Abstract
The correlated-factors theorem states that the harmonic average of a product of factors is given exactly by the result of a differential operator ${\mathit{e}}^{\mathit{W}}$ acting on the product of the averages of the factors. The theorem is true both classically and quantum mechanically. It is proved and the form of the operator W is found. The theorem suggests an approximation for calculating the free energy of anharmonic solids. As an example, the approximate free energy is used to calculate the specific heat, magnetization, susceptibility, and the dependence of the phase-transition temperature on the coupling constant for the ${\mathrm{\ensuremath{\varphi}}}^{4}$ model on a square lattice. By comparing with Monte Carlo results, this approximation is found to be significantly more accurate than the correlated Einstein model, self-consistent phonon theory, and mean-field theory.
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