Hydrodynamical behavior of symmetric exclusion with slow bonds

Details Hydrodynamical behavior of symmetric exclusion with slow bonds Hydrodynamical behavior of symmetric exclusion with slow bonds

2010-10-22

arxiv.org/abs/1010.4769v2

Mathematics

Probability

Accepted for publication in the Annales de l'Institut Henri Poincar\'e: Probability and Statistics

Scientific paper

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 $N^{-\beta}$, with $\beta\in[0,\infty)$. 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 $\beta$. If $\beta\in [0,1)$, the hydrodynamic limit is given by the usual heat equation. If $\beta=1$, 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 $\beta\in(1,\infty)$, it is given by the heat equation with Neumann's boundary conditions, meaning no passage through the slow bonds in the continuum.

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