  We consider three one-dimensional continuous-time Markov processes on a
lattice, each of which models the conduction of heat: the family of Brownian
Energy Processes with parameter $m$, a Generalized Brownian Energy Process, and
the Kipnis-Marchioro-Presutti process. The hydrodynamic limit of each of these
three processes is a parabolic equation, the linear heat equation in the case
of the BEP$(m)$ and the KMP, and a nonlinear heat equation for the GBEP($a$).
We prove the hydrodynamic limit rigorously for the BEP$(m)$, and give a formal
derivation for the GBEP($a$).
  We then formally derive the pathwise large-deviation rate functional for the
empirical measure of the three processes. These rate functionals imply
gradient-flow structures for the limiting linear and nonlinear heat equations.
We contrast these gradient-flow structures with those for processes describing
the diffusion of mass, most importantly the class of Wasserstein gradient-flow
systems. The linear and nonlinear heat-equation gradient-flow structures are
each driven by entropy terms of the form $-\log \rho$; they involve dissipation
or mobility terms of order $\rho^2$ for the linear heat equation, and a
nonlinear function of $\rho$ for the nonlinear heat equation.
