The cylindrical Ising model between two temperatures is used to explore the heat conduction for any temperatures interval. To this end, the standard Q2R dynamics is improved by mixing with Kadanoff-Swift moves. This new dynamics, recovering old results in their domains of validity, proves highly efficient in exploring the steady heat transport also very far from equilibrium, between two arbitrary temperatures. From an ansatz avoiding any reference to quasi equilibrium or local temperature, we can also consistently define a generalized diffusivity, independently of the Green-Kubo formula. Finallly, we investigate the relevance, for both thermodynamical and geometrical-dynamical observables, of the non-equilibrium. The main point is the existence of an energy band, starting just below the critical energy, where the fluctuations of observables are remarkably wider for a system undergoing an heat flux with respect to thermalized or close systems. Such an effect vanishes in the termodynamic limit, being non trivially related to the mesoscopic size of the magnetic lattice. |