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The Gradient Flow Approach to Hydrodynamic Limits for the Simple Exclusion Process

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From Particle Systems to Partial Differential Equations III

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 162))

Abstract

We present a new approach to prove the macroscopic hydrodynamic behaviour for interacting particle systems, and as an example we treat the well-known case of the symmetric simple exclusion process (SSEP). More precisely, we characterize any possible limit of its empirical density measures as solutions to the heat equation by passing to the limit in the gradient flow structure of the particle system.

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Notes

  1. 1.

    This is also the assumption used to make Yau’s relative entropy method work, see [21].

References

  1. Adams, S., Dirr, N., Peletier, M.A., Zimmer, J.: From a large-deviations principle to the Wasserstein gradient flow: a new micro-macro passage. Commun. Math. Phys. 307, 791–815 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Adams, S., Dirr, N., Peletier, M. A., Zimmer, J.: Large deviations and gradient flows. Phil. Trans. R. Soc. A 371(2005), 0341 (2013)

    Google Scholar 

  3. Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zurich, 2nd edn. Birkhauser Verlag, Basel (2008)

    MATH  Google Scholar 

  4. Ambrosio, L., Savaré, G., Zambotti, L.: Existence and stability for Fokker-Planck equations with log-concave reference measure. Probab. Theory Relat. Fields 145, 517–564 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. De Giorgi, E., Marino, A., Tosques, M.: Problems of evolution in metric spaces and maximal decreasing curve. Att. Acc. Naz. Linc. R. Cl. Sci. Fis. Mat. Nat. 8(68)(3), 180–187 (1980)

    Google Scholar 

  6. Erbar, M., Fathi, M., Laschos, V., Schlichting, A.: Gradient flow structure for McKean-Vlasov equations on discrete spaces. arXiv:1601.08098

  7. Erbar, M., Maas, J.: Ricci curvature of finite Markov chains via convexity of the entropy. Arch. Ration. Mech. Anal. 206(3), 997–1038 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fathi, M.: A gradient flow approach to large deviations for diffusion processes. Journal de Mathématiques Pures et Appliquées (2014). arXiv:1405.3910

  9. Fathi, M., Maas, J.: Entropic Ricci curvature bounds for discrete interacting systems. Ann. Appl. Probab. (2015)

    Google Scholar 

  10. Guo, M.Z., Papanicolaou, G.C., Varadhan, S.R.S.: Nonlinear diffusion limit for a system with nearest neighbor interactions. Commun. Math. Phys. 118, 31–59 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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 

  12. Kosygina, E.: The behavior of the specific entropy in the hydrodynamic scaling limit. Ann. Probab. 29(3), 1086–1110 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Maas, J.: Gradient flows of the entropy for finite Markov chains. J. Funct. Anal. 261(8), 2250–2292 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Maas, J., Mielke, A.: Gradient structures for chemical reactions with detailed balance: I. Modeling and large-volume limit (2015)

    Google Scholar 

  15. Mariani, M.: A Gamma-convergence approach to large deviations. Preprint arXiv:1204.0640v1 (2012)

  16. Mielke, A.: Geodesic convexity of the relative entropy in reversible Markov chains. Calc. Var. Partial. Differ. Equ. 48(1–2), 1–31 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial. Differ. Equ. 26(1–2), 101–174 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Sandier, E., Serfaty, S.: Gamma-convergence of gradient flows with applications to Ginzburg-Landau. Commun. Pure Appl. Math 57(12), 1627–1672 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Serfaty, S.: Gamma-convergence of gradient flows on Hilbert and metric spaces and applications. Discret. Contin. Dyn. Syst. A 31(4), 1427–1451 (2011)

    Google Scholar 

  20. Vazquez, J.L.: The Porous Medium Equation: Mathematical Theory. Oxford Mathematical Monographs. A Clarendon Press Publication, Oxford (2006)

    Book  Google Scholar 

  21. Yau, H.T.: Relative entropy and hydrodynamics of Ginzburg-Landau models. Lett. Math. Phys. 22, 63–80 (1991)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

M.F. : I would like to thank Hong Duong, Matthias Erbar, Vaios Laschos and André Schlichting for discussions on convergence of gradient flows. Part of this work was done while I was staying at the Hausdorff Institute for Mathematics in Bonn, whose support is gratefully acknowledged. I also benefited from funding from GDR MOMAS and from NSF FRG grant DMS-1361185. M.S.: This work has been supported by the French Ministry of Education through the grant ANR (EDNHS), and also by CAPES (Brazil) and IMPA (Instituto de Matematica Pura e Aplicada, Rio de Janeiro) through a post-doctoral fellowship.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Berkeley, USA

    Max Fathi

  2. Équipe MEPHYSTO, INRIA Lille Nord Europe, 40 avenue du Halley, 59650 Villeneuve d’Ascq, Lille, France

    Marielle Simon

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  1. Max Fathi
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  2. Marielle Simon
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Correspondence to Marielle Simon .

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Editors and Affiliations

  1. University of Minho Department of Mathematics, Braga, Portugal

    Patrícia Gonçalves

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

    Ana Jacinta Soares

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Fathi, M., Simon, M. (2016). The Gradient Flow Approach to Hydrodynamic Limits for the Simple Exclusion Process. In: Gonçalves, P., Soares, A. (eds) From Particle Systems to Partial Differential Equations III. Springer Proceedings in Mathematics & Statistics, vol 162. Springer, Cham. https://doi.org/10.1007/978-3-319-32144-8_8

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