Effect of initial conditions on current fluctuations in non-interacting active particles

Abstract

We investigate the effect of initial conditions on the fluctuations of the integrated density current across the origin ($x=0$) up to a given time $t$ in a one-dimensional system of non-interacting run-and-tumble particles. Each particle has initial probabilities $f^+$ and $f^-$ to move with an initial velocity $+v$ and $-v$ respectively, where $v>0$. We derive exact results for the variance (second cumulant) of the current for quenched and annealed averages over the initial conditions for the magnetization and the density fields associated with the particles. We show that at large times, the variance displays a $\sqrt{t}$ behavior, with a prefactor contingent on the specific density initial conditions used. However, at short times, the variance displays either linear $t$ or quadratic $t^2$ behavior, which depends on the combination of magnetization and density initial conditions, along with the fraction $f^+$ of particles in the positive velocity state at $t=0$. Intriguingly, if $f^+=0$, the variance displays a short time $t^2$ behavior with the same prefactor irrespective of the initial conditions for both fields.