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 \(\tau\) and inverse temperatures \(B\) and \(\beta\). In the limit in which one bath becomes extremely slow (\(\tau \to \infty\)), such models amount to a paramagnet and to a one-dimensional ferromagnet, in contact with a single bath. We show that these systems reach a stationary state in a finite time for any choice of \(B\) and \(\beta\). We determine the non-equilibrium fluctuation-dissipation relation between the autocorrelation and the response function in such 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.