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

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 =₏^-1 and density ; 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 Q₈ (T, L) across the ith bond in a time interval T1/D grows first subdiffusively and then diffusively (linearly) with T: Q₈^2T^ with =1/2 for 1/DL^2/D and =1 for TL^2/D, where D (, ) is the collective- or bulk-diffusion coefficient; at small times T1/D, exponent depends on the details. Remarkably, regardless of the model details, the scaled bond-current fluctuations DQ₈^2 (T, L) /2 (y) as a function of scaled variable y=DT/L^2 collapse onto a universal scaling curve W (y), where (, ) is the collective particle mobility. In the limit of small density and tumbling rate, , 0, with =/ fixed, there exists a scaling law: The scaled mobility ^a (, ) /^ (0) () as a function of collapses onto a scaling curve H (), where a=1 and 2 in models I and II, respectively, and ^ (0) is the mobility in the limiting case of a symmetric simple exclusion process; notably, the scaling function H () is model dependent. For model II (LLG), we calculate exactly, within a truncation scheme, both the scaling functions, W (y) and H (). 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 ₏^1/2 diverging with persistence time ₏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).