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 ${\mathit{scrH}}_{\mathit{e}\mathit{f}\mathit{f}}$ , dependent on the Planck constant ħ and on the temperature T=${\mathrm{\beta }}^{\mathrm{-}1}$, 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 ${\mathit{e}}_{\mathit{e}\mathit{f}\mathit{f}}^{\mathrm{-}\mathrm{\beta }\mathit{s}\mathit{c}\mathit{r}\mathit{H}}$. 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.