Shock profiles for the asymmetric simple exclusion process in one dimension
Shock profiles for the asymmetric simple exclusion process in one dimension
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DOI：
10.1007/bf02770758
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发表时间：
1997-07
影响因子：
1.6
通讯作者：
B. Derrida;J. Lebowitz;E. Speer;E. Speer
中科院分区：
文献类型：
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作者：
B. Derrida;J. Lebowitz;E. Speer;E. Speer
一维晶格上的非对称简单排斥过程(ASEP)是一个粒子系统，它们分别以速率p和1-p(herep>；1/2)跳跃到其右侧和左侧的相邻空位。该系统在适当的宏观时空尺度上用无粘性Burgers型方程描述，后者具有从左密度ρ-到右密度ρ+，ρ-<；ρ+的不连续跃迁激波解，并随速度(2p−1)(1−ρ+−p−)传播。在微观系统中，我们可以通过引入第二类粒子来跟踪激波的位置，第二类粒子被激波吸引并与激波一起运动。在本文中，我们从这样一个粒子出发，得到了ASEP中激波解的时不变度量。从该粒子测得的晶格格位的平均密度以指数速率n→±∞接近p±，其特征长度与p时无关。对于不对称性的一个特定值，给定byp/(1−p)=p+(1−p−)/p−(1−p+)，度量是伯努利，密度ρ−在左边，密度在右边。在弱不对称极限2p−1→0时，激波的微观宽度发散为(2p+1)-1。因此，定常测量实质上是伯努利测量的叠加，对应于由粘性Burgers方程描述的密度分布与第二类粒子位置的明确分布的卷积。
The asymmetric simple exclusion process (ASEP) on a one-dimensional lattice is a system of particles which jump at ratespand 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 lengthwhich becomes independent ofpwhen. 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.