Entropy Geometry and Normalized Means on Infinite-Dimensional Hamiltonian Manifolds

We propose a geometric--analytic framework for equilibrium statistical mechanics on infinite-dimensional Hamiltonian systems. In situations where no suitable $\sigma$-additive invariant measure is available, we use \emph{normalized means}, which generalize probability measures and normalized integrals. This construction yields entropy and free-energy functionals on weak symplectic Fr\'echet manifolds and gives existence and uniqueness of exponential-family equilibrium states under explicit admissibility and separation assumptions. These states are stationary under Hamiltonian flows preserving both the reference mean and the equilibrium weight, and satisfy a classical Poisson--KMS identity when the reference mean is Poisson invariant. Under a local exponential regularity assumption, the logarithmic partition functional is smooth and convex, with Hessian given by the covariance form. It is strictly convex modulo thermodynamically null directions and, through Legendre--Fenchel duality, induces a concave entropy on the domain of extensive variables. We illustrate the framework with $H^s$-geodesic equations on current groups $\operatorname{Map}(M,G)$ and diffeomorphism groups $\operatorname{Diff}(M)$, including hydrodynamic and field-theoretic examples.