Equilibrium properties of the Vlasov functional: The generalized Poisson-Boltzmann-Emden equation

This article investigates in a systematic way the properties of the classical continuous mean-field theory governed by the generalized Poisson-Boltzmann-Emden equation $\rho \mathrm{}\left(x\right)=A\exp [-\beta \int {\Lambda }^{}\mathrm{dy}\rho \mathrm{}\left(y\right)V(x-y)\mathrm{}]$ together with the associated variational problem ${infop}_{\rho }\frac{1}{2}\int {\Lambda }^{}\mathrm{dx}\int {\Lambda }^{}\mathrm{dy}\rho \mathrm{}\left(x\right)\mathrm{}\rho \mathrm{}\left(y\right)V(x-y)\mathrm{}+kT\int {\Lambda }^{}\mathrm{dx}\rho \mathrm{}\left(x\right)\mathrm{}\ln \rho \mathrm{}\left(x\right)\mathrm{}$. Origins of the theory are traced back. Past studies (freezing theories, electrostatic and self-gravitating systems) are relocated in a broader framework. New results concerning the thermodynamic limit, phase transitions, metastability, and the shape of density profiles are provided. In particular, the question of ground states (in relationship to condensation and wetting phenomena) is illustrated by numerous explicit solutions.