On a non-linear sigma model of knotted relaxed states far from thermodynamic equilibrium in plasma physics and beyond

TL;DR

This paper models steady-state, knotted structures in plasma physics using a non-linear sigma model, revealing stability conditions, linking magnetic helicity to Gauss linking number, and deriving bounds for skyrmion writing.

Contribution

It introduces a Faddeev-Niemi non-linear sigma model framework for knotted plasma states, establishing stability criteria and connecting topological invariants to physical properties.

Findings

Model describes filamentary structures with minimal resistance.

Gauss linking number relates to magnetic helicity.

Derived lower bound on beam current for skyrmion creation.

Abstract

We show that a Faddeev-Niemi non-linear sigma model describes in the long wavelength limit a wide class of steady-state, knotted physical systems far from thermodynamic equilibrium which are stable against perturbations of temperature and interact weakly with the external world. In these systems temperature gradients are negligible, inertial effects are negligible in comparison with diffusion effects, entropy is mainly produced through Joule and-or viscous heating, the macroscopic state is described by specifying a unit vector at each point, and the Gauss linking number of this unit vector is lower than a threshold. In fluids and plasmas, the model describes filamentary structures which adjust themselves in order to offer minimum resistance to the medium embedding them and to the electric currents (if any) flowing across them; in the latter case, Gauss linking number is related to magnetic helicity. Both n and the relative velocity of the filament with respect to the medium are approximately Double Beltrami vector fields. We derive a stability criterion for a double helix. Moreover, a similar discussion describes the recently discovered writing process of skyrmions in a magnetic film with the help of a beam of polarised electrons. We derive a lower bound on the value of beam current required to write a skyrmion.