A Variational Method in Out of Equilibrium Physical Systems

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.

💡 Research Summary

The paper introduces a novel variational principle tailored for out‑of‑equilibrium dynamical systems by embedding the concept of maximum entropy directly into the action functional. Starting from the conventional Lagrangian framework, the author augments it with an entropy‑production term and associated Lagrange multipliers that enforce the usual conservation constraints (mass, charge, momentum). Performing the variation yields a pair of first‑order differential equations for the generalized coordinates and their conjugate momenta. Remarkably, these equations retain the canonical symplectic structure familiar from Hamiltonian mechanics, thereby establishing a formal bridge between classical mechanics, fluid dynamics, and thermodynamics even in the presence of dissipative processes.

When the formalism is applied to a rotating reference frame, additional terms appear in the equations of motion that involve the vector potential (A) (which may represent gravitational, electromagnetic, or combined gravito‑electromagnetic fields) and the angular velocity (\omega) of the frame. The resulting extra force density takes the form (\epsilon_{ijk}A_j\omega_k), which the author interprets as a “topological torsion current.” This term embodies a coupling between the rotation of the medium and the underlying potential fields, and it generates a non‑conservative circulation that is absent in standard inertial formulations.

Building on this, the author derives a modified Umov‑Poynting theorem for rotating gravito‑electromagnetic systems. The conventional energy‑flux vector is supplemented by a contribution proportional to (\Omega\cdot J), where (\Omega) denotes the rotation vector and (J) the torsion current. This extra term accounts for the exchange of energy between rotational kinetic energy and field energy, providing a more complete energy balance for systems such as rotating plasmas or atmospheric vortices.

A key outcome of the theory is a generalized equilibrium condition for a rotating plasma. By balancing the pressure gradient, Lorentz force, and centrifugal force, the author obtains an explicit criterion that links plasma pressure, magnetic field strength, and angular velocity. This criterion can be used to assess the stability of magnetically confined rotating plasmas and suggests new pathways for controlling confinement through deliberate rotation.

The paper then showcases four concrete applications that illustrate the practical relevance of the formalism:

1. Bennett Pinch: The torsion current amplifies the self‑magnetic pinch effect, leading to a tighter current filament and enhanced plasma compression beyond what classical Bennett theory predicts.

2. Vacuum Arcs: In high‑current vacuum arcs, the rotation‑induced torsion modifies the electric field distribution, reducing the arc voltage and increasing the power transfer efficiency.

3. Hurricane Power Extraction: By modeling the hurricane as a rotating gravito‑electromagnetic system, the author quantifies a theoretical maximum power that could be harvested from the vortex, highlighting the role of the (\epsilon_{ijk}A_j\omega_k) term in converting rotational kinetic energy into usable electrical energy.

4. Radiative Heating of Planetary Atmospheres: The analysis shows that angular‑momentum transport, mediated by the torsion current, can significantly enhance the radiative heating rates in planetary atmospheres, offering a new mechanism to explain observed temperature anomalies.

In the concluding discussion, the author emphasizes that the entropy‑augmented variational approach provides a unified language for describing dissipative, rotating, and field‑coupled phenomena. Because the underlying symplectic structure is preserved, the framework is compatible with established numerical schemes (e.g., symplectic integrators) while extending their applicability to non‑conservative contexts. The paper suggests that future work could explore extensions to relativistic fluids, magnetohydrodynamic turbulence, and engineered energy‑harvesting devices that exploit the topological torsion current. Overall, the study offers a compelling synthesis of thermodynamic irreversibility and classical variational mechanics, opening new avenues for both theoretical insight and technological innovation.