Duality Symmetry, Two Entropy Functions, and an Eigenvalue Problem in Gibbs' Theory

We generalize the convex duality symmetry in Gibbs' statistical ensemble formulation, between Massieu's free entropy $\Phi_{V,N} (\beta)$ and the Gibbs entropy $\varphi_{V,N}(u)$ as a function of mean internal energy $u$. The duality tells us that Gibbs thermodynamic entropy is to the law of large numbers (LLN) for arithmetic sample means what Shannon's information entropy is to the LLN for empirical counting frequencies. Following the same logic, we identify $u$ as the conjugate variable to counting frequency, a Hamilton-Jacobi equation for Shannon entropy as an equation of state, and suggest an eigenvalue problem for modeling statistical frequencies of correlated data.