Geodesic
Metabasins -- a state space aggregation for highly disordered energy landscapes
Teoriâ slučajnyh processov, Tome 18 (2012) no. 1, pp. 3-44.

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Glass-forming systems, characterized by a highly disordered energy landscape, have been studied in physics by a simulation-based state space aggregation. Choosing a path-independent approach within the framework of finite Markov chains, this article provides an aggregation procedure which, at an appropriate aggregation level, leads to the definition of certain metastates, called metabasins in Physics (for their properties see the Introduction). Roughly speaking, this will be the finest aggregation such that transitions back to an already visited (meta-)state are very unlikely within a moderate time frame.
Keywords: Metastability, metabasins, Markov chain aggregation, disordered systems, exit time, Metropolis algorithm. 
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G. Alsmeyer; A. Winkler. Metabasins -- a state space aggregation for highly disordered energy landscapes. Teoriâ slučajnyh processov, Tome 18 (2012) no. 1, pp. 3-44. http://geodesic.mathdoc.fr/item/THSP_2012_18_1_a0/

[1] V. S. Barbu, N. Limnios, Semi-Markov chains and hidden semi-Markov models toward applications: Their use in reliability and DNA analysis, Lecture Notes in Statistics, 191, Springer, New York, 2008 | MR | Zbl

[2] J. Beltrán, C. Landim, “Metastability of reversible finite state markov processes”, Stochastic Processes and their Applications, 121:8 (2011), 1633 – 1677 | DOI | MR | Zbl

[3] J. P. Bouchaud, “Weak ergodicity breaking and aging in disordered systems”, Journal de Physique I France, 2 (1992), 1705–1713 | DOI

[4] A. Bovier, “Metastability: a potential theoretic approach”, Proceedings of the International Congress of Mathematicians (Madrid, Spain, 2006), v. III, Europen Mathematical Society, Zürich, 2006, 499–518 | MR | Zbl

[5] A. Bovier, M. Eckhoff, V. Gayrard, M. Klein, “Metastability in stochastic dynamics of disordered mean-field models”, Probab. Theory Related Fields, 119:1 (2001), 99–161 | DOI | MR | Zbl

[6] M. Cassandro, A. Galves, E. Olivieri, M. E. Vares, “Metastable behavior of stochastic dynamics: a pathwise approach”, J. Statist. Phys., 35:5-6 (1984), 603–634 | DOI | MR | Zbl

[7] J. N. Darroch, E. Seneta, “On quasi-stationary distributions in absorbing discrete-time finite Markov chains”, J. Appl. Probability, 2:1 (1965), 88–100 | DOI | MR | Zbl

[8] M. I. Freidlin, A. D. Wentzell, Random perturbations of dynamical systems, Translated from the 1979 Russian original by Joseph Szücs, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 260, second edition, Springer-Verlag, New York, 1998 | DOI | MR | Zbl

[9] G. Frobenius, “Über Matrizen aus nicht negativen Elementen”, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaft, 1912, 456–477 | Zbl

[10] F. R. Gantmacher, Applications of the theory of matrices, Translated by J. L. Brenner, with the assistance of D. W. Bushaw and S. Evanusa, Interscience Publishers, Inc., New York, 1959 | MR | Zbl

[11] M. Goldstein, “Viscous liquids and the glass transition: A potential energy barrier picture”, The Journal of Chemical Physics, 51:9 (1969), 3728–3739 | DOI

[12] A. Heuer, “Exploring the potential energy landscape of glass-forming systems: from inherent structures via metabasins to macroscopic transport”, J. Phys.: Condens. Matter, 20:37 (2008), 373101 | DOI

[13] T. R. Kirkpatrick, D. Thirumalai, P. G. Wolynes, “Scaling concepts for the dynamics of viscous liquids near an ideal glassy state”, Phys. Rev. A, 40:2 (1989), 1045–1054 | DOI

[14] T. M. Liggett, Interacting particle systems, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 276, Springer-Verlag, New York, 1985 | DOI | MR | Zbl

[15] T. Okushima, T. Niiyama, K. S. Ikeda, Y. Shimizu, “Graph-based analysis of kinetics on multidimensional potential-energy surfaces”, Phys. Rev. E, 80:3 (2009), 036112 | DOI | MR

[16] E. Olivieri, E. Scoppola, “Markov chains with exponentially small transition probabilities: first exit problem from a general domain. {II}. The general case”, J. Statist. Phys., 84:5-6 (1996), 987–1041 | DOI | MR | Zbl

[17] P. K. Pollett, Quasi-stationary distributions: A bibliography, Version 2010 } {\tt http://www.maths.uq.edu.au/p̃kp/papers/qsds/qsds.html

[18] O. Rubner, A. Heuer, “From elementary steps to structural relaxation: A continuous-time random-walk analysis of a supercooled liquid”, Phys. Rev. E, 78:1 (2008), 011504 | DOI

[19] S. Sastry, P. G. Debenedetti, F. H. Stillinger, “Signatures of distrinct dynamical regimes in the energy landscape of a glass-forming liquid”, Nature, 393 (1998), 554–557 | DOI

[20] E. Scoppola, “Renormalization group for Markov chains and application to metastability”, J. Statist. Phys., 73:1-2 (1993), 83–121 | DOI | MR | Zbl

[21] E. Scoppola, “Metastability for Markov chains: a general procedure based on renormalization group ideas”, Probability and phase transition (Cambridge, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 420, Kluwer Acad. Publ., Dordrecht, 1994, 303–322 | MR | Zbl

[22] N. G. van Kampen, Stochastic Processes in Physics and Chemistry, 1st edition, North Holland, 1987