Anomalous current fluctuations and mobility-driven clustering

Abstract

We study steady-state current fluctuations in hardcore lattice gases on a ring of $L$ sites, where $N$ particles perform symmetric, {\it extended-ranged} hopping. The hop length is a random variable depending on a length scale $l_0$ (hopping range) and the inter-particle gap. The systems have mass-conserving dynamics with global density $\rho = N/L$ fixed, but violate detailed balance. We consider two analytically tractable cases: (i) $l_0 = 2$ (finite-ranged) and (ii) $l_0 \to \infty$ (infinite-ranged); in the latter, the system undergoes a clustering or condensation transition below a critical density $\rho_c$. In the steady state, we compute, exactly within a closure scheme, the variance $\langle Q^2(T) \rangle_c = \langle Q^2(T) \rangle - \langle Q(T) \rangle^2$ of the cumulative (time-integrated) current $Q(T)$ across a bond $(i,i+1)$ over a time interval $[0, T]$. We show that for $l_0 \to \infty$, the scaled variance of the time-integrated bond current, or equivalently, the mobility diverges at $\rho_c$. That is, near criticality, the mobility $\chi(\rho) = \lim_{L \to \infty} [\lim_{T \to \infty} L \langle Q^2(T, L) \rangle_c / 2T] \sim (\rho - \rho_c)^{-1}$ has a simple-pole singularity, thus providing a dynamical characterization of the condensation transition, previously observed in a related mass aggregation model by Majumdar et al.\ [{\it Phys.\ Rev.\ Lett.\ {\bf 81}, 3691 (1998)}]. At the critical point $\rho = \rho_c$, the variance has a scaling form $\langle Q^2(T, L) \rangle_c = L^{\gamma} {\cal W}(T/L^{z})$ with $\gamma = 4/3$ and the dynamical exponent $z = 2$. Thus, near criticality, the mobility {\it diverges} while the diffusion coefficient remains {\it finite}, {\it unlike} in equilibrium systems with short-ranged hopping, where diffusion coefficient usually {\it vanishes} and mobility remains finite.