Open two-species exclusion processes with integrable boundaries

We give a complete classification of integrable Markovian boundary conditions for the asymmetric simple exclusion process with two species (or classes) of particles. Some of these boundary conditions lead to non-vanishing particle currents for each species. We explain how the stationary state of all these models can be expressed in a matrix product form, starting from two key components, the Zamolodchikov-Faddeev and Ghoshal-Zamolodchikov relations. This statement is illustrated by studying in detail a specific example, for which the matrix Ansatz (involving 9 generators) is explicitly constructed and physical observables (such as currents, densities) calculated.

[2]  Evans,et al.  Spontaneous symmetry breaking in a one dimensional driven diffusive system. , 1995, Physical review letters.

[3]  B. Derrida.  AN EXACTLY SOLUBLE NON-EQUILIBRIUM SYSTEM : THE ASYMMETRIC SIMPLE EXCLUSION PROCESS , 1998 .

[4]  How algebraic Bethe ansatz works for integrable model , 1996, hep-th/9605187.

[5]  N. Rajewsky,et al.  Exact solution of an exclusion process with three classes of particles and vacancies , 1999 .

[6]  Luigi Cantini,et al.  Algebraic Bethe ansatz for the two species ASEP with different hopping rates , 2007, 0710.4083.

[7]  Masaru Uchiyama.  Two-species asymmetric simple exclusion process with open boundaries , 2008 .

[8]  Y. Kohayakawa,et al.  Invariant measures for a two-species asymmetric process , 1994 .

[9]  S. Belliard,et al.  Heisenberg XXX Model with General Boundaries: Eigenvectors from Algebraic Bethe Ansatz , 2013, 1309.6165.

[10]  T. Liggett.  Stochastic models of interacting systems , 1997 .

[11]  Bernard Derrida,et al.  Nonequilibrium Statistical Mechanics in One Dimension: The asymmetric exclusion model: exact results through a matrix approach , 1997 .

[12]  On Some Classes of Open Two-Species Exclusion Processes , 2010, 1008.4721.

[13]  B. Derrida.  Microscopic versus macroscopic approaches to non-equilibrium systems , 2010, 1012.1136.

[14]  C. Landim,et al.  Macroscopic Fluctuation Theory for Stationary Non-Equilibrium States , 2001, cond-mat/0108040.

[15]  K. Mallick,et al.  Shocks in the asymmetry exclusion model with an impurity , 1996 .

[16]  M. R. Evans,et al.  Bethe ansatz solution for a defect particle in the asymmetric exclusion process , 1999 .

[17]  E. Ragoucy,et al.  Integrable approach to simple exclusion processes with boundaries. Review and progress , 2014, 1408.5357.

[18]  Correlations in the multispecies TASEP and a conjecture by Lam , 2014, 1404.6679.

[19]  Diffusion algebras , 2001, cond-mat/0103603.

[20]  N. Crampé.  Algebraic Bethe ansatz for the totally asymmetric simple exclusion process with boundaries , 2014, 1411.7954.

[21]  B. Derrida,et al.  Exact solution of the totally asymmetric simple exclusion process: Shock profiles , 1993 .

[22]  Gilles Schaeffer,et al.  A combinatorial approach to jumping particles , 2005, J. Comb. Theory, Ser. A.

[23]  The Two Species Totally Asymmetric Simple Exclusion Process , 1994 .

[24]  The complex story of simple exclusion , 1996 .

[25]  Omer Angel.  The stationary measure of a 2-type totally asymmetric exclusion process , 2006, J. Comb. Theory, Ser. A.

[26]  P. Ferrari,et al.  MICROSCOPIC STRUCTURE OF TRAVELLING WAVES IN THE ASYMMETRIC SIMPLE EXCLUSION PROCESS , 1991 .

[27]  James B. Martin,et al.  Stationary distributions of multi-type totally asymmetric exclusion processes , 2005, math/0501291.

[28]  B. Aneva,et al.  Deformed coherent and squeezed states of multiparticle processes , 2003 .

[29]  C. Landim,et al.  Macroscopic fluctuation theory , 2014, 1404.6466.

[30]  S. Redner,et al.  A Kinetic View of Statistical Physics , 2010 .

[31]  Svante Linusson,et al.  Continuous multi-line queues and TASEP , 2015, 1501.04417.

[32]  THE KARDAR-PARISI-ZHANG,et al.  The Kardar-Parisi-Zhang Equation and Universality Class , 2011 .

[33]  Nobuyuki Ikeda,et al.  Ito's Stochastic Calculus and Probability Theory , 1996 .

[34]  Kazumitsu Sakai,et al.  Spectrum of a multi-species asymmetric simple exclusion process on a ring , 2009, 0904.1481.

[35]  E. Sklyanin.  Boundary conditions for integrable quantum systems , 1988 .

[36]  T. Kriecherbauer,et al.  A pedestrian's view on interacting particle systems, KPZ universality and random matrices , 2008, 0803.2796.

[37]  B. Derrida,et al.  Exact solution of a 1d asymmetric exclusion model using a matrix formulation , 1993 .

[38]  H. Stanley,et al.  Phase Transitions and Critical Phenomena , 2008 .

[39]  H. Spohn.  Large Scale Dynamics of Interacting Particles , 1991 .

[40]  M. Wadati,et al.  Stationary State of Integrable Systems in Matrix Product Form. , 1997 .

[41]  Herbert Spohn,et al.  Nonequilibrium steady states of stochastic lattice gas models of fast ionic conductors , 1984 .

[42]  S. Belliard.  Modified algebraic Bethe ansatz for XXZ chain on the segment - I: Triangular cases , 2014, 1408.4840.

[43]  E. Ragoucy,et al.  Algebraic Bethe Ansatz for Open XXX Model with Triangular Boundary Matrices , 2012, 1209.4269.

[44]  C. Arita.  Exact Analysis of Two-Species Totally Asymmetric Exclusion Process with Open Boundary Condition , 2006 .

[45]  L. D. Faddeev,et al.  Quantum Completely Integrable Models in Field Theory , 1995 .

[46]  R. A. Blythe,et al.  Nonequilibrium steady states of matrix-product form: a solver's guide , 2007, 0706.1678.

[47]  T. Chou,et al.  Non-equilibrium statistical mechanics: from a paradigmatic model to biological transport , 2011, 1110.1783.

[48]  Chikashi Arita,et al.  Phase transitions in the two-species totally asymmetric exclusion process with open boundaries , 2006 .

[49]  Beate Schmittmann,et al.  Statistical mechanics of driven diffusive systems , 1995 .

[50]  K. Mallick,et al.  The matrix product solution of the multispecies partially asymmetric exclusion process , 2008, 0812.3293.

[51]  V. Karimipour.  Multispecies asymmetric simple exclusion process and its relation to traffic flow , 1999 .

[52]  P. Ferrari.  Microscopic shocks in one dimensional driven systems , 1991 .

[53]  G. Schütz.  1 – Exactly Solvable Models for Many-Body Systems Far from Equilibrium , 2001 .

[54]  Reaction - diffusion processes, critical dynamics and quantum chains , 1993, hep-th/9302112.

[55]  Vladimir Privman,et al.  Nonequilibrium Statistical Mechanics in One Dimension: Experimental Results , 1997 .

[56]  C. Arita.  Remarks on the multi-species exclusion process with reflective boundaries , 2011, 1112.5585.

[57]  F. Essler,et al.  Exact spectral gaps of the asymmetric exclusion process with open boundaries , 2006, cond-mat/0609645.

[58]  Exact solution of asymmetric diffusion with second-class particles of arbitrary size , 2000, cond-mat/0007390.

[59]  Krug,et al.  Boundary-induced phase transitions in driven diffusive systems. , 1991, Physical review letters.

[60]  E. R. Speer,et al.  Shock profiles for the asymmetric simple exclusion process in one dimension , 1997 .

[61]  Alexander B. Zamolodchikov,et al.  Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models , 1979 .

[62]  B. Derrida,et al.  of Statistical Mechanics : Theory and Experiment Non-equilibrium steady states : fluctuations and large deviations of the density and of the current , 2007 .

[63]  Fabian H.L. Essler,et al.  Representations of the quadratic algebra and partially asymmetric diffusion with open boundaries , 1995 .

[64]  P. Ferrari,et al.  Matrix Representation of the Stationary Measure for the Multispecies TASEP , 2008, 0807.0327.

[65]  Arvind Ayyer,et al.  On the Two Species Asymmetric Exclusion Process with Semi-Permeable Boundaries , 2008, 0807.2423.

[66]  A. Borodin,et al.  THE KARDAR-PARISI-ZHANG EQUATION AND UNIVERSALITY CLASS , 2013 .

[67]  H. Thorisson.  Coupling, stationarity, and regeneration , 2000 .

[68]  Boundary S matrix and boundary state in two-dimensional integrable quantum field theory , 1993, hep-th/9306002.