Family of Probability Distributions Derived from Maximal Entropy Principle with Scale Invariant Restrictions

TL;DR

This paper derives a family of probability distributions from the maximum entropy principle with scale invariance, applicable to non-equilibrium thermodynamic systems and intermittency models, linking thermodynamics and stochastic processes.

Contribution

It introduces a new family of probability distributions based on maximal entropy with scale invariance, applicable to non-equilibrium thermodynamics and intermittency phenomena.

Findings

Derived a general stationary distribution for non-equilibrium systems.

Identified a singularity with physical interpretation in intermittency models.

Established a connection between entropy-based distributions and stochastic processes.

Abstract

Using statistical thermodynamics, we derive a general expression of the stationary probability distribution for thermodynamic systems driven out of equilibrium by several thermodynamic forces. The local equilibrium is defined by imposing the minimum entropy production and the maximum entropy principle under the scale invariance restrictions. The obtained probability distribution presents a singularity that has immediate physical interpretation in terms of the intermittency models. The derived reference probability distribution function is interpreted as time and ensemble average of the real physical one. A generic family of stochastic processes describing noise-driven intermittency, whose stationary density distribution coincides exactly with the one resulted from entropy maximization, is presented.