Solvable multi-species extensions of the drop-push model

Farinaz Roshani, Mohammad Khorrami

Published Jan 1, 2003• 7 citations

Abstract

A family of multispecies drop-push system on a one-dimensional lattice is investigated. It is shown that this family is solvable in the sense of the Bethe ansatz, provided a nonspectral matrix equation is satisfied. The large-time behavior of the conditional probabilities, and the dynamics of the particle-type change are also investigated. PACS numbers: 05.40.-a, 02.50.Ga

Related papers

Asymmetric one-dimensional exclusion processes: a two-parameter discrete-time exactly solvable model

Farinaz Roshani, Mohammad Khorrami

A two-parameter family of discrete-time asymmetric exclusion processes on a one-dimensional lattice is considered. Among these processes are the asymmetric simple exclusion model and the drop–push model. Using the Bethe ansatz, the exact solution to the master equation and from that the drift rates of the two-particle sector are obtained.

Journal of Physics Condensed MatterMar 25, 20059 citations

Integrability of Multispecies Long-Range Swap Models with Species-Dependent Interpolation

Eunghyun Lee

arXiv:2604.12136Apr 13, 20260 citations

Exact solution of a one-parameter family of asymmetric exclusion processes

M. Alimohammadi, Vahid Karimipour, Mohammad Khorrami

arXiv:cond-mat/9801136Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsJun 1, 199855 citations

Integrability of the Multi-Species TASEP with Species-Dependent Rates

Eunghyun Lee

Assume that each species l has its own jump rate bl in the multi-species totally asymmetric simple exclusion process. We show that this model is integrable in the sense that the Bethe ansatz method is applicable to obtain the transition probabilities for all possible N-particle systems with up to N different species.

arXiv:2105.14612SymmetryAug 27, 20217 citations

On the integrability of zero-range chipping models with factorized steady states

Alexander M. Povolotsky

arXiv:1308.3250Journal of Physics A Mathematical and TheoreticalNov 1, 2013113 citations

Algebraic Bethe ansatz for the two species ASEP with different hopping rates

Luigi Cantini A

An ASEP with two species of particles and different hopping rates is considered on a ring. Its integrability is proved, and the nested algebraic Bethe ansatz is used to derive the Bethe equations for states with arbitrary numbers of particles of each type, generalizing the results of Derrida and Evans [10]. We also present formulae for the total velocity of particles of a given type and their limit given the large size of the system and the finite densities of the particles.

arXiv:0710.4083Journal of Physics A Mathematical and TheoreticalFeb 19, 200838 citations

Solvable multi-species extensions of the drop-push model

Farinaz Roshani, Mohammad Khorrami

Published Jan 1, 2003• 7 citations

OpenAlex

Abstract

Related papers

Asymmetric one-dimensional exclusion processes: a two-parameter discrete-time exactly solvable model

Farinaz Roshani, Mohammad Khorrami

Journal of Physics Condensed MatterMar 25, 20059 citations

Integrability of Multispecies Long-Range Swap Models with Species-Dependent Interpolation

Eunghyun Lee

We introduce a class of multispecies exclusion processes with long-range swap interactions, incorporating species-dependent interpolation between TASEP-type and drop--push-type dynamics: each species $i$ is assigned a parameter $μ_i$ governing same-species interactions, resulting in a heterogeneous system in which different species follow distinct microscopic interaction mechanisms. In contrast to previously studied integrable multispecies models, where species dependence typically enters through jump rates, the present framework allows the interaction mechanism itself to depend on the species. Our main result establishes integrability of the model in the binary parameter regime $μ_i \in \{0,1\}$ for arbitrary species compositions. In the continuous parameter regime $μ_i \in (0,1)$ , we identify several nontrivial classes of species compositions for which integrability is preserved. We further extend the model to include bidirectional motion, going beyond totally asymmetric dynamics. Using the coordinate Bethe ansatz, we prove two-particle reducibility and derive the associated scattering matrix, which is shown to satisfy the Yang--Baxter equation. The resulting scattering matrix exhibits genuinely species-dependent diagonal entries.We introduce a class of multispecies exclusion processes with long-range swap interactions, incorporating species-dependent interpolation between TASEP-type and drop--push-type dynamics: each species $i$ is assigned a parameter $μ_i$ governing same-species interactions, resulting in a heterogeneous system in which different species follow distinct microscopic interaction mechanisms. In contrast to previously studied integrable multispecies models, where species dependence typically enters through jump rates, the present framework allows the interaction mechanism itself to depend on the species. Our main result establishes integrability of the model in the binary parameter regime $μ_i \in \{0,1\}$ for arbitrary species compositions. In the continuous parameter regime $μ_i \in (0,1)$, we identify several nontrivial classes of species compositions for which integrability is preserved. We further extend the model to include bidirectional motion, going beyond totally asymmetric dynamics. Using the coordinate Bethe ansatz, we prove two-particle reducibility and derive the associated scattering matrix, which is shown to satisfy the Yang--Baxter equation. The resulting scattering matrix exhibits genuinely species-dependent diagonal entries.

arXiv:2604.12136Apr 13, 20260 citations

Exact solution of a one-parameter family of asymmetric exclusion processes

M. Alimohammadi, Vahid Karimipour, Mohammad Khorrami

We define a family of asymmetric processes for particles on a one-dimensional lattice, depending on a continuous parameter $\ensuremath{\lambda}\ensuremath{\in}[0,1]$, interpolating between the completely asymmetric processes (for $\ensuremath{\lambda}=1)$ and the $n=1$ drop-push models (for $\ensuremath{\lambda}=0)$. For arbitrary $\ensuremath{\lambda}$, the model describes an exclusion process, in which a particle pushes its right neighboring particles to the right, with rates depending on the number of these particles. Using the Bethe ansatz, we obtain the exact solution of the master equation.We define a family of asymmetric processes for particles on a one-dimensional lattice, depending on a continuous parameter $\ensuremath{\lambda}\ensuremath{\in}[0,1]$, interpolating between the completely asymmetric processes (for $\ensuremath{\lambda}=1)$ and the $n=1$ drop-push models (for $\ensuremath{\lambda}=0)$. For arbitrary $\ensuremath{\lambda}$, the model describes an exclusion process, in which a particle pushes its right neighboring particles to the right, with rates depending on the number of these particles. Using the Bethe ansatz, we obtain the exact solution of the master equation.

arXiv:cond-mat/9801136Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsJun 1, 199855 citations

Integrability of the Multi-Species TASEP with Species-Dependent Rates

Eunghyun Lee

arXiv:2105.14612SymmetryAug 27, 20217 citations

On the integrability of zero-range chipping models with factorized steady states

Alexander M. Povolotsky

Conditions of integrability of general zero range chipping models with factorized steady state, which were proposed in [Evans, Majumdar, Zia 2004 J. Phys. A 37 L275], are examined. We find a three-parametric family of hopping probabilities for the models solvable by the Bethe ansatz, which includes most of known integrable stochastic particle models as limiting cases. The solution is based on the quantum binomial formula for two elements of an associative algebra obeying generic homogeneous quadratic relations, which is proved as a byproduct. We use the Bethe ansatz to solve an eigenproblem for the transition matrix of the Markov process. On its basis we conjecture an integral formula for the Green function of evolution operator for the model on an infinite lattice and derive the Bethe equations for the spectrum of the model on a ring.Conditions of integrability of general zero range chipping models with factorized steady state, which were proposed in [Evans, Majumdar, Zia 2004 J. Phys. A 37 L275], are examined. We find a three-parametric family of hopping probabilities for the models solvable by the Bethe ansatz, which includes most of known integrable stochastic particle models as limiting cases. The solution is based on the quantum binomial formula for two elements of an associative algebra obeying generic homogeneous quadratic relations, which is proved as a byproduct. We use the Bethe ansatz to solve an eigenproblem for the transition matrix of the Markov process. On its basis we conjecture an integral formula for the Green function of evolution operator for the model on an infinite lattice and derive the Bethe equations for the spectrum of the model on a ring.

arXiv:1308.3250Journal of Physics A Mathematical and TheoreticalNov 1, 2013113 citations

Algebraic Bethe ansatz for the two species ASEP with different hopping rates

Luigi Cantini A

arXiv:0710.4083Journal of Physics A Mathematical and TheoreticalFeb 19, 200838 citations

We introduce a class of multispecies exclusion processes with long-range swap interactions, incorporating species-dependent interpolation between TASEP-type and drop--push-type dynamics: each species

i

is assigned a parameter

μ_i

governing same-species interactions, resulting in a heterogeneous system in which different species follow distinct microscopic interaction mechanisms. In contrast to previously studied integrable multispecies models, where species dependence typically enters through jump rates, the present framework allows the interaction mechanism itself to depend on the species. Our main result establishes integrability of the model in the binary parameter regime

μ_i \in \{0,1\}

for arbitrary species compositions. In the continuous parameter regime

μ_i \in (0,1)

, we identify several nontrivial classes of species compositions for which integrability is preserved. We further extend the model to include bidirectional motion, going beyond totally asymmetric dynamics. Using the coordinate Bethe ansatz, we prove two-particle reducibility and derive the associated scattering matrix, which is shown to satisfy the Yang--Baxter equation. The resulting scattering matrix exhibits genuinely species-dependent diagonal entries.

n=1

drop-push models (for $\ensuremath{\lambda}=0)$. For arbitrary $\ensuremath{\lambda}$, the model describes an exclusion process, in which a particle pushes its right neighboring particles to the right, with rates depending on the number of these particles. Using the Bethe ansatz, we obtain the exact solution of the master equation.