  A few decades after Hill's work on nano-thermodynamics, the development of a
thermodynamic framework, to account consistently for the fluctuations of small
systems due to their interactions with the surrounding environment, is still
underway. Here we discuss how, in a small system, the interaction energy with
the environment may be described through a conjugate pair of intensive and
extensive variables, giving rise to a Gibbs thermodynamics with an additional
thermodynamic degree of freedom. The relevant thermodynamic potentials help
describe the equilibrium conditions and the material properties that measure
the susceptibility of the system to the interaction with the environment. The
resulting generalized ensembles, which describe the non-negligible, small
system fluctuations, are shown to be equivalent. Away from the average
thermodynamic state, the availability describes the distance between system and
the environment, in terms of the maximum work extractable from the fluctuation,
and a proper definition of entropy extends the thermodynamics to the generic
fluctuating state. From the latter state, entropic forces arise to restore the
average thermodynamic state. Our framework unifies and extends the ensemble
thermodynamics by Hill and the recent advances in statistical mechanics under
strong coupling and reduces to classical macroscopic thermodynamics when the
system is large. The particular case of a small ideal gas system is discussed
in detail and the example of a single particle immersed in a bath is revisited
in the light of the new formalism.
