Anomalous field-induced growth of fluctuations in dynamics of a biased intruder moving in a quiescent medium

Abstract

We present exact results on the dynamics of a biased, by an external force ${\bf F}$, intruder (BI) in a two-dimensional lattice gas of unbiased, randomly moving hard-core particles. Going beyond the usual analysis of the force-velocity relation, we study the probability distribution $P({\bf R}_n)$ of the BI displacement ${\bf R}_n$ at time {\it n}. We show that despite the fact that the BI drives the gas to a non-equilibrium steady-state, $P({\bf R}_n)$ converges to a Gaussian distribution as $n \to \infty$. We find that the variance $σ_x^2$ of $P({\bf R}_n)$ along ${\bf F}$ exhibits a weakly superdiffusive growth $σ_x^2 \sim ν_1 \, n \, \ln(n)$, and a usual diffusive growth, $σ_y^2 \sim ν_2 \, n$, in the perpendicular direction. We determine $ν_1$ and $ν_2$ exactly for arbitrary bias, in the lowest order in the density of vacancies, and show that $ν_1 \sim |{\bf F}|^2$ for small bias, which signifies that superdiffusive behaviour emerges beyond the linear-response approximation. Monte Carlo simulations confirm our analytical results, and reveal a striking field-induced superdiffusive behavior $σ_x^2 \sim n^{3/2}$ for infinitely long 2D stripes and 3D capillaries.