q-Zero Range has Random Walking Shocks

Abstract

but no other surprises in the zero range world. We check all nearest neighbour 1-dimensional asymmetric zero range processes for random walking product shock measures as demonstrated already for a few cases in the literature. We find the totally asymmetric version of the celebrated q-zero range process as the only new example besides an already known model of doubly infinite occupation numbers and exponentially increasing jump rates. We also examine the interaction of shocks, which appears somewhat more involved for q-zero range than in the already known cases.

A reverse duality for the ASEP with open boundaries

“…On macroscopic scale this phenomenon corresponds to a coalescence of shocks [35]. Also other particle-conserving models with random-walking shocks are known [3,5,7,41,45] 2. 4 In fact, it was suggested even earlier in [45] that the shocks whose invariant distribution in the finite system is described by the MPM remain stable during the stochastic time evolution and perform a random walk dynamics.…”

Similarity revisited: shock random walks in the asymmetric simple exclusion process with open boundaries

“…However, no auxiliary sites need to be introduced. Shock random walks on the infinite lattice are known to appear not only in the ASEP [41] and its n-species generalization with lower-class particles [18,63], but also in other conservative interacting particle systems [64,65]. From a physics perspective, this is an interesting phenomenon not only as a model for traffic jams but also for another reason: The stability of the shock on microscopic scale during the time evolution elucidates the fine structure of the macroscopic shock discontinuity that appears in the large-scale hydrodynamic description [66].…”