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arXiv:1802.10170

A simple method to obtain the all order quantum corrected Bose-Einstein distribution

Anirban Bose

Published Feb 16, 2018• 0 citations
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Abstract

A simple method has been introduced to derive the all order quantum corrected Bose-Einstein distribution as the solution of the Wigner equation. The process is a perturbative one where the Bose-Einstein distribution has been taken as the unperturbed solution. This solution has been applied to calculate the number density of the bosons at finite temperature. The study may be important to investigate the properties of bosons and bose condensates at finite temperature. This process can also be applied to obtain the quantum corrected Fermi distribution.

DOI
10.48550/arxiv.1802.10170

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A simple method has been introduced to derive the all order quantum corrected Bose-Einstein distribution as the solution of the Wigner equation. The process is a perturbative one where the Bose-Einstein distribution has been taken as the unperturbed solution. This solution has been applied to calculate the number density of the bosons at finite temperature. The study may be important to investigate the properties of bosons and bose condensates at finite temperature. This process can also be applied to obtain the quantum corrected Fermi distribution.
Abstract A method has been introduced to derive the all order Bose–Einstein distribution as the solution of the Wigner equation. The process is a perturbative one where the Bose–Einstein distribution has been taken as the unperturbed solution. This solution has been applied to calculate the number density of the bosons at finite temperature. The study may be important to investigate the properties of bosons at finite temperature. This process can also be applied to obtain the all order Fermi distribution.
In a previous article \cite{kn:anirban1} a method has been introduced to derive the all order Bose-Einstein distribution of the non interacting Bosons as the solution of the Wigner equation. The process was a perturbative one where the Bose-Einstein distribution was taken as the unperturbed solution. In this article it is shown that the same formalism is also applicable in the case of interacting Bosons. The formalism has been applied to calculate the quantum second virial coefficient of the Bosons interacting pairwise via Lenard-Jones potential and compared with the previous result.
The recently proposed method for investigating the large-order behaviour of the perturbative expansion is applied to a quantum thermodynamical system of interacting bosons in one space dimension. The method makes essential use of the classical solutions for the total Hamiltonian and of the oscillations about them. The contribution of these oscillations is calculated by using a canonical transformation related to the inverse scattering method, the relevant equation being the nonlinear Schrodinger equation. The leading-order corrections to the Bose-Einstein distribution are obtained explicitly.
We propose an efficient stochastic method to implement numerically the Bogolubov approach to study finite-temperature Bose-Einstein condensates. Our method is based on the Wigner representation of the density matrix describing the non-condensed modes and a Brownian motion simulation to sample the Wigner distribution at thermal equilibrium. Allowing it to sample any density operator Gaussian in the field variables, our method is very general and it applies both to the Bogolubov and to the Hartree-Fock Bogolubov approach, in the equilibrium case as well as in the time-dependent case. We think that our method can be useful to study trapped Bose-Einstein condensates in two or three spatial dimensions without rotational symmetry properties, as in the case of condensates with vortices, where the traditional Bogolubov approach is difficult to implement numerically due to the need to diagonalize very big matrices.
At low temperatures the De Broglie wavelength of the gas particles becomes on the order of their average separation, and the effects of their indistinguishability become important. In the absence of a phase transition in the gas, the quantum mechanical Wigner distribution function for a dilute gas of fermions or bosons satisfies the Uehling-Uhlenbeck equation. This equation satisfies an H- theorem with equilibrium solutions being ideal boson or ideal fermion distributions. Navier-Stokes equations can be derived by standard methods. A low temperature gas of weakly interacting bosons undergoes a Bose-Einstein condensation with a macroscopically occupied ground state. A different approach is required to describe the non-equilibrium processes in such a situation. A kinetic equation can be derived for the Bogoliubov excitations in the gas at very low temperatures. The associated hydrodynamic equations are the Landau-Khaltnikov, two fluid equations, and explicit expressions are obtained for the six associated transport coefficients.
Motivated by the asynchronous finite differences method utilized for the calculation of the most probable distributions of finite particle number systems, this study employs numerical variation and central difference techniques to provide more precise estimations regarding these distributions. Specifically, three novel finite distributions are derived and applied to Bose–Einstein condensation, revealing that the critical condition (nλ3 = 2.612) may be relaxed in finite particle number scenarios. Moreover, maintaining density as a constant is anticipated to result in a higher critical temperature compared to infinite number systems. Notably, the obtained condensate number on the zero-energy level surpasses that of predictions generated by canonical distributions.