Shock profiles for the asymmetric simple exclusion process in one dimension

B. Derrida, J. L. Lebowitz, E. R. Speer

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The asymmetric simple exclusion process (ASEP) on a one-dimensional lattice is a system of particles which jump at rates p and 1 - p (here p > 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+ - ρ-). 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 site n, measured from this particle, approaches ρ± at an exponential rate as n → ± ∞, with a characteristic length which becomes independent of p when p/(1 - p) > √ρ+(1 - ρ-)/ρ-(1 - ρ+). For a special value of the asymmetry, given by p/(1 - p) = ρ+(1 - ρ-)/ ρ-(1 - ρ+), the measure is Bernoulli, with density ρ- on the left and ρ+ 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.

Original languageEnglish (US)
Pages (from-to)135-167
Number of pages33
JournalJournal of Statistical Physics
Issue number1-2
StatePublished - Oct 1997

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


  • Asymmetric simple exclusion process
  • Burgers equation
  • Second class particles
  • Shock profiles
  • Weakly asymmetric limit


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