Emergence and breaking of duality symmetry in the generalized thermodynamic fundamental relations*

Thermodynamics provides a universal framework to describe the relationship among macroscopic properties of equilibrated systems. Such a framework has been extended to nanoscale systems by C. Jarzynski, U. Seifert, and T. Hill. Using the theory of large deviation, we find that Hill's nano-thermodynamic can be generalized to systems beyond equilibrium. Generalized thermodynamic results such as entropy, free entropy, fundamental equation, and Hill-Gibbs-Duhem (HGD) equation emerge in the statistics of repeated measurements of an arbitrary stationary system with a priori probability distribution.  Moreover, in the repeated measurement limit, a duality symmetry arises between entropy and free entropy -- yielding the fundamental equation and HGD equation as a duality pair.  If one further introduces the thermodynamic limit (i.e., for macroscopic systems entropy becoming an Eulerian 1-st order homogeneous function of all extensive variables), the duality symmetry is broken, and the HGD equation reduces to the Gibbs-Duhem equation. The new framework provides an operational approach to derive thermodynamic-like relations for arbitrary stationary systems and offers a unique perspective of the relationship between Hill's nano-thermodynamics and thermodynamics.