Abstract

The use of the divergence theorem to continuous probability densities with compact support leads to the conjugate variables theorem (CVT), an identity with a free, differentiable vector field. The CVT can be used to derive the equipartition theorem of classical statistical mechanics, as well as several formulas such as Rugh's temperature. In this work we augment this identity by including constraints of the form g(x) = G and considering a Bayesian update from the state of knowledge I to (G, I). The versatile identity that follows not only contains the CVT but also the fluctuation-dissipation theorem, and we show several examples of its use: performing coordinate transformations, computing distributions of sums of variables and densities of states.