Nature of the Condensate in Mass Transport Models

Abstract

We study the phenomenon of real space condensation in the steady state of a class of one dimensional mass transport models. We derive the criterion for the occurrence of a condensation transition and analyse the precise nature of the shape and the size of the condensate in the condensed phase. We find two distinct condensate regimes: one where the condensate is gaussian distributed and the particle number fluctuations scale normally as L 1/2 where L is the system size, and a second regime where the particle number fluctuations become anomalously large and the condensate peak is non-gaussian. We interpret these results within the framework of sums of random variables.PACS numbers: 02.50.Ey, Condensation transitions are ubiquitous in nature. For systems in thermal equilibrium, clustering is well understood in terms of the competition between entropy and energy (typically associated with attractive interactions). More exotic are condensations in systems with no interactions, e.g., Bose-Einstein's free (quantum) particles. Less understood are such transitions in non-equilibrium systems, in some of which even the concept of energy is dubious. For example, condensation is known to occur in many mass transport models, defined only by a set of rules of evolution, with no clear 'attraction' between the masses [1][2][3][4][5][6]. The relevance of these models lies in their applicability to a broad variety of phenomena, e.g., traffic flow [7]force propagation through granular media How such transitions arise is especially intriguing for one dimensional (d = 1) systems with local dynamical rules. A well known example is the Zero-Range Process (ZRP) [2,12,13] in which masses hop from site to (the next) site according to some transfer rule. In the steady state, a finite fraction of the total mass 'condenses' onto a single site when ρ, the global mass density, is increased beyond a certain critical value: ρ c . The system goes from a fluid phase, where the mass at each site hovers around ρ, to a condensed phase, where a fluid of density ρ c coexists with a condensate containing all the 'excess' mass.Though condensation in these systems share interesting analogies [3,6] with the traditional Bose-Einstein condensation, there are important differences. For example, here condensation occurs in real space and in all dimensions. Moreover these systems are non-equilibrium in the sense that they are defined by the dynamics, generally lack a Hamiltonian and the stationary state is not specified by the usual Gibbs-Boltzmann distribution. There are two major problems that one faces in the analysis of condensation phenomenon in these systems. First, the stationary state itself often is very difficult to determine and secondly, even if it is known such as in ZRP, the analysis of condensation has so far been possible only within a grand canonical enemble (GCE) where one is already in the thermodynamic (L → ∞) limit. While the GCE approach correctly predicts when a condensation transition can happen and even the value of the critical densit...

Canonical Analysis of Condensation in Factorised Steady States

“…In this paper we will analyse in detail the structure of the condensate, elucidating when it occurs, its shape and its fluctuations. A short communication of some of our results has been given in [11].…”

Condensation transition in polydisperse hard rods

“…Below the critical density the distribution typically decays exponentially for large mass, indicating a fluid phase. At the critical density the decay of the single-site mass distribution is slower, typically it decays as a power law or sometimes a stretched exponential distribution, indicating a critical fluid [15,16]. Above the critical density a bump in single-site mass distribution emerges and corresponds to a single site containing the excess mass above the critical value.…”

Zero-range process with finite compartments: Gentile's statistics and glassiness

“…The finite site capacity implies that a) in the limit of large system size the grand canonical analysis becomes exact for any density of particles (in contrast to the infinite-capacity ZRP where the grand-canonical ensemble often fails to describe the condensed phase [6,7]), b) we are able to treat analytically a broader class of hopping rates (than in the infinitecapacity case), c) the dynamics of condensate growth exhibits a dynamical self-blocking which significantly prolongs relaxation times (the entropic effect known from models of glasses [8][9][10][11][12][13][14][15]). * rjabov.a@gmail.com…”