  We study the equilibrium temperature distribution in a model for strongly
magnetized plasmas in dimension two and higher. Provided the magnetic field is
sufficiently structured (integrable in the sense that it is fibered by
co-dimension one invariant tori, on most of which the field lines ergodically
wander) and the effective thermal diffusivity transverse to the tori is small,
it is proved that the temperature distribution is well approximated by a
function that only varies across the invariant surfaces. The same result holds
for "nearly integrable" magnetic fields up to a "critical" size. In this case,
a volume of non-integrability is defined in terms of the temperature defect
distribution and related the non-integrable structure of the magnetic field,
confirming a physical conjecture of Paul-Hudson-Helander. Our proof crucially
uses a certain quantitative ergodicity condition for the magnetic field lines
on full measure set of invariant tori, which is automatic in two dimensions for
magnetic fields without null points and, in higher dimensions, is guaranteed by
a Diophantine condition on the rotational transform of the magnetic field.
