On exclusion type inhomogeneous interacting particle systems

AbstractFor a large class of inhomogeneous interacting particle systems (IPS) on a lattice wedevelop a rigorous method for mapping them onto homogeneous IPS. Our novel approachprovides a direct way of obtaining the statistical properties of such inhomogeneous systemsby studying the far simpler homogeneous systems. In the cases when the latter can be solvedexactly our method yields an exact solution for the statistical properties of an inhomogeneousIPS. This approach is illustrated by studies of three of IPS, namely those with particles ofdifferent sizes, or with varying (between particles) maximal velocities, or accelerations. Key words: interacting particle system, exclusion process, parallel dynamics, traffic flow model. 1 Introduction A very substantial progress on the understanding of statistical properties of lattice interactingparticle systems (IPS) (see an excellent review in [5] and numerous references therein) hasbeen achieved mainly for continuous time systems describing interactions of identical particles.Only recently results related to non homogeneous particle systems started to show up. Someof them analyze the presence of a single inhomogeneity (like a street light in [4]) or spatiallyvarying hopping rates (see, e.g. [7, 6]), while in some other papers the situation when a singleparticle occupies several lattice sites were considered (see, e.g. [3]). In each of these papers theauthors developed new (and quite complicated) constructions or approximations to deal withthe inhomogeneities. Our strategy is based on a completely different idea, namely we reduce theanalysis of an inhomogeneous problem to a homogeneous one for which a solution is much simpleror even already known. Returning to the original setting one is able to recover the completestatistical description for the inhomogeneous system. In distinction to complicated mean fieldapproximations the exact constructions that we use are surprisingly simple and straightforward.We study three new situations when the analysis of an inhomogeneous particle system can bereduced to a homogeneous one. The first of them is the case when the particles differ in size,i.e. each particle occupies several lattice sites and, what is more important, the sizes of differentparticles might differ. It is worth note that the ability to deal with particles of different sizes isvery important from the point of view of applications to dynamics of traffic flows (see a reviewwith numerous references in [8]), where ordinary vehicles and buses or tracks are clearly ofdifferent sizes, or to various biological models like ion channels and mRNA translation whereribosomes and large molecules or vesicles might be of very different lengths. The only knownresult related to such systems [3] describing the motion of identical ‘long’ particles is based on amean field approximation. Note also that the difference in particle sizes represents a fundamentalobstacle for the application of one of the basic tools of the IPS theory – the coupling method.