Matrix Ansatz for the Fluctuations of the Current in the ASEP with Open Boundaries

Abstract

The Asymmetric Simple Exclusion Process is one of the most extensively studied models in non-equilibrium statistical mechanics. The macroscopic particle current produced in its steady state is directly related to the breaking of detailed balance, and is therefore a physical quantity of particular interest. In this paper, we build a matrix product Ansatz which allows to access the exact statistics of the fluctuations of that current for finite sizes, as well as the probabilities of configurations conditioned on the mean current. We also show how this Ansatz can be used for the periodic ASEP, and how it translates in the language of the XXZ spin chain.

20 pages

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• Bethe Ansatz and Q-operator for the open ASEP
• Spectrum of the totally asymmetric simple exclusion process on a periodic lattice -- first excited states
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• Algebraic Bethe ansatz for the totally asymmetric simple exclusion process with boundaries
• Large deviations in the symmetric simple exclusion process with slow boundaries
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• Theoretical approaches to the steady-state statistical physics of interacting dissipative units
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• KPZ fluctuations in finite volume
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• Matrix product construction for Koornwinder polynomials and fluctuations of the current in the open ASEP
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• General limit distributions for sums of random variables with a matrix product representation
• Large deviations for correlated random variables described by a matrix product ansatz
• Rare-event trajectory ensemble approach to study dynamical phase transitions in the zero temperature Glauber model
• Approach to stationarity for the KPZ fixed point with boundaries
• Mapping current and activity fluctuations in exclusion processes: consequences and open questions