The generalized exponential family in statistical physics: a framework for microcanonical phase transitions

Statistical models belonging to a generalized exponential family automatically satisfy the variational principle, which is a stronger statement than the maximum entropy principle. It implies a dual structure which in thermodynamics is known since long in the form of Legendre transforms and their inverses. The entropy function in the variational principle is essentially unique. For convenience, let us call it the stable entropy function. For the special case of the $q$-exponential family this is Tsallis' entropy function with parameter $2-q$.

Statistical models belonging to the $q$-exponential family occur frequently in statistical physics. In particular, the configurational probability distribution of any model of classical mechanics, when considered as a function of the total energy, belongs to the $q$-exponential family, with a parameter $q$ which tends to 1 when the number of degrees of freedom tends to infinity. It is well-known from the canonical case of the Boltzmann-Gibbs distribution that the variational principle implies a property of stability, which roughly means that phase transitions cannot occur in systems with a finite number of degrees of freedom. This stability holds also for models belonging to a generalized exponential family.

If the stable entropy function is replaced by an increasing function of itself then the maximum entropy principle is still satisfied, but the variational principle is violated. In particular, if Tsallis' entropy is replaced by that of Rényi, then the stability property gets lost. In physical models the lack of stability means that phase transitions may occur. We show that, when Rényi's entropy function is used, the simple model of the pendulum exhibits a first order phase transition between small angle librational motion at low values of the energy and full rotational motion at high energies.