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We consider three one-dimensional continuous-time Markov processes on alattice, each of which models the conduction of heat: the family of BrownianEnergy Processes with parameter $m$, a Generalized Brownian Energy Process, andthe Kipnis-Marchioro-Presutti process. The hydrodynamic limit of each of thesethree processes is a parabolic equation, the linear heat equation in the caseof 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 formalderivation for the GBEP($a$).  We then formally derive the pathwise large-deviation rate functional for theempirical measure of the three processes. These rate functionals implygradient-flow structures for the limiting linear and nonlinear heat equations.We contrast these gradient-flow structures with those for processes describingthe diffusion of mass, most importantly the class of Wasserstein gradient-flowsystems. The linear and nonlinear heat-equation gradient-flow structures areeach driven by entropy terms of the form $-\log \rho$; they involve dissipationor mobility terms of order $\rho^2$ for the linear heat equation, and anonlinear function of $\rho$ for the nonlinear heat equation.

Large Deviations in Stochastic Heat-Conduction Processes Provide a Gradient-Flow Structure for Heat Conduction