  We consider one dimensional weakly asymmetric boundary driven models of heat
conduction. In the cases of a constant diffusion coefficient and of a quadratic
mobility we compute the quasi-potential that is a non local functional obtained
by the solution of a variational problem. This is done using the dynamic
variational approach of the macroscopic fluctuation theory \cite{MFT}. The case
of a concave mobility corresponds essentially to the exclusion model that has
been discussed in \cite{Lag,CPAM,BGLa,ED}. We consider here the convex case
that includes for example the Kipnis-Marchioro-Presutti (KMP) model and its
dual (KMPd) \cite{KMP}. This extends to the weakly asymmetric regime the
computations in \cite{BGL}. We consider then, both microscopically and
macroscopically, the limit of large external fields. Microscopically we discuss
some possible totally asymmetric limits of the KMP model. In one case the
totally asymmetric dynamics has a product invariant measure. Another possible
limit dynamics has instead a non trivial invariant measure for which we give a
duality representation. Macroscopically we show that the quasi-potentials of
KMP and KMPd, that for any fixed external field are non local, become local in
the limit. Moreover the dependence on one of the external reservoirs
disappears. For models having strictly positive quadratic mobilities we obtain
instead in the limit a non local functional having a structure similar to the
one of the boundary driven asymmetric exclusion process.
