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Metastability: A Brief Introduction Through Three Examples

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Book cover World Women in Mathematics 2018

Part of the book series: Association for Women in Mathematics Series ((AWMS,volume 20))

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Abstract

Metastability is a very frequent phenomenon in nature. It also finds many applications in science and engineering. A noticeable basic feature is the presence of “quasi-equilibria states” and relatively sudden transitions between them. The goal of this short expository note is to discuss some aspects of the stochastic modeling of metastability, usually done through the consideration of special stochastic processes. This includes a “pathwise approach” developed since the 1980s. Thought as an invitation to the readership, three examples are quickly reviewed, starting with a class of reaction-diffusion equations subject to a small stochastic noise, for which the theory of large deviations has been a very useful tool, and further precision achieved through the help of potential theoretical techniques. We present then brief summaries of results on the Harris contact process on suitable finite graphs, and a quick discussion of stochastic dynamics for the well-known Ising model. The first can be thought as an oversimplified model for the propagation of an infection, and the second has been used in the context of magnetization. From a probabilistic analysis and technical viewpoint, the Ising model enjoys time-reversibility, which provides useful tools, while the contact process is non-reversible.

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Notes

  1. 1.

    V  is the set of vertices and \(\mathcal {E}\) denotes the set of unordered edges.

  2. 2.

    Right continuous, with left limits.

  3. 3.

    We write h∕2 instead of h in (19) just to match with the notation in [30, 55].

  4. 4.

    This is not the situation described in the previous page where h → 0 and the volume must grow. It is much simpler and opened the door to a huge amount of work in the mathematical description.

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Acknowledgements

M. E. Vares acknowledges support of CNPq (grant 305075/2016-0) and FAPERJ (grant E-26/203.048/2016).

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Authors and Affiliations

  1. Instituto Venezolano de Investigaciones Científicas, Caracas, Venezuela

    Stella Brassesco

  2. Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brazil

    Maria Eulalia Vares

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  1. Stella Brassesco
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  2. Maria Eulalia Vares
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Correspondence to Maria Eulalia Vares .

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Editors and Affiliations

  1. Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Brazil

    Carolina Araujo

  2. Department of Mathematics, University of Wisconsin–Madison, Madison, WI, USA

    Georgia Benkart

  3. Department of Mathematics, University of Western Australia, Crawley, WA, Australia

    Cheryl E. Praeger

  4. Department of Mathematics, Boğaziçi University, Bebek, Istanbul, Turkey

    Betül Tanbay

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Brassesco, S., Vares, M.E. (2019). Metastability: A Brief Introduction Through Three Examples. In: Araujo, C., Benkart, G., Praeger, C., Tanbay, B. (eds) World Women in Mathematics 2018. Association for Women in Mathematics Series, vol 20. Springer, Cham. https://doi.org/10.1007/978-3-030-21170-7_3

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