Shock profiles for the asymmetric simple exclusion process in one dimension |
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Authors: | B Derrida J L Lebowitz E R Speer |
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Institution: | (1) Laboratoire de Physique Statistique, Ecole Normale Supérieure, 75005 Paris, France;(2) Institut des Hautes Etudes Scientifiques, F-91440 Bures-sur-Yvette, France;(3) Department of Mathematics, Rutgers University, 08903 New Brunswick, New Jersey |
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Abstract: | The asymmetric simple exclusion process (ASEP) on a one-dimensional lattice is a system of particles which jump at ratesp and 1-p (herep > 1/2) to adjacent empty sites on their right and left respectively. The system is described on suitable macroscopic spatial
and temporal scales by the inviscid Burgers’ equation; the latter has shock solutions with a discontinuous jump from left
density ρ- to right density ρ+, ρ-< ρ +, which travel with velocity (2p−1 )(1−ρ+−p
−). In the microscopic system we may track the shock position by introducing a second class particle, which is attracted to
and travels with the shock. In this paper we obtain the time-invariant measure for this shock solution in the ASEP, as seen
from such a particle. The mean density at lattice siten, measured from this particle, approachesp
± at an exponential rate asn→ ±∞, witha characteristic length which becomes independent ofp when
. For a special value of the asymmetry, given byp/(1−p)=p
+(1−p
−)/p
−(1−p
+), the measure is Bernoulli, with densityρ
− on the left andp
+ on the right. In the weakly asymmetric limit, 2p−1 → 0, the microscopic width of the shock diverges as (2p+1)-1. The stationary measure is then essentially a superposition of Bernoulli measures, corresponding to a convolution of a density
profile described by the viscous Burgers equation with a well-defined distribution for the location of the second class particle. |
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Keywords: | Asymmetric simple exclusion process weakly asymmetric limit shock profiles second class particles Burgers equation |
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