Microscopic structure of travelling wave solutions in a class of stochastic interacting particle systems

Abstract

We obtain exact travelling wave solutions for three families of stochastic one-dimensional nonequilibrium lattice models with open boundaries. These solutions describe the diffusive motion and microscopic structure of (i) of shocks in the partially asymmetric exclusion process with open boundaries, (ii) of a lattice Fisher wave in a reaction-diffusion system, and (iii) of a domain wall in non-equilibrium Glauber-Kawasaki dynamics with magnetization current. For each of these systems we define a microscopic shock position and calculate the exact hopping rates of the travelling wave in terms of the transition rates of the microscopic model. In the steady state a reversal of the bias of the travelling wave marks a first-order non-equilibrium phase transition, analogous to the Zel'dovich theory of kinetics of first-order transitions. The stationary distributions of the exclusion process with $n$ shocks can be described in terms of $n$-dimensional representations of matrix product states.

27 pages

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Cited by in corpus (40)

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• Construction of a Coordinate Bethe Ansatz for the asymmetric simple exclusion process with open boundaries
• Eigenvectors of open XXZ and ASEP models for a class of non-diagonal boundary conditions
• Matrix Coordinate Bethe Ansatz: Applications to XXZ and ASEP models
• Slowest relaxation mode of the partially asymmetric exclusion process with open boundaries
• On the Two Species Asymmetric Exclusion Process with Semi-Permeable Boundaries
• Exact shock measures and steady-state selection in a driven diffusive system with two conserved densities
• Shocks in the asymmetric exclusion process with internal degree of freedom
• Matrix Product States of Three Families of One-Dimensional Interacting Particle Systems
• Matrix Product Steady States as Superposition of Product Shock Measures in 1D Driven Systems
• Multiple Shocks in a Driven Diffusive System with Two Species of Particles
• Temporal evolution of product shock measures in TASEP with sublattice-parallel update
• Entropy Production of Nonequilibrium Steady States with Irreversible Transitions
• Reaction fronts in stochastic exclusion models with three-site interactions
• An exactly soluble noisy traveling wave equation appearing in the problem of directed polymers in a random medium
• A reverse duality for the ASEP with open boundaries
• Phase transitions in systems possessing Shock solutions
• Determinant representation for some transition probabilities in the TASEP with second class particles
• Exact Shock Profile for the ASEP with Sublattice-Parallel Update
• The Study of Shocks in Three-States Driven-Diffusive Systems: A Matrix Product Approach
• Construction of a matrix product stationary state from solutions of finite size system
• Ergodicity breaking in one-dimensional reaction-diffusion systems
• Multi shocks in Reaction-diffusion models
• On Effective Stochastic Generators for Conditioned Dynamics at an Atypical Reaction-Diffusion Current
• On matrix product ansatz for Asymmetric Simple Exclusion Process with open boundary in the singular case
• Statistics of sums of correlated variables described by a matrix product ansatz
• On the phase transition in the sublattice TASEP with stochastic blockage
• Particle current fluctuations in a variant of asymmetric Glauber model
• Exact Solution of a Reaction-Diffusion Model with Particle Number Conservation
• Repelling Random Walkers in a Diffusion-Coalescence System
• Finite-dimensional representation of the quadratic algebra of a generalized coagulation-decoagulation model
• Equivalence of a one-dimensional driven-diffusive system and an equilibrium two-dimensional walk model
• Shock in a Branching-Coalescing Model with Reflecting Boundaries
• Weak pinning and long-range anticorrelated motion of phase boundaries in driven diffusive systems
• The Fokker-Planck equation, and stationary densities
• Discontinuous Phase Transition in an Exactly Solvable One-Dimensional Creation-Annihilation System
• Connection between matrix-product states and superposition of Bernoulli shock measures
• Effective dynamics in an asymmetric death-branching process
• Relaxation time in a non-conserving driven-diffusive system with parallel dynamics
• Multi-species reaction-diffusion models admitting shock solutions