On Thermodynamic Information

Information based thermodynamic logic is revisited. It consists of two parts: Part A applies the modern theory of probability in which an arbitrary convex function φis employed as an analytic “device” to express information as statistical dependency contained in the topological sub-σ-algebra structure. Via thermo-doubling, Fenchel-Young equality (FYE) that consists of φ(x) and its conjugate ψ(y) establishes the notion of equilibrium between x and y through duality symmetry and the principle of maximum entropy/minimum free energy. Part B deals with a given set of repetitive measurements, where an inherent convex function emerges via the mathematics of large deviations. Logarithm-based Shannon entropy with φ(x)=-\log x figures prominently for i.i.d. sample statistics. Information can be a measure of the agreement between a statistical observation and its theoretical models. Maximum likelihood principle arises here and FYE provides a thermodynamic energetic narrative of recurrent data.