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arXiv:1411.7954 · Journal of Physics A Mathematical and Theoretical

Algebraic Bethe ansatz for the totally asymmetric simple exclusion process with boundaries

Nicolas Crampé

Published Jan 28, 2015• 40 citations
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Abstract

We study the one-dimensional totally asymmetric simple exclusion process in contact with two reservoirs including also a fugacity at one boundary. The eigenvectors and the eigenvalues of the corresponding Markov matrix are computed using the modified algebraic Bethe ansatz, a method introduced recently to study the spin chain with non-diagonal boundaries. We provide in this case a proof of this method.

DOI
10.1088/1751-8113/48/8/08ft01

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We study the one-dimensional totally asymmetric simple exclusion process in contact with two reservoirs including also a fugacity at one boundary. The eigenvectors and the eigenvalues of the corresponding Markov matrix are computed using the modified algebraic Bethe ansatz, a method introduced recently to study the spin chain with non-diagonal boundaries. We provide in this case a proof of this method.
We study the one-dimensional asymmetric simple exclusion process (ASEP) with generic open boundaries (including current-counting deformation), and obtain the exact solutions of this ASEP via the off-diagonal Bethe ansatz method. In particular, numerical results for the small size asymmetric simple exclusion process indicate that the spectrum obtained by the Bethe ansatz equations is complete. Moreover, we present the eigenvalue of the totally asymmetric exclusion process and the corresponding Bethe ansatz equations.
The asymmetric simple exclusion process with open boundaries, which is a very simple model of out-of-equilibrium statistical physics, is known to be integrable. In particular, its spectrum can be described in terms of Bethe roots. The large deviation function of the current can be obtained as well by diagonalizing a modified transition matrix, which is still integrable: the spectrum of this new matrix can also be described in terms of Bethe roots for special values of the parameters. However, due to the algebraic framework used to write the Bethe equations in previous works, the nature of the excitations and the full structure of the eigenvectors remained unknown. This paper explains why the eigenvectors of the modified transition matrix are physically relevant, gives an explicit expression for the eigenvectors and applies it to the study of atypical currents. It also shows how the coordinate Bethe ansatz developed for the excitations leads to a simple derivation of the Bethe equations and of the validity conditions of this ansatz. All the results obtained by de Gier and Essler are recovered and the approach gives a physical interpretation of the exceptional points. The overlap of this approach with other tools such as the matrix ansatz is also discussed. The method that is presented here may be not specific to the asymmetric exclusion process and may be applied to other models with open boundaries to find similar exceptional points.
We study the long time asymptotics of the relaxation dynamics of the totally asymmetric simple exclusion process on a ring.Evaluating the asymptotic amplitudes of the local currents by the algebraic Bethe ansatz method, we find that the relaxation times starting from the step and alternating initial conditions are governed by different eigenvalues of the Markov matrix.In both cases, the scaling exponents of the leading asymptotic amplitudes with respect to the total number of sites are found to be -1.We also study the asymptotics of correlation functions such as the emptiness formation probability.
In non-equilibrium statistical mechanics, the Asymmetric Simple Exclusion Process (ASEP) serves as a paradigmatic example. We investigate the spectral characteristics of the ASEP, focusing on the spectral boundary of its generator matrix. We examine finite ASEP chains of length LLL, under periodic (pbc) and open boundary conditions (obc). Notably, the spectral boundary exhibits LLL spikes for pbc and L+1L+1L+1 spikes for obc. Treating the ASEP generator as an interacting non-Hermitian fermionic model, we extend the model to have tunable interaction. In the non-interacting case, the analytically computed many-body spectrum shows a spectral boundary with prominent spikes. For pbc, we use the coordinate Bethe ansatz to interpolate between the noninteracting case to the ASEP limit, and show that these spikes stem from clustering of Bethe roots. The robustness of the spikes in the spectral boundary is demonstrated by linking the ASEP generator to random matrices with trace correlations or, equivalently, random graphs with distinct cycle structures, both displaying similar spiked spectral boundaries.
In nonequilibrium statistical mechanics, the asymmetric simple exclusion process (ASEP) serves as a paradigmatic example. We investigate the spectral characteristics of the ASEP, focusing on the spectral boundary of its generator matrix. We examine finite ASEP chains of length L, under periodic boundary conditions (PBCs) and open boundary conditions (OBCs). Notably, the spectral boundary exhibits L spikes for PBCs and L+1 spikes for OBCs. Treating the ASEP generator as an interacting non-Hermitian fermionic model, we extend the model to have tunable interaction. In the noninteracting case, the analytically computed many-body spectrum shows a spectral boundary with prominent spikes. For PBCs, we use the coordinate Bethe ansatz to interpolate between the noninteracting case to the ASEP limit and show that these spikes stem from clustering of Bethe roots. The robustness of the spikes in the spectral boundary is demonstrated by linking the ASEP generator to random matrices with trace correlations or, equivalently, random graphs with distinct cycle structures, both displaying similar spiked spectral boundaries.