# American Institute of Mathematical Sciences

November  2009, 8(6): 1895-1916. doi: 10.3934/cpaa.2009.8.1895

## Rigorous derivation of the Landau equation in the weak coupling limit

 1 Massachusetts Institute of Technology, 77 Mass. Ave., Cambridge, MA 02139, United States

Received  August 2008 Revised  April 2009 Published  August 2009

We examine a family of microscopic models of plasmas, with a parameter $\alpha$ comparing the typical distance between collisions to the strength of the grazing collisions. These microscopic models converge in distribution, in the weak coupling limit, to a velocity diffusion described by the linear Landau equation (also known as the Fokker-Planck equation). The present work extends and unifies previous results that handled the extremes of the parameter $\alpha$ to the whole range $(0, 1/2]$, by showing that clusters of overlapping obstacles are negligible in the limit. Additionally, we study the diffusion coefficient of the Landau equation and show it to be independent of the parameter.
Citation: Kay Kirkpatrick. Rigorous derivation of the Landau equation in the weak coupling limit. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1895-1916. doi: 10.3934/cpaa.2009.8.1895
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