After general considerations on the properties of distribution functions (DF) for steady state stellar systems, I will talk about the choice of the DF. In the 3D gravitational context, I deal with a very old Chandrasekhar's conjecture (1942): What are the properties of a stellar system which DF depends only on the mean particle energy E (The inverse but unphysical problem is trivial), and I will use a rigourous theorem from Gidas, Ni and Nirenberg, 1981 to answer. In the 2D gravitational context, I will present a recent work (Aly and Perez, 1999) in which we obtain DF and gravitational potential for an unbounded system from thermodynamical consideration. An application of such a potential to orbits in thin stellars disks will be presented.