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

Abstract

We investigate steady-state current fluctuations in two models of hardcore run-and-tumble particles (RTPs) on a periodic one-dimensional lattice of $L$ sites, for arbitrary tumbling rate $\ensuremath{\gamma}={\ensuremath{\tau}}_{p}^{\ensuremath{-}1}$ and density $\ensuremath{\rho}$; model I consists of standard hardcore RTPs, while model II is an analytically tractable variant of model I, called a long-ranged lattice gas (LLG). We show that, in the limit of $L$ large, the fluctuation of cumulative current ${Q}_{i}(T,L)$ across the $i\mathrm{th}$ bond in a time interval $T\ensuremath{\gg}1/D$ grows first subdiffusively and then diffusively (linearly) with $T$: $\ensuremath{\langle}{Q}_{i}^{2}\ensuremath{\rangle}\ensuremath{\sim}{T}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}=1/2$ for $1/D\ensuremath{\ll}T\ensuremath{\ll}{L}^{2}/D$ and $\ensuremath{\alpha}=1$ for $T\ensuremath{\gg}{L}^{2}/D$, where $D(\ensuremath{\rho},\ensuremath{\gamma})$ is the collective- or bulk-diffusion coefficient; at small times $T\ensuremath{\ll}1/D$, exponent $\ensuremath{\alpha}$ depends on the details. Remarkably, regardless of the model details, the scaled bond-current fluctuations $D\ensuremath{\langle}{Q}_{i}^{2}(T,L)\ensuremath{\rangle}/2\ensuremath{\chi}L\ensuremath{\equiv}\mathcal{W}(y)$ as a function of scaled variable $y=DT/{L}^{2}$ collapse onto a universal scaling curve $\mathcal{W}(y)$, where $\ensuremath{\chi}(\ensuremath{\rho},\ensuremath{\gamma})$ is the collective particle mobility. In the limit of small density and tumbling rate, $\ensuremath{\rho},\ensuremath{\gamma}\ensuremath{\rightarrow}0$, with $\ensuremath{\psi}=\ensuremath{\rho}/\ensuremath{\gamma}$ fixed, there exists a scaling law: The scaled mobility ${\ensuremath{\gamma}}^{a}\ensuremath{\chi}(\ensuremath{\rho},\ensuremath{\gamma})/{\ensuremath{\chi}}^{(0)}\ensuremath{\equiv}\mathcal{H}(\ensuremath{\psi})$ as a function of $\ensuremath{\psi}$ collapses onto a scaling curve $\mathcal{H}(\ensuremath{\psi})$, where $a=1$ and 2 in models I and II, respectively, and ${\ensuremath{\chi}}^{(0)}$ is the mobility in the limiting case of a symmetric simple exclusion process; notably, the scaling function $\mathcal{H}(\ensuremath{\psi})$ is model dependent. For model II (LLG), we calculate exactly, within a truncation scheme, both the scaling functions, $\mathcal{W}(y)$ and $\mathcal{H}(\ensuremath{\psi})$. 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 $\ensuremath{\xi}\ensuremath{\sim}{\ensuremath{\tau}}_{p}^{1/2}$ diverging with persistence time ${\ensuremath{\tau}}_{p}\ensuremath{\gg}1$. Overall, our theory is in excellent agreement with simulations and complements the prior findings [T. Chakraborty and P. Pradhan, Phys. Rev. E 109, 024124 (2024)].

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