Statistical Mechanics of the Self-gravitating gas with two or more kinds of Particles

We study the statistical mechanics of the self-gravitating gas at thermal equilibrium with two kinds of particles. We start from the partition function in the canonical ensemble which we express as a functional integral over the densities of the two kinds of particles for a large number of particles. The system is shown to possess an infinite volume limit when (N_1,N_2,V)->infty, keeping N_1/V^{1/3} and N_2/V^{1/3} fixed. The saddle point approximation becomes here exact for (N_1,N_2,V)->infty.It provides a nonlinear differential equation on the particle densities. For the spherically symmetric case, we compute the densities as functions of two dimensionless physical parameters: eta_1=G m_1^2 N_1/[V^{1/3} T] and eta_2=G m_2^2 N_2/[V^{1/3} T] (where G is Newton's constant, m_1 and m_2 the masses of the two kinds of particles and T the temperature). According to the values of eta_1 and eta_2 the system can be either in a gaseous phase or in a highly condensed phase.The gaseous phase is stable for eta_1 and eta_2 between the origin and their collapse values. The gas is inhomogeneous and the mass M(R) inside a sphere of radius R scales with R as M(R) propto R^d suggesting a fractal structure. The value of d depends in general on eta_1 and eta_2 except on the critical line for the canonical ensem- ble where it takes the universal value d simeq 1.6 for all values of N_1/N_2. The equation of state is computed.It is found to be locally a perfect gas equation of state. Thermodynamic functions are computed as functions of eta_1 and eta_2. They exhibit a square root Riemann sheet with the branch points on the critical canonical line. This treatment is further generalized to the self-gravitating gas with n-types of particles.