Estudos das funções entrópicas

Abstract

This work conducts a mathematical characterization of the set B of Boltzmann functions, formed by strictly increasing real functions that satisfy the multiplicative additivity property f (xy) = f (x) + f (y). The study investigates the analytical and topological properties of this set and establishes a formal connection with the informational formulation of Shannon’s entropy. Using methods from real analysis and probability theory, it is demonstrated that the functions in B are analytic and of class C ∞ , and that the set is convex and connected, although it is not open, closed, or compact in the topology of uniform convergence on compact sets. The fundamental characterization reveals that the domain of B is the interval (0, ∞) and that B = {k ln x | k > 0}. From this basis, it is proven that Shannon’s entropy, when restricted to independent random variables, inherits the mathematical structure of B through a one-to-one correspondence. It is concluded that the logarithmic form of entropy is uniquely determined by its monotonicity and additivity, and that Shannon’s formulation generalizes that of Boltzmann-Gibbs by accommodating systems with arbitrary correlations, unifying the foundations of statistical thermodynamics and information theory.