Nonlinear theory of quantum Brownian motion

TL;DR

This paper develops a nonlinear quantum Brownian motion theory using a thermodynamically enhanced nonlinear Schrödinger equation, leading to a quantum generalization of Einstein's law and insights into decoherence times.

Contribution

It introduces a novel nonlinear quantum Brownian motion framework based on a thermodynamic extension of the Schrödinger equation, connecting quantum and classical diffusion behaviors.

Findings

Reproduces key results from quantum and classical physics.

Derives a quantum generalization of Einstein's law of Brownian motion.

Shows decoherence time scales with the square of the thermal de Broglie wavelength.

Abstract

A nonlinear theory of quantum Brownian motion in classical environment is developed based on a thermodynamically enhanced nonlinear Schrodinger equation. The latter is transformed via the Madelung transformation into a nonlinear quantum Smoluchowski-like equation, which is proven to reproduce key results from quantum and classical physics. The application of the theory to a free quantum Brownian particle results in a nonlinear dependence of the position dispersion on time, being quantum generalization of the Einstein law of Brownian motion. It is shown that the time of decoherence for the transition from quantum to classical diffusion is proportional to the square of the thermal de Broglie wavelength divided by the Einstein diffusion constant.