Quantum Diffusion with Drift and the Einstein Relation I

TL;DR

This paper analyzes the behavior of a quantum particle on a lattice under external force and thermal reservoirs, demonstrating non-equilibrium steady state formation, diffusive motion, and the Einstein relation linking drift and diffusion.

Contribution

It establishes the existence of a non-equilibrium steady state for a quantum particle with drift and diffusion, confirming the Einstein relation in this quantum setting.

Findings

The particle reaches a non-equilibrium steady state with a non-zero drift.

The particle's motion around the drift is diffusive.

The diffusion constant and drift velocity satisfy the Einstein relation.

Abstract

We study the dynamics of a quantum particle hopping on a simple cubic lattice and driven by a constant external force. It is coupled to an array of identical, independent thermal reservoirs consisting of free, massless Bose fields, one at each site of the lattice. When the particle visits a site x of the lattice it can emit or absorb field quanta of the reservoir at x. Under the assumption that the coupling between the particle and the reservoirs and the driving force are sufficiently small, we establish the following results: The ergodic average over time of the state of the particle approaches a non-equilibrium steady state (NESS) describing a non-zero mean drift of the particle. Its motion around the mean drift is diffusive, and the diffusion constant and the drift velocity are related to one another by the Einstein relation.