  Quantum mechanics and classical statistical mechanics are two physical
theories that share several analogies in their mathematical apparatus and
physical foundations. In particular, classical statistical mechanics is
hallmarked by the complementarity between two descriptions that are unified in
thermodynamics: (i) the parametrization of the system macrostate in terms of
mechanical macroscopic observables $I=\{I^{i}\}$; and (ii) the dynamical
description that explains the evolution of a system towards the thermodynamic
equilibrium. As expected, such a complementarity is related to the uncertainty
relations of classical statistical mechanics $\Delta I^{i}\Delta \eta_{i}\geq
k$. Here, $k$ is the Boltzmann's constant, $\eta_{i}=\partial
\mathcal{S}(I|\theta)/\partial I^{i}$ are the restituting generalized forces
derived from the entropy $\mathcal{S}(I|\theta)$ of a closed system, which is
found in an equilibrium situation driven by certain control parameters
$\theta=\{\theta^{\alpha}\}$. These arguments constitute the central
ingredients of a reformulation of classical statistical mechanics from the
notion of complementarity. In this new framework, Einstein postulate of
classical fluctuation theory $dp(I|\theta)\sim\exp[\mathcal{S}(I|\theta)/k]dI$
appears as the correspondence principle between classical statistical mechanics
and thermodynamics in the limit $k\rightarrow0$, while the existence of
uncertainty relations can be associated with the non-commuting character of
certain operators.
