  We study the macroscopic profiles of temperature and angular momentum in the
stationary state of chains of rotors under a thermo-mechanical forcing applied
at the boundaries. These profiles are solutions of a system of diffusive
partial differential equations with boundary conditions determined by the
thermo-mechanical forcing. Instead of expensive Monte Carlo simulations of the
underlying microscopic dynamics, we perform extensive numerical computations
based on a finite difference method for the system of partial differential
equations describing the macroscopic steady state. We first present a formal
derivation of these stationary equations based on a linear response argument
and local equilibrium assumptions. We then study various properties of the
solutions to these equations. This allows to characterize the regime of
parameters leading to uphill energy diffusion -- a situation in which the
energy flows in the direction of the gradient of temperature -- and to identify
regions of parameters corresponding to a negative energy conductivity (i.e. a
positive linear response of the energy current to a gradient of temperature).
The macroscopic equations we derive are consistent with some previous results
obtained by numerical simulation of the microscopic physical system, which
confirms their validity.
