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A Particle System with Explosions: Law of Large Numbers for the Density of Particles and the Blow-Up Time

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Abstract

Consider a system of independent random walks in the discrete torus with creation-annihilation of particles and possible explosion of the total number of particles in finite time. Rescaling space and rates for diffusion/creation/annihilation of particles, we obtain a strong law of large numbers for the density of particles in the supremum norm. The limiting object is a classical solution to the semilinear heat equation ∂ t u=∂ xx u+f(u). If f(u)=u p, 1<p≤3, we also obtain a law of large numbers for the explosion time.

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Acknowledgements

We want to thank Pablo Ferrari, Milton Jara and Mariela Sued for fruitful discussions.

PG is partially supported by UBACyT 20020090100208, ANPCyT PICT No. 2008-0315 and CONICET PIP 2010-0142 and 2009-0613. TF acknowledges support from ANPCyT Argentina through a post-doctoral fellowship.

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Authors and Affiliations

  1. Universidade Federal da Bahia, Salvador, Brazil

    Tertuliano Franco

  2. Departamento de Matemática, Fac. Cs. Exactas y Naturales, UBA and IMAS-CONICET, Buenos Aires, Argentina

    Pablo Groisman

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  1. Tertuliano Franco
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  2. Pablo Groisman
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Correspondence to Pablo Groisman.

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Franco, T., Groisman, P. A Particle System with Explosions: Law of Large Numbers for the Density of Particles and the Blow-Up Time. J Stat Phys 149, 629–642 (2012). https://doi.org/10.1007/s10955-012-0621-8

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  • DOI: https://doi.org/10.1007/s10955-012-0621-8

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