Thermodynamic explanation of Landau damping by reduction to hydrodynamics

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.