Microscopic energy flows in disordered Ising spin systems

An efficient microcanonical dynamics has been recently introduced for Ising spin models embedded in a generic connected graph even in the presence of disorder, i.e. with the spin couplings chosen from a random distribution. Such a dynamics allows a coherent definition of local temperatures also when open boundaries are coupled to thermostats, imposing an energy flow. Within this framework, here we introduce a consistent definition for local energy currents and we study their dependence on the disorder. In the linear response regime, when the global gradient between thermostats is small, we also define local conductivities following a Fourier discretized picture. Then, we work out a linearized 'mean-field approximation', where local conductivities are supposed to depend on local couplings and temperatures only. We compare the approximated currents with the exact results of the nonlinear system, showing the reliability range of the mean-field approach, which proves very good at high temperatures and not so efficient in the critical region. In the numerical studies we focus on the disordered cylinder but our results could be extended to an arbitrary, disordered spin model on generic discrete structures.