Time-dependent properties of run-and-tumble particles. II.: Current fluctuations

Abstract

We investigate steady-state current fluctuations in two models of run-and-tumble particles (RTPs) on a ring of $L$ sites, for \textit{arbitrary} tumbling rate $\gamma=\tau_p^{-1}$ and density $\rho$; model I consists of standard hardcore RTPs, while model II is an analytically tractable variant of model I, called long-ranged lattice gas (LLG). We show that, in the limit of $L$ large, the fluctuation of cumulative current $Q_i(T, L)$ across $i$th bond in a time interval $T \gg 1/D$ grows first {\it subdiffusively} and then {\it diffusively} (linearly) with $T$, where $D$ is the bulk diffusion coefficient. Remarkably, regardless of the model details, the scaled bond-current fluctuations $D \langle Q_i^2(T, L) \rangle/2 \chi L \equiv {\cal W}(y)$ as a function of scaled variable $y=DT/L^2$ collapse onto a {\it universal} scaling curve ${\cal W}(y)$, where $\chi(\rho,\gamma)$ is the collective particle {\it mobility}. In the limit of small density and tumbling rate $\rho, \gamma \rightarrow 0$ with $\psi=\rho/\gamma$ fixed, there exists a scaling law: The scaled mobility $\gamma^{a} \chi(\rho, \gamma)/\chi^{(0)} \equiv {\cal H} (\psi)$ as a function of $\psi$ collapse onto a scaling curve ${\cal H}(\psi)$, where $a=1$ and $2$ in models I and II, respectively, and $\chi^{(0)}$ is the mobility in the limiting case of symmetric simple exclusion process (SSEP). For model II (LLG), we calculate exactly, within a truncation scheme, both the scaling functions, ${\cal W}(y)$ and ${\cal H}(\psi)$. We also calculate spatial correlation functions for the current, and compare our theory with simulation results of model I; for both models, the correlation functions decay exponentially, with correlation length $\xi \sim \tau_p^{1/2}$ diverging with persistence time $\tau_p \gg 1$. Overall our theory is in excellent agreement with simulations and complements the findings of Ref. {\it arXiv:2209.11995}.