Multi-species Asymmetric Exclusion Research Articles

Transition probability and total crossing events in the multi-species asymmetric exclusion process

We present explicit formulas for total crossing events in the multi-species asymmetric exclusion process (r-ASEP) with underlying symmetry. In the case of the two-species TASEP these can be derived using an explicit expression for the general transition probability on in terms of a multiple contour integral derived from a nested Bethe ansatz approach. For the general r-ASEP we employ a vertex model approach within which the probability of total crossing can be derived from partial symmetrisation of an explicit high rank rainbow partition function. In the case of r-TASEP, the total crossing probability can be show to reduce to a multiple integral over the product of r determinants. For two-TASEP we additionally derive convenient formulas for cumulative total crossing probabilities using Bernoulli-step initial conditions for particles of type 2 and type 1 respectively.

Matrix product and sum rule for Macdonald polynomials

We present a new, explicit sum formula for symmetric Macdonald polynomials Pλ and show that they can be written as a trace over a product of (infinite dimensional) matrices. These matrices satisfy the Zamolodchikov– Faddeev (ZF) algebra. We construct solutions of the ZF algebra from a rank-reduced version of the Yang–Baxter algebra. As a corollary, we find that the normalization of the stationary measure of the multi-species asymmetric exclusion process is a Macdonald polynomial with all variables set equal to one.

Integrable Stochastic Dualities and the Deformed Knizhnik–Zamolodchikov Equation

Abstract We present a new method for obtaining duality functions in multi-species asymmetric exclusion processes (mASEP), from solutions of the deformed Knizhnik–Zamolodchikov (KZ) equations. Our method reproduces, as a special case, duality functions for the self-dual single species ASEP on the integer lattice.

Koornwinder polynomials and the stationary multi-species asymmetric exclusion process with open boundaries

We prove that the normalisation of the stationary state of the multi-species asymmetric simple exclusion process (mASEP) is a specialisation of a Koornwinder polynomial. As a corollary we obtain that the normalisation of mASEP factorises as a product over multiple copies of the two-species ASEP.

Matrix product formula for Macdonald polynomials

We derive a matrix product formula for symmetric Macdonald polynomials. Our results are obtained by constructing polynomial solutions of deformed Knizhnik–Zamolodchikov equations, which arise by considering representations of the Zamolodchikov–Faddeev and Yang–Baxter algebras in terms of t-deformed bosonic operators. These solutions are generalized probabilities for particle configurations of the multi-species asymmetric exclusion process, and form a basis of the ring of polynomials in n variables whose elements are indexed by compositions. For weakly increasing compositions (anti-dominant weights), these basis elements coincide with non-symmetric Macdonald polynomials. Our formulas imply a natural combinatorial interpretation in terms of solvable lattice models. They also imply that normalizations of stationary states of multi-species exclusion processes are obtained as Macdonald polynomials at q = 1.

A multi-species asymmetric exclusion model with an impurity

A multi-species generalization of the asymmetric simple exclusion process has been considered in the presence of a single impurity on a ring. The model describes particles hopping in one direction with stochastic dynamics and hard-core exclusion condition. The ordinary particles hop forward with their characteristic hopping rates and fast particles can overtake slow ones with a relative rate. The impurity, which is the slowest particle in the ensemble of particles on the ring, hops in the same direction of the ordinary particles with its intrinsic hopping rate and can be overtaken by ordinary particles with a rate which is not necessarily a relative rate. We will show that the phase diagram of the model can be obtained exactly. It turns out that the phase structure of the model depends on the density distribution function of the ordinary particles on the ring so that it can have either four phases or only one. The mean speed of impurity and also the total current of the ordinary particles are explicitly calculated in each phase. Using Monte Carlo simulation, the density profile of the ordinary particles is also obtained. The simulation data confirm all of the analytical calculations.

Multi-species asymmetric exclusion process in ordered sequential update

A multi-species generalization of the asymmetric simple exclusion process (ASEP) is studied in ordered sequential and sub-lattice parallel updating schemes. In this model particles hop with their own specific probabilities to their rightmost empty site and fast particles overtake slow ones with a definite probability. Using Matrix Product Ansatz (MPA), we obtain the relevant algebra, and study the uncorrelated stationary state of the model both for an open system and on a ring. A complete comparison between the physical results in these updates and those of random sequential introduced in [20,21] is made.