Tracer particle diffusion in a system with hardcore interacting particles

TL;DR

This paper develops an exact theoretical framework for tracer particle diffusion in systems with hardcore interacting crowders, extending previous models to various lattice types and validating results through simulations.

Contribution

It introduces a formal expansion for tracer density and provides a closed-form approximation for diffusion constants on multiple lattice types, broadening prior work.

Findings

Derived an exact formal expansion for tracer density.

Extended diffusion constant calculations to b.c.c. and f.c.c. lattices.

Validated analytical results with simulations in 2D and 3D.

Abstract

In this study, inspired by the work of K. Nakazato and K. Kitahara [Prog. Theor. Phys. 64, 2261 (1980)], we consider the theoretical problem of tracer particle diffusion in an environment of diffusing hardcore interacting crowder particles. The tracer particle has a different diffusion constant from the crowder particles. Based on a transformation of the generating function, we provide an exact formal expansion for the tracer particle probability density, valid for any lattice in the thermodynamic limit. By applying this formal solution to dynamics on regular Bravais lattices we provide a closed form approximation for the tracer particle diffusion constant which extends the Nakazato and Kitahara results to include also b.c.c. and f.c.c. lattices. Finally, we compare our analytical results to simulations in two and three dimensions.