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

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 nonequilibrium one particle wigner distribution in the relaxation-time approximation and under near-equilibrium conditions. PACS numbers: 05.30.-d, 05.20.Dd

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

“…Inserting W 2n and W 4n from equations ( 6) and (10) respectively and collecting the Λ 4 order coecients of dierent powers of p and separately equating them to zero, a set of first order dierential equations emerge [45].…”

“…An introduction to the theory along with a Monte Carlo method for the simulation of time-dependent quantum systems of fermions evolving in a phase-space has been presented [43]. In a previous work [45] of the author the Wigner equation has been solved perturbatively for any arbitrary order without taking into account the quantum statistics. In this article quantum statistics is also incorporated to ensure the completeness of the problem.…”

“…In this article quantum statistics is also incorporated to ensure the completeness of the problem. Moreover, in that previous article [45] the derivation of the solution of a particular order requires the exact solution of the previous order. If the solution of a particular order is correct up to that order but not exact https://doi.org/10.1088/1742-5468/ab43d1…”

Bernstein-Greene-Kruskal approach for the quantum Vlasov equation

“…This includes the original article by Wigner [1], where the lowest order quantum correction to the Maxwell-Boltzmann equilibrium was evaluated. Nevertheless, already in [1] the possibility of series solutions up to arbitrary order has been proposed, see also [13]. In addition, the role of the energy as an useful dynamical variable has been identified, for a certain class of solutions of the stationary one-dimensional Wigner-Moyal equation and Wigner-Poisson system not restricted to the semiclassical case [14].…”

Bernstein-Greene-Kruskal approach for the quantum Vlasov equation

“…This includes the original article by Wigner [1], where the lowest-order quantum correction to the Maxwell-Boltzmann equilibrium was evaluated. Nevertheless, already in [1] the possibility of series solutions up to arbitrary order has been proposed, see also [13]. In addition, the role of the energy as a useful dynamical variable has been identified, for a certain class of solutions of the stationary one-dimensional Wigner-Moyal equation and Wigner-Poisson system not restricted to the semiclassical case [14].…”