Skip to main content
SpringerLink
Log in

Equilibrium Fluctuations for the Slow Boundary Exclusion Process

  • Conference paper
  • First Online:
From Particle Systems to Partial Differential Equations (PSPDE 2015)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 209))

Included in the following conference series:

Abstract

We prove that the equilibrium density fluctuations of the symmetric simple exclusion process in contact with slow boundaries is given by an Ornstein–Uhlenbeck process with Dirichlet, Robin or Neumann boundary conditions depending on the range of the parameter that rules the slowness of the boundaries.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Baldasso, R., Menezes, O., Neumann, A., Souza, R.R.: Exclusion process with slow boundary. J. Stat. Phys. 167, 1112–1142 (2017)

    Google Scholar 

  2. Birkhoff, G., Rota, G.-C.: Ordinary Differential Equations, 4th edn. Wiley, New York (1989)

    MATH  Google Scholar 

  3. Blythe, R.A., Evans, M.R.: Nonequilibrium steady states of matrix-product form: a solver’s guide. J. Phys. A: Mathe. Theor. 40(46), R333 (2007)

    Google Scholar 

  4. Bodineau, T., Derrida, B., Lebowitz, J.L.: A diffusive system driven by a battery or by a smoothly varying field. J. Stat. Phys. 140(4), 648–675 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Boyce, W., DiPrima, R.: Elementary Differential Equations and Boundary Value Problems, 9th edn. Wiley, New York (2009)

    Google Scholar 

  6. De Masi, A., Presutti, E., Tsagkarogiannis, D., Vares, M.E.: Current reservoirs in the simple exclusion process. J. Stat. Phys. 144(6), 1151–1170 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. De Masi, A., Presutti, E., Tsagkarogiannis, D., Vares, M.E.: Non equilibrium stationary state for the sep with births and deaths. J. Stat. Phys. 147(3), 519–528 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. De Masi, A., Presutti, E., Tsagkarogiannis, D., Vares, M.E.: Truncated correlations in the stirring process with births and deaths. Electron. J. Probab. 17(6), 1–35 (2012)

    MathSciNet  MATH  Google Scholar 

  9. De Masi, A., Ferrari, P., Presutti, E.: Symmetric simple exclusion process with free boundaries. Probab. Theory Relat. Fields 161(1), 155–193 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Derrida, B.: Non-equilibrium steady states: fluctuations and large deviations of the density and of the current. J. Stat. Mech. 2007(7), P07023 (2007)

    Google Scholar 

  11. Derrida, B., Janowsky, S.A., Lebowitz, J.L., Speer, E.R.: Exact solution of the totally asymmetric simple exclusion process: shock profiles. J. Stat. Phys. 73(5), 813–842 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  12. Farfán, J.: Hydrostatics, statical and dynamical large deviations of boundary driven gradient symmetric exclusion processes. Ph.D. thesis, 2008

    Google Scholar 

  13. Franco, T., Gonçalves, P., Neumann, A.: Hydrodynamical behavior of symmetric exclusion with slow bonds. Ann. Inst. H. Poincaré Probab. Stat. 49(2), 402–427 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Franco, T., Gonçalves, P., Neumann, A.: Phase transition in equilibrium fluctuations of symmetric slowed exclusion. Stoch. Process. Appl. 123(12), 4156–4185 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Franco, T., Gonçalves, P., Neumann, A.: Phase transition of a heat equation with Robin’s boundary conditions and exclusion process. Trans. Am. Math. Soc. 367(9), 6131–6158 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Franco, T., Gonçalves, P., Neumann, A.: Non-equilibrium and stationary fluctuations of a slowed boundary symmetric exclusion, arXiv e-prints (2016)

    Google Scholar 

  17. Holley, R.A., Stroock, D.W.: Generalized Ornstein-Uhlenbeck processes and infinite particle branching brownian motions. Publ. Res. Inst. Math. Sci. 14(3), 741–788 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kipnis, C., Landim, C.: Scaling limits of interacting particle systems. Grundlehren der Mathematischen Wissenschaften. [Fundamental Principles of Mathematical Sciences] vol. 320. Springer, Berlin (1999)

    Google Scholar 

  19. Landim, C., Milanés, A., Olla, S.: Stationary and nonequilibrium fluctuations in boundary driven exclusion processes. Markov Process Relat. Fields 14(2), 165–184 (2008)

    MathSciNet  MATH  Google Scholar 

  20. Mitoma, I.: Tightness of probabilities on \({C}([0, 1 ]; Y^{\prime })\) and \({D}([0, 1 ]; Y^{\prime })\). Ann. Probab. 11(4), 989–999 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  21. Reed, M., Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis, 1st edn. Academic Press, New York (1981)

    Google Scholar 

Download references

Acknowledgements

A. N. was supported through a grant “L’ORÉAL - ABC - UNESCO Para Mulheres na Ciência”. P. G. thanks FCT/Portugal for support through the project UID/MAT/04459/2013. T. F. was supported by FAPESB through the project Jovem Cientista-9922/2015. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovative programme (grant agreement No 715734).

Author information

Authors and Affiliations

  1. UFBA, Instituto de Matemática, Campus de Ondina, Av. Adhemar de Barros, S/N. CEP, 40170-110, Salvador, Brazil

    Tertuliano Franco

  2. Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001, Lisboa, Portugal

    Patrícia Gonçalves

  3. UFRGS, Instituto de Matemática, Campus do Vale, Av. Bento Gonçalves, 9500, 91509-900, Porto Alegre, Brazil

    Adriana Neumann

Authors
  1. Tertuliano Franco
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Patrícia Gonçalves
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Adriana Neumann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Patrícia Gonçalves .

Editor information

Editors and Affiliations

  1. Center for Mathematics, Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Lisboa, Portugal

    Patrícia Gonçalves

  2. Centre of Mathematics, University of Minho, Braga, Portugal

    Ana Jacinta Soares

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Franco, T., Gonçalves, P., Neumann, A. (2017). Equilibrium Fluctuations for the Slow Boundary Exclusion Process. In: Gonçalves, P., Soares, A. (eds) From Particle Systems to Partial Differential Equations. PSPDE 2015. Springer Proceedings in Mathematics & Statistics, vol 209. Springer, Cham. https://doi.org/10.1007/978-3-319-66839-0_9

Download citation

Publish with us

Policies and ethics

Search

Navigation