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Large Deviations from the Hydrodynamic Limit

  • Chapter
Scaling Limits of Interacting Particle Systems

Part of the book series: Grundlehren der mathematischen Wissenschaften ((GL,volume 320))

  • 1999 Accesses

Abstract

In Chapters 4 and 5 we proved a law of large numbers for the empirical density of reversible interacting particle systems. A natural development of the theory is to investigate the large deviations from the hydrodynamic limit.

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Comments and References

  • Kipnis, C., Olla, S., Varadhan, S.R.S. (1989): Hydrodynamics and large deviations for simple exclusion processes, Comm. Pure Appl. Math. 42, 115–137

    MathSciNet  MATH  Google Scholar 

  • Donsker, M.D., Varadhan, S.R.S. (1989): Large deviations from a hydrodynamic scaling limit. Comm. Pure Appl. Math. 49, 243–270

    MathSciNet  Google Scholar 

  • Landim, C. (1992): Occupation time large deviations for the symmetric simple exclusion process. Ann. Probab. 20, 206–231

    Article  MathSciNet  MATH  Google Scholar 

  • Benois, O., Kipnis, C., Landim, C. (1995): Large deviations for mean zero asymmetric zero-range processes in infinite volume. Stoch. Proc. Appl. 55, 65–89

    Article  MathSciNet  MATH  Google Scholar 

  • Yau, H.T. (1994): Metastability of Ginzburg—Landau model with a conservation law. J. Stat. Phys. 74, 705–742

    Article  MATH  Google Scholar 

  • Landim, C., Yau, H.T. (1995): Large deviations of interacting particle systems in infinite volume. Comm. Pure Appl. Math. 48, 339–379

    MathSciNet  MATH  Google Scholar 

  • Jona-Lasinio, G., Landim, C., Vares, M.E. (1993): Large deviations for a reaction-diffusion model. Probab. Th. Rel. Fields 97, 339–361

    Article  MathSciNet  MATH  Google Scholar 

  • Landim, C. (1991c): An overview on large deviations of interacting particle systems. Ann. Inst. H. Poincaré, Physique Théorique 55, 615–635

    MathSciNet  MATH  Google Scholar 

  • Jona-Lasinio, G. (1991): Stochastic reaction diffusion equations and interacting particle systems. Ann. Inst. H. Poincaré, Physique Théorique 55, 751–758

    MathSciNet  MATH  Google Scholar 

  • Jona-Lasinio, G. (1992): Structure of hydrodynamic fluctutations in interacting particle systems. In F. Guerra, M. I. Loffredo and C. Marchioro, editors, Probabilistic Methods in Mathematical Physics, pages 262–263, World Scientific, Singapore

    Google Scholar 

  • Quastel, J. (1995a): Large deviations from a hydrodynamical scaling limit for a nongradient system. Ann. Probab. 23, 724–742

    Article  MathSciNet  MATH  Google Scholar 

  • Quastel, J., Yau, H.T. (1997): Lattice gases, large deviations and the incompressible NavierStokes equation, Stoch. Proc. Appl. 42, 31–37

    Google Scholar 

  • Onsager, L., Machlup, S. (1953): Fluctuation and irreversible processes I, II. Phys. Rev. 91, 1505–1512, 1512–1515

    Google Scholar 

  • Eyink, G.L. (1990): Dissipation and large thermodynamic fluctuations. J. Stat. Phys. 61, 533–572

    Article  MathSciNet  Google Scholar 

  • Gabrielli, D., Jona-Lasinio, G., Landim, C., Vares, M.E. (1997): Microscopic reversibility and thermodynamic fluctuations. In C. Cercignani, G. Jona—Lasinio, G. Parisi and L. A. Radicati di Brozolo, editors, Boltzmann’s Legacy 150 Years After His Birth. volume 131 of Atti dei Convegni Licei, pages 79–88, Accademia Nazionale dei Lincei, Roma

    Google Scholar 

  • De Masi, A., Ferrari, P. A., Lebowitz, J.L. (1986): Reaction-diffusion equations for interacting particle systems, J. Stat. Phys. 44, 589–644

    Article  MATH  Google Scholar 

  • Eyink, G.L., Lebowitz, J.L., Spohn, H. (1996): Hydrodynamics and fluctuations outside of local equilibrium: driven diffusive systems. J. Stat. Phys. 83, 385–472

    Article  MathSciNet  MATH  Google Scholar 

  • Cassandro, M., Galves, A., Olivieri, E. Vares, M. E. (1984): Metastable behavior of stochastic dynamics: a pathwise approach. J. Stat. Phys. 35, 603–628

    MathSciNet  MATH  Google Scholar 

  • Penrose, O., Lebowitz, J.L. (1987): Towards a rigorous molecular theory of metastability. In E. W. Montroll and J. L. Lebowitz, editors, Fluctutation Phenomena, second edition. North-Holland Physics Publishing, Amsterdam

    Google Scholar 

  • Olivieri, E., Vares, M.E. (1997): Large deviations and Metastability. To be published by Cambridge Universtiy Press, Cambridge

    Google Scholar 

  • Comets, F. (1987): Nucleation for a long-range magnetic model. Ann. Inst. H. Poincaré, Probabilités 23, 135–178

    MathSciNet  MATH  Google Scholar 

  • Vares, M.E. (1991): On long time behavior of a class of reaction—diffusion models. Ann. Inst. H. Poincaré, Physique Théorique 55, 601–613

    MathSciNet  MATH  Google Scholar 

  • De Masi, A., Presutti, E., Vares, M.E. (1986): Escape from the unstable equilibrium in a random process with infinitely many interacting particles. J. Stat. Phys. 44, 645–696

    Article  MATH  Google Scholar 

  • De Masi, A., Pellegrinotti, A., Presutti, E., Vares, E. (1994): Spatial patterns when phases separate in an interacting particle system. Ann. Probab. 22, 334–371

    Article  MathSciNet  MATH  Google Scholar 

  • Calderoni, P., Pellegrinotti, A., Presutti, E., Vares, M.E. (1989): Transient bimodality in interacting particle systems. J. Stat. Phys. 55, 523–577

    Article  MATH  Google Scholar 

  • De Masi, A., Presutti, E. (1991): Mathematical methods for hydrodynamic limits, volume 1501 of Lecture Notes in Mathematics, Springer-Verlag, New York

    Google Scholar 

  • Giacomin, G. (1994): Phase separation and random domain patterns in a stochastic particle model. Stoch. Proc. Appl. 51, 25–62

    Article  MathSciNet  MATH  Google Scholar 

  • Giacomin, G. (1995): Onset and structure of interfaces in a Kawasaki + Glauber interacting particle system. Probab. Th. Rel. Fields 103, 1–24

    Article  MathSciNet  MATH  Google Scholar 

  • Bellman, R., Harris, T.E. (1951): Recurrence times for the Ehrenfest model. Pacific J. Math. 1, 179–193

    MathSciNet  MATH  Google Scholar 

  • Harris, T.E. (1952): First passage and recurrence distributions. Trans. Amer. Math. Soc. 73, 471–486

    Article  MathSciNet  MATH  Google Scholar 

  • Aldous, D.J. (1982): Markov chains with almost exponential hitting times. Stoch. Proc. Appl. 13, 305–310

    Article  MathSciNet  MATH  Google Scholar 

  • Aldous, D.J. (1989): Probability Approximations via the Poisson Clumping Heuristics. Volume 77 of Applied Mathematical Sciences. Springer-Verlag, New York

    Google Scholar 

  • Aldous, D.J., Brown, M. (1992): Inequalities for rare events in time reversible Markov chains I. In M. Shaked and Y. L. Tong, editors, Stochastic Inequalities,volume 22 of IMS Lecture Notes, pages 1–16. IMS

    Google Scholar 

  • Aldous, D.J., Brown, M. (1993): Inequalities for rare events in time reversible Markov chains II. Stoch. Proc. Appl. 44, 15–25

    Article  MathSciNet  MATH  Google Scholar 

  • Korolyuk, D.V., Silvestrov, D.S. (1984): Entry times into asymptotically receding domains for ergodic Markov chains. Theory Probab. Appl. 28, 432–442

    MATH  Google Scholar 

  • Cogburn, R. (1985): On the distribution of first passage and return times for small sets. Ann. Probab. 13, 1219–1223

    Article  MathSciNet  MATH  Google Scholar 

  • Lebowitz, J.L., Schonmann, R.H. (1987): On the asymptotics of occurence times of rare events for stochastic spin systems. J. Stat. Phys. 48, 727–751

    Article  MathSciNet  MATH  Google Scholar 

  • Galves, A., Martinelli, F., Olivieri, E. (1989): Large density fluctuations for the one dimensional supercritical contact process. J. Stat. Phys. 55, 639–648

    Article  MathSciNet  MATH  Google Scholar 

  • Ferrari, P.A., Galves, A., Landim, C. (1994): Exponential waiting times for a big gap in a one-dimensional zero range process. Ann. Probab. 22, 284–288

    Article  MathSciNet  MATH  Google Scholar 

  • Ferrari, P.A., Galves, A., Liggett, T.M. (1995): Exponential waiting time for filling a large interval in the symmetric simple exclusion process. Ann. Inst. H. Poincaré, Probabilités 31, 155–175

    MathSciNet  MATH  Google Scholar 

  • Asselah, A., Dai Pra, P. (1997): Sharp estimates for the occurrence of rare events for symmetric simple exclusion. Stoch. Proc. Appl. 71, 259–273

    Article  MATH  Google Scholar 

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Kipnis, C., Landim, C. (1999). Large Deviations from the Hydrodynamic Limit. In: Scaling Limits of Interacting Particle Systems. Grundlehren der mathematischen Wissenschaften, vol 320. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03752-2_11

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  • DOI: https://doi.org/10.1007/978-3-662-03752-2_11

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-08444-7

  • Online ISBN: 978-3-662-03752-2

  • eBook Packages: Springer Book Archive

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