The zero-range process, in which particles hop between sites on a lattice under conserving dynamics, is a prototypical model for studying real-space condensation. Within this model the system is critical only at the transition point. Here we consider a non-conserving zero-range process which is shown to exhibit generic critical phases which exist in a range of creation and annihilation parameters. The model also exhibits phases characterized by mesocondensates, each of which contains a subextensive number of particles. A detailed phase diagram, delineating the various phases, is derived.
Abstract. The Zero-Range Process, in which particles hop between sites on a lattice under conserving dynamics, is a prototypical model for studying real-space condensation. Within this model the system is critical only at the transition point. Here we consider a non-conserving Zero-Range Process which is shown to exhibit generic critical phases which exist in a range of creation and annihilation parameters. The model also exhibits phases characterised by mesocondensates each of which contains a subextensive number of particles. A detailed phase diagram, delineating the various phases, is derived. PACS numbers: 89.75.-k, 05.70.Ln, 05.40.-aCriticality and Condensation in a Non-conserving Zero Range Process 2 1.
We study a nonconserved one-dimensional stochastic process which involves two species of particles
A and
B. The particles diffuse asymmetrically and react in pairs as $A\ensuremath{\emptyset}\ensuremath{\leftrightarrow}AA\ensuremath{\leftrightarrow}BA\ensuremath{\leftrightarrow}A\ensuremath{\emptyset}$ and $B\ensuremath{\emptyset}\ensuremath{\leftrightarrow}BB\ensuremath{\leftrightarrow}AB\ensuremath{\leftrightarrow}B\ensuremath{\emptyset}$. We show that the stationary state of the model can be calculated exactly by using matrix product techniques. The model exhibits a phase transition at a particular point in the phase diagram which can be related to a condensation transition in a particular zero-range process. We determine the corresponding critical exponents and provide a heuristic explanation for the unusually strong corrections to scaling seen in the vicinity of the critical point.
We study a disordered, driven zero range process which models a closed system of attractive particles that hop with site-dependent rates and whose steady state shows a condensation transition with increasing density. We characterize the dynamical properties of the mass fluctuations in the steady state in one dimension both analytically and numerically and show that there is a dynamic phase transition in the density-disorder plane. We also determine the form of the scaling function which describes the growth of the condensate as a function of time, starting from a uniform density distribution.
We consider the phenomenon of condensation of a globally conserved quantity $H={\ensuremath{\sum}}_{i=1}^{N}{\ensuremath{\epsilon}}_{i}$ distributed on
N sites, occurring when the density
h=H/N exceeds a critical density
hc. We numerically study the dependence of the participation ratio ${Y}_{2}=\ensuremath{\langle}{\ensuremath{\epsilon}}_{i}^{2}\ensuremath{\rangle}/(N{h}^{2})$ on the size
N of the system and on the control parameter $\ensuremath{\delta}=(h\ensuremath{-}{h}_{c})$, for various models: (i) a model with two conservation laws, derived from the discrete nonlinear Schr\"odinger equation; (ii) the continuous version of the zero-range process class, for different forms of the function $f(\ensuremath{\epsilon})$ defining the factorized steady state. Our results show that various localization scenarios may appear for finite
N and close to the transition point. These scenarios are characterized by the presence or the absence of a minimum of
Y2 when plotted against
N and by an exponent $\ensuremath{\gamma}\ensuremath{\ge}2$ defined through the relation ${N}^{*}\ensuremath{\simeq}{\ensuremath{\delta}}^{\ensuremath{-}\ensuremath{\gamma}}$, where
N∗ separates the delocalized region ($N\ensuremath{\ll}{N}^{*}$,
Y2 vanishes with increasing
N) from the localized region ($N\ensuremath{\gg}{N}^{*}$,
Y2 is approximately constant). We finally compare our results with the structure of the condensate obtained through the single-site marginal distribution.