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=Alnρp=Alnρ. A logotrope can be viewed as a limiting form of polytrope (p=Kργp=Kργ, γ=1+1/nγ=1+1/n), with index γ=0γ=0 or n=-1n=-1. In the language of generalized thermodynamics, it corresponds to a Tsallis distribution with index q=0q=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-αr-α, with α=2n/(n-1)α=2n/(n-1) for n<-d/2n<-d/2, and α=2d/(d+2)α=2d/(d+2) for -d/2⩽n<0-d/2⩽n<0.