Phase separation in disordered exclusion models

Abstract

The e ect of quenched disorder in the one-dimensional asymmetric exclusion process is reviewed. Both particlewise and sitewise disorder generically induce phase separation in a range of densities. In the particlewise case the existence of stationary product measures in the homogeneous phase implies that the critical density can be computed exactly, while for sitewise disorder only bounds are available. The coarsening of phase-separated domains starting from a homogeneous initial condition is addressed using scaling arguments and extremal statistics considerations. Some of these results have been obtained previously in the context of directed polymers subject to columnar disorder.

Driven diffusive systems with disorder

“…Thus the mass at all such slow sites, except for the very slowest, shows a nonmonotonic variation in time. The mass at the slowest (condensate) site grows as M 1 ∼ t β , with β = (n + 1)/(n + 2), a result first derived for a model with deterministic dynamics [18], but which remains true even in the stochastic model under consideration [5,6,15]. This translates into a statement about the growth of headway lengths ξ in the particle model…”

“…The coarsening properties of this model were obtained by Krug [5] who pointed out that bottleneck stretch lengths are much smaller than their separations, implying that we may mentally replace stretches by single bonds that allow a maximum current J l ∼ λ 1 2 l to flow through. Following through, we find that the coarsening length is given by [5] …”

“…Interestingly, it is possible to obtain upper and lower bounds [5] on the value of the threshold density for phase separation ρ c = 1/2 − ∆ in terms of the ratio of bond strengths r = u 2 /u 1 < 1 and the fraction f of weak bonds. An upper bound is obtained on noting that the maximum current that can flow through a stretch of slow bonds decreases as the stretch length increases.…”

“…In the phase-separated regime, the kinetics of approach to the steady state involves a coarsening process [5]. Large-density stretches accumulate behind bottleneck regions which are stretches of consecutive weak bonds.…”

“…Furthermore, disorder can also induce variations of the density on a macroscopic scale, associated with phase separation [5] . The models considered here share this feature, but in some cases the phase separated state has quite unusual properties.…”