The Asymmetric Simple Exclusion Process with Multiple Shocks

Abstract

We consider the one dimensional totally asymmetric simple exclusion process with initial product distribution with densities $0\le {\rho }_0<{\rho }_1<...<{\rho }_n\le 1$ in $(-\infty,c_1\ve^{-1})$, $[c_1\ve^{-1},c_2ε^{-1}),...,[c_n \ve^{-1}, + \infty)$, respectively. The initial distribution has shocks (discontinuities) at ${\epsilon }^{-1}{c}_k$, k=1,...,n and we assume that in the corresponding macroscopic Burgers equation the n shocks meet in $r^{\ast }$ at time $t^{\ast }$. The microscopic position of the shocks is represented by second class particles whose distribution in the scale ${\epsilon }^{-1/2}$ is shown to converge to a function of n independent Gaussian random variables representing the fluctuations of these particles ``just before the meeting''. We show that the density field at time $\ve^{-1}t^*$, in the scale $\ve^{-1/2}$ and as seen from $\ve^{-1}r^*$ converges weakly to a random measure with piecewise constant density as $\ve \to 0$; the points of discontinuity depend on these limiting Gaussian variables. As a corollary we show that, as $\epsilon \to 0$, the distribution of the process at site $ε^{-1}r^*+\ve^{-1/2}a$ at time ${\epsilon }^{-1}{t}^{\ast }$ tends to a non trivial convex combination of the product measures with densities ${\rho }_k$, the weights of the combination being explicitly computable.

20 pages, one figure

Cited by in corpus (9)

• Critical phenomena and universal dynamics in one-dimensional driven diffusive systems with two species of particles
• Microscopic structure of travelling wave solutions in a class of stochastic interacting particle systems
• Random walk of second class particles in product shock measures
• A reverse duality for the ASEP with open boundaries
• Traffic disruption and recovery in road networks
• Multiple shocks in bricklayers' model
• TASEP fluctuations with soft-shock initial data
• Self-duality and shock dynamics in the $n$-component priority ASEP
• Multi-species reaction-diffusion models admitting shock solutions