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The Bose Gas and Asymmetric Simple Exclusion Process on the Half-Line

Published Dec 14, 2013• 0 citations
OpenAlex

Abstract

one-dimensional bosons interacting via a delta-function potential with particles confined to the positive half-line; and (2) the transition probability for the one-dimensional asymmetric simple exclusion process (ASEP) with particles confined to the nonnegative integers. These are both for systems with a finite number of particles. The formulas are analogous to ones obtained earlier for the Bose gas and ASEP on the line and integers, respectively. We use coordinate Bethe Ansatz appropriately modified to account for confinement of the particles to the half-line. As in the earlier work, the proof for the ASEP is less straightforward than for the Bose gas.

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one-dimensional bosons interacting via a delta-function potential with particles confined to the positive half-line; and (2) the transition probability for the one-dimensional asymmetric simple exclusion process (ASEP) with particles confined to the nonnegative integers. These are both for systems with a finite number of particles. The formulas are analogous to ones obtained earlier for the Bose gas and ASEP on the line and integers, respectively. We use coordinate Bethe Ansatz appropriately modified to account for confinement of the particles to the half-line. As in the earlier work, the proof for the ASEP is less straightforward than for the Bose gas.
Abstract The one-dimensional asymmetric simple exclusion process (ASEP), where N hard-core particles hop forward with rate 1 and backward with rate q < 1 , is considered on a periodic lattice of L site. Using KPZ universality and previous results for the totally asymmetric model q = 0 , precise conjectures are formulated for asymptotics at finite density ρ = N / L of ASEP eigenstates close to the stationary state. The conjectures are checked with high precision using extrapolation methods on finite size Bethe ansatz numerics. For weak asymmetry 1 − q ∼ 1 / L , double extrapolation combined with an integer relation algorithm gives an exact expression for the spectral gap up to 10th order in the asymmetry.
The one-dimensional totally asymmetric simple exclusion process, a Markov process describing classical hard-core particles hopping in the same direction, is considered on a periodic lattice of L sites. The relaxation to the nonequilibrium steady state, which occurs on the time scale t∼L^{3/2} for large L, is studied for the half-filled system with N=L/2 particles. Using large L asymptotics of Bethe ansatz formulas for the eigenstates, exact expressions depending explicitly on the rescaled time t/L^{3/2} are obtained for the average and two-point function of the local density, and for the current fluctuations for simple (stationary, flat and step) initial conditions, relating previous results for the infinite system to stationary large deviations. The final formulas have a nice interpretation in terms of a functional integral with the action of a scalar field in a linear potential.
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.
We consider the spectrum of the totally asymmetric simple exclusion process on a periodic lattice of L sites. The first eigenstates have an eigenvalue with real part scaling as for large L with finite density of particles. Bethe ansatz shows that these eigenstates are characterized by four finite sets of positive half-integers, or equivalently by two integer partitions. Each corresponding eigenvalue is found to be equal to the value at its saddle point of a function indexed by the four sets. Our derivation of the large L asymptotics relies on a version of the Euler–Maclaurin formula with square root singularities at both ends of the summation range.