  Based on the properties of exponential distribution families we analyze the
Fisher information of the Gibbs canonical ensemble to construct a new state
function for simple systems with no mechanical work. Such a function possesses
nice symmetry properties with respect to Legendre transform and provides a
connection with previous alternative formulations of thermodynamics, most
notably the work by Biot, Serrin and Frieden and collaborators. Logical
extensions to systems with mechanical work may similarly consider generalized
Gibbs ensembles.
