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arXiv:2212.11646 · Journal of Physics A Mathematical and Theoretical

Integrability of two-species partially asymmetric exclusion processes

Ivan Lobaskin, M. R. Evans, Kirone Mallick

Published Mar 17, 2023• 6 citations
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Abstract

Abstract We work towards the classification of all one-dimensional exclusion processes with two species of particles that can be solved by a nested coordinate Bethe ansatz (BA). Using the Yang–Baxter equations, we obtain conditions on the model parameters that ensure that the underlying system is integrable. Three classes of integrable models are thus found. Of these, two classes are well known in literature, but the third has not been studied until recently, and never in the context of the BA. The Bethe equations are derived for the latter model as well as for the associated dynamics encoding the large deviation of the currents.

DOI
10.1088/1751-8121/acc55b

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Abstract We work towards the classification of all one-dimensional exclusion processes with two species of particles that can be solved by a nested coordinate Bethe ansatz (BA). Using the Yang–Baxter equations, we obtain conditions on the model parameters that ensure that the underlying system is integrable. Three classes of integrable models are thus found. Of these, two classes are well known in literature, but the third has not been studied until recently, and never in the context of the BA. The Bethe equations are derived for the latter model as well as for the associated dynamics encoding the large deviation of the currents.
We consider a process in which there are two types of particles, A and B, on an infinite one-dimensional lattice. The particles hop to their adjacent sites, like the totally asymmetric exclusion process (ASEP), and have also the following interactions: A+B-->B+B and B+A-->B+B, which all occur with equal rate. We study this process by imposing four boundary conditions on the ASEP master equation. It is shown that this model is integrable, in the sense that its N-particle S matrix is factorized into a product of two-particle S matrices and, more importantly, the two-particle S matrix satisfies the quantum Yang-Baxter equation. Using the coordinate Bethe-ansatz, the N-particle wave functions and the two-particle conditional probabilities are found exactly. Further, by imposing four reasonable physical conditions on two-species diffusion-reaction processes (where the most important ones are the equality of the reaction rates and the conservation of the number of particles in each reaction), we show that among the 4096 types of interactions which have these properties and can be modeled by a master equation and an appropriate set of boundary conditions, there are only 28 independent interactions which are integrable. We find all these interactions and also their corresponding wave functions. Some of these may be new solutions of the quantum Yang-Baxter equation.
Within the Matrix Product Formalism we have already introduced a multi-species exclusion process ( Phys. Rev. E , 59 (1999) 25, cond-mat/9809193), in which different particles hop with different rates and fast particles stochastically overtake slow ones. In this letter we show that on an open chain, the master equation of this process can be exactly solved via the coordinate Bethe ansatz. It is shown that the N -body S -matrix of this process is factorized into a product of two-body S -matrices, which in turn satisfy the quantum Yang-Baxter equation (QYBE). This solution is, to our knowledge, a new solution of QYBE.
We give a complete classification of integrable Markovian boundary conditions for the asymmetric simple exclusion process with two species (or classes) of particles. Some of these boundary conditions lead to non-vanishing particle currents for each species. We explain how the stationary state of all these models can be expressed in a matrix product form, starting from two key components, the Zamolodchikov–Faddeev and Ghoshal–Zamolodchikov relations. This statement is illustrated by studying in detail a specific example, for which the matrix ansatz (involving nine generators) is explicitly constructed and physical observables (such as currents, densities) calculated.
The exact solution of the asymmetric exclusion problem and several of its generalizations is obtained by a matrix product ansatz. Due to the similarity of the master equation and the Schrodinger equation at imaginary times the solution of these problems reduces to the diagonalization of a one dimensional quantum Hamiltonian. Initially, we present the solution of the problem when an arbitrary mixture of molecules, each of then having an arbitrary size (s = 0; 1; 2; ...) in units of lattice spacing, diffuses asymmetrically on the lattice. The solution of the more general problem where we have the diffusion of particles belonging to N distinct classes of particles (c = 1; ... ; N), with hierarchical order and arbitrary sizes, is also presented. Our matrix product ansatz asserts that the amplitudes of an arbitrary eigenfunction of the associated quantum Hamiltonian can be expressed by a product of matrices. The algebraic properties of the matrices defining the ansatz depend on the particular associated Hamiltonian. The absence of contradictions in the algebraic relations defining the algebra ensures the exact integrability of the model. In the case of particles distributed in N > 2 classes, the associativity of this algebra implies the Yang-Baxter relations of the exact integrable model.
Abstract We introduce a class of multispecies exclusion processes in which each species follows its own intra-species interaction mechanism. In contrast to previously studied integrable multispecies models—such as the multispecies TASEP, where the intra-species interaction rule is identical across all species—our framework allows the intra-species rule to vary across species, while inter-species interactions obey a long-range swap rule. We prove that Bethe ansatz integrability is preserved in the binary regime, where each species follows either TASEP-type or drop-push-type dynamics, by establishing the Yang–Baxter structure of the associated scattering matrices. Furthermore, we identify nontrivial families of species compositions through a continuous interpolation between these dynamics that remain integrable, and extend the model to include bidirectional motion. These results show that Bethe ansatz integrability extends to a broad class of multispecies systems with species-dependent interaction mechanisms, providing a framework for deriving exact transition probabilities and analyzing nonequilibrium dynamics.