Towards a new approach of quantum dissipation in simple chemical systems

TL;DR

This paper introduces a novel approach to quantum dissipation in small systems, linking thermodynamics, path integrals, and equations of motion to describe non-equilibrium dynamics and relaxation processes.

Contribution

It presents a new framework for quantum dissipation that does not rely on system-plus-reservoir models, using algebraic and path integral methods to derive equations of motion.

Findings

Derived a quantum Smoluchowski equation with a physically meaningful probability solution.

Calculated a time-dependent chemical rate that can be non-monotonic and relate to Kramers' rate.

Established a connection between thermodynamic equilibrium and underlying quantum dynamics.

Abstract

We suggest a new approach for describing quantum dissipation in a small systems for which the system-plus-reservoir approach is not relevant. We first analyze the fact that equilibrium thermodynamics may reveal the existence of an underlying dynamics. This is true in the algebraic approach of quantum mechanics via the Tomita-Takesaki theorem. A similar result is obtained if we start from the path integral expression of the partition function, an equation of motion is introduced. In both cases a parameter has to be identified with the physical time. Several arguments are presented showing that it is so. By investigating the dynamics for short times we introduce an equilibrium condition from which a natural unit of time appears. The equation of motion is extended to non equilibrium situations for which a H-theorem can be derived. The equation of motion can be transformed into a quantum Smoluchovski equation, its solution is a probability having a clear physical meaning. The relaxation of this probability towards its equilibrium value requires a time during which a thermodynamic description is possible or not. A standard bistable model is investigated and the chemical rate $k(t)$ is calculated as a function of time, it appears to be a non monotonic function of time. In some conditions the stationary value of $k(t)$ can be identified with the Kramers result. Finally we compare our approach which is naturally irreversible with an analysis based on the Schr\"{o}dinger equation which is reversible. From all these results it appears that our equation of motion or its Smoluchovski version appear as a realistic starting point for describing quantum dissipation in small systems.