Classical long-range interacting N-particle configurations and its applications

We consider classical N-particle system with arbitrary central pair potential. Mechanical equilibrium condition in spherically-symmetric case leads to a nonlinear integro-differential equation for concentration n(r). For special state equation $p=an^{\gamma}(1+bn)$, original integro-differential equation transforms into purely integral one. Its solution (under b=0) is written through a row over interaction parameter. Physical conditions for its convergency is discussed. For power-like potential kernal of integral operator is calculated in apparent kind. The cases of Coulomb and harmonic potentials are considered separately up to a third order. General scheme of application of the theory for some astrophysical and cosmological problems is presented. Model system with spherical potential pits is considered. By perturbation theory first virial correction $\delta n$ is calculated.