  Recently, we have derived a generalization of the known canonical fluctuation
relation $k_{B}C=\beta^{2}< \delta U^{2} >$ between heat capacity $C$ and
energy fluctuations, which can account for the existence of macrostates with
negative heat capacities $C<0$. In this work, we presented a panoramic overview
of direct implications and connections of this fluctuation theorem with other
developments of statistical mechanics, such as the extension of canonical Monte
Carlo methods, the geometric formulations of fluctuation theory and the
relevance of a geometric extension of the Gibbs canonical ensemble that has
been recently proposed in the literature.
