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01 October 2020
  » arxiv » 1010.4769

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Hydrodynamical behavior of symmetric exclusion with slow bonds
Tertuliano Franco ; Patricia Goncalves ; Adriana Neumann ;
Date 22 Oct 2010
AbstractWe 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 $N^{-eta}$, with $etain[0,infty)$. 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 $eta$. If $etain [0,1)$, the hydrodynamic limit is given by the usual heat equation. If $eta=1$, it is given by a parabolic equation involving an operator $frac{d}{dx}frac{d}{dW}$, where $W$ is the Lebesgue measure on the torus plus the sum of the Dirac measure supported on each macroscopic point related to the slow bond. If $etain(1,infty)$, it is given by the heat equation with Neumann’s boundary conditions, meaning no passage through the slow bonds in the continuum.
Source arXiv, 1010.4769
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