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The Brownian motion as the limit of a deterministic system of hardspheres
, 2013
"... We provide a rigorous derivation of the brownian motion as the hydrodynamic limit of systems of hardspheres as the number of particles N goes to infinity and their diameter ε simultaneously goes to 0, in the fast relaxation limit Nεd−1 → ∞ (with a suitable scaling of the observation time and leng ..."
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We provide a rigorous derivation of the brownian motion as the hydrodynamic limit of systems of hardspheres as the number of particles N goes to infinity and their diameter ε simultaneously goes to 0, in the fast relaxation limit Nεd−1 → ∞ (with a suitable scaling of the observation time and length). As suggested by Hilbert in his sixth problem, we use the linear Boltzmann equation as an intermediate level of description for one tagged particle in a gas close to global equilibrium. Our proof relies on the fundamental ideas of Lanford. The main novelty here is the detailed study of the branching process, leading to explicit estimates on pathological collision trees.
Approaches to derivation of the Boltzmann equation with hard sphere collisions
, 2013
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The Boltzmann–Grad Limit of a Hard Sphere System: Analysis of the Correlation Error
"... In memory of Oscar Erasmus Lanford III Abstract. We present a quantitative analysis of the Boltzmann–Grad (low–density) limit of a hard sphere system. We introduce and study a set of functions (correlation errors) measuring the deviations in time from the statistical independence of particles (propa ..."
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In memory of Oscar Erasmus Lanford III Abstract. We present a quantitative analysis of the Boltzmann–Grad (low–density) limit of a hard sphere system. We introduce and study a set of functions (correlation errors) measuring the deviations in time from the statistical independence of particles (propagation of chaos). In the context of the BBGKY hierarchy, a correlation error of order k measures the event where k tagged particles are connected by a chain of interactions preventing the factorization. We prove that, provided k is not too large, such an error flows to zero with the hard spheres diameter ε, for short times, as εγk, for some γ> 0. This requires a new analysis of many–recollision events, and improves previous estimates of high order
4 Exponential approach to, and properties of, a nonequilibrium steady state in a dilute gas
, 2014
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Proceedings of Symposia in Pure Mathematics Kinetic
"... limits of dynamical systems ..."
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