System response time and optimal path

In this work we study, in general terms, the optimal path for an out-of-equilibrium process from a geometric perspective constructed from a probability density function belonging to the exponential family (PDFE). With this objective we have constructed the speed of entropy production for a system out of equilibrium. This speed of entropy production is not determined with respect to the clock time t but rather with respect to a dimensionless time ξ t associated with a characteristic response time of each system λ t . In this context, we demonstrate that the constant entropy production speed condition leads to the system moving on a geodesic of the manifold constructed from the probability density function. In the last part of this work we apply our general results to study the Ornstein–Uhlenbeck process (OU), which is a well-known stochastic process that strictly speaking describes the velocity of a Brownian particle under the influence of friction. As the system evolves we find that dimensionless time ξ t passes slower than clock time t .