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Conservative Interacting Particles System with Anomalous Rate of Ergodicity

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Abstract

We analyse certain conservative interacting particle system and establish ergodicity of the system for a family of invariant measures. Furthermore, we show that convergence rate to equilibrium is exponential. This result is of interest because it presents counterexample to the standard assumption of physicists that conservative system implies polynomial rate of convergence. The system in question is stochastic rather than deterministic.

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References

  1. Anderson, W.J.: Continuous-Time Markov Chains: An Applications-Oriented Approach. Springer Series in Statistics: Probability and Its Applications, vol. 7, Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  2. Barbato, D., Flandoli, F., Morandin, F.: Anomalous dissipation in a stochastic inviscid dyadic model. Ann. Appl. Probab. (to appear)

  3. Barbato, D., Flandoli, F., Morandin, F.: Uniqueness for a stochastic inviscid dyadic model. Proc. Am. Math. Soc. 138, 2607–2617 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bernardin, C.: Hydrodynamics for a system of harmonic oscillators perturbed by a conservative noise. Stoch. Process. Appl. 117, 487–513 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bernardin, C.: Superdiffusivity of asymmetric energy model in dimensions 1 and 2. J. Math. Phys. 49, 103301 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  6. Bertini, L., Zegarliński, B.: Coercive inequalities for Gibbs measures. J. Funct. Anal. 162, 257–286 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bertini, L., Zegarliński, B.: Coercive inequalities for Kawasaki dynamics: the product case. Markov Process. Relat. Fields 5, 125–162 (1999)

    MATH  Google Scholar 

  8. Bogachev, V.I.: Gaussian Measures. Mathematical Surveys and Monographs, vol. 62 (1998)

    MATH  Google Scholar 

  9. Dobrushin, R.L.: Markov processes with a large number of locally interacting components: existence of a limit process and its ergodicity. Probl. Inf. Transm. 7, 149–164 (1971)

    Google Scholar 

  10. Flandoli, F., Ga̧tarek, D.: Martingale and stationary solutions for stochastic Navier-Stokes equations. Probab. Theory Relat. Fields 102, 367–391 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fritz, J., Nagy, K., Olla, S.: Equilibrium fluctuations for a system of harmonic oscillators with conservative noise. J. Stat. Phys. 122, 399–415 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. Giardinà, C., Kurchan, J., Redig, F.: Duality and exact correlations for a model of heat conduction. J. Math. Phys. 48, 103301 (2007)

    Article  MathSciNet  Google Scholar 

  13. Giardinà, C., Kurchan, J., Redig, F., Vafayi, K.: Duality and hidden symmetries in interacting particle systems. J. Stat. Phys. 135, 25–55 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. Holevo, A.S.: On dissipative stochastic equations in a Hilbert space. Probab. Theory Relat. Fields 104, 483–500 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Inglis, J., Neklyudov, M., Zegarlinski, B.: Ergodicity for infinite particle systems with locally conserved quantities. Infin. Dimens. Anal. Quantum Probab. (2011, to appear)

  16. Janvresse, E., Landim, C., Quastel, J., Yau, H.T.: Relaxation to equilibrium of conservative dynamics. I. Zero-range processes. Ann. Probab. 27, 325–360 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kato, T.: On the semi-groups generated by Kolmogoroff’s differential equations. J. Math. Soc. Jpn. 6, 1–15 (1954)

    Article  MATH  Google Scholar 

  18. Krylov, N., Rozovskii, B.: Stochastic Evolution Equations. Itogi Nauki i Tekhniki, Seria Sovremiennyie Problemy Matematiki, vol. 14 (1979) (in Russian)

    Google Scholar 

  19. Landim, C., Yau, H.T.: Convergence to equilibrium of conservative particle systems on ℤd. Ann. Probab. 31, 115–147 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Liggett, T.: Interacting Particle Systems. Springer, Berlin (2004)

    Google Scholar 

  21. Pardoux, E.: Stochastic partial differential equations and filtering of diffusion processes. Stochastics 3, 127–167 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  22. Spitzer, F.: Random processes defined through the interaction of an infinite particle system. In: Lecture Notes in Math., vol. 89, pp. 201–223. Springer, Berlin (1969)

    Google Scholar 

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Author information

Author notes

  1. M. Neklyudov

    Present address: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076, Tübingen, Germany

Authors and Affiliations

  1. Dipartimento di Matematica Applicata, Universitàdi Pisa, Pisa, Italy

    F. Flandoli

  2. Department of Mathematics, University of York, York, UK

    Z. Brzeźniak & M. Neklyudov

  3. CNRS, Toulouse, France

    B. Zegarliński

Authors
  1. Z. Brzeźniak
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  2. F. Flandoli
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  3. M. Neklyudov
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  4. B. Zegarliński
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Corresponding author

Correspondence to M. Neklyudov.

Additional information

Supported by EPSRC GR/R90994/01 & EP/D05379X/1.

B. Zegarliński is on leave from Imperial College London.

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Brzeźniak, Z., Flandoli, F., Neklyudov, M. et al. Conservative Interacting Particles System with Anomalous Rate of Ergodicity. J Stat Phys 144, 1171 (2011). https://doi.org/10.1007/s10955-011-0327-3

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  • DOI: https://doi.org/10.1007/s10955-011-0327-3

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