## Abstract

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 language | English (US) |
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Pages (from-to) | 135-167 |

Number of pages | 33 |

Journal | Journal of Statistical Physics |

Volume | 89 |

Issue number | 1-2 |

DOIs | |

State | Published - Oct 1997 |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

## Keywords

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