Quantum thermodynamics in classical phase space

We present a theoretical approach that permits the reproduction of the quantum-thermodynamic properties of a variety of physical systems with many degrees of freedom, over the whole temperature range. The method is based on the introduction of an effective classical Hamiltonian H(eff), dependent on the Planck constant hBAR and on the temperature T = beta--1, by means of which classical-like formulas for the thermodynamic quantities can be written. For instance, the partition function is expressed by the usual phase-space integral of e(-beta-H(eff)). The effective Hamiltonian is the generalization of a previous effective potential. The latter was obtained in the case of standard Hamiltonians, i.e., with a separated quadratic kinetic energy, and has been successfully used in a number of applications. The starting point of the method is the path-integral expression for the unnormalized density operator, and we exploit Feynman's idea of classifying paths by the equivalence relation of having the same average phase-space point. The contribution of each class of paths to the density matrix is approximated within a generalized self-consistent harmonic approximation (SCHA). We show that the framework is consistent with the usual SCHA, which, however, only holds at low temperatures, as well as with the semiclassical high-temperature expansion by Wigner and Kirkwood. The practical implementation of the method is made straightforward by a further approximation of low quantum coupling.