Finite-dimensional representations of the quadratic algebra: Applications to the exclusion process

Mallick, Kirone; Sandow, Sven

doi:10.1088/0305-4470/30/13/008

Cited by 70 publications

(117 citation statements)

References 10 publications

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“…Curves of this type were first noted in the more general case γ, δ = 0 in [202] and finite dimensional representations were catalogued in [203]. where the parameters a and b are given in (D.6), and κ is a constant usually chosen so that V |W = 1.…”

Section: D1 Representations Of Pasep Algebramentioning

confidence: 99%

Nonequilibrium steady states of matrix-product form: a solver's guide

Blythe

Evans²

2007

J. Phys. A: Math. Theor.

639

1,014

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Abstract. We consider the general problem of determining the steady state of stochastic nonequilibrium systems such as those that have been used to model (among other things) biological transport and traffic flow. We begin with a broad overview of this class of driven diffusive systems-which includes exclusion processes-focusing on interesting physical properties, such as shocks and phase transitions. We then turn our attention specifically to those models for which the exact distribution of microstates in the steady state can be expressed in a matrix product form. In addition to a gentle introduction to this matrix product approach, how it works and how it relates to similar constructions that arise in other physical contexts, we present a unified, pedagogical account of the various means by which the statistical mechanical calculations of macroscopic physical quantities are actually performed. We also review a number of more advanced topics, including nonequilibrium free energy functionals, the classification of exclusion processes involving multiple particle species, existence proofs of a matrix product state for a given model and more complicated variants of the matrix product state that allow various types of parallel dynamics to be handled. We conclude with a brief discussion of open problems for future research.

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Section: D1 Representations Of Pasep Algebramentioning

confidence: 99%

Nonequilibrium steady states of matrix-product form: a solver's guide

Blythe

Evans²

2007

J. Phys. A: Math. Theor.

639

1,014

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“…In this approach one defines a product measure with matrix entries D m rather than c-numbers as stationary weights for finding a given site in state m. The matrices D m to together with a set of auxiliary matrices [131,132] have to satisfy algebraic relations which are obtained from requiring the matrix product state to satisfy the stationarity condition (3.2). This leads to algebras with quadratic relations [20,78,127,134,135,136,137,138].…”

Section: Polymer Dynamicsmentioning

confidence: 99%

Critical phenomena and universal dynamics in one-dimensional driven diffusive systems with two species of particles

Schütz

2003

J. Phys. A: Math. Gen.

137

136

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Recent work on stochastic interacting particle systems with two particle species (or single-species systems with kinematic constraints) has demonstrated the existence of spontaneous symmetry breaking, long-range order and phase coexistence in nonequilibrium steady states, even if translational invariance is not broken by defects or open boundaries. If both particle species are conserved, the temporal behaviour is largely unexplored, but first results of current work on the transition from the microscopic to the macroscopic scale yield exact coupled nonlinear hydrodynamic equations and indicate the emergence of novel types of shock waves which are collective excitations stabilized by the flow of microscopic fluctuations. We review the basic stationary and dynamic properties of these systems, highlighting the role of conservation laws and kinetic constraints for the hydrodynamic behaviour, the microscopic origin of domain wall (shock) stability and the coarsening dynamics of domains during phase separation.

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“…Finite-dimensional representations of the PASEP have widely been studied in [18,19]. It is known that the PASEP has also an N -dimensional representation provided that…”

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confidence: 99%

Matrix product steady states as superposition of product shock measures in 1D driven systems

Jafarpour

Masharian

2007

J. Stat. Mech.

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It is known that exact traveling wave solutions exist for families of (n + 1)-states stochastic one-dimensional non-equilibrium lattice models with open boundaries provided that some constraints on the reaction rates are fulfilled. These solutions describe the diffusive motion of a product shock or a domain wall with the dynamics of a simple biased random walker. The steady state of these systems can be written in terms of linear superposition of such shocks or domain walls. These steady states can also be expressed in a matrix product form. We show that in this case the associated quadratic algebra of the system has always a twodimensional representation with a generic structure. A couple of examples for n = 1 and n = 2 cases are presented.

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Finite-dimensional representations of the quadratic algebra: Applications to the exclusion process

Cited by 70 publications

References 10 publications

Nonequilibrium steady states of matrix-product form: a solver's guide

Nonequilibrium steady states of matrix-product form: a solver's guide

Critical phenomena and universal dynamics in one-dimensional driven diffusive systems with two species of particles

Matrix product steady states as superposition of product shock measures in 1D driven systems

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