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

We consider the totally asymmetric simple exclusion process (TASEP) on the periodic chain in the presence of a single impurity site that is inaccessible to other particles and therefore acts as a static defect. Particles are allowed to advance any distance l ≥ 1 on the right with a probability that decays as l−(1+σ), where σ > 1. Despite the long range of the hopping, we find the same type of phase transition that occurs in the standard short-range TASEP with a defect site where a defect induces a macroscopic shock in the stationary state. In particular, our model displays two main features characteristic of the short-range TASEP with defect site: a growth of the shock width with system size L as L1/2 or L1/3, depending on the existence of the particle–hole symmetry, and the power-law decay in density profiles of the shock phase. However, unlike the profiles in the short-range case, we find that the latter are well reproduced by the mean-field approximation, which enables us to derive the analytical expression for the σ-dependent exponent ν = σ − 1 of this power-law decay and the point σc = 4/3 at which the transition takes place.