Zero-Range Process with Open Boundaries

Abstract

We calculate the exact stationary distribution of the one-dimensional zero-range process with open boundaries for arbitrary bulk and boundary hopping rates. When such a distribution exists, the steady state has no correlations between sites and is uniquely characterized by a space-dependent fugacity which is a function of the boundary rates and the hopping asymmetry. For strong boundary drive the system has no stationary distribution. In systems which on a ring geometry allow for a condensation transition, a condensate develops at one or both boundary sites. On all other sites the particle distribution approaches a product measure with the finite critical density ρ c . In systems which do not support condensation on a ring, strong boundary drive leads to a condensate at the boundary. However, in this case the local particle density in the interior exhibits a complex algebraic growth in time. We calculate the bulk and boundary growth exponents as a function of the system parameters.

Emergence of dynamic phases in the presence of different kinds of open boundaries in stochastic transport with short-range interactions

“…[23] and numerically reproduced for the ZRP and a short-range interaction transport model with smooth condensates in this and our own previous work [24] we expect to some extent a similar phase diagram. Therefore, to identify the phases, we continue to use the time series of the total number of particles M (t), its scaling exponent α and the bulk system particle density ρ bulk introduced in the previous section.…”

“…That is, a single site contains a finite fraction 1 − ρ crit /ρ of all particles, where ρ crit = 1/(b − 2) is the critical density that is assumed on average in the rest of the system. Effects of open, driven boundaries on this model have been studied and discussed by Levine et al [23]. For the ZRP, we will use the parameter b = 5 that results in a critical density of ρ crit = 1/3.…”

“…For symmetric hopping, the argument is similar but results in a diagonal transition line to the aggregate condensate phases for p in ≥ p out because particles may leave directly after entering. For a more detailed discussion of this argument, also with respect to partially asymmetric hopping for the ZRP, we refer to the original work of Levine et al [23].…”

“…Using these observations of the scaling of transition times to the aggregate condensate phase, we reproduced the value of the critical influx rate p in,crit = 1 for the ZRP (see Ref. [23]) as well as determined the scaling exponent κ, c.f. Table 1.…”

Emergent motion of condensates in mass-transport models

“…The fugacity of the second site is then z 2 ≡ J 2 / w 2 J 2 > 1, and therefore its occupation distribution (5) cannot be normalized. This means that as long as site 1 is highly occupied, site 2 tends to accumulate particles, implying that its occupation too becomes macroscopic (of order L) for a long period of time [31,32]. We call such a site with fugacity z i > 1 supercritical.…”