  The Weyl-Wigner representation of quantum mechanics allows one to map the
density operator in a function in phase space - the Wigner function - which
acts like a probability distribution. In the context of statistical mechanics,
this mapping makes the transition from the classical to the quantum regimes
very clear, because the thermal Wigner function tends to the Boltzmann
distribution in the high temperature limit. We approximate this quantum phase
space representation of the canonical density operator for general temperatures
in terms of classical trajectories, which are obtained through a Wick rotation
of the semiclassical approximation for the Weyl propagator. A numerical scheme
which allows us to apply the approximation for a broad class of systems is also
developed. The approximation is assessed by testing it against systems with one
and two degrees of freedom, which shows that, for a considerable range of
parameters, the thermodynamic averages are well reproduced.
