Kinetic equation for a weakly relativistic system with gravitational interaction

Kinetic equations of the type of those of Vlasov and Landau are obtained for systems with gravitational interaction by the use of the weakly relativistic BBGKY hierarchy. Allowance is made for the $\varepsilon$ and $\varepsilon^2$ orders of magnitude in the dimensionless coupling constant $\varepsilon$. It is shown that the collision integral in the weakly relativistic Landau equation vanishes identically when the momentum distribution function $A(1-p^2/m^2c^2)\times\exp(-\gamma p^2-3\gamma p^4/4m^3c^2)$ is substituted. Here, $A$ and $\gamma$ are constants, $c$ is the velocity of light, $m$ is the rest mass of the particle, $\mathbf p=m(d\mathbf q/dt)$, $\mathbf q$, is the coordinate of the particle, and $t$ is the time of an observer in an inertial frame of reference.