Vlasov Theory in Newtonian Gravity: Approach to Equilibrium, Phase Transitions, Gravothermal Catastrophe, and the Expanding Universe

Abstract

The Vlasov, or collisionless Boltzmann, equation provides an important tool for understanding the behavior of gravitational systems. In this review I will describe four applications of Vlasov theory that have contributed to this understanding, and provide comparisons with dynamical simulations for each. (1) The first gravitational computer simulations modeled one‐dimensional systems that, formally consist of infinite parallel mass planes moving perpendicular to their surface. Since the Vlasov limit is singular, in common with the Fermi–Pasta–Ulam system, the approach to equilibrium is problematic. I will describe how Vlasov theory was used to ascertain the approach to equilibrium of a planar system consisting of two different mass components. (2) Observations of galaxy positions strongly suggest that matter in the universe has a type of fractal geometry. I will use the time dependent Vlasov equation to show how stretching occurs in the phase space of an expanding universe, resulting in the formation of “voids” and “clusters”. (3) If the singularity of the gravitational potential is screened, the system can undergo a phase transition. I will review how Vlasov theory can be used to completely describe the central features of transitions in different ensembles. (4) Finally I will demonstrate that metastable Vlasov states play an important role in the rate of collapse of a spherical gravitational system.