Matrix product steady states as superposition of product shock measures in 1D driven systems

2007/07/31 by F H Jafarpour, S R Masharian · 25 citations
Mathematics · Physics and Astronomy · #Domain (mathematical analysis) #Lattice (music) #Matrix multiplication #Product (mathematics) #Quadratic equation #Quantum many-body systems #Representation (politics) #Steady state (chemistry) #Stochastic processes and statistical mechanics #Superposition principle #Theoretical and Computational Physics #cond-mat.stat-mech

paper · pdf · doi:10.1088/1742-5468/2007/10/p10013

published in Journal of Statistical Mechanics Theory and Experiment 2007(10), P10013 (Institute of Physics) · 12 pages, 1 figure

arxiv created 2007/10/24 · openalex publication_date 2007/10/24 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

It is known that exact traveling wave solutions exist for families of ( n +1)-states stochastic one-dimensional non-equilibrium lattice models with open boundaries provided that some constraints on the reaction rates are fulfilled. These solutions describe the diffusive motion of a product shock or a domain wall with the dynamics of a simple biased random walker. The steady state of these systems can be written in terms of linear superposition of such shocks or domain walls. These steady states can also be expressed in a matrix product form. We show that, in this case, the associated quadratic algebra of the system always has a two-dimensional representation with a generic structure. A couple of examples for the n = 1 and 2 cases are presented.

Citations

• Finite-dimensional representations of the quadratic algebra: Applications to the exclusion process
• Microscopic structure of travelling wave solutions in a class of stochastic interacting particle systems
• Matrix product states of three families of one-dimensional interacting particle systems
• Shocks in the asymmetric exclusion process with internal degree of freedom