Occupation probabilities and fluctuations in the asymmetric simple inclusion process

Abstract

* O. Hirschberg and S. Reuveni had equal contribution to this workThe Asymmetric Simple Inclusion Process (ASIP), a lattice-gas model of unidirectional transport and aggregation, was recently proposed as an 'inclusion' counterpart of the Asymmetric Simple Exclusion Process (ASEP). In this paper we present an exact closed-form expression for the probability that a given number of particles occupies a given set of consecutive lattice sites. Our results are expressed in terms of the entries of Catalan's trapezoids -number arrays which generalize Catalan's numbers and Catalan's triangle. We further prove that the ASIP is asymptotically governed by: (i) an inverse square root law of occupation; (ii) a square root law of fluctuation; and (iii) a Rayleigh law for the distribution of inter-exit times. The universality of these results is discussed.

An ASIP model with general gate opening intervals

“…Our results will become somewhat simpler in the special case in which the next gate opening is of gate j with a fixed probability q j , i.e., irrespective of the index of the previous gate opening. Notice that the original ASIP model of [5][6][7][8][9] also has this property, as there the gate openings are governed by independent Poisson processes. We work out this special case of fixed gate opening probabilities q j in an example in Sect.…”

“…From a statistical physics perspective, the ASIP is a reaction-diffusion model for unidirectional transport with coagulation. Our model significantly generalizes the model of [5][6][7][8][9], allowing us to more accurately represent those stochastic processes. It also allows one to represent movements of ships, crowds, or cars.…”

“…The ASIP model as introduced and studied in [5][6][7][8][9] is a generic model that may represent many different stochastic processes in chemistry, physics, and everyday life. From a queueing perspective, it is a series of queues with unlimited batch service.…”

“…(iii) The LST and mean of the draining time (the time from an arbitrary moment when the system is in steady state and the inflow is stopped, until the first moment thereafter that the system becomes empty). Occupation probabilities were considered in [9]. Closed-form results were obtained for the probabilities that the total occupation of 'lattice intervals' of m sites, sites k to k + m − 1, is equal to l, l = 0, 1, 2, .…”

“…The asymmetric inclusion process (ASIP), introduced and analyzed in [5][6][7][8][9], is a one-dimensional lattice of n sites (queues), where particles (for example, customers) arrive randomly into the first site (Q 1 ), stay there ('served') for a random time, continue moving simultaneously and unidirectionally from site to site while staying for a random time in each site, until finally exiting the last site (Q n ) and leaving the system. The ASIP defines the missing link between the celebrated Tandem Jackson Network (TJN) and the Asymmetric Exclusion Process (ASEP) [1][2][3] which plays the role of a paradigm in nonequilibrium statistical mechanics.…”