Extended variational method in statistical mechanics

Through cumulant expansions of the free energy and the susceptibility, a new variational procedure is proposed with the purpose of improving the standard variational method in equilibrium statistical mechanics. The procedure is tested for two types of classical anharmonic single oscillators, namely, those whose elastic potential is proportional to $x^{2n}(n=1,2,\dots )$ and those of the type $a{x}^2+b{x}^4$, whose exact free energy, specific heat, and susceptibility are herein established. Although convergence problems (similar to those appearing in the asymptotic series) exist (at least for the free energy) in the limit of high perturbative orders, great improvement (typically of the order of 40) with respect to the standard variational method is obtained in all the physically meaningful situations, and a quite satisfactory description is provided (with a "single shot") for both limits $T\to 0$ and $T\to \infty$ simultaneously.