  We develop further the study of a system in contact with a multibath having
different temperatures at widely separated timescales. We consider those
systems that do not thermalize in finite times when in contact with an ordinary
bath but may do so in contact with a multibath. Thermodynamic integration is
possible, thus allowing one to recover the stationary distribution on the basis
of measurements performed in a `multi-reversible' transformation. We show that
following such a protocol the system is at each step described by a
generalization of the Boltzmann-Gibbs distribution, that has been studied in
the past. Guerra's bound interpolation scheme for spin-glasses is closely
related to this: by translating it into a dynamical setting, we show how it may
actually be implemented in practice. The phase diagram plane of temperature vs
"number of replicas", long studied in spin-glasses, in our approach becomes
simply that of the two temperatures the system is in contact with. We suggest
that this representation may be used to directly compare phenomenological and
mean-field inspired models.Finally, we show how an approximate out of
equilibrium probability distribution may be inferred experimentally on the
basis of measurements along an almost reversible transformation.
