The Landau Equation

Abstract

A weakly coupled gas/fluid is governed on the macroscopic scale by the Landau equation

$$ \frac{\partial }{{\partial t}}{{f}_{t}}\left( {x,v} \right) + v\cdot {{\nabla }_{x}}{{f}_{t}}\left( {x,v} \right)\,{\text{ = }}\alpha \sum\limits_{{i,j = 1}}^{3} {\frac{\partial }{{\partial vi}}} \int {{{d}^{3}}} w\left[ {{{{\left| {v - w} \right|}}^{{ - 3}}}\left( {{{{\left| {v - w} \right|}}^{2}}{{\delta }_{{ij}}} - {{{\left( {v - w} \right)}}_{i}}{{{\left( {v - w} \right)}}_{j}}} \right) \times \left( {\frac{\partial }{{\partial {{v}_{j}}}} - \frac{\partial }{{\partial {{v}_{j}}}}} \right){{f}_{t}}\left( {x,w} \right){{f}_{t}}\left( {x,v} \right)} \right]. $$

(6.1)

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