Current fluctuations in finite-sized one-dimensional non-interacting passive and active systems

TL;DR

This paper analyzes particle effusion fluctuations in finite one-dimensional systems for passive and active particles, revealing how initial conditions and boundary setups influence fluctuation behavior over time.

Contribution

It provides analytic results for fluctuations in finite systems, demonstrating the convergence of annealed and quenched averages and characterizing their time-dependent behavior.

Findings

Fluctuations in passive particles grow as √t and decay as 1/√t after a certain time.

Active particles show linear growth and similar decay in fluctuations.

Annealed and quenched fluctuations become equal beyond a system-size dependent timescale.

Abstract

We investigate the problem of effusion of particles initially confined in a finite one-dimensional box of size $L$. We study both passive as well active scenarios, involving non-interacting diffusive particles and run-and-tumble particles, respectively. We derive analytic results for the fluctuations in the number of particles exiting the boundaries of the finite confining box. The statistical properties of this quantity crucially depend on how the system is prepared initially. Two common types of averages employed to understand the impact of initial conditions in stochastic systems are annealed and quenched averages. It is well known that for an infinitely extended system, these different initial conditions produce quantitatively different fluctuations, even in the infinite time limit. We demonstrate explicitly that in finite systems, annealed and quenched fluctuations become equal beyond a system-size dependent timescale, $t \sim L^2$. For diffusing particles, the fluctuations exhibit a $\sqrt{t}$ growth at short times and decay as $1/\sqrt{t}$ for time scales, $t \gg L^2/D$, where $D$ is the diffusion constant. Meanwhile, for run-and-tumble particles, the fluctuations grow linearly at short times and then decay as $1/\sqrt{t}$ for time scales, $t \gg L^2/D_{\text{eff}}$, where $D_{\text{eff}}$ represents the effective diffusive constant for run-and-tumble particles. To study the effect of confinement in detail, we also analyze two different setups (i) with one reflecting boundary and (ii) with both boundaries open.