Nonlinear thermodynamical formalism
Jérôme Buzzi, Benoît Kloeckner, Renaud Leplaideur
Abstract
We define a nonlinear thermodynamical formalism which translates into dynamical system theory the statistical mechanics of generalized mean-field models, extending the investigation of the quadratic case in one or more potentials by Leplaideur and Watbled. We prove a variational principle for the nonlinear pressure and we characterize the nonlinear equilibrium measures and relate them to specific classical equilibrium measures. In this non-linear thermodynamical formalism, as for mean-field theories of statistical mechanics, several kind of phase transitions appear, some of which cannot happen in the linear case. Our techniques can deal with known cases (Curie–Weiss and Potts models) as well as with new examples (metastable phase transition). Finally, we apply some of these ideas to the classical, linear setting proving that freezing phase transitions can occur over any zero-entropy invariant compact subset of the phase space.