Asymmetric Simple Exclusion Process with Open Boundaries and Quadratic Harnesses

We show that the joint probability generating function of the stationary measure of a finite state asymmetric exclusion process with open boundaries can be expressed in terms of joint moments of Markov processes called quadratic harnesses. We use our representation to prove the large deviations principle for the total number of particles in the system. We use the generator of the Markov process to show how explicit formulas for the average occupancy of a site arise for special choices of parameters. We also give similar representations for limits of stationary measures as the number of sites tends to infinity.

[1]  Infinitesimal generators of q-Meixner processes , 2013, 1309.3528.

[2]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .

[3]  Marcin Świeca,et al.  Zonal polynomials and a multidimensional quantum Bessel process , 2015 .

[4]  W. Schoutens.  Stochastic processes and orthogonal polynomials , 2000 .

[5]  B Derrida,et al.  Exact free energy functional for a driven diffusive open stationary nonequilibrium system. , 2002, Physical review letters.

[6]  q-Gaussian Processes: Non-commutative and Classical Aspects , 1996, funct-an/9604010.

[7]  Eytan Domany,et al.  An exact solution of a one-dimensional asymmetric exclusion model with open boundaries , 1992 .

[8]  P. Szabłowski.  Moments of q-Normal and conditional q-Normal distributions , 2015, 1506.07970.

[9]  Free martingale polynomials , 2001, math/0112194.

[10]  R. A. Blythe,et al.  Exact solution of a partially asymmetric exclusion model using a deformed oscillator algebra , 2000 .

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

[12]  Dennis Stanton,et al.  Moments of AskeyWilson polynomials , 2014 .

[13]  B. Derrida,et al.  Large Deviation Functional of the Weakly Asymmetric Exclusion Process , 2003 .

[14]  Jang Soo Kim,et al.  Moments of Askey-Wilson polynomials , 2012, J. Comb. Theory A.

[15]  Horacio González Duhart Muñoz de Cote.  Large Deviations for Boundary Driven Exclusion Processes , 2015 .

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

[17]  F. Spitzer.  Interaction of Markov processes , 1970 .

[18]  S. Sandow,et al.  Partially asymmetric exclusion process with open boundaries. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[19]  Combinatorics of the asymmetric exclusion process on a semi-infinite lattice , 2012, 1204.1114.

[20]  J. Zimmer,et al.  The Semi-infinite Asymmetric Exclusion Process: Large Deviations via Matrix Products , 2014, 1411.3270.

[21]  W. Bryc,et al.  Quadratic harnesses, q-commutations, and orthogonal martingale polynomials , 2005, math/0504194.

[22]  Matthieu Josuat-Vergès,et al.  Rook placements in Young diagrams and permutation enumeration , 2008, Adv. Appl. Math..

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

[24]  Zentrum Mathematik,et al.  Phase transitions in nonequilibrium stochastic particle systems with local conservation laws , 2004 .

[25]  THE BI-POISSON PROCESS: A QUADRATIC HARNESS , 2005, math/0510208.

[26]  Sylvie Corteel,et al.  A Markov Chain on Permutations which Projects to the PASEP , 2006 .

[27]  Current Fluctuations in the One-Dimensional Symmetric Exclusion Process with Open Boundaries , 2003, cond-mat/0310453.

[28]  E. R. Speer,et al.  Exact Large Deviation Functional of a Stationary Open Driven Diffusive System: The Asymmetric Exclusion Process , 2002 .

[29]  T. Liggett.  Ergodic theorems for the asymmetric simple exclusion process , 1975 .

[30]  B. Derrida,et al.  Free energy functional for nonequilibrium systems: an exactly solvable case. , 2001, Physical review letters.

[31]  Sylvie Corteel,et al.  Formulae for Askey-Wilson moments and enumeration of staircase tableaux , 2010, 1007.5174.

[32]  Matthieu Josuat-Vergès,et al.  Combinatorics of the Three-Parameter PASEP Partition Function , 2009, Electron. J. Comb..

[33]  W. Bryc,et al.  BI-POISSON PROCESS , 2004, math/0404241.

[34]  Tomohiro Sasamoto,et al.  One-dimensional partially asymmetric simple exclusion process with open boundaries: Orthogonal polynomials approach , 1999 .

[35]  Philippe Biane,et al.  Processes with free increments , 1998 .

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

[37]  E. Domany,et al.  Phase transitions in an exactly soluble one-dimensional exclusion process , 1993, cond-mat/9303038.

[38]  PR ] 2 3 N ov 2 01 4 INFINITESIMAL GENERATORS FOR POLYNOMIAL , 2014 .

[39]  M. Wadati,et al.  Correlation Function of Asymmetric Simple Exclusion Process with Open Boundaries , 2004, cond-mat/0410015.

[40]  W. Bryc,et al.  Askey--Wilson polynomials, quadratic harnesses and martingales , 2008, 0812.0657.

[41]  The classical bi-Poisson process: An invertible quadratic harness ☆ , 2005, math/0508383.