Tracer dynamics in the active random average process

TL;DR

This paper analyzes how non-Markovian active dynamics influence tracer particle fluctuations in a one-dimensional random average process, revealing sub-diffusive growth and specific ratio relations between different initial conditions.

Contribution

It introduces a model with active, persistent motion in the RAP, deriving analytical results for variance growth and correlations, and confirms findings through simulations.

Findings

Variance of tracer position grows as √t with different coefficients for initial conditions.

Ratio of coefficients for annealed and quenched initial conditions is √2, consistent with other systems.

Active persistence affects higher-order correlations, altering fluctuation behavior.

Abstract

We investigate the dynamics of tracer particles in the random average process (RAP), a single-file system in one dimension. In addition to the position, every particle possesses an internal spin variable $\sigma (t)$ that can alternate between two values, $\pm 1$, at a constant rate $\gamma$. Physically, the value of $\sigma (t)$ dictates the direction of motion of the corresponding particle and for finite $\gamma$, every particle performs a non-Markovian active dynamics. Herein, we study the effect of this non-Markovianity in the fluctuations and correlations of the positions of tracer particles. We analytically show that the variance of the position of a tagged particle grows sub-diffusively as $\sim \zeta_{\text{q}} \sqrt{t}$ at large times for the quenched uniform initial condition. While this sub-diffusive growth is identical to that of the Markovian/non-persistent RAP, the coefficient $\zeta_{\text{q}} $ is rather different and bears the signature of the persistent motion of active particles through higher point correlations (unlike in the Markovian case). Similarly, for the annealed (steady state) initial condition, we find that the variance scales as $\sim \zeta_{\text{a}} \sqrt{t}$ at large times with coefficient $\zeta_{\text{a}} $ once again different from the non-persistent case. Although $\zeta_{\text{q}}$ and $\zeta_{\text{a}} $ both individually depart from their Markov counterparts, their ratio $\zeta_{\text{a}} / \zeta_{\text{q}}$ is still equal to $\sqrt{2}$, a condition observed for other diffusive single-file systems. This condition turns out to be true even in the strongly active regimes as corroborated by extensive simulations and calculations. Finally, we study the correlation between the positions of two tagged particles in both quenched uniform and annealed initial conditions. We verify all our analytic results by extensive numerical simulations.