Current reversals in driven lattice gases and Brownian motion

TL;DR

This paper addresses how interactions induce current reversals in driven lattice gases and extends the framework to continuous-space Brownian motion in periodic landscapes. It develops a general, symmetry-based approach rooted in particle-hole symmetry: if the time-dependent driving changes sign under translations in time and/or space, the time-averaged current satisfies $\bar{J}(\rho)=-\bar{J}(1-\rho)$ in stationary states. The authors formulate a lattice-gas model with arbitrary pair interactions, derive exact relations between particle and hole dynamics, and demonstrate the reversal mechanism with traveling-wave drivings through kinetic Monte Carlo and BASEP Brownian simulations, including amplitude modulations and multi-wave drivings. The results hold in higher dimensions and predict reversals in continuous-space dynamics, offering a predictive framework and experimental pathways for observing current reversals in colloidal systems and related setups.

Abstract

Particle currents flowing against an external driving are a fascinating phenomenon in both single-particle and interacting many-particle systems. Underlying physical mechanisms of such current reversals are not fully understood yet. Predicting their appearance is difficult, in particular for interaction-induced ones that emerge upon changes of the particle density. We here derive conditions on external time-dependent drivings, under which current reversals occur in lattice gases with arbitrary pair interactions. Our derivation is based on particle-hole symmetry and shows that current reversals must emerge if the time-varying driving potential changes sign after a translation in time and/or space. Our treatment includes nonstationary dynamics and time-dependent spatially averaged currents in nonequilibrium steady states. It gives insight also into possible occurrences of current reversals in continuous-space dynamics, which we demonstrate for hardcore interacting particles driven across a periodic potential by a traveling wave.

Figures (4)

• Figure 1: Current reversal for nonstationary dynamics in a lattice gas under driving by a sinusoidal traveling wave [Eq. \ref{['eq:w-sinus']}] with $A=3$, $\lambda=4$ and $\tau=2$ and site-exclusion interaction ($V=0$), demonstrating Eq. \ref{['eq:J-relation-nonstationary-1']}. For an initial state in a system $\mathcal{S}$ with $N$ particles occupying the first $N$ lattice sites $i=1,\ldots,N$ (density $\rho=N/L$), and an initial state in a system $\tilde{\mathcal{S}}$ with $L-N$ particles occupying the last $L-N$ lattice sites $i=L-N+1,\ldots,L$ (density $1-\rho$), the link current $J_{N,N+1}(t+\tau/2)$ in system $\tilde{\mathcal{S}}$ is the negative of the link current $J_{N,N+1}(t)$ in system $\mathcal{S}$.
• Figure 2: Current reversal for time-dependent spatially averaged currents in the stationary state for the same driving as in Fig. \ref{['fig:link_currents_nonstationary']}, demonstrating Eq. \ref{['eq:Jrelation-spatialaveraged']}. When averaging over one spatial period $\lambda$ of the sinusoidal traveling wave in Eq. \ref{['eq:w-sinus']}, $\bar{J}(t,\rho)=-\bar{J}(t,1-\rho)$ in systems with densities $\rho=0.2$ and $\rho=0.8$.
• Figure 3: Demonstration of current reversals for stationary dynamics according to Eq. \ref{['eq:Jrelation-time-averaged']} in lattice gases under driving by a sinusoidal traveling wave for (a), (b) constant amplitude $A=3$, (c), (d) time-modulated amplitude given in Eq. \ref{['eq:Amod-time']}, and (e), (f) superposition of two traveling waves with spatially modulated amplitudes given in Eq. \ref{['eq:two-wave-driving']}. In the left panels (a), (c), and (e), particles interact with site exclusion only ($V=0$), while in the right panels (b), (d), and (f), an additional strong repulsive interaction $V=10$ is present. The wavelength $\lambda=4$ and time period $\tau=2$ in (a)-(d) are the same as in Fig. \ref{['fig:link_currents_nonstationary']}. In (e) and (f), $\lambda=8$ and $\tau=2$.
• Figure 4: Current reversal in one-dimensional Brownian motion of hard spheres with diameter $\sigma$ in a sinusoidal potential with amplitude $U_0/2$ and under driving by a sinusoidal traveling wave with same amplitude, according to the time-dependent potential in Eq. \ref{['eq:U-Brownian']}. Parameters are $\sigma=\lambda=1$, $U_0=6k_{\rm B}T$, and $v=0.05\, k_{\rm B}T\mu/\lambda$ ($\mu$: particle mobility).