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

We characterize collective diffusion of hardcore run-and-tumble particles (RTPs) by explicitly calculating the bulk-diffusion coefficient $D(\rho ,\gamma )$ for arbitrary density $\rho$ and tumbling rate $\gamma$, in systems on a $d$-dimensional periodic lattice. We study two minimal models of RTPs: Model I is the standard version of hardcore RTPs introduced in [Phys. Rev. E 89, 012706 (2014)], whereas model II is a long-ranged lattice gas (LLG) with hardcore exclusion, an analytically tractable variant of model I. We calculate the bulk-diffusion coefficient analytically for model II and numerically for model I through an efficient Monte Carlo algorithm; notably, both models have qualitatively similar features. In the strong-persistence limit $\gamma \to 0$ (i.e., dimensionless ratio $r_0\gamma /v\to 0$), 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: (i) persistence length $l_p=v/\gamma$ and (ii) a "mean free path," being a measure of the average empty stretch or gap size in the hopping direction. We find that 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. As a result, the density relaxation is governed by a nonlinear diffusion equation with anomalous spatiotemporal scaling. In the thermodynamic limit, we show that the bulk-diffusion coefficient—for $\rho ,\gamma \to 0$ with $\rho /\gamma$ fixed—has a scaling form $D(\rho ,\gamma )=D^{\left(0\right)}F(\rho av/\gamma )$, where $a\sim r_0^{d-1}$ is particle cross section and $D^{\left(0\right)}$ is proportional to the diffusion coefficient of noninteracting particles; the scaling function $F\left(\psi \right)$ is calculated analytically for model II (LLG) and numerically for model I. Our arguments are independent of dimensions and microscopic details.