Exact solution of an exclusion process with three classes of particles and vacancies

Abstract. We construct matrix product steady state for a class of interacting particle systems where particles do not obey hardcore exclusion, meaning each site can occupy any number of particles subjected to the global conservation of total number of particles in the system. To represent the arbitrary occupancy of the sites, the matrix product ansatz here requires an infinite set of matrices which in turn leads to an algebra involving infinite number of matrix equations. We show that these matrix equations, in fact, can be reduced to a single functional relation when the matrices are parametric functions of the representative occupation number. We demonstrate this matrix formulation in a class of stochastic particle hopping processes on a one dimensional periodic lattice where hop rates depend on the occupation numbers of the departure site and its neighbors within a finite range; this includes some well known stochastic processes like, totally asymmetric zero range process, misanthrope process, finite range process and partially asymmetric versions of the same processes but with different rate functions depending on the direction of motion.

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“…(9) with hop rate there replaced by Eq. (25). We find that this algebra is satisfied by the following representation of matrices,…”

Section: Finite Range Process Withmentioning

confidence: 86%

Matrix product states for interacting particles without hardcore constraints

Chatterjee¹,

Mohanty²

2017

J. Phys. A: Math. Theor.

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“…Meanwile, the matrix product solution for the totally asymmetric case (q = 0) was constructed in [60] and hierarchical structure for increasing N was elucidated. We note that previously a solution for the N = 3 case was given by by Mallick, Mallick and Rajewsky [68]. In these solutions the 'matrices' are in fact tensor products of the fundamental matrices δ = D − 1, = D − 1 and A and obey more complicated relations than (124-125), involving an auxiliary set of 'hat' operators.…”

Section: Multispecies Generalisationmentioning

confidence: 87%

Combinatorial mappings of exclusion processes

Wood¹,

Blythe²,

Evans³

2020

J. Phys. A: Math. Theor.

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We review various combinatorial interpretations and mappings of stationary-state probabilities of the totally asymmetric, partially asymmetric and symmetric simple exclusion processes (TASEP, PASEP, SSEP respectively). In these steady states, the statistical weight of a configuration is determined from a matrix product, which can be written explicitly in terms of generalised ladder operators. This lends a natural association to the enumeration of random walks with certain properties.Specifically, there is a one-to-many mapping of steady-state configurations to a larger state space of discrete paths, which themselves map to an even larger state space of number permutations. It is often the case that the configuration weights in the extended space are of a relatively simple form (e.g., a Boltzmann-like distribution). Meanwhile, various physical properties of the nonequilibrium steady state-such as the entropy-can be interpreted in terms of how this larger state space has been partitioned.These mappings sometimes allow physical results to be derived very simply, and conversely the physical approach allows some new combinatorial problems to be solved. This work brings together results and observations scattered in the combinatorics and statistical physics literature, and also presents new results. The review is pitched at statistical physicists who, though not professional combinatorialists, are competent and enthusiastic amateurs.

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“…This problem was considered originally for the case N = 2 as a model to describe shocks [25]- [27] in nonequilibrium. The stationary properties of the N = 2 [31] and N = 3 [43] models can also be studied through a matrix product ansatz. In …”

Section: The Asymmetric Diffusion Model With N Classes Of Particlementioning

confidence: 99%

The exact solution of the asymmetric exclusion problem with particles of asrbitrary size: matrix product ansatz

Alcaraz¹,

Lazo²

2003

Braz. J. Phys.

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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 Schrödinger 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. I IntroductionThe representation of interacting stochastic particle dynamics in terms of quantum spin systems produced interesting and fruitful interchanges between the areas of equilibrium and nonequilibrium statistical mechanics. The connection between these areas follows from the similarity between the master equation describing the time-fluctuations on the nonequilibrium stochastic problem and the quantum fluctuations of the equilibrium quantum spin chains [1]- [20].Unlike the area of nonequilibrium interacting systems, where very few models are fully solvable, there exists a huge family of quantum chains appearing in equilibrium problems that are exactly integrable. The machinery that allows the exact solutions of these quantum chains comes from the Bethe ansatz in its several formulations (see [21]-[24] for reviews). The above mentioned mathematical connection between equilibrium and nonequilibrium revealed that some quantum chains related to interacting stochastic problems are exactly solvable through the Bethe ansatz. The simplest example is the problem of asymmetric diffusion of hard-core particles on the one dimensional lattice (see [16,17,20] for reviews). The time fluctuations of this last model are governed by a time evolution operator that coincides with the exact integrable anisotropic Heisenberg chain, or the so called, XXZ quantum chain, in its ferromagnetically ordered regime. A generalization of this stochastic problem where exact integrability is also known [25]-[27] is the case where there are N (N = 1, 2, ...) classes of particles hierarchically ordered and diffusing asymmetrically on the lattice. The quantum chain related to this problem is known in the literature as the anisotropic Sutherland model [28...

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Exact solution of an exclusion process with three classes of particles and vacancies

Cited by 54 publications

References 31 publications

Matrix product states for interacting particles without hardcore constraints

Matrix product states for interacting particles without hardcore constraints

Combinatorial mappings of exclusion processes

The exact solution of the asymmetric exclusion problem with particles of asrbitrary size: matrix product ansatz

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