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arXiv:1005.3780 · Physical Review E

Asymmetric exclusion model with impurities

Matheus J. Lazo, Anderson A. Ferreira

Published May 20, 2010• 6 citations
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Abstract

An integrable asymmetric exclusion process with impurities is formulated. The model displays the full spectrum of the stochastic asymmetric XXZ chain plus new levels. We derive the Bethe equations and calculate the spectral gap for the totally asymmetric diffusion at half filling. While the standard asymmetric exclusion process without impurities belongs to the KPZ universality class with an exponent 3/2, our model has a scaling exponent 5/3.

DOI
10.1103/physreve.81.050104

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An integrable asymmetric exclusion process with impurities is formulated. The model displays the full spectrum of the stochastic asymmetric XXZ chain plus new levels. We derive the Bethe equations and calculate the spectral gap for the totally asymmetric diffusion at half filling. While the standard asymmetric exclusion process without impurities belongs to the KPZ universality class with an exponent 3/2, our model has a scaling exponent 5/3.
We formulate a new integrable asymmetric exclusion process with N − 1 = 0, 1, 2, ... kinds of impurities and with hierarchically ordered dynamics. The model we proposed displays the full spectrum of the simple asymmetric exclusion model plus new levels. The first excited state belongs to these new levels and displays unusual scaling exponents. We conjecture that, while the simple asymmetric exclusion process without impurities belongs to the KPZ universality class with dynamical exponent , our model has a scaling exponent . In order to check the conjecture, we solve numerically the Bethe equation with N = 3 and N = 4 for the totally asymmetric diffusion and found the dynamical exponents and in these cases.
Abstract. We derive the Bethe ansatz equations describing the complete spectrum of the transition matrix of the partially asymmetric exclusion process with the most general open boundary conditions. By analysing these equations in detail for the cases of totally asymmetric and symmetric diffusion, we calculate the finite-size scaling of the spectral gap, which characterizes the approach to stationarity at large times. In the totally asymmetric case we observe boundary induced crossovers between massive, diffusive and KPZ scaling regimes. We further study higher excitations, and demonstrate the absence of oscillatory behaviour at large times on the “coexistence line”, which separates the massive low and high density phases. In the maximum current phase, oscillations are present on the KPZ scale t ∝ L−3/2. While independent of the boundary parameters, the spectral gap as well as the oscillation frequency in the maximum current phase have different values compared to the totally asymmetric exclusion process with periodic boundary conditions. We discuss a possible interpretation of our results in terms of an effective domain wall theory.
We derive the Bethe ansatz equations describing the complete spectrum of the transition matrix of the partially asymmetric exclusion process with the most general open boundary conditions. By analysing these equations in detail for the cases of totally asymmetric and symmetric diffusion, we calculate the finite-size scaling of the spectral gap, which characterizes the approach to stationarity at large times. In the totally asymmetric case we observe boundary induced crossovers between massive, diffusive and KPZ (Kardar–Parisi–Zhang) scaling regimes. We further study higher excitations, and demonstrate the absence of oscillatory behaviour at large times on the ‘coexistence line’, which separates the massive low and high density phases. In the maximum current phase, oscillations are present on the KPZ scale . While independent of the boundary parameters, the spectral gap as well as the oscillation frequency in the maximum current phase have different values compared to the totally asymmetric exclusion process with periodic boundary conditions. We discuss a possible interpretation of our results in terms of an effective domain wall theory.
A one-dimensional asymmetric diffusion model on a periodic chain is proposed. The model is defined in terms of the master equation. In contrast with the asymmetric simple exclusion process, particles are not subject to the exclusion interaction: each lattice site can accommodate more than one particle. The model is solved by the Bethe ansatz method and the resulting Bethe equation is analyzed in the thermodynamic limit. The finite size correction of the energy gap is calculated to be ${O(L}^{\ensuremath{-}3/2}),$ where L denotes the length of the chain.