An effective hamiltonian for quantum statistical mechanics

After having expressed the partition function of a general hamiltonian system in path-integral form, we introduce a variational method in order to derive an effective hamiltonian function. The latter can be introduced in a classical like phase-space integral, which gives an approximation of the partition function and yields the exact quantum result in the case of quadratic systems. The method starts from the Feynman-Jensen inequality, whose validity in the most general case is, until now, not well established. Application is made to the one-dimensional Double Sine-Gordon field.