Logotropic distributions

Abstract

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, $p=A\ln \rho$. A logotrope can be viewed as a limiting form of polytrope ($p=K{\rho }^{\gamma }$, $\gamma =1+1/n$), with index $\gamma =0$ 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 $r^{-\alpha }$, with $\alpha =\frac{2n}{n-1}$ for $n<-\frac{d}{2}$, and $\alpha =\frac{2d}{d+2}$ for $-\frac{d}{2}\le n<0$.

References in corpus (3)

• Dynamical stability of collisionless stellar systems and barotropic stars: the nonlinear Antonov first law
• Nonlinear mean-field Fokker-Planck equations and their applications in physics, astrophysics and biology
• Jeans type instability for a chemotactic model of cellular aggregation

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