Abstract
The content of these notes is based on a series of lectures given by the first author at HIM, Bonn, in May 2019. They provide the material for a short introductory course on effective equations for classical particle systems. They concern the basic equations in kinetic theory, written by Boltzmann and Landau, describing rarefied gases and weakly interacting plasmas respectively. These equations can be derived formally, under suitable scaling limits, taking classical particle systems as a starting point. A rigorous proof of this limiting procedure is difficult and still largely open. We discuss some mathematical problems arising in this context.
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Notes
- 1.
Note that this is not the conventional form for the Boltzmann equation and usually the factor (v − v 1) ⋅ n is rewritten in terms of ω, which amounts to introduce the differential cross-section.
- 2.
The Landau equation was obtained from the Boltzmann equation for cutoffed Coulomb potential (truncated both at short and large distances). Actually the word “Coulomb” is frequently used for the Landau equation with kernel singularity \(\frac {1}{|U|}\) (see (38)), which is somehow misleading. In fact as we have seen, this singularity is always present.
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Pulvirenti, M., Simonella, S. (2021). A Brief Introduction to the Scaling Limits and Effective Equations in Kinetic Theory. In: Albi, G., Merino-Aceituno, S., Nota, A., Zanella, M. (eds) Trails in Kinetic Theory. SEMA SIMAI Springer Series, vol 25. Springer, Cham. https://doi.org/10.1007/978-3-030-67104-4_6
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