Nonlinear macroscopic transport equations in many-body systems from microscopic exclusion processes

TL;DR

This paper derives nonlinear macroscopic transport equations from microscopic exclusion processes, addressing complex many-body interactions in inhomogeneous media and introducing models for non-ideal fluids with applications in biological and physical systems.

Contribution

It introduces a generalized nonlinear advection-diffusion equation and a modified diffusion equation for non-ideal fluids, linking microscopic exclusion processes to macroscopic transport models.

Findings

Derived a nonlinear advection-diffusion equation for inhomogeneous media.

Developed a modified diffusion equation for non-ideal fluids of hard spheres.

Demonstrated the formalism's ability to recover known results in specific applications.

Abstract

Describing particle transport at the macroscopic or mesoscopic level in non-ideal environments poses fundamental theoretical challenges in domains ranging from inter and intra-cellular transport in biology to diffusion in porous media. Yet, often the nature of the constraints coming from many-body interactions or reflecting a complex and confining environment are better understood and modeled at the microscopic level. In this paper we investigate the subtle link between microscopic exclusion processes and the mean-field equations that ensue from them in the continuum limit. We derive a generalized nonlinear advection diffusion equation suitable for describing transport in a inhomogeneous medium in the presence of an external field. Furthermore, taking inspiration from a recently introduced exclusion process involving agents with non-zero size, we introduce a modified diffusion equation appropriate for describing transport in a non-ideal fluid of $d$-dimensional hard spheres. We consider applications of our equations to the problem of diffusion to an absorbing sphere in a non-ideal self-crowded fluid and to the problem of gravitational sedimentation. We show that our formalism allows one to recover known results. Moreover, we introduce the notions of point-like and extended crowding, which specify distinct routes for obtaining macroscopic transport equations from microscopic exclusion processes.