Hydrodynamical behavior of symmetric exclusion with slow bonds

Franco, Tertuliano; Gonçalves, Patrícia; Neumann, Adriana

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

[1] T. Bodineau, B. Derrida and J. L. Lebowitz. A diffusive system driven by a battery or by a smoothly varying field. J. Stat. Phys. 140 (2010) 648-675. | MR | Zbl

[2] L. Evans. Partial Differential Equations. Graduate Studies in Mathematics 19. American Mathematical Society, Providence, RI, 1998. | MR | Zbl

[3] A. Faggionato. Bulk diffusion of 1D exclusion process with bond disorder. Markov Process. Related Fields 13 (2007) 519-542. | MR | Zbl

[4] A. Faggionato, M. Jara and C. Landim. Hydrodynamic behavior of one dimensional subdiffusive exclusion processes with random conductances. Probab. Theory Related Fields 144 (2009) 633-667. | MR | Zbl

[5] T. Franco and C. Landim. Hydrodynamic limit of gradient exclusion processes with conductances. Arch. Ration. Mech. Anal. 195 (2010) 409-439. | MR | Zbl

[6] T. Franco, A. Neumann and C. Landim. Large deviations for the one-dimensional exclusion process with a slow bond. Unpublished manuscript, 2010.

[7] T. Franco, A. Neumann and G. Valle. Hydrodynamic limit for a type of exclusion processes with slow bonds in dimension $\ge 2$. J. Appl. Probab. 48 (2011) 333-351. | MR | Zbl

[8] P. Gonçalves, C. Landim and C. Toninelli. Hydrodynamic limit for a particle system with degenerate rates. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009) 887-909. | Numdam | MR | Zbl

[9] M. Jara. Hydrodynamic limit of particle systems in inhomogeneous media. In Dynamics, Games and Science II. M. Peixoto, A. Pinto and D. Rand (Eds). Springer, Berlin, 2011. | MR

[10] C. Kipnis and C. Landim. Scaling Limits of Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 320. Springer, Berlin, 1999. | MR | Zbl

[11] O. A. Ladyzhenskaya. The Boundary Value Problems of Mathematical Physic. Applied Math. Sci. 49. Springer, New York, 1985. | MR | Zbl

[12] G. Leoni. A First Course in Sobolev Spaces. Graduate Studies in Mathematics 105. American Mathematical Society, Providence, RI, 2009. | MR | Zbl

[13] T. Seppäläinen. Hydrodynamic profiles for the totally asymmetric exclusion process with a slow bond. J. Stat. Phys. 102 (2001) 69-96. | MR | Zbl

[14] C. Stone. Limit theorems for random walks, birth and death processes, and diffusion processes. Ill. J. Math. 7 (1963) 638-660. | MR | Zbl

[15] F. Valentim. Hydrodynamic limit of gradient exclusion processes with conductances. Ann. Inst. Henri Poincaré Probab. Stat. To appear. | Numdam | MR | Zbl