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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

One-dimensional asymmetric diffusion model without exclusion

Tomohiro Sasamoto, Miki Wadati

Published Oct 1, 1998• 35 citations
Publisher

Abstract

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.

DOI
10.1103/physreve.58.4181

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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),{O(L}^{\ensuremath{-}3/2}),O(L\ensuremath−3/2), where L denotes the length of the chain.
We define a family of asymmetric processes for particles on a one-dimensional lattice, depending on a continuous parameter $\ensuremath{\lambda}\ensuremath{\in}[0,1]$, interpolating between the completely asymmetric processes (for $\ensuremath{\lambda}=1)$ and the n=1n=1n=1 drop-push models (for $\ensuremath{\lambda}=0)$. For arbitrary $\ensuremath{\lambda}$, the model describes an exclusion process, in which a particle pushes its right neighboring particles to the right, with rates depending on the number of these particles. Using the Bethe ansatz, we obtain the exact solution of the master equation.
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.
A generalization of the simple exclusion asymmetric model is introduced. In this model an arbitrary mixture of molecules with distinct sizes s=0,1,2, ..., in units of lattice space, diffuses asymmetrically on the lattice. A related surface growth model is also presented. Variations of the distribution of the molecules sizes may change the excluded volume almost continuously. We solve the model exactly through the Bethe ansatz and the dynamical critical exponent z is calculated from the finite-size corrections of the mass gap of the related quantum chain. Our results show that for an arbitrary distribution of molecules, the dynamical critical behavior is on the Kardar-Parizi-Zhang universality.
The exact solution of the asymmetric exclusion problem with first- and scond-class particles is presented. In this model the particles (size 1) of both classes are located at lattice points, and diffuse with equal asymmetric rates, but particles in the first class do not distinguish those in the second class from holes (empty sites). We generalize and solve exactly this model by considering molecules in the first and second class with sizes s1 and s2 (s1, s2 = 0, 1, 2, ...), in units of lattice spacing, respectively. The solution is derived by a Bethe ansatz of nested type. We give a simple pedagogical presentation of the Bethe ansatz solution of the problem which can easily be followed by a reader not specialized in exactly integrable models.
The exact solution of the asymmetric exclusion problem with N distinct classes of particles (c = 1, 2, ..., N), with hierarchical order is presented. In this model the particles (size 1) are located at lattice points, and diffuse with equal asymmetric rates, but particles in a class c do not distinguish those in the classes c' > c from holes (empty sites). We generalize and solve exactly this model by considering the molecules in each distinct class c = 1, 2, ..., N with sizes s c (s c = 0, 1, 2, ...), in units of the lattice spacing. The solution is derived via a Bethe ansatz of nested type.