Zero-range processes with multiple condensates: statics and dynamics

Abstract

The steady-state distributions and dynamical behaviour of zero-range processes with hopping rates which are non-monotonic functions of the site occupation are studied. We consider two classes of non-monotonic hopping rates. The first results in a condensed phase containing a large (but subextensive) number of mesocondensates each containing a subextensive number of particles. The second results in a condensed phase containing a finite number of extensive condensates. We study the scaling behaviour of the peak in the distribution function corresponding to the condensates in both cases. In studying the dynamics of the condensate we identify two timescales: one for creation, the other for evaporation of condensates at a given site. The scaling behaviour of these timescales is studied within the Arrhenius law approach and by numerical simulations.

Emergent motion of condensates in mass-transport models

“…A primary role in these studies was played by the zero-range process (ZRP), an exactly solvable model in which particles hop between sites with rates that depend only on the number of particles in the departure site [9][10][11]. Extensions and variations of the ZRP have been used to study the emergence of multiple condensates [12], first-order condensation transitions [13,14], and the effect of interactions [15] and disorder [16] on condensation. Moreover, one-dimensional phase separation transitions in exclusion processes and other driven diffusive systems can quite generally be understood by a mapping on ZRPs [17].…”

Emergence of particle clusters in a one-dimensional model: connection to condensation processes

“…Also, the system respects detailed balance with respect to the equilibrium distribution (2) so, for large times, the system should converge to that distribution. Moreover, in the limit a max → 0 (assuming now that > 0), this dynamical MC method converges to the solution of the Langevin equation (3). However, we note that the results presented here are far from the limit a max → 0, in particular, this limit may require a max while our numerical results have a max…”

“…If the move is not accepted then the particle is returned to its original position x i . After each attempted move, the time is incremented by a 2 max /(6D 0 N ) so that in the absence of interparticle forces, the diffusion constant for the MC dynamics matches that of the Langevin equation (3).…”

“…We have β → 1 − as N → ∞, and since β = 1 is the limit of stability of the model, one may expect to see nontrivial behaviour in this limit. A similar construction was used to define interacting particle systems with multiple condensates [3], and in models with discontinuous condensation [17]. Now consider an equilibrium configuration of the model, and a random point within the system.…”

“…Even in much simpler model systems such as exclusion processes or zero-range processes in one dimension, unexpected phenomena continue to surprise physicists and mathematicians, including condensation [1,2,3,4,5,6,7] and unusual fluctuation phenomena [8,9,10,11,12].…”