Phase transitions in one-dimensional nonequilibrium systems
TL;DR: In this paper, the authors discuss the properties of an energy function which may allow phase transitions and phase ordering in one-dimensional systems and give an overview of the phase transitions which have been studied in nonequilibrium systems.
Abstract: The phenomenon of phase transitions in one-dimensional systems is discussed. Equilibrium systems are reviewed and some properties of an energy function which may allow phase transitions and phase ordering in one dimension are identified. We then give an overview of the one-dimensional phase transitions which have been studied in nonequilibrium systems. A particularly simple model, the zero-range process, for which the steady state is known exactly as a product measure, is discussed in some detail. Generalisations of the model, for which a product measure still holds, are also discussed. We analyse in detail a condensation phase transition in the model and show how conditions under which it may occur may be related to the existence of an effective long-range energy function. It is also shown that even when the conditions for condensation are not fulfilled one can still observe very sharp crossover behaviour and apparent condensation in a finite system. Although the zero-range process is not well known within the physics community, several nonequilibrium models have been proposed that are examples of a zero-range process, or closely related to it, and we review these applications here.
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TL;DR: In this paper, a short review of matrix ansatz, the additivity principle or macroscopic fluctuation theory, developed recently in the theory of non-equilibrium phenomena is given.
Abstract: These lecture notes give a short review of methods such as the matrix ansatz, the additivity principle or the macroscopic fluctuation theory, developed recently in the theory of non-equilibrium phenomena. They show how these methods allow us to calculate the fluctuations and large deviations of the density and the current in non-equilibrium steady states of systems like exclusion processes. The properties of these fluctuations and large deviation functions in non-equilibrium steady states (for example, non-Gaussian fluctuations of density or non-convexity of the large deviation function which generalizes the notion of free energy) are compared with those of systems at equilibrium.
859 citations
TL;DR: The general problem of determining the steady state of stochastic nonequilibrium systems such as those used to model biological transport and traffic flow is considered, and a unified, pedagogical account of the various means by which the statistical mechanical calculations of macroscopic physical quantities are actually performed is presented.
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.
701 citations
Cites background from "Phase transitions in one-dimensiona..."
...This state of affairs applies, in fact, to all 1d spin systems with short-range interactions [70]....
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TL;DR: In this article, the authors present a review of universality classes in nonequilibrium systems defined on regular lattices and discuss the most important critical exponents and relations, as well as the field-theoretical formalism used in the text.
Abstract: This article reviews our present knowledge of universality classes in nonequilibrium systems defined on regular lattices. The first section presents the most important critical exponents and relations, as well as the field-theoretical formalism used in the text. The second section briefly addresses the question of scaling behavior at first-order phase transitions. In Sec. III the author looks at dynamical extensions of basic static classes, showing the effects of mixing dynamics and of percolation. The main body of the review begins in Sec. IV, where genuine, dynamical universality classes specific to nonequilibrium systems are introduced. Section V considers such nonequilibrium classes in coupled, multicomponent systems. Most of the known nonequilibrium transition classes are explored in low dimensions between active and absorbing states of reaction-diffusion-type systems. However, by mapping they can be related to the universal behavior of interface growth models, which are treated in Sec. VI. The review ends with a summary of the classes of absorbing-state and mean-field systems and discusses some possible directions for future research.
698 citations
Cites background from "Phase transitions in one-dimensiona..."
...There are also other nonequilibrium phase transitions, for example, in lattice gases with currents sEvans et al., 1995, 1998; Kolomeisky et al., 1998; Evans, 2000d or in traffic models sChowdhury et al., 2000d, but in these systems the critical universality classes have not yet been explored....
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698 citations
TL;DR: In this article, the authors review recent progress on the zero-range process, a model of interacting particles which hop between the sites of a lattice with rates that depend on the occupancy of the departure site.
Abstract: We review recent progress on the zero-range process, a model of interacting particles which hop between the sites of a lattice with rates that depend on the occupancy of the departure site. We discuss several applications which have stimulated interest in the model such as shaken granular gases and network dynamics; we also discuss how the model may be used as a coarse-grained description of driven phase-separating systems. A useful property of the zero-range process is that the steady state has a factorized form. We show how this form enables one to analyse in detail condensation transitions, wherein a finite fraction of particles accumulate at a single site. We review condensation transitions in homogeneous and heterogeneous systems and also summarize recent progress in understanding the dynamics of condensation. We then turn to several generalizations which also, under certain specified conditions, share the property of a factorized steady state. These include several species of particles; hop rates which depend on both the departure and the destination sites; continuous masses; parallel discrete-time updating; non-conservation of particles and sites.
603 citations
References
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Book•
31 Dec 1985
TL;DR: The construction, and other general results are given in this paper, with values in [0, ] s. The voter model, the contact process, the nearest-particle system, and the exclusion process.
Abstract: The Construction, and Other General Results.- Some Basic Tools.- Spin Systems.- Stochastic Ising Models.- The Voter Model.- The Contact Process.- Nearest-Particle Systems.- The Exclusion Process.- Linear Systems with Values in [0, ?)s.
4,365 citations
"Phase transitions in one-dimensiona..." refers background in this paper
...Various models are reviewed elsewhere in this volume[43] so here I just sketch the basic behaviour by referring to a particular model, the contact process [44] [45]....
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...The zero-range process is also closely related to the more widely known asymmetric exclusion process [10] [44] as we shall describe below....
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...Since then the mathematical achievements have been to prove existence theorems, invariant measures and hydrodynamic limits [44] [51]....
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Book•
01 Jan 1966
TL;DR: In this paper, the Basic Limit Theorem of Markov Chains and its applications are discussed and examples of continuous time Markov chains are presented. But they do not cover the application of continuous-time Markov chain in matrix analysis.
Abstract: Preface. Elements of Stochastic Processes. Markov Chains. The Basic Limit Theorem of Markov Chains and Applications. Classical Examples of Continuous Time Markov Chains. Renewal Processes. Martingales. Brownian Motion. Branching Processes. Stationary Processes. Review of Matrix Analysis. Index.
3,881 citations
08 Mar 2018
TL;DR: In this article, the authors describe how phase transitions occur in practice in practice, and describe the role of models in the process of phase transitions in the Ising Model and the Role of Models in Phase Transition.
Abstract: Introduction * Scaling and Dimensional Analysis * Power Laws in Statistical Physics * Some Important Questions * Historical Development * Exercises How Phase Transitions Occur In Principle * Review of Statistical Mechanics * The Thermodynamic Limit * Phase Boundaries and Phase Transitions * The Role of Models * The Ising Model * Analytic Properties of the Ising Model * Symmetry Properties of the Ising Model * Existence of Phase Transitions * Spontaneous Symmetry Breaking * Ergodicity Breaking * Fluids * Lattice Gases * Equivalence in Statistical Mechanics * Miscellaneous Remarks * Exercises How Phase Transitions Occur In Practice * Ad Hoc Solution Methods * The Transfer Matrix * Phase Transitions * Thermodynamic Properties * Spatial Correlations * Low Temperature Expansion * Mean Field Theory * Exercises Critical Phenomena in Fluids * Thermodynamics * Two-Phase Coexistence * Vicinity of the Critical Point * Van der Waals Equation * Spatial Correlations * Measurement of Critical Exponents * Exercises Landau Theory * Order Parameters * Common Features of Mean Field Theories * Phenomenological Landau Theory * Continuous Phase Transitions * Inhomogeneous Systems * Correlation Functions * Exercises Fluctuations and the Breakdown of Landau Theory * Breakdown of Microscopic Landau Theory * Breakdown of Phenomenological Landau Theory * The Gaussian Approximation * Critical Exponents * Exercises Scaling in Static, Dynamic and Non-Equilibrium Phenomena * The Static-Scaling Hypothesis * Other Forms of the Scaling Hypothesis * Dynamic Critical Phenomena * Scaling in the Approach to Equilibrium * Summary The Renormalisation Group * Block Spins * Basic Ideas of the Renormalisation Group * Fixed Points * Origin of Scaling * RG in Differential Form * RG for the Two Dimensional Ising Model * First Order Transitions and Non-Critical Properties * RG for the Correlation Function * Crossover Phenomena * Correlations to Scaling * Finite Size Scaling Anomalous Dimensions Far From Equilibrium * Introduction * Similarity Solutions * Anomalous Dimensions in Similarity Solutions * Renormalisation * Perturbation Theory for Barenblatts Equation * Fixed Points * Conclusion Continuous Symmetry * Correlation in the Ordered Phase * Kosterlitz-Thouless Transition Critical Phenomena Near Four Dimensions * Basic Idea of the Epsilon Expansion * RG for the Gaussian Model * RG Beyond the Gaussian Approximation * Feyman Diagrams * The RG Recursion Relations * Conclusion
2,245 citations
"Phase transitions in one-dimensiona..." refers methods in this paper
...A more mathematical way of addressing the question of phase transition in 1d is to use the transfer matrix technique [ 16 ]....
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Book•
01 Jan 1991
TL;DR: In this article, the authors present a model of a Tracer Particle in a Fluid with Hard Core Exclusion (TPE) and a Brownian Particle with hard core exclusion.
Abstract: Scales.- Outline.- I Classical Particles.- 1. Dynamics.- 1.1 Newtonian Dynamics.- 1.2 Boundary Conditions.- 1.3 Dynamics of Infinitely Many Particles.- 2. States of Equilibrium and Local Equilibrium.- 2.1 Equilibrium Measures, Correlation Functions.- 2.2 The Infinite Volume Limit.- 2.3 Local Equilibrium States.- 2.4 Local Stationarity.- 2.5 The Static Continuum Limit.- 3. The Hydrodynamic Limit.- 3.1 Propagation of Local Equilibrium.- 3.2 Hydrodynamic Equations.- 3.3 The Hard Rod Fluid.- 3.4 Steady States.- 4. Low Density Limit: The Boltzmann Equation.- 4.1 Low Density (Boltzmann-Grad) Limit.- 4.2 BBGKY Hierarchy for Hard Spheres and Collision Histories.- 4.3 Convergence of the Scaled Correlation Functions.- 4.4 The Boltzmann Hierarchy.- 4.5 Time Reversal.- 4.6 Law of Large Numbers, Local Poisson.- 4.7 The H-Function.- 4.8 Extensions.- 5. The Vlasov Equation.- 6. The Landau Equation.- 7. Time Correlations and Fluctuations.- 7.1 Fluctuation Fields.- 7.2 The Green-Kubo Formula.- 7.3 Transport for the Hard Rod Fluid.- 7.4 The Fluctuating Boltzmann Equation.- 7.5 The Fluctuating Vlasov Equation.- 8. Dynamics of a Tracer Particle.- 8.1 Brownian Particle in a Fluid.- 8.2 The Stationary Velocity Process.- 8.3 Brownian Motion (Hydrodynamic) Limit.- 8.4 Large Mass Limit.- 8.5 Weak Coupling Limit.- 8.6 Low Density Limit.- 8.7 Mean Field Limit.- 8.8 External Forces and the Einstein Relation.- 8.9 Self-Diffusion.- 8.10 Corrections to Markovian Limits.- 9. The Role of Probability, Irreversibility.- II Stochastic Lattice Gases.- 1. Lattice Gases with Hard Core Exclusion.- 1.1 Dynamics.- 1.2 Stochastic Reversibility.- 1.3 Invariant Measures, Ergodicity, Domains of Attraction.- 1.4 Driven Lattice Gases.- 1.5 Standard Models.- 2. Equilibrium Fluctuations.- 2.1 Density Correlations and Bulk Diffusion.- 2.2 The Green-Kubo Formula.- 2.3 Currents.- 2.4 The Gradient Condition.- 2.5 Linear Response, Conductivity.- 2.6 Steady State Transport.- 2.7 State of Minimal Entropy Production.- 2.8 Bounds on the Conductivity.- 2.9 The Field of Density Fluctuations.- 2.10 Scaling Limit for the Density Fluctuation Field (Proof).- 2.11 Critical Dynamics.- 3. Nonequilibrium Dynamics for Reversible Lattice Gases.- 3.1 The Nonlinear Diffusion Equation.- 3.2 Hydrodynamic Limit (Proof).- 3.3 Low Temperatures.- 3.4 Weakly Driven Lattice Gases.- 3.5 Nonequilibrium Fluctuations.- 3.6 Local Equilibrium States and Minimal Entropy Production.- 3.7 Large Deviations.- 4. Nonequilibrium Dynamics of Driven Lattice Gases.- 4.1 Hyperbolic Equation of Conservation Type.- 4.2 Asymmetric Exclusion Dynamics.- 4.3 Fluctuation Theory.- 5. Beyond the Hydrodynamic Time Scale.- 5.1 Navier-Stokes Correction for Driven Lattice Gases.- 5.2 Local Structure of a Shock.- 5.2.1 Macroscopic Equation with Fluctuations.- 5.2.2 Shock in a Random Frame of Reference.- 5.2.3 Shock in Higher Dimensions.- 6. Tracer Dynamics.- 6.1 Two Component Systems.- 6.2 Tracer Diffusion.- 6.3 Convergence to Brownian Motion.- 6.4 Nearest Neighbor Jumps in One Dimension: The Case of Vanishing Self-Diffusion.- 7. Stochastic Models with a Single Conservation Law Other than Lattice Gases.- 7.1 Lattice Gases Without Hard Core/Zero Range Dynamics.- 7.2 Interacting Brownian Particles.- 7.3 Ginzburg-Landau Dynamics.- 8. Non-Hydrodynamic Limit Dynamics.- 8.1 Kinetic Limit.- 8.2 Mean Field Limit.- References.- List of Mathematical Symbols.
1,946 citations
"Phase transitions in one-dimensiona..." refers background in this paper
...Since then the mathematical achievements have been to prove existence theorems, invariant measures and hydrodynamic limits [44] [51]....
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Book•
13 May 1999TL;DR: In this paper, Driven lattice gases: simulations are used to model the contact process of a lattice gas with a reaction and a contact process with a particle reaction model.
Abstract: Preface 1. Introduction 2. Driven lattice gases: simulations 3. Driven lattice gases: theory 4. Lattice gases with reaction 5. Catalysis models 6. The contact process 7. Models of disorder 8. Conflicting dynamics 9. Particle reaction models Bibliography Index.
1,137 citations