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Journal of Physics Condensed Matter

Asymmetric one-dimensional exclusion processes: a two-parameter discrete-time exactly solvable model

Farinaz Roshani, Mohammad Khorrami

Published Mar 25, 2005• 9 citations
Publisher

Abstract

A two-parameter family of discrete-time asymmetric exclusion processes on a one-dimensional lattice is considered. Among these processes are the asymmetric simple exclusion model and the drop–push model. Using the Bethe ansatz, the exact solution to the master equation and from that the drift rates of the two-particle sector are obtained.

DOI
10.1088/0953-8984/17/14/016

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A two-parameter family of discrete-time asymmetric exclusion processes on a one-dimensional lattice is considered. Among these processes are the asymmetric simple exclusion model and the drop–push model. Using the Bethe ansatz, the exact solution to the master equation and from that the drift rates of the two-particle sector are obtained.
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.
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.
We consider the totally asymmetric exclusion process in discrete time with generalized updating rules. We introduce a control parameter into the interaction between particles. Two particular values of the parameter correspond to known parallel and sequential updates. In the whole range of its values the interaction varies from repulsive to attractive. In the latter case the particle flow demonstrates an apparent jamming tendency not typical for the known updates. We solve the master equation for N particles on the infinite lattice by the Bethe ansatz. The non-stationary solution for arbitrary initial conditions is obtained in a closed determinant form.
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