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

By Richard A. Blythe, M. Evans. 2007

586 citations recorded by the source corpus.

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.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/232efad1b928aa5056c3c279d5e23f44c0fc1f83

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In Condensed Matter Physics

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Non-equilibrium statistical mechanics: from a paradigmatic model to biological transport (2011, unknown): “Detailed properties of the phase diagram are reviewed in [96, 122, 123].”
Stochastic models of intracellular transport (2013, background): “Another issue in active transport involves exclusion effects due to multiple motors moving along the same filament track (Blythe and Evans, 2007; Chou et al., 2011; Schadschneider et al., 2010).”
A pedestrian's view on interacting particle systems, KPZ universality and random matrices (2008, background): “With open boundary conditions allowing for the injection and extraction of particles, both the SSEP and the ASEP display nontrivial invariant measures characterized by long-ranged correlations and the possibility of phase transitions [20, 38].”
Lattice models of nonequilibrium bacterial dynamics (2010, methodology): “In the latter category we can think of lattice transport models such as the exclusion [17] or zero-range [18] processes, for which some exact analytical results can be found, as well as methods to characterize fluctuations and large deviations in nonequilibrium states [19].”