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Journal of Statistical Mechanics Theory and Experiment

A method to obtain the all order quantum corrected Bose–Einstein distribution from the Wigner equation

Anirban Bose

Published Nov 1, 2019• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1742-5468/ab43d1

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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.
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.
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.
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.
Abstract Chapter 11 presents a statistical mechanical treatment of the quantum ideal gases, i.e., the ideal Boltzmann, fermion, and boson gases. The discussion begins with the microscopic description and the solution of the eigenvalue problem for a quantum ideal gas. It is argued that calculating the partition function is most readily accomplished in the grand canonical ensemble using a second-quantized formulation. When the particles are distinguishable, the equation of state is identical to that of a classical ideal gas. For fermions and bosons, however, the problem of computing thermodynamic properties is significantly more complex and can only be solved exactly in certain limits. Away from these limits approximations are needed and are discussed in detail. The relevant distributions - the Fermi-Dirac and Bose-Einstein distributions are derived. The local density approximation of density functional theory is derived for the ideal electron gas. For the ideal boson gas, the phenomenon of Bose-Einstein condensation is discussed