Hydrodynamical behavior of symmetric exclusion with slow bonds

Franco, Tertuliano; Gonçalves, Patrícia; Neumann, Adriana

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 2, pp. 402-427.

Nous considérons le processus d’exclusion dans le tore discret uni-dimensionnel avec $N$ points, où tous les liens ont conductance un, sauf pour un nombre fini de liens lents qui ont conductance $N^{-\beta }$, avec $\beta \in [0,\infty )$. Nous prouvons que l’évolution en temps de la densité empirique de particules, après un changement d’échelle diffusif, a un comportement différent selon la valeur du paramètre $\beta$. Si $\beta \in [0,1)$, la limite hydrodynamique est donnée par l’équation de la chaleur usuelle. Si $\beta =1$, la limite est donnée par une équation parabolique avec un opérateur $\frac{\mathrm{d}}{\mathrm{d}x}\frac{\mathrm{d}}{\mathrm{d}W}$, où $W$ est la mesure de Lebesgue sur le tore plus la somme des masses de Dirac en chaque point macroscopique relatif à un lien lent. Si $\beta \in (1,\infty )$, la limite est donnée par l’équation de la chaleur avec conditions au bord de Neumann, et ceci traduit l’absence de passage par les liens lents dans le continu.

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 $\frac{\mathrm{d}}{\mathrm{d}x}\frac{\mathrm{d}}{\mathrm{d}W}$, 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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