Condensed Matter > Statistical Mechanics
[Submitted on 24 Sep 2022 (v1), last revised 16 Feb 2024 (this version, v4)]
Title:Time-dependent properties of run-and-tumble particles: Density relaxation
View PDFAbstract: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.
Submission history
From: Tanmoy Chakraborty [view email][v1] Sat, 24 Sep 2022 12:35:27 UTC (422 KB)
[v2] Tue, 11 Apr 2023 16:21:43 UTC (432 KB)
[v3] Tue, 26 Sep 2023 08:12:52 UTC (1,125 KB)
[v4] Fri, 16 Feb 2024 12:33:06 UTC (1,014 KB)
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