Geometro Thermodynamics of The Boltzmann-Gibbs and Pareto Distributions

The formalism of geometrothermodynamics is used to derive Riemannian metrics for the equilibrium spaces of thermodynamic systems described by the Botzmann-Gibbs and Pareto distributions, which are widely used in econophysics. We thus show that the corresponding equilibrium spaces are, in fact, Riemannian differential manifolds for which we can calculate the respective curvature. In the case of the Boltzmann-Gibbs distribution, we show that the curvature vanishes so that the manifold is globally flat. In the case of the Pareto distribution, the curvature is non-zero and singularities are present for a wide range of values of the Pareto parameter. We thus show that it is possible to characterize these distributions by using the geometric properties of the corresponding equilibrium manifolds. Keywords - Boltzmann-Gibbs Distribution, Pareto distribution, Econophysics, Geometrothermodynamics.