An exact calculation of the free energy in systems with separable interactions

Abstract

By deriving an upper and a lower bound we give a rigorous calculation of the free energy, in the thermodynamic limit, for a general class of model systems, characterized by a hamiltonian that contains a one-particle part and separable two-particle operators. The result is an expression for the free energy which is of the molecular-field type. The upper bound is obtained by a variational type of argument. Using the Trotter product formula and a well-known integral representation the partition function can be expressed as a multidimensional integral of a function e - NG . In the derivation of the lower bound we have employed Laplace's method. The absolute minimum of the function G can be obtained using Hölder's inequality for operators. In addition the second derivatives of G at the minimum are investigated in detail.