Thermalization with a multibath: an investigation in simple models

We study analytically and numerically a couple of paradigmatic spin models, each described in terms of two sets of variables attached to two different thermal baths with characteristic timescales T and τ and inverse temperatures B and β . In the limit in which one bath becomes extremely slow ( τ → ∞ ), such models amount to a paramagnet and to a one-dimensional ferromagnet in contact with a single bath. Our study is also motivated by analogies with disordered systems where widely separated timescales associated with different effective temperatures emerge. We show that these systems reach a stationary state in a finite time for any choice of B and β . We determine the non-equilibrium fluctuation-dissipation relation between the autocorrelation and the response function in such a state and, from that, we discuss if and how thermalization with the two baths occurs and the emergence of a non-trivial fluctuation-dissipation ratio.