Description of instabilities in Uhlenbeck–Ornstein process

Abstract

The method of Riemannian geometry has been successful in the context of equilibrium thermodynamics. In this work, we extend this approach to non-equilibrium processes. As a geometric-differential frame of non-equilibrium systems, we consider in our study the geometric properties of a manifold associated with simple but typical non-equilibrium models. We consider a Uhlenbeck–Ornstein process and the formal structure of the probability density function solution of the Fokker–Planck equation. We propose a general geometric strategy for the construction of macroscopic potentials in non-equilibrium problems. This macroscopic potential is a function of the transport coefficient and is associated with system instabilities.