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Metastability and Ageing in Stochastic Dynamics

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Dynamics and Randomness II

Part of the book series: Nonlinear Phenomena and Complex Systems ((NOPH,volume 10))

Abstract

In these notes I review recent results on metastability and ageing in stochastic dynamics. The first part reviews a somewhat novel approach to the computation of key quantities such as mean exit times in metastable systems and small eigenvalues of the generator of metastable Markov chain developed over the last years with M. Eckhoff, V. Gayrard and M. Klein. This approach is based on extensive use of potential theoretic ideas and allows, at least in the case of reversible dynamics, to get very accurate results with comparatively little effort. This methods have also been used in recent joint work with G. Ben Arous and V. Gayrard on the dynamics of the random energy model. The second part of these lectures is devoted to a review of this work.

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References

  1. Ben Arous, G., Bovier, A. and Gayrard, V. (2002) Aging in the Random Energy Model, Phys. Rev. Letts., Vol. 88, pp. 087201.

    Article  Google Scholar 

  2. Ben Arous, G., Bovier, A. and Gayrard, V. (2003) Glauber Dynamics of the Random Energy Model. 1. Metastable Motion on the Extreme States, Commun. Math. Phys., Vol. 235, pp. 379–425.

    Article  MathSciNet  MATH  Google Scholar 

  3. Ben Arous, G., Bovier, A. and Gayrard, V. (2003) Glauber Dynamics of the Random Energy Model. 2. Aging below the Critical Temperature, Commun. Math. Phys., Vol. 236, pp. 1–54.

    Article  MathSciNet  MATH  Google Scholar 

  4. Ben Arous, G., Bovier, A. and Gayrard, V. Random Walks on the Hypercube, in preparation.

    Google Scholar 

  5. Ben Arous, G. (2002) Aging and Spin Glasses, in Proceedings of the International Congress of Mathematicians 2002, Beijing, China, (Li, Ta, Tsien et al. Eds.), China: Higher Education Press, Vol.3, pp. 3–14.

    Google Scholar 

  6. Bouchaud, J.P., Cugliandolo, L., Kurchan, J., Mézard, M. (1998) Out-of-Equilibrium Dynamics in Spin-Glasses and other Glassy Systems, in Spin-Glasses and Random Fields (A.P. Young, Ed.), World Scientific, Singapore.

    Google Scholar 

  7. Bouchaud, J.P. and Dean, D. (1995) Aging on Parisi’s Tree, J. Phys, I, France, Vol. 5, pp. 265.

    Article  Google Scholar 

  8. Bovier, A., Eckhoff, M., Gayrard, V. and Klein, M. (2001) Metastability in Stochastic Dynamics of Disordered Mean-Field Models, Probab. Theor. Rel. Fields, Vol. 119, pp. 99–161.

    Article  MathSciNet  MATH  Google Scholar 

  9. Bovier, A., Eckhoff, M., Gayrard, V. and Klein, M. (2002) Metastability and Low-Lying Spectra in Reversible Markov Chains, Commun. Math. Phys., Vol. 228, pp. 219–255.

    Article  MathSciNet  MATH  Google Scholar 

  10. Bovier, A., Eckhoff, M., Gayrard, V. and Klein, M. (2002) Metastability in Reversible Diffusion Processes I. Sharp Asymptotics for Capacities and Exit Times, Preprint.

    Google Scholar 

  11. Bovier, A., Gayrard, V. and Klein, M. (2002) Metastability in Reversible Diffusion Processes II. Precise Asymptotics for Small Eigenvalues, Preprint.

    Google Scholar 

  12. Bovier, A. and Manzo, F. (2002) Metastability in Glauber Dynamics in the Low-Temperature Limit: beyond Exponential Asymptotics, J. Statist. Phys., Vol. 107, pp. 757–779.

    Article  MathSciNet  MATH  Google Scholar 

  13. Burke, C.J. and Rosenblatt, M. (1958) A Markovian Function of a Markov Chain, Ann. Math. Statist, Vol. 29, pp. 1112–1122.

    Article  MathSciNet  MATH  Google Scholar 

  14. Cassandro, M., Galves, A. and Picco, P. (1991) Dynamical Phase Transitions in Disordered Systems: the Study of a Random Walk Model, Ann. Inst. H. Poincare Phys. Thor., Vol.55, pp. 689–705.

    MathSciNet  MATH  Google Scholar 

  15. Cugliandolo, L. (2002) Dynamics of Glassy Systems, Les Houches Lecture Notes, cond-mat/0210312.

    Google Scholar 

  16. Cugliandolo, L. and Kurchan, J. (1999) Thermal Properties of Slow Dynamics, Physica, Vol. A 263, pp. 242–253.

    Google Scholar 

  17. Davies, E.B. (1982) Metastable States of Symmetric Markov Semigroups. I. Proc. Loud. Math. Soc. Ill, Ser., Vol. 45, pp. 133–150.

    Article  MATH  Google Scholar 

  18. Davies, E.B. (1982) Metastable States of Symmetric Markov Semigroups. II. J. Lond. Math. Soc. II, Ser., Vol. 26, pp. 541–556.

    Article  MATH  Google Scholar 

  19. Davies, E.B. (1983) Spectral Properties of Metastable Markov Semigroups, J. Fund. Anal, Vol. 52, pp. 315–329.

    Article  MATH  Google Scholar 

  20. den Hollander, W.T.F. and Shuler, K.E. (1992) Random Walks in a Random Field of Decaying Traps, J. Statist Phys., Vol.67, pp. 13–31.

    Article  MathSciNet  MATH  Google Scholar 

  21. Derrida, B. (1980) Random Energy Model: Limit of a Family of Disordered Models, Phys. Rev. Letts., Vol. 45, pp. 79–82.

    Article  MathSciNet  Google Scholar 

  22. Derrida, B. (1981) Random Energy Model: An Exactly Solvable Model of Disordered Systems, Phys. Rev. B, Vol. 24, pp. 2613–2626.

    Article  MathSciNet  Google Scholar 

  23. Diaconis, P. (1988) Applications of Noncommutative Fourier Analysis to Probability Problems, Ecole d’Ete de Probability de Saint-Flour XV-XVII, 1985–87, Lecture Notes in Math., Vol.1362, Springer Verlag, Berlin, pp. 51–100.

    Google Scholar 

  24. Doob, J.L. (1984) Classical Potential Theory and its Probabilistic Counterpart, Grundlehren der Mathematischen Wissenschaften Vol.262, Springer Verlag, Berlin.

    Book  MATH  Google Scholar 

  25. Doyle, P.G. and Snell, J.L. (1984) Random Walks and Electrical Networks, Carus Mathematical Monographs, Vol.22, Mathematical Association of America, Washington, DC.

    Google Scholar 

  26. Donsker, M.D. and Varadhan, S.R.S. (1976) On the Principal Eigenvalue of Second-Order Elliptic Differential Operators, Comm. Pure Appl. Math., Vol.29, pp. 595–621.

    Article  MathSciNet  MATH  Google Scholar 

  27. Eisele, Th. (1983) On a Third Order Phase Transition, Commun. Math. Phys., Vol. 90, pp. 125–159.

    Article  MathSciNet  MATH  Google Scholar 

  28. Eyring, H. (1935) The Activated Complex in Chemical Reactions, J. Chem. Phys., Vol.3, pp. 107–115.

    Article  Google Scholar 

  29. Eyring, H. and Polanyi, M. (1931) Z. Physik. Chemie., Vol.B12, pp. 279.

    Google Scholar 

  30. Fontes, L.R.G., Isopi, M., Kohayakawa, Y. and Picco, P. (2001) The Spectral Gap of the REM under Metropolis Dynamics, Ann. Appl. Probab., Vol. 8, pp. 917–943.

    MathSciNet  Google Scholar 

  31. Preidlin, M.L and Wentzell, A.D. (1984) Random Perturbations of Dynamical Systems, Springer Verlag, Berlin-Heidelberg-New York.

    Book  Google Scholar 

  32. Galvez, A., Martinez, S. and Picco, P. (1989) Fluctuations in Derrida’s Random Energy and Generalized Random Enery Models, J. Stat. Phys., Vol. 54, pp. 515–529.

    Article  Google Scholar 

  33. Gaveau, B. and Schulman, L.S. (1998) Theory of Nonequilibrium First-Order Phase Transitions for Stochastic Dynamics, J. Math. Phys., Vol. 39, pp. 1517–1533.

    Article  MathSciNet  MATH  Google Scholar 

  34. Gayrard, V. (1992) Thermodynamic Limit of the g-State Potts-Hopfield Model with Infinitely many Patterns, J. Statist. Phys., Vol. 68, pp. 977–1011.

    Article  MathSciNet  MATH  Google Scholar 

  35. Glasstone, S., Laidler, K.J. and Eyring, H. (1941) The Theory of Rate Processes, McGraw-Hill, New York.

    Google Scholar 

  36. Holley, R.A., Kusuoka, S., and Stroock, S.W. (1989) Asymptotics of the Spectral Gap with Applications to the Theory of Simulated Annealing, J. Fund. Anal., Vol. 83, pp. 333–347.

    Article  MathSciNet  MATH  Google Scholar 

  37. Huisinga, W., Meyn, S. and Schiitte, Ch. (2002) Phase Transitions and Metastability for Markovian and Molecular Systems, FU Berlin, Preprint.

    Google Scholar 

  38. Kato, T. (1976) Perturbation Theory for Linear Operators, Second edition, Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York.

    Book  MATH  Google Scholar 

  39. Kakutani, S. (1941) Markov Processes and the Dirichlet Problem, Proc. Jap. Acad. Vol.21, pp. 227–233.

    MathSciNet  Google Scholar 

  40. Kemeny, J.G. and Snell, J.L. (1960) Finite Markov Chains, D. van Nostrand Company, Princeton.

    MATH  Google Scholar 

  41. Kemperman, J.H.B. (1961) The Passage Problem for a Stationary Markov Chain, Statistical Research Monographs, Vol.I, The University of Chicago Press.

    Google Scholar 

  42. Kolokoltsov, V.N. (2000) Semiclassical Analysis for Diffusions and Stochastic Processes, Springer Verlag, Berlin.

    MATH  Google Scholar 

  43. Kolokoltsov, V.N. and Makarov, K.A. (1996) Asymptotic Spectral Analysis of a Small Diffusion Operator and the Life Times of the Corresponding Diffusion Process, Russian J. Math. Phys., Vol.4, pp. 341–360.

    MathSciNet  Google Scholar 

  44. Koch, H. and Piasko, J. (1989) Some Rigorous Results on the Hopfield Neural Network Model, J. Statist Phys., Vol.55, pp. 903–928.

    Article  MathSciNet  MATH  Google Scholar 

  45. Kramers, H.A. (1940) Brownian Motion in a Field of Force and the Diffusion Model of Chemical Reactions, Physica, Vol.7, pp. 284–304.

    Article  MathSciNet  MATH  Google Scholar 

  46. Mathieu, P. (1995) Spectra, Exit Times and Long Times Asymptotics in the Zero White Noise Limit, Stock Rep., Vol. 55, pp. 1–20.

    MathSciNet  MATH  Google Scholar 

  47. Mathieu, P. and Picco, P. (2000) Convergence to Equilibrium for Finite Markov Processes with Application to the Random Energy Model, CPT-2000/P.39, Preprint.

    Google Scholar 

  48. Matthews, P. (1987) Mixing Rates for a Random Walk on the Cube, SIAM J. Algebraic Discrete Methods, Vol. 8, pp. 746–752.

    Article  MathSciNet  MATH  Google Scholar 

  49. Miclo, L. (1995) Comportement de Spectres d’Operateurs de Schrodinger a Basse Temperature, Bull Set. Math., Vol.119, pp. 529–553.

    MathSciNet  MATH  Google Scholar 

  50. Matkowsky, B.J. and Schuss, Z. (1979) The Exit Problem: a New Approach to Diffusion Across Potential Barriers, SIAM J. Appl. Math., Vol. 36, pp. 604–623.

    Article  MathSciNet  MATH  Google Scholar 

  51. Maier, R.S. and Stein, D.L. (1997) Limiting Exit Location Distributions in the Stochastic Exit Problem, SIAM J. Appl Math., Vol.57, pp. 752–790.

    Article  MathSciNet  MATH  Google Scholar 

  52. Olivieri, E. and Picco, P. (1991) On the Existence of Thermodynamics for the Random Energy Model, Commun. Math. Phys., Vol. 96, pp. 125–144.

    Article  MathSciNet  Google Scholar 

  53. Reed, M. and Simon, B. (1978) Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, New York-London.

    MATH  Google Scholar 

  54. Ruelle, D. (1987) A Mathematical Reformulation of Derrida’s REM and GREM, Commun. Math. Phys., Vol. 108, pp. 225–239.

    Article  MathSciNet  MATH  Google Scholar 

  55. Saloff-Coste, L. (1997) Lectures on Finite Markov Chains, Lectures on Probability Theory and Statistics, Saint-Flour, 1996, Lecture Notes in Math., Springer Verlag, Berlin, Vol.1665, pp. 301–413.

    Google Scholar 

  56. Scoppola, E. (1995) Renormalization and Graph Methods for Markov Chains, Advances in Dynamical Systems and Quantum Physics, Capri, 1993, World Sci. Publishing, River Edge, NJ, pp. 260–281.

    Google Scholar 

  57. Sznitman, A.-S. (1998) Brownian Motion, Obstacles and Random Media, Springer Monographs in Mathematics, Springer Verlag, Berlin.

    Book  MATH  Google Scholar 

  58. Soardi, P.M. (1994) Potential Theory on Infinite Networks, LNM 1590, Springer Verlag, Berlin.

    Google Scholar 

  59. Talkner, P. (1987) Mean First Passage Times and the Lifetime of a Metastable State, Z. Phys., Vol.B 68, pp. 201–207.

    Article  MathSciNet  Google Scholar 

  60. Wentzell, A.D. (1972) On the Asymptotic Behaviour of the Greatest Eigenvalue of a Second Order Elliptic Differential Operator with a Small Parameter in the Higher Derivatives, Soviet Math. Doci, Vol. 13, pp. 13–17.

    Google Scholar 

  61. Wentzell, A.D. (1973) Formulas for Eigenfunctions and Eigenmeasures that are Connected with a Markov Process, Teor. Verojatnost. i Primenen., Vol.18, pp. 329.

    Google Scholar 

  62. Wigner, E.P. (1938) Trans. Faraday Soc. Vol.34, pp. 29.

    Article  Google Scholar 

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Authors and Affiliations

  1. Weierstrass-Institut für Angewandte Analysis und Stochastik, Mohrenstrasse 39, D-10117, Berlin, Germany

    Anton Bovier

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  1. Anton Bovier
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Editors and Affiliations

  1. Universidad de Chile, Santiago, Chile

    Alejandro Maass , Servet Martínez  & Jaime San Martín ,  & 

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Bovier, A. (2004). Metastability and Ageing in Stochastic Dynamics. In: Maass, A., Martínez, S., Martín, J.S. (eds) Dynamics and Randomness II. Nonlinear Phenomena and Complex Systems, vol 10. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-2469-6_2

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  • DOI: https://doi.org/10.1007/978-1-4020-2469-6_2

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  • Print ISBN: 978-90-481-6565-0

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