Higher orders of self-similar approximations for thermodynamic potentials

The problem of calculating thermodynamic potentials in statistical mechanics by using the method of self-similar approximations developed by the authors is considered. An emphasis is made on the search for a regular procedure of defining higher-order terms providing a good accuracy and stability of the method. It is shown how the renormalized perturbation theory can be reformulated to the language of dynamical theory. Then, instead of sequences of approximants one can study trajectories of cascades whose fixed-point attractors play the role of the limits for the corresponding sequences of perturbative terms. As an illustration of the procedure the free energy of a zero-dimensional ϕ4 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 parameters ranging from zero to infinity.