  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.
