Landau damping is the tendency of solutions to the Vlasov equation towards spatially homogeneous distribution functions. The distribution functions, however, approach the spatially homogeneous manifold only weakly, and Boltzmann entropy is not changed by the Vlasov equation. On the other hand, density and kinetic energy density, which are integrals of the distribution function, approach spatially homogeneous states strongly, which is accompanied by growth of the hydrodynamic entropy. Such a behavior can be seen when the Vlasov equation is reduced to the evolution equations for density and kinetic energy density by means of the Ehrenfest reduction.

Department of Mathematics FNSPE Czech Technical University Prague Trojanova 13 120 00 Prague Czech Republic

École Polytechnique de Montréal C P 6079 suc Centre ville Montréal QC H3C 3A7 Canada

Mathematical Institute Faculty of Mathematics and Physics Charles University Sokolovská 83 186 75 Prague Czech Republic

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Gradient and GENERIC time evolution towards reduced dynamics

Dynamic Maximum Entropy Reduction