A Quantum Theory of Thermodynamic Relaxation

Roumen Tsekov

doi:10.3390/i2020066

Abstract

A new approach to quantum Markov processes is developed and the corresponding Fokker-Planck equation is derived. The latter is examined to reproduce known results from classical and quantum physics. It was also applied to the phase-space description of a mechanical system thus leading to a new treatment of this problem different from the Wigner presentation. The equilibrium probability density obtained in the mixed coordinate-momentum space is a reasonable extension of the Gibbs canonical distribution.

Highlights

Despite the great progress in contemporary quantum statistical physics [1], there are still problems in the applicability of the developed concept to complex systems such as in chemistry
The theory of quantum relaxation is much less elaborated than that of the equilibrium. This fact is not surprising since the same situation holds in classical physics
The most general approach to quantum dissipation consists in the projection of the Universe dynamics to those of a system and environment

Summary

Hβ a

Ψβa is the temperature dependent wave function and the probability density is defined as the square of the wave function. The modified Schrödinger equation can be rewritten in the form. Representing the static force balance between the system and its environment. The thermal force functional φa = −kT ∂a ln ρe plays a key role in the description of thermodynamic relaxation [2,20,21]. The time evolution of the probability density should necessarily satisfy the continuity equation. Which follows from the von Neuman equation and can be adopted as definition of the velocity v in the system probability space. In the frame of linear non-equilibrium thermodynamics [2], the following two equations hold φa = −kT ∂a ln ρ − L−a1v + o(v2 )

Ha ψ β a

In the latter case we have

Hamiltonian operator