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

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.

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.

Classification : 60K35, 26A24, 35K55
Keywords: hydrodynamic limit, exclusion process, slow bonds

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