Heat conduction in a two-dimensional Ising model

A cylindrical Ising model between thermostats is used to explore the heat conduction for any temperature interval. The standard Q2R and Creutz dynamics, previously used by Saito, Takesue and Miyashita, fail below the critical temperature, limiting the analysis to high temperatures intervals. We introduce improved dynamics by removing limitations due to the chessboard-like refresh, and by supplementing the Q2R rule with Kadanoff-Swift moves. These new dynamics not only prove highly efficient in recovering old results in their domains of validity, but also allow exploration of steady heat transport between two arbitrary temperatures, i.e. very far from equilibrium. From an ansatz avoiding references to quasi equilibrium or to local temperature, and from comparison with numerical simulations, we can consistently define a generalized diffusivity. Its dependence on the energy density may be evaluated without any recourse to the Green-Kubo formula.