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arXiv:1205.0471 · Journal of Statistical Mechanics Theory and Experiment

Asymmetric exclusion model with several kinds of impurities

Matheus J. Lazo, Anderson A. Ferreira

Published May 16, 2012• 2 citations
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Abstract

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.

DOI
10.1088/1742-5468/2012/05/p05017

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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.
The spectrum of Hamiltonian (Markov matrix) of a multi-species asymmetric simple exclusion process on a ring is studied. The dynamical exponent concerning the relaxation time is found to coincide with the one-species case. It implies that the system belongs to the Kardar-Parisi-Zhang or Edwards-Wilkinson universality classes depending on whether the hopping rate is asymmetric or symmetric, respectively. Our derivation exploits a poset structure of the particle sectors, leading to a new spectral duality and inclusion relations. The Bethe ansatz integrability is also demonstrated.
The spectrum of the Hamiltonian (Markov matrix) of a multi-species asymmetric simple exclusion process on a ring is studied. The dynamical exponent concerning the relaxation time is found to coincide with the one-species case. It implies that the system belongs to the Kardar–Parisi–Zhang or Edwards–Wilkinson universality classes depending on whether the hopping rate is asymmetric or symmetric, respectively. Our derivation exploits a poset structure of the particle sectors, leading to a new spectral duality and inclusion relations. The Bethe ansatz integrability is also demonstrated.
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 relaxation dynamics of the one-dimensional totally asymmetric simple exclusion process on a ring is considered in the case of step initial condition. Analyzing the time evolution of the local particle densities and currents by the Bethe ansatz method, we examine their full relaxation dynamics. As a result, we observe peculiar behaviors, such as the emergence of a ripple in the density profile and the existence of the excessive particle currents. Moreover, by making a finite-size scaling analysis of the asymptotic amplitudes of the local densities and currents, we find the scaling exponents with respect to the total number of sites to be -3/2 and -1, respectively.
The applicability of the concepts of finite-size scaling and universality to nonequilibrium phase transitions is considered in the framework of the one-dimensional totally asymmetric simple-exclusion process with open boundaries. In the thermodynamic limit there are boundary-induced transitions both of the first and second order between steady-state phases of the model. We derive finite-size scaling expressions for the current near the continuous phase transition and for the local density near the first-order transition under different stochastic dynamics and compare them to establish the existence of universal functions. Next we study numerically the finite-size behavior of the Lee-Yang zeros of the normalization factor for the different steady-state probabilities.