| We consider closures of the one-dimensional Vlasov-Poisson system, in which a system of fluid equations is generated from the kinetic theory of particles interacting through a mean field. Closure may be understood as the restriction of the dynamics of the probability density, a function of space, velocity, and time, to a form parametrized by functions of space and time alone. This generates a set of relations among the moments of the conditional probability density at a given time and point in space. For closures which exactly reproduce Vlasov-Poisson dynamics, these relations are subject to the requirement that various functions of the moments remain zero. Inexact closures may be understood as arising from Vlasov-Poisson dynamics with a source which constrains the motion of the probability density to the specified form. This source, whose moments are the previously mentioned functions, represents the emergence of fine scale structure in the probability density with respect to the closure. |
