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Self-gravitating systems have negative specific heats, thus if heat is allowed to flow between two of them, the hotter one loses heat and gets yet hotter while the colder gains heat and gets yet colder. Evolution is thus away from equilibrium. When a single isothermal sphere within a non-conducting box is sufficiently centrally condensed a similar instability arises between the central parts and the outer parts; as a result no equilibrium states exist for an isothermal sphere of energy E (<0) and mass M within a spherical box of radius greater than 0. 335 GM2 / (− E). This is Antonov's discovery that no state of locally maximal entropy exists for stellar systems of given energy and mass contained within a rigid sphere of radius larger than this. The instability is distinct from that found by Ebert which is similar to the Schönberg–Chandrasekhar limit in stars and relates to isothermal spheres at fixed temperature. In fact there are four distinct critical points for instability of isothermal spheres which are related to the turning points of the four total thermodynamic free energies by Poincar^'s theory of linear series of equilibria.
Lynden–Bell et al. (Thu,) studied this question.