Anomalous current fluctuations and mobility-driven clustering

We study steady-state current fluctuations in hardcore lattice gases on a one-dimensional ring of L sites, where N particles perform symmetric, extended-ranged hopping, with the hop length being a random variable depending on a length scale l 0 (hopping range) and the inter-particle gap. Notably, the systems have mass-conserving dynamics with global density ρ = N / L fixed, but violate detailed balance. We consider two analytically tractable cases: (i) l 0 = 2 (finite-ranged) and (ii) l 0 → ∞ (infinite-ranged); in the latter case, the system undergoes a clustering or condensation transition below a critical density ρ c . In the steady state, we calculate, exactly within a closure scheme, the variance ⟨ Q 2 ( T ) ⟩ c = ⟨ Q 2 ( T ) ⟩ − ⟨ Q ( T ) ⟩ 2 of cumulative (time-integrated) current Q ( T ) across a bond ( i , i + 1 ) and in a time interval [ 0 , T ] . We show that, when l 0 → ∞ , the scaled variance of time-integrated bond current or, equivalently, the mobility, diverges at the critical density. That is, near criticality, the mobility χ ( ρ ) = lim L → ∞ [ lim T → ∞ L ⟨ Q 2 ( T , L ) ⟩ c / 2 T ] ∼ ( ρ − ρ c ) − 1 has a simple-pole singularity, thus providing a dynamical characterization of the condensation transition, which was observed previously in a related mass aggregation model by Majumdar et al ( Phys. Rev. Lett. 81 , 3691 (1998)). We find that, at the critical point ρ = ρ c , the variance has a scaling form ⟨ Q 2 ( T , L ) ⟩ c = L γ W ( T / L z ) where γ = 4 / 3 , z = 2 and W ( y ) ∼ y 2 / 3 for y ≪ 1 small; far from criticality ρ > ρ c , we have γ = 1, z = 2 and the scaling function W ( y ) ∼ y 1 / 2 for y ≪ 1 small. For y ≫ 1 large, W ( y ) ∼ y in both cases. The dynamic exponent z = 2 implies diffusive relaxation, both far from and near criticality. In other words, at the critical point, the mobility diverges and the diffusion coefficient remains finite , unlike in equilibrium systems with short-ranged hopping, where the bulk-diffusion coefficient usually vanishes , while the mobility remains finite.