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)--> infinity, keeping N(1)/V(1/3) and N(2)/V(1/3) fixed. The saddle point approximation becomes here exact for (N1,N2,V)--> infinity. It provides a nonlinear differential equation for the densities of each kind of particle. For the spherically symmetric case, we compute the densities as functions of two dimensionless physical parameters: eta(1)=Gm(2)(1)N(1)/V(1/3)T and eta(2)=Gm(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. We have thus generalized the well-known isothermal sphere for two kinds of particles. The gas is inhomogeneous and the mass M(R) inside a sphere of radius R scales with R as M(R) proportional to 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 ensemble in the (eta(1),eta(2)) plane where it takes the universal value d approximately 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. The thermodynamic functions (free energy, energy, entropy) are expressed and plotted as functions of eta(1) and eta(2). They exhibit a square root Riemann sheet with the branch points on the critical canonical line. The behavior of the energy and the specific heat at the critical line is computed. This treatment is further generalized to the self-gravitating gas with n types of particles.