Dynamics of Condensation in the Totally Asymmetric Inclusion Process

Abstract

We study the dynamics of condensation of the inclusion process on a one-dimensional periodic lattice in the thermodynamic limit, generalising recent results on finite lattices for symmetric dynamics. Our main focus is on totally asymmetric dynamics which have not been studied before, and which we also compare to exact solutions for symmetric systems. We identify all relevant dynamical regimes and corresponding time scales as a function of the system size, including a coarsening regime where clusters move on the lattice and exchange particles, leading to a growing average cluster size. Suitable observables exhibit a power law scaling in this regime before they saturate to stationarity following an exponential decay depending on the system size. Our results are based on heuristic derivations and exact computations for symmetric systems, and are supported by detailed simulation data. arXiv:1311.4814v2 [cond-mat.stat-mech]

Occupancy correlations in the asymmetric simple inclusion process

“…In the ASEP particles are subject to exclusion interactions that keep them singled apart, whereas in the ASIP particles are subject to inclusion interactions that coalesce them into inseparable clusters. To avoid confusion we note that there is also a different model with the name ASIP [32,33] which will not be discussed here.…”

Coupling and Hydrodynamic Limit for the Inclusion Process

“…1 We similarly couple two independent random walk particles absorbed at left end 0 and right end N + 1 and two S I P particles absorbed at left end 0 and right end N + 1. The random walk jumps until absorption are performed together, and the inclusion jumps are performed only by the inclusion particles.…”

“…The attractive interaction as opposed to exclusion in the well-known exclusion process [12] is responsible for interesting phenomena. In the limit of small diffusion, condensation phenomena occur [1,8,9], and there is an analogue of Liggett's comparison inequality which leads to "propagation of positive correlations" for appropriately chosen initial product measures [7].…”

“…[3,4] for two good reference on this approach based on v-functions. The construction of a good coupling between exclusion walkers and independent random walkers relies quite strongly on partic-B Alex Opoku alexop2000@yahoo.com 1 Delft Institute of Applied Mathematics, Technische Universiteit Delft, Mekelweg 4, 2628 CD Delft, The Netherlands ular properties of the exclusion process, in particular monotonicity, which in turn leads to a decreasing number of discrepancies (second class particles) in the course of time. The SIP is not monotone because of the attractive interaction, and therefore presents new challenges both for coupling and for a strong hydrodynamic limit behavior.…”

A simple non-equilibrium, statistical-physics toy model of thin-film growth

“…These models share a common feature: the steady-state probability of a microstate can be expressed analytically as a function of transition rates which define the dynamics of the model. Examples of such systems are the zero-range process (ZRP) [2,3,4], closely related to its equilibrium counterpart: balls-in-boxes model (B-in-B) [5], the asymmetric simple exclusion process (ASEP) [6] and its totally asymmetric version (TASEP) [7], asymmetric inclusion process (ASIP) [8,9,10] and many variations on these two models [11,12,13,14,15]. In all these models, particles jump between sites of a one-or higher-dimensional lattice and the dynamics is defined by specifying the hopping rates of the particles.…”