  We introduce a thermal dynamics for the Ising ferromagnet where the energy
variations occurring within the system exhibit a diffusive character typical of
thermalizing agents such as e.g. localized excitations. Time evolution is
provided by a walker hopping across the sites of the underlying lattice
according to local probabilities depending on the usual Boltzmann weight at a
given temperature. Despite the canonical hopping probabilities the walker
drives the system to a stationary state which is not reducible to the canonical
equilibrium state in a trivial way. The system still exhibits a magnetic phase
transition occurring at a finite value of the temperature larger than the
canonical one. The dependence of the model on the density of walkers realizing
the dynamics is also discussed. Interestingly the differences between the
stationary state and the Boltzmann equilibrium state decrease with increasing
number of walkers.
