Consequences of coarse-grained Vlasov equations

The Vlasov equation is analyzed for coarse-grained distributions resembling a finite width of test particles as used in numerical implementations. It is shown that this coarse-grained distribution obeys a kinetic equation similar to the Vlasov equation, but with additional terms. These terms give rise to entropy production indicating dissipative features due to a nonlinear mode coupling. The interchange of coarse graining and dynamical evolution is discussed with the help of an exactly solvable model for the self-consistent Vlasov equation and practical consequences are worked out. By calculating analytically the stationary solution of a general Vlasov equation we can show that a sum of modified Boltzmann-like distributions is approached dependent on the initial distribution. This behavior is independent of degeneracy and only controlled by the width of test particles. The condition for approaching a stationary solution is derived and it is found that the coarse graining energy given by the momentum width of test particles should be smaller than a quarter of the kinetic energy. Observable consequences of this coarse graining are: (i) spatial correlations in observables, (ii) too large radii of clusters or nuclei in self-consistent Thomas–Fermi treatments, (iii) a structure term in the response function resembling vertex correction correlations or internal structure effects and (iv) a modified centroid energy and higher damping width of collective modes.