The cluster expansion for the self-gravitating gas and the thermodynamic limit

We develop the cluster expansion and the Mayer expansion for the self-gravitating thermal gas and prove the existence and stability of the thermodynamic limit N,V→∞ with N/V13 fixed. The essential (dimensionless) variable is here η≡Gm2N(V13T) (which is kept fixed in the thermodynamic limit). We succeed in this way to obtain the expansion of the grand canonical partition function in powers of the fugacity. The corresponding cluster coefficients behave in the thermodynamic limit as (ηN)j−1cj, where cj are pure numbers. They are expressed as integrals associated to tree cluster diagrams. A bilinear recurrence relation for the coefficients cj is obtained from the mean field equations in the Abel's form. In this way the large j behaviour of the cj is calculated. This large j behaviour provides the position of the nearest singularity which corresponds to the critical point (collapse) of the self-gravitating gas in the grand canonical ensemble. Finally, we discuss why other attempts to define a thermodynamic limit for the self-gravitating gas fail.