Scaling properties of the asymmetric exclusion process with long-range hopping

Abstract

The exclusion process in which particles may jump any distance l ≥ 1 with the probability that decays as l −(1+σ) is studied from coarse-grained equation for density profile in the limit when the lattice spacing goes to zero. For 1 < σ < 2, the usual diffusion term of this equation is replaced by the fractional one, which affects dynamical-scaling properties of the late-time approach to the stationary state. When applied to an open system with totally asymmetric hopping, this approach gives two results: first, it accounts for the σ-dependent exponent that characterizes the algebraic decay of density profile in the maximum-current phase for 1 < σ < 2, and second, it shows that in this region of σ the exponent is of the mean-field type.

Impurity-induced shocks in the asymmetric exclusion process with long-range hopping

“…We recover this result also in the long-range model, in accordance with our earlier result [22] that the density fluctuations in the infinite system travel with the velocity v g = λ(σ)(1 − 2ρ). In figure 2 we plot the width ξ L , obtained by the Gaussian fit of the discrete derivative τ n+1 − τ n vs. system size L for densities ρ = 0.5 (figure 2a) and 0.55 (figure 2b).…”

“…The best collapsing fit is achieved for ν ≈ 0.6 and ν ≈ 0.8 respectively, suggesting that ν is equal to σ − 1. On the other hand, for σ > 2 one generally expects the short-range regime to set in [22] with a σ-independent ν. Indeed, figure 3b for σ = 2 and σ = 2.3 shows good agreement between the scaled profiles if one assumes that ν(σ) = ν(2) = 1.…”

Fractional Fick’s law for the boundary driven exclusion process with long jumps

“…Observe that the reservoirs add and remove particles on all the sites of the lattice Λ N , and not only at the boundaries, but with rates which decrease as the distance from the corresponding reservoir increases. The same kind of reservoirs is used in [32]. The bulk dynamics (i.e.…”

“…In this paper we are interested in the case where p(·) has a long tail, proportional to | · | −(1+γ) for γ > 1. Curiously it is only very recently that the investigation of the exclusion process with long jumps started ( [4,14,15,13,28,32]).…”

Nonlocal asymmetric exclusion process on a ring and conformal invariance

“…Among the extensions of ASEP, some consider nonlocal hopping. The model examined in [11] is an obvious generalization of ASEP, hoppings don't take place only on empty neighboring sites but also on empty sites at a distance k with a probability P (k) ∼ k −(1+a) . The dynamic critical exponent changes accordingly z = min(a, 3/2).…”