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Article overview
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Crossover to the stochastic Burgers equation for the WASEP with a slow bond | | Tertuliano Franco
; Patricia Gonçalves
; Marielle Simon
; | | Date: |
22 Jun 2015 | | Abstract: | We consider the weakly asymmetric simple exclusion process in the presence of
a slow bond and starting from the invariant state, namely the Bernoulli product
measure of parameter $
hoin(0,1)$. The rate of passage of particles to the
right (resp. left) is
$frac1{vphantom{n^eta}2}+frac{a}{2n^{vphantom{eta}gamma}}$ (resp.
$frac1{vphantom{n^eta}2}-frac{a}{2n^{vphantom{eta}gamma}}$) except at
the bond of vertices ${-1,0}$ where the rate to the right (resp. left) is
given by $frac{alpha}{2n^eta}+frac{a}{2n^{vphantom{eta}gamma}}$ (resp.
$frac{alpha}{2n^eta}-frac{a}{2n^{vphantom{eta}gamma}}$). Above,
$alpha>0$, $gammageq etageq 0$, $ageq 0$. For $eta<1$, we show that
the limit density fluctuation field is an Ornstein-Uhlenbeck process defined on
the Schwartz space $mathcal{S}(mathbb{R})$ for the strength asymmetry
$an^{2-gamma}$ if $gamma>frac12$, while for $gamma = frac12$ it is an
energy solution of the stochastic Burgers equation. For $gamma>eta=1$, it is
an Ornstein-Uhlenbeck process associated to the heat equation with Robin’s
boundary conditions. The case $eta=gamma=1$ remains open, being conjectured
its behaviour. For $gammageqeta> 1$, the limit density fluctuation field is
an Ornstein-Uhlenbeck process associated to the heat equation with Neumann’s
boundary conditions. | | Source: | arXiv, 1506.6560 | | Services: | Forum | Review | PDF | Favorites | |
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