Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-20T17:47:41.201Z Has data issue: false hasContentIssue false
  • English
  • Français

Hydrodynamic Limit for a Type of Exclusion Process with Slow Bonds in Dimension d ≥ 2

Published online by Cambridge University Press:  14 July 2016

Tertuliano Franco ,
Adriana Neumann  and
Glauco Valle
Tertuliano Franco*
Affiliation:
Instituto Nacional de Matemática Pura e Aplicada
Adriana Neumann*
Affiliation:
Instituto Nacional de Matemática Pura e Aplicada
Glauco Valle*
Affiliation:
Universidade Federal do Rio de Janeiro
*
∗Postal address: Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botanico, 22460-320 Rio de Janeiro, Brazil.
∗Postal address: Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botanico, 22460-320 Rio de Janeiro, Brazil.
∗∗∗Postal address: Departamento de Métodos Estatísticos do Instituto de Matemática, Caixa Postal 68530, 21495-970 Rio de Janeiro, Brazil.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let Λ be a connected closed region with smooth boundary contained in the d-dimensional continuous torus Td. In the discrete torus N-1TdN, we consider a nearest-neighbor symmetric exclusion process where occupancies of neighboring sites are exchanged at rates depending on Λ in the following way: if both sites are in Λ or Λc, the exchange rate is 1; if one site is in Λ and the other site is in Λc, and the direction of the bond connecting the sites is ej, then the exchange rate is defined as N-1 times the absolute value of the inner product between ej and the normal exterior vector to ∂Λ. We show that this exclusion-type process has a nontrivial hydrodynamical behavior under diffusive scaling and, in the continuum limit, particles are not blocked or reflected by ∂Λ. Thus, the model represents a system of particles under hard-core interaction in the presence of a permeable membrane which slows down the passage of particles between two complementary regions.

Keywords

Hydrodynamic limitexclusion processnonhomogeneous media
Type
Research Article
Information
Journal of Applied Probability , Volume 48 , Issue 2 , June 2011 , pp. 333 - 351
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Chambolle, A. and Thouroude, G. (2009). Homogenization of interfacial energies and construction of plane-like minimizers in periodic media through a cell problem. Netw. Heterog. Media 4, 127–152.Google Scholar
[2] Evans, L. C. (1998). Partial Differential Equations (Graduate Stud. Math. 19). American Mathematical Society, Providence, RI.Google Scholar
[3] Faggionato, A. (2007). Bulk diffusion of 1D exclusion process with bond disorder. Markov Process. Relat. Fields 13, 519–542.Google Scholar
[4] Faggionato, A. (2010). Hydrodynamic limit of symmetric exclusion processes in inhomogeneous media. Preprint. Available at {http://arxiv.org/abs/1003.5521v1}.Google Scholar
[5] Faggionato, A., Jara, M. and Landim, C. (2008). Hydrodynamic behavior of one dimensional subdiffusive exclusion processes with random conductances. Prob. Theory Relat. Fields 144, 633–667.CrossRefGoogle Scholar
[6] Franco, T. and Landim, C. (2010). Hydrodynamic limit of gradient exclusion processes with conductances. Arch. Ration. Mech. Anal. 195, 409–439.Google Scholar
[7] Jara, M. (2009), Hydrodynamic limit of particle systems in inhomogeneous media. Preprint. Available at {http://arxiv.org/abs/0908.4120v1}.Google Scholar
[8] Kipnis, C. and Landim, C. (1999). Scaling Limits of Interacting Particle Systems (Fundamental Principles Math. Sci. 320). Springer, Berlin.CrossRefGoogle Scholar
[9] Nagy, K. (2002). {Symmetric random walk in random environment}. Periodica Math. Hung. 45, 101–120.Google Scholar
[10] Valentim, F. J. (2009). Hydrodynamic limit of gradient exclusion processes with conductances on {Z}d . Preprint. Available at {http://arxiv.org/pdf/0903.4993v1}.Google Scholar
[11] Zeidler, E. (1995). Applied Functional Analysis (Appl. Math. Sci. 108). Springer, New York.Google Scholar