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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• Density Profile of the One-Dimensional Partially Asymmetric Simple Exclusion Process with Open Boundaries
• Yang-Lee zeros for a nonequilibrium phase transition
• Modelling one-dimensional driven diffusive systems by the Zero-Range Process
• Why spontaneous symmetry breaking disappears in a bridge system with PDE-friendly boundaries
• On the ubiquity of matrix-product states in one-dimensional stochastic processes with boundary interactions
• Long-range attraction between probe particles mediated by a driven fluid
• Nonequilibrium Criticality at Shock Formation in Steady States
• Nonequilibrium stationary states and equilibrium models with long range interactions
• Stochastic Ballistic Annihilation and Coalescence
• One-Dimensional Partially Asymmetric Simple Exclusion Process on a Ring with a Defect Particle
• Bethe ansatz solution for crossover scaling functions of the asymmetric XXZ chain and the Karder--Parisi--Zhang-type growth model
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• Matrix Product States of Three Families of One-Dimensional Interacting Particle Systems
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• Phase diagram and edge effects in the ASEP with bottlenecks
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• Dynamics of Fluctuations in Quantum Simple Exclusion Processes
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• Introduction to stochastic models of transportation flows. Part I
• Fluctuations in meta-population exclusion processes
• Maximizing Protein Translation Rate in the Nonhomogeneous Ribosome Flow Model: A Convex Optimization Approach
• Ion transport through confined ion channels in the presence of immobile charges
• Chebyshev type lattice path weight polynomials by a constant term method
• Phase transitions in cellular automata for cargo transport and kinetically constrained traffic
• Tradeoff of generalization error in unsupervised learning
• Inhomogeneous MPA and exact steady states of boundary driven spin chains at large dissipation
• Classical Non-equilibrium Phases from a Quantum Perspective: Quasi-adiabatic evolution and Local Ansatz for Solving The Equilibrium State of Local Stochastic Dynamics
• Excluded Volume Effect in Queueing Theory
• Exact steady states and fragmentation-induced relaxation in the no-passing asymmetric simple exclusion process
• Controllability analysis and control synthesis for the ribosome flow model
• Negative feedback and oscillations in a model for mRNA translation
• Boundary Perturbation Effects in Quantum Systems with Conserved Energy and Continuous Symmetry
• Localization of a microtubule organizing center by kinesin motors
• Maximizing average throughput in oscillatory biological synthesis systems: an optimal control approach
• Limit fluctuations of stationary measure of totally asymmetric simple exclusion process with open boundaries on the coexistence line
• Matrix product solution of the stationary states of two-species open zero range processes
• Limits of open ASEP stationary measures near a boundary
• A deterministic model for non-monotone relationship between translation of upstream and downstream open reading frames
• Particle hopping on a ladder: exact solution using multibalance
• Jamming of molecular motors as a tool for transport cargos along microtubules
• Ribosome Flow Model with Extended Objects
• TASEP and Planar Binary Trees: A Combinatorial Approach
• Steady-state dynamics of exclusion process with local reversible association of particles
• On the Diffusion of Sticky Particles in 1D
• Nonequilibrium phases and quantum correlations in synthetic transport models
• A hopping mechanism for cargo transport by molecular motors in crowded microtubules
• Intrinsic Step Jamming in Nanometer-Scale KPZ-like Rough Surfaces under Interface-Limited Crystal Growth and Retreat
• A stochastic model solvable without integrability
• Collective excitations in jammed states: ultrafast defect propagation and finite-size scaling
• Bi-orthogonal Polynomial Sequences and the Asymmetric Simple Exclusion Process
• Relaxation time in a non-conserving driven-diffusive system with parallel dynamics
• On the Structure of the Initiation and Elongation Rates that Maximize Protein Production in the Ribosome Flow Model
• Invariant measures of discrete interacting particle systems: Algebraic aspects