Biased Tracer Diffusion in Hard-Core Lattice Gases: Some Notes on the Validity of the Einstein Relation

TL;DR

This paper investigates biased tracer diffusion in lattice gases, demonstrating that the Einstein relation generally holds even with anomalous diffusion, supported by analytical and simulation results.

Contribution

It provides exact and approximate analytical results on tracer particle mobility and diffusion, confirming the Einstein relation in various lattice gas models.

Findings

Einstein relation holds despite anomalous diffusion.

Analytical results confirmed by Monte Carlo simulations.

Biased tracer causes density profile perturbations.

Abstract

In this presentation we overview some recent results on biased tracer diffusion in lattice gases. We consider both models in which the gas particles density is explicitly conserved and situations in which the lattice gas particles undergo continuous exchanges with a reservoir, which case is appropriate, e.g., to adsorbed monolayers in contact with the vapor phase. For all these models we determine, in some cases exactly and in other ones - using a certain decoupling approximation, the mean displacement of a tracer particle (TP) driven by a constant external force in a dynamical background formed by the lattice gas particles whose transition rates are symmetric. Evaluating the TP mean displacement explicitly we are able to define the TP mobility, which allows us to demonstrate that the Einstein relation between the TP mobility and the diffusivity generally holds, despite the fact that in some cases diffusion is anomalous. For models treated within the framework of the decoupling approximation, our analytical results are confirmed by Monte Carlo simulations. Perturbance of the lattice gas particles distribution due to the presence of a biased TP and the form of the particle density profiles are also discussed.