting finite system at any temperatures is established in the framework of the Bogoliubov theory. The temperature dependence of the condensate fraction and specific heat with different particles and interactions is obtained numerically. By defining two different transition temperatures of the finite systems, the effect of interactions on the transition temperatures yields the different and nonmonotonous behaviors. The finite-size scaling condensate fraction at the transition temperature for the systems is also discussed. It is shown that the calculated finite-size scaling is universally independent on the various system sizes and transition temperatures.
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The classical-field approximation to cold weakly interacting bosons allows for a unified treatment of condensed and uncondensed parts of the system. Until now, however, the quantitative predictions were limited by a dependence of the results on a grid chosen for numerical implementation of the method. In this paper, we propose replacing this unphysical ambiguity by an additional postulate: the temperature of the gas at thermal equilibrium should be the same as that of an ideal Bose gas with the same fraction of condensed atoms. As it turns out, with this additional assumption, nearly all atoms are within the classical fields, thus the method applies to the whole system.
We review our version of the classical field approximation to the dynamics of a finite temperature Bose gas. In the case of a periodic box potential, we investigate the role of the high momentum cut-off, essential in the method. In particular, we show that the cut-off going to infinity limit decribes the particle number going to infinity with the scattering length going to zero. In this weak interaction limit, the relative population of the condensate tends to unity. We also show that the cross-over energy, at which the probability distribution of the condensate occupation changes its character, grows with a growing scattering length. In the more physical case of the condensate in the harmonic trap we investigate the dissipative dynamics of a vortex. We compare the decay time and the velocities of the vortex with the available analytic estimates.
We study fluctuations of the number of Bose condensed atoms in a weakly interacting homogeneous and trapped gases. For a homogeneous system we apply the particle-number-conserving formulation of the Bogoliubov theory and calculate the condensate fluctuations within the canonical and the microcanonical ensembles. We demonstrate that, at least in the low-temperature regime, predictions of the particle-number-conserving and traditional, nonconserving theory are identical, and lead to the anomalous scaling of fluctuations. Furthermore, the microcanonical fluctuations differ from the canonical ones by a quantity which scales normally in the number of particles, thus predictions of both ensembles are equivalent in the thermodynamic limit. We observe a similar behavior for a weakly interacting gas in a harmonic trap. This is in contrast to the trapped, ideal gas, where microcanonical and canonical fluctuations are different in the thermodynamic limit.
We study condensate statistics and thermodynamics of weakly interacting Bose gas with a fixed total number NNN of particles in a cubic box. We find the exact recursion relation for the canonical ensemble partition function. Using this relation, we calculate the distribution function of condensate particles for N=200N=200N=200. We also calculate the distribution function based on multinomial expansion of the characteristic function. Similar to the ideal gas, both approaches give exact statistical moments for all temperatures in the framework of Bogoliubov model. We compare them with the results of unconstraint canonical ensemble quasiparticle formalism and the hybrid master equation approach. The present recursion relation can be used for any external potential and boundary conditions. We investigate the temperature dependence of the first few statistical moments of condensate fluctuations as well as thermodynamic potentials and heat capacity analytically and numerically in the whole temperature range.