  We show how statistical thermodynamics can be formulated in situations in
which thermodynamics applies, while equilibrium statistical mechanics does not.
A typical case is, in the words of Landau and Lifshitz, that of partial (or
incomplete) equilibrium. One has a system of interest in equilibrium with the
environment, and measures one of its quantities, for example its specific heat,
by raising the temperature of the environment. However, within the observation
time the global system settles down to a state of apparent equilibrium, so that
the measured value of the specific heat is different from the equilibrium one.
In such cases formulae for quantities such as the effective specific heat
exist, which are provided by Fluctuation Dissipation theory. However, what is
lacking is a proof that internal energy exists, i.e., that the fundamental
differential form delta Q-delta W (difference between the heat absorbed by the
system and the work performed by it) is closed. Here we show how the
coefficients of the fundamental form can be expressed in such a way that the
closure property of the form becomes manifest, so that the first principle is
proven. We then show that the second principle too follows, and indeed as a
consequence of microscopic time--reversibility. The treatment is given in a
classical Hamiltonian setting. One has a global time--independent Hamiltonian
system constituted by the system of interest and two auxiliary ones controlling
temperature and pressure, and the occurring of a process due to a change in the
thermodynamic parameters is implemented by a suitable choice of the measure for
the initial data.
