Extended variational method in statistical mechanics

Abstract

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 " (n =1,2, . . .) and those of the type ax +bx, 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 meaningfulwituations, and a quite satisfactory description is provided (with a "single shot") for both limits T~O and T~oo simultaneously.

On the Thermodynamical‐Statistical Consistency of a Variational Approximation for the Free Energy

“…It is worth noticing that (9) for the approximate specific heat obtained through the Bogolyubov inequality cannot satisfy a relation such as (4) and the approximate isothermal susceptibility (8) cannot clearly satisfy a fluctuation-dissipation relation. The present approximation seems to be a more realistic approach than the Bogolyubov method because it can produce thermodynamical relations similar to the exact ones (cf.…”

“…Attempts to improve the results obtained through the Bogolyubov method have been made [4,5] but they neither preserve the thermodynamical-statistical consistency ((2) and (3)) nor produce a proper approximate ensemble. These attempts may be considered as particular cases of a more general upper bound for the free energy F Among several possibilities we may consider meaning that V is divided in M sets of N , terms, and we may choose D as I n order to ( D ) , = 1 we assume that…”

A New Effective Ensemble Approach for the Free Energy

“…Therefore, A is formally a density matrix associated with an effective Hamiltonian H A defined in (6) and there is a corresponding effective ensemble that has the same partition function of H,, (5). Thermal averages in this effective ensemble may be calculated as modified averages in the H, ensemble,…”

“…Perreira et al[5] employed the Bogolyubov principle combining cluster division and series approximation. Tsallis and da Silva[6] proposed what they called an extended variational method, obtained from a cumulant expansion and studied two types of classical anharmonic single oscillators. It has already been used t o calculate critical temperatures of the Ising ferromagnet [ 7 3. trivial expression shown in Section 2 allows the calculation of thermal functionsof a system described by a Hamiltonian H related t o thermal averages calculated in a simpler ensemble associated with a Hamiltonian H,, but that calculation may not be easier.…”

A generalization of the cumulant expansion. Application to a scale-invariant probabilistic model

“…. Relation (4) is a convenient re-writting of the relations given in [1,2]. An alternative connection between cumulants and moments can be seen in [3].…”