Shock profiles for the asymmetric simple exclusion process in one dimension View Full Text


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Article Info

DATE

1997-10

AUTHORS

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

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. More... »

PAGES

135-167

References to SciGraph publications

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  • 1987-04. Hydrodynamic equations for attractive particle systems on ℤ in JOURNAL OF STATISTICAL PHYSICS
  • 1987-10. Cellular automata and statistical mechanical models in JOURNAL OF STATISTICAL PHYSICS
  • 1994-06. Shock fluctuations in the asymmetric simple exclusion process in PROBABILITY THEORY AND RELATED FIELDS
  • 1988-06. Shocks in the asymmetric exclusion process in PROBABILITY THEORY AND RELATED FIELDS
  • 1991-10. Hydrodynamic limit for attractive particle systems on 417-1417-1417-1 in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1992-03. Shock fluctuations in asymmetric simple exclusion in PROBABILITY THEORY AND RELATED FIELDS
  • 1990. Mathematical Methods in Kinetic Theory in NONE
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  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/bf02770758

    DOI

    http://dx.doi.org/10.1007/bf02770758

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1021788079


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        "description": "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\u2019 equation; the latter has shock solutions with a discontinuous jump from left density \u03c1- to right density \u03c1+, \u03c1-< \u03c1 +, which travel with velocity (2p\u22121 )(1\u2212\u03c1+\u2212p\u2212). 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\u00b1 at an exponential rate asn\u2192 \u00b1\u221e, witha characteristic length which becomes independent ofp when. For a special value of the asymmetry, given byp/(1\u2212p)=p+(1\u2212p\u2212)/p\u2212(1\u2212p+), the measure is Bernoulli, with density\u03c1\u2212 on the left andp+ on the right. In the weakly asymmetric limit, 2p\u22121 \u2192 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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    44 schema:description 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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