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

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.