Logotropic Distributions

In all spatial dimensions d, we study the static and dynamical properties of a generalized Smoluchowski equation which describes the evolution of a gas obeying a logotropic equation of state, p = A ln ρ. A logotrope can be viewed as a limiting form of polytrope (p = Kρ γ , γ = 1 + 1/n), with index γ = 0 or n = −1. In the language of generalized thermodynamics, it corresponds to a Tsallis distribution with index q = 0. We solve the dynamical logotropic Smoluchowski equation in the presence of a fixed external force deriving from a quadratic potential, and for a gas of particles subjected to their mutual gravitational force. In the latter case, the collapse dynamics is studied for any negative index n, and the density scaling function is found to decay as r −α , with α = 2n n−1 for n < − d 2 , and α = 2d d+2 for − d 2 ≤ n < 0.