Flow Characteristics in a Crowded Transport Model

TL;DR

This paper models particle transport with size exclusion using nonlinear drift-diffusion equations, deriving boundary conditions from microscopic processes, analyzing flow phases, and providing numerical methods for complex geometries.

Contribution

It introduces Robin-type boundary conditions derived from microscopic exclusion processes and analyzes flow phases in nonlinear drift-diffusion models.

Findings

Existence of solutions for stationary equations.

Identification of three flow phases including a maximal current phase.

Development of a numerical approach for multi-dimensional problems.

Abstract

The aim of this paper is to discuss the appropriate modelling of in- and outflow boundary conditions for nonlinear drift-diffusion models for the transport of particles including size exclusion and their effect on the behaviour of solutions. We use a derivation from a microscopic asymmetric exclusion process and its extension to particles entering or leaving on the boundaries. This leads to specific Robin-type boundary conditions for inflow and outflow, respectively. For the stationary equation we prove the existence of solutions in a suitable setup. Moreover, we investigate the flow characteristics for small diffusion, which yields the occurence of a maximal current phase in addition to well-known one-sided boundary layer effects for linear drift-diffusion problems. In a one-dimensional setup we provide rigorous estimates in terms of $\epsilon$, which confirm three different phases. Finally, we derive a numerical approach to solve the problem also in multiple dimensions. This provides further insight and allows for the investigation of more complicated geometric setups.