# Brownian Motion of a Classical Particle in Quantum Environment

## TL;DR

This paper derives a generalized Klein-Kramers equation for classical particles in quantum environments, introduces quantum temperature and friction operators, and extends to quantum particles using Bohm's potential.

## Contribution

It presents a novel derivation of the Klein-Kramers and Smoluchowski equations incorporating quantum effects and introduces quantum operators for temperature and friction.

## Key findings

- Derived Klein-Kramers equation in quantum environment
- Extended Smoluchowski equation for quantum particles
- Incorporated Bohm quantum potential into Brownian motion model

## Abstract

The Klein-Kramers equation, governing the Brownian motion of a classical particle in quantum environment under the action of an arbitrary external potential, is derived. Quantum temperature and friction operators are introduced and at large friction the corresponding Smoluchowski equation is obtained. Introducing the Bohm quantum potential, this Smoluchowski equation is extended to describe the Brownian motion of a quantum particle in quantum environment.

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Source: https://tomesphere.com/paper/1701.05032