A simple method to obtain the equilibrium solution of Wigner-Boltzmann Equation with all higher order quantum corrections

TL;DR

This paper introduces a straightforward recursive method to derive the equilibrium solution of the Wigner-Boltzmann equation incorporating all higher-order quantum corrections, offering a compact exponential-polynomial form useful for quantum system analysis.

Contribution

A novel recursive approach for obtaining the equilibrium Wigner solution with all quantum corrections, differing from traditional methods and enabling applications to non-equilibrium scenarios.

Findings

Derives a compact exponential-polynomial form of the solution.

Provides a recursive relation for quantum correction coefficients.

Applicable to both closed and open quantum systems.

Abstract

A simple method has been introduced to furnish the equilibrium solution of the Wigner equation for all order of the quantum correction. This process builds up a recursion relation involving the coefficients of the different power of the velocity. The technique greatly relies upon the proper guess work of the trial solution and is different from the Wigner s original work. The solution is in a compact exponential form with a polynomial of velocity in the argument and returns the Wigner s form when expansion of the exponential factor is carried out. The study keeps its importance in studying various close as well as open quantum mechanical system. In addition, this solution may be employed to obtain the non equilibrium one particle wigner distribution in the relaxation-time approximation and under near-equilibrium conditions.