A Variational Method in Out of Equilibrium Physical Systems

TL;DR

This paper develops a variational principle for out-of-equilibrium systems using maximum entropy, leading to new equations and insights into rotating systems, plasma equilibrium, and applications in astrophysics and engineering.

Contribution

It introduces a novel variational framework for out-of-equilibrium systems, revealing a symplectic structure and deriving new equations for rotating and gravito-electromagnetic systems.

Findings

Derived a set of differential equations with symplectic structure

Formulated an extended equation of motion for rotating systems

Applied the method to analyze devices and planetary atmospheres

Abstract

A variational principle is further developed for out of equilibrium dynamical systems by using the concept of maximum entropy. With this new formulation it is obtained a set of two first-order differential equations, revealing the same formal symplectic structure shared by classical mechanics, fluid mechanics and thermodynamics. In particular, it is obtained an extended equation of motion for a rotating dynamical system, from where it emerges a kind of topological torsion current of the form $\epsilon_{ijk} A_j \omega_k$, with $A_j$ and $\omega_k$ denoting components of the vector potential (gravitational or/and electromagnetic) and $\omega$ is the angular velocity of the accelerated frame. In addition, it is derived a special form of Umov-Poynting's theorem for rotating gravito-electromagnetic systems, and obtained a general condition of equilibrium for a rotating plasma. The variational method is then applied to clarify the working mechanism of some particular devices, such as the Bennett pinch and vacuum arcs, to calculate the power extraction from an hurricane, and to discuss the effect of transport angular momentum on the radiactive heating of planetary atmospheres. This development is seen to be advantageous and opens options for systematic improvements.