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Hydrodynamical behavior of symmetric exclusion with slow bonds

Tertuliano Franco;  Patrícia Gonçalves;  Adriana Neumann — 2013

Annales de l'I.H.P. Probabilités et statistiques

We consider the exclusion process in the one-dimensional discrete torus with N points, where all the bonds have conductance one, except a finite number of slow bonds, with conductance N - β , with β ∈ [ 0 , ∞ ) . We prove that the time evolution of the empirical density of particles, in the diffusive scaling, has a distinct behavior according to the range of the parameter β . If β ∈ [ 0 , 1 ) , the hydrodynamic limit is given by the usual heat equation. If β = 1 , it is given by a parabolic equation involving an operator d d x d d W , where W ...

Collision probabilities in the rarefaction fan of asymmetric exclusion processes

Pablo A. Ferrari;  Patricia Gonçalves;  James B. Martin — 2009

Annales de l'I.H.P. Probabilités et statistiques

We consider the one-dimensional asymmetric simple exclusion process (ASEP) in which particles jump to the right at rate ∈(1/2, 1] and to the left at rate 1−, interacting by exclusion. In the initial state there is a finite region such that to the left of this region all sites are occupied and to the right of it all sites are empty. Under this initial state, the hydrodynamical limit of the process converges to the rarefaction fan of the associated Burgers equation. In particular suppose that the...

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