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Relaxation Height in Energy Landscapes: An Application to Multiple Metastable States

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Abstract

The study of systems with multiple (not necessarily degenerate) metastable states presents subtle difficulties from the mathematical point of view related to the variational problem that has to be solved in these cases. We prove sufficient conditions to identify multiple metastable states. Since this analysis typically involves non-trivial technical issues, we give different conditions that can be chosen appropriately depending on the specific model under study. We show how these results can be used to attack the problem of multiple metastable states via the use of the modern approaches to metastability. We finally apply these general results to the Blume–Capel model for a particular choice of the parameters for which the model happens to have two multiple not degenerate in energy metastable states. We estimate in probability the time for the transition from the metastable states to the stable state. Moreover we identify the set of critical configurations that represent the minimal gate for the transition.

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Notes

  1. Since the energy landscape is reversible, the energy of the state ω n is implicitly taken into account in (2.2), indeed (2.1) implies H(ω n )≤Δ(ω n−1,ω n )+H(ω n−1).

  2. The centers of the unit squares forming a connected polyomino are a nearest neighbor connected subset of ℤ2. As in [1] here a polyomino is not necessarily connected, different definitions, see for instance [6], can be found in the literature.

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Acknowledgements

This work has been partially done at Eurandom during the Stochastic Activity Month, February 2012. ENMC wants to express his thanks to the organizers, R. Fernandez, R. van der Hofstad, and M. Heydenreich, for the invitation and to Eurandom for the kind hospitality. ENMC also thanks R. Cerf for clarifying discussions on polyominoes. FRN thanks F. den Hollander and A. Troiani for many interesting discussions related to the topic of the paper.

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Authors and Affiliations

  1. Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sapienza Università di Roma, via A. Scarpa 16, 00161, Rome, Italy

    Emilio N. M. Cirillo

  2. Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    Francesca R. Nardi

  3. Eurandom, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    Francesca R. Nardi

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  1. Emilio N. M. Cirillo
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  2. Francesca R. Nardi
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Correspondence to Emilio N. M. Cirillo.

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Cirillo, E.N.M., Nardi, F.R. Relaxation Height in Energy Landscapes: An Application to Multiple Metastable States. J Stat Phys 150, 1080–1114 (2013). https://doi.org/10.1007/s10955-013-0717-9

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  • DOI: https://doi.org/10.1007/s10955-013-0717-9

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