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 γ=τₚ^-1 and density ρ; 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 Qᵢ(T,L) across the ith bond in a time interval T1/D grows first subdiffusively and then diffusively (linearly) with T: Qᵢ²~T^α with α=1/2 for 1/DTL²/D and α=1 for TL²/D, where D(ρ,γ) is the collective- or bulk-diffusion coefficient; at small times T1/D, exponent α depends on the details. Remarkably, regardless of the model details, the scaled bond-current fluctuations DQᵢ²(T,L)/2χL≡W(y) as a function of scaled variable y=DT/L² collapse onto a universal scaling curve W(y), where χ(ρ,γ) is the collective particle mobility. In the limit of small density and tumbling rate, ρ,γ→0, with ψ=ρ/γ fixed, there exists a scaling law: The scaled mobility γᵃχ(ρ,γ)/χ⁽⁰⁾≡H(ψ) as a function of ψ collapses onto a scaling curve H(ψ), where $a=1$ and 2 in models I and II, respectively, and χ⁽⁰⁾ is the mobility in the limiting case of a symmetric simple exclusion process; notably, the scaling function H(ψ) is model dependent. For model II (LLG), we calculate exactly, within a truncation scheme, both the scaling functions, W(y) and H(ψ). 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 ξ~τₚ1/2 diverging with persistence time τₚ1. 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)].