Virial theorem and dynamical evolution of self-gravitating Brownian particles in an unbounded domain: II. Inertial models
Résumé
We propose a general kinetic and hydrodynamic description of self-gravitating Brownian particles in $d$ dimensions. We go beyond usual approximations by considering inertial effects and finite $N$ effects while previous works use a mean-field approximation valid in a proper thermodynamic limit ($N\rightarrow +\infty$) and consider an overdamped regime ($\xi \rightarrow +\infty$). We recover known models in some particular cases of our general description. We derive the expression of the Virial theorem for self-gravitating Brownian particles and study the linear dynamical stability of isolated clusters of particles and uniform systems by using technics introduced in astrophysics. We investigate the influence of the equation of state, of the dimension of space and of the friction coefficient on the dynamical stability of the system. We obtain the exact expression of the critical temperature $T_{c}$ for a multi-components self-gravitating Brownian gas in $d=2$. We also consider the limit of weak frictions $\xi\rightarrow 0$ and derive the orbit-averaged-Kramers equation.