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 (N1, N2, V ) → ∞, keeping N1/V 1 3 and N2/V 1 3 fixed. The saddle point approximation becomes here exact for (N1, N2, V ) → ∞. It provides a nonlinear differential equation on the densities of each kind of particles. For the spherically symmetric case, we compute the densities as functions of two dimensionless physical parameters: η1 = Gm1N1 V 1 3 T and η2 = Gm2N2 V 1 3 T (where G is Newton’s constant, m1 and m2 the masses of the two kinds of particles and T the temperature). According to the values of η1 and η2 the system can be either in a gaseous phase or in a highly condensed phase. The gaseous phase is stable for η1 and η2 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) ∝ Rd suggesting a fractal structure. The value of d depends in general on η1 and η2 except on the critical line for the canonical ensemble in the (η1, η2) plane where it takes the universal value d ≃ 1.6 for all values of N1/N2. 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 η1 and η2. They exhibit a square root Riemann sheet with the branch points on the critical canonical line. The behaviour 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.