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Diffusivity of Lattice Gases

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Abstract

We consider one-component lattice gases with local dynamics and a stationary product Bernoulli measure on \({\mathbb{Z}^d}\). We study the scaling exponents of the space-time correlations of the system in equilibrium at a given density. We consider a variance-like quantity computed from the correlations called the diffusivity (connected to the Green–Kubo formula) and give rigorous upper and lower bounds on it that depend on the dimension and the local behavior of the macroscopic flux function. Our results identify the cases in which the system scales superdiffusively; these cases have been predicted before, using non-rigorous scaling arguments. Our main tool is the resolvent method: the estimates are the result of a careful analysis of a complicated variational problem.

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Authors and Affiliations

  1. Departments of Mathematics and Statistics, University of Toronto, Toronto, Canada

    Jeremy Quastel

  2. Department of Mathematics, University of Wisconsin-Madison, Madison, USA

    Benedek Valkó

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  1. Jeremy Quastel
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Correspondence to Jeremy Quastel.

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Communicated by J. Fritz

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Quastel, J., Valkó, B. Diffusivity of Lattice Gases. Arch Rational Mech Anal 210, 269–320 (2013). https://doi.org/10.1007/s00205-013-0651-7

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