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 $-\frac{d}{2}\le n<0$.

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• Cosmology with a stiff matter era
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• The Logotropic Dark Fluid as a unification of dark matter and dark energy
• Landau equation for self-gravitating classical and quantum particles: Application to dark matter
• Kinetic and hydrodynamic models of chemotactic aggregation
• Critical dynamics of self-gravitating Langevin particles and bacterial populations
• Comparison between the Logotropic and $\Lambda$CDM models at the cosmological scale
• Critical mass of bacterial populations and critical temperature of self-gravitating Brownian particles in two dimensions
• High resolution modeling of [CII], [CI], [OIII] and CO line emission from the ISM and CGM of a star forming galaxy at z ~ 6.5
• Kinetic theory of collisionless relaxation for systems with long-range interactions
• Phase separation of bacterial colonies in a limit of high degradation. Analogy with Jupiter's great red spot
• Generalized Euler, Smoluchowski and Schrödinger equations admitting self-similar solutions with a Tsallis invariant profile
• A new logotropic model based on a complex scalar field with a logarithmic potential
• Critical mass of bacterial populations in a generalized Keller-Segel model. Analogy with the Chandrasekhar limiting mass of white dwarf stars
• Elastic Planetoids