HIGHER ORDERS OF SELF-SIMILAR APPROXIMATIONS FOR THERMODYNAMIC POTENTIALS

The problem of calculating thermodynamic potentials in statistical mec hanics by using the method of self-similar approximations developed by the authors is considered. An emphasis is made on the search for a re gular procedure of defining higher-order terms providing a good accura cy and stability of the method. It is shown how the renormalized pertu rbation theory can be reformulated to the language of dynamical theory . Then, instead of sequences of approximants one can study trajectorie s of cascades whose fixed-point attractors play the role of the limits for the corresponding sequences of perturbative terms. As an illustra tion of the procedure the free energy of a zero-dimensional phi4 model is calculated up to the third order of the self-similar approximation whose precision is found to be greater than 0.1% for all coupling par ameters ranging from zero to infinity.