Time-dependent properties of run-and-tumble particles: Density relaxation

TL;DR

This paper investigates the collective diffusion behavior of hardcore run-and-tumble particles on lattices, revealing how persistence and density influence diffusion coefficients and density relaxation, with results applicable across dimensions.

Contribution

It provides an explicit analytical and numerical characterization of the bulk-diffusion coefficient for two minimal models of RTPs, highlighting the interplay of persistence and interactions.

Findings

Diffusion coefficient varies as a power law with density, from $ ho^{-2}$ at high density to constant at low density.

Density relaxation follows a nonlinear diffusion equation with anomalous scaling.

Scaling form of the diffusion coefficient is derived and validated for different models.

Abstract

We characterize collective diffusion of hardcore run-and-tumble particles (RTPs) by explicitly calculating the bulk-diffusion coefficient $D(\rho, \gamma)$ in two minimal models on a $d$ dimensional periodic lattice for arbitrary density $\rho$ and tumbling rate $\gamma$. We focus on two models: Model I is the standard version of hardcore RTPs [Phys. Rev. E \textbf{89}, 012706 (2014)], whereas model II is a long-ranged lattice gas (LLG) with hardcore exclusion - an analytically tractable variant of model I; notably, both models are found to have qualitatively similar features. In the strong-persistence limit $\gamma \rightarrow 0$ (i.e., dimensionless $r_0 \gamma /v \rightarrow 0$), with $v$ and $r_{0}$ being the self-propulsion speed and particle diameter, respectively, the fascinating interplay between persistence and interaction is quantified in terms of two length scales - mean gap, or "mean free path", and persistence length $l_{p}=v/ \gamma$. Indeed, for a small tumbling rate, the bulk-diffusion coefficient varies as a power law in a wide range of density: $D \propto \rho^{-\alpha}$, with exponent $\alpha$ gradually crossing over from $\alpha = 2$ at high densities to $\alpha = 0$ at low densities. Thus, the density relaxation is governed by a nonlinear diffusion equation with anomalous spatiotemporal scaling. Moreover, in the thermodynamic limit, we show that the bulk-diffusion coefficient - for $\rho,\gamma \rightarrow 0$ with $\rho/\gamma$ fixed - has a scaling form $D(\rho, \gamma) = D^{(0)}\mathcal{F}(\psi=\rho a v/\gamma)$, where $a\sim r_{0}^{d-1}$ is particle cross-section and $D^{(0)}$ is proportional to the diffusivity of noninteracting particles; the scaling function $\mathcal{F}(\psi)$ is calculated analytically for model I and numerically for model II. Our arguments are independent of dimensions and microscopic details.