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arXiv:0704.0869 · Journal of Physics A Mathematical and Theoretical

Connected operators for the totally asymmetric exclusion process

O. Golinelli, Kirone Mallick

Published Oct 16, 2007• 4 citations
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Abstract

We fully elucidate the structure of the hierarchy of the connected operators that commute with the Markov matrix of the Totally Asymmetric Exclusion Process (TASEP). We prove for the connected operators a combinatorial formula that was conjectured in a previous work. Our derivation is purely algebraic and relies on the algebra generated by the local jump operators involved in the TASEP. Keywords: Non-Equilibrium Statistical Mechanics, ASEP, Exact Results, Algebraic Bethe Ansatz.

DOI
10.1088/1751-8113/40/44/004

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We fully elucidate the structure of the hierarchy of the connected operators that commute with the Markov matrix of the Totally Asymmetric Exclusion Process (TASEP). We prove for the connected operators a combinatorial formula that was conjectured in a previous work. Our derivation is purely algebraic and relies on the algebra generated by the local jump operators involved in the TASEP. Keywords: Non-Equilibrium Statistical Mechanics, ASEP, Exact Results, Algebraic Bethe Ansatz.
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The algebraic structure underlying the totally asymmetric exclusion process is studied by using the Bethe Ansatz technique. From the properties of the algebra generated by the local jump operators, we explicitly construct the hierarchy of operators (called generalized hamiltonians) that commute with the Markov operator. The transfer matrix, which is the generating function of these operators, is shown to represent a discrete Markov process with long-range jumps. We give a general combinatorial formula for the connected hamiltonians obtained by taking the logarithm of the transfer matrix. This formula is proved using a symbolic calculation program for the first ten connected operators.
We consider a multispecies generalization of the totally asymmetric simple exclusion process (TASEP) with the simple hopping rule: for the α and βth-class particles (α < β), the transition αβ → βα occurs with a rate independent from the values α and β. Ferrari and Martin (2007 Ann. Prob. 35 807) obtained the stationary state of this model thanks to a combinatorial algorithm, which was subsequently interpreted as a matrix product representation by Evans et al (2009 J. Stat. Phys. 135 217). This 'matrix ansatz' shows that the stationary state of the multispecies TASEP with N classes of particles ( N -TASEP) can be constructed algebraically by the action of an operator on the ( N − 1)-TASEP stationary state. Besides, Arita et al ( 2009 J. Phys. A. Math Theor. 45 345002 ) analyzed the spectral structure of the Markov matrix: they showed that the set of eigenvalues of the N -TASEP contains those of the ( N − 1)-TASEP and that the various spectral inclusions can be encoded in a hierarchical set-theoretic structure known as the Hasse diagram. Inspired by these works, we define nontrivial operators that allow us to construct eigenvectors of the N -TASEP by lifting the eigenvectors of the ( N − 1)-TASEP. This goal is achieved by generalizing the matrix product representation and the Ferrari–Martin algorithm. In particular, we show that the matrix ansatz is not only a convenient tool to write the stationary state but in fact intertwines Markov matrices of different values of N .
We derive, using the algebraic Bethe ansatz, a generalized matrix product ansatz for the asymmetric exclusion process (ASEP) on a one-dimensional periodic lattice. In this matrix product ansatz, the components of the eigenvectors of the ASEP Markov matrix can be expressed as traces of products of non-commuting operators. We derive the relations between the operators involved and show that they generate a quadratic algebra. Our construction provides explicit finite-dimensional representations for the generators of this algebra.