The thermal statistics of quasi-probabilities' analogs in phase space

We focus attention upon the thermal statistics of the classical analogs of quasi-probabilities's (QP) in phase space for the important case of quadratic Hamiltonians. We consider the three more important OPs: 1) Wigner's, $P$-, and Husimi's. We show that, for all of them, the ensuing semiclassical entropy is a function {\it only} of the fluctuation product $\Delta x \Delta p$. We ascertain that {\it the semi-classical analog of the $P$-distribution} seems to become un-physical at very low temperatures. The behavior of several other information quantifiers reconfirms such an assertion in manifold ways. We also examine the behavior of the statistical complexity and of thermal quantities like the specific heat.