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The Asymmetric Simple Exclusion Process with Multiple Shocks | | Pablo A. Ferrari
; L. Renato G. Fontes
; M. Eulalia Vares
; | | Date: |
29 Nov 1999 | | Journal: | Ann. Inst. Henry Poincare, Probabilités et Statistiques 36, 2 (2000) 109-126 | | Subject: | Probability; Mathematical Physics MSC-class: 60K35; 82C | math.PR math-ph math.MP | | Abstract: | We consider the one dimensional totally asymmetric simple exclusion process with initial product distribution with densities $0 leq
ho_0 <
ho_1 <...<
ho_n leq 1$ in $(-infty,c_1ve^{-1})$, $[c_1ve^{-1},c_2epsilon^{-1}),...,[c_n ve^{-1}, + infty)$, 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 $r^*$ at time $t^*$. 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 $ve o 0$; the points of discontinuity depend on these limiting Gaussian variables. As a corollary we show that, as $epsilon o 0$, the distribution of the process at site $epsilon^{-1}r^*+ve^{-1/2}a$ at time $epsilon^{-1}t^*$ tends to a non trivial convex combination of the product measures with densities $
ho_k$, the weights of the combination being explicitly computable. | | Source: | arXiv, math.PR/9911237 | | Services: | Forum | Review | PDF | Favorites | |
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