Modelling quenched disorder with multiple thermal baths

We discuss the relation between a system with quenched disorder in contact with a thermal bath at inverse temperature B and a companion one where the originally frozen disorder is annealed by thermalisation with a second thermal reservoir at a different temperature β. In the limit in which the disorder variables become extremely slow and (Formula presented.) the two systems are expected to share the same static properties. Focusing on dynamical aspects, we discuss the non-equilibrium fluctuation–dissipation (FD) relation in the annealed system and the associated concept of effective temperature in four different models, which are studied numerically and/or analytically. We show that such effective temperature not only may bear a signature of the temperatures of the two reservoirs, but it may also reveal, in some cases, the FD relation of the corresponding quenched model in the ageing state after a temperature quench.