Two-species Exclusion Process Research Articles

Clustering and finite size effects in a two-species exclusion process

Clustering and finite size effects in a two-species exclusion process

Macroscopic Behaviour in a Two-Species Exclusion Process Via the Method of Matched Asymptotics

We consider a two-species simple exclusion process on a periodic lattice. We use the method of matched asymptotics to derive evolution equations for the two population densities in the dilute regime, namely a cross-diffusion system of partial differential equations for the two species’ densities. First, our result captures non-trivial interaction terms neglected in the mean-field approach, including a non-diagonal mobility matrix with explicit density dependence. Second, it generalises the rigorous hydrodynamic limit of Quastel (Commun Pure Appl Math 45(6):623–679, 1992), valid for species with equal jump rates and given in terms of a non-explicit self-diffusion coefficient, to the case of unequal rates in the dilute regime. In the equal-rates case, by combining matched asymptotic approximations in the low- and high-density limits, we obtain a cubic polynomial approximation of the self-diffusion coefficient that is numerically accurate for all densities. This cubic approximation agrees extremely well with numerical simulations. It also coincides with the Taylor expansion up to the second-order in the density of the self-diffusion coefficient obtained using a rigorous recursive method.

Macdonald–Koornwinder moments and the two-species exclusion process

Introduced in the late 1960’s (Macdonald et al. in Biopolymers 6:1–25, 1968; Spitzer in Adv Math 5:246–290, 1970), the asymmetric exclusion process (ASEP) is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice with open boundaries. It has been known for awhile that there is a tight connection between the partition function of the ASEP and moments of Askey–Wilson polynomials (Uchiyama et al. in J Phys A 37(18):4985–5002, 2004; Corteel and Williams in Duke Math J 159(3):385–415, 2011; Corteel et al. in Trans Am Math Soc 364(11):6009–6037, 2012), a family of orthogonal polynomials which are at the top of the hierarchy of classical orthogonal polynomials in one variable. On the other hand, Askey–Wilson polynomials can be viewed as a specialization of the multivariate Macdonald–Koornwinder polynomials (also known as Koornwinder polynomials), which in turn give rise to the Macdonald polynomials associated to any classical root system via a limit or specialization (van Diejen in Compos Math 95(2):183–233, 1995). In light of the fact that Koornwinder polynomials generalize the Askey–Wilson polynomials, it is natural to ask whether one can find a particle model whose partition function is related to Koornwinder polynomials. In this article we answer this question affirmatively, by showing that Koornwinder moments at $$q=t$$ are closely connected to the partition function for the two-species exclusion process.

Interference of two codirectional exclusion processes in the presence of a static bottleneck: A biologically motivated model.

We develop a two-species exclusion process with a distinct pair of entry and exit sites for each species of rigid rods. The relatively slower forward stepping of the rods in an extended bottleneck region, located in between the two entry sites, controls the extent of interference of the codirectional flow of the two species of rods. The relative positions of the sites of entry of the two species of rods with respect to the location of the bottleneck are motivated by a biological phenomenon. However, the primary focus of the study here is to explore the effects of the interference of the flow of the two species of rods on their spatiotemporal organization and the regulations of this interference by the extended bottleneck. By a combination of mean-field theory and computer simulation, we calculate the flux of both species of rods and their density profiles as well as the composite phase diagrams of the system. If the bottleneck is sufficiently stringent, then some of the phases become practically unrealizable, although not ruled out on the basis of any fundamental physical principle. Moreover, the extent of suppression of flow of the downstream entrants by the flow of the upstream entrants can also be regulated by the strength of the bottleneck. We speculate on the possible implications of the results in the context of the biological phenomenon that motivated the formulation of the theoretical model.

Spontaneous pulsing states in an active particle system

We study a two-lane two-species exclusion process inspired by Lin et al (2011 J. Stat. Mech. P09027) that exhibits a non-equilibrium pulsing phase. Particles move on two parallel one-dimensional tracks, with one open and one reflecting boundary. The particle type defines the hopping direction. When only particles hopping towards the open end are allowed to change lane, the system exhibits a phase transition from a low density phase to a pulsing phase depending on the ratio between particle injection and type-changing rate. This phase transition can be observed in the stochastic model as well as in a mean-field description. In the low density phase, the density profile can be predicted analytically. The pulsing phase is characterised by a fast filling of the system and—once filled—by a slowly backwards moving front separating a decreasing dense region and an expanding low density region. The hopping of the front on the discrete lattice is found to create density oscillations, both, in time and space. By means of a stability analysis we can predict the structure of the dense region during the emptying process, characterised by exponentially damped perturbations, both at the open end and near the moving front.