Nonequilibrium Steady States of Matrix Product Form: A Solver's Guide

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.

127 pages, 31 figures, invited topical review for J. Phys. A (uses IOP class file)

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Cited by in corpus (34)

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• Sedimentation, trapping, and rectification of dilute bacteria
• Construction of a Coordinate Bethe Ansatz for the asymmetric simple exclusion process with open boundaries
• Solvable quantum nonequilibrium model exhibiting a phase transition and a matrix product representation
• Matrix representation of the stationary measure for the multispecies TASEP
• DMRG-study of current and activity fluctuations near non-equilibrium phase transitions
• Matrix product solution of the multispecies partially asymmetric exclusion process
• Active Absorbing State Phase Transition Beyond Directed Percolation : A Class of Exactly Solvable Models
• Phase diagram and edge effects in the ASEP with bottlenecks
• Slowest relaxation mode of the partially asymmetric exclusion process with open boundaries
• A dynamically extending exclusion process
• Spectrum in multi-species asymmetric simple exclusion process on a ring
• Phase diagram of two-lane driven diffusive systems
• Cumulants of the current in the weakly asymmetric exclusion process
• Current Fluctuations in the exclusion process and Bethe Ansatz
• Algebraic Bethe Ansatz for the two species ASEP with different hopping rates
• Asymmetric Exclusion Processes with Disorder: Effect of Correlations
• Non-equilibrium phase transitions in tubulation by molecular motors
• Parallel Coupling of Symmetric and Asymmetric Exclusion Processes
• Fluctuations and skewness of the current in the partially asymmetric exclusion process
• Non-equilibrium processes: driven lattice gases, interface dynamics, and quenched disorder effects on density profiles and currents
• Reversibility, heat dissipation and the importance of the thermal environment in stochastic models of nonequilibrium steady states
• Disordered driven lattice gases with boundary reservoirs and Langmuir kinetics
• Ultra-discrete Optimal Velocity Model: a Cellular-Automaton Model for Traffic Flow and Linear Instability of High-Flux Traffic
• Continued Fractions and the Partially Asymmetric Exclusion Process
• Temporal evolution of product shock measures in TASEP with sublattice-parallel update
• Smoothly-varying hopping rates in driven flow with exclusion
• Dynamic Singularities in Cooperative Exclusion
• Tridiagonal Symmetries of Models of Nonequilibrium Physics
• Repelling Random Walkers in a Diffusion-Coalescence System
• Finite-dimensional representation of the quadratic algebra of a generalized coagulation-decoagulation model
• One-transit paths and steady-state of a non-equilibrium process in a discrete-time update
• A Family of Exactly-Solvable Driven-Diffusive Systems in One-dimension
• Connection between matrix-product states and superposition of Bernoulli shock measures