Geometric Thermodynamics of Cycles: Curvature and Local Thermodynamic Response

TL;DR

This paper develops a unified geometric framework for classical thermodynamic cycles, linking cycle geometry, response functions, and nonequilibrium work relations through a canonical two-form on the thermodynamic manifold.

Contribution

It introduces a single geometric structure that unifies the description of thermodynamic cycles in different representations and connects cycle geometry to local thermodynamic response and stochastic thermodynamics.

Findings

Work and heat laws are projections of a canonical two-form.

Cycle work relates to the mixed curvature of the energy surface.

Framework links classical cycle geometry to nonequilibrium work relations.

Abstract

Classical thermodynamics contains familiar geometric relations associated with cyclic processes, most notably the identification of mechanical work with the area enclosed by a trajectory in the $(P,V)$ plane. We show that the area laws for work and reversible heat arise as projections of a single canonical two--form defined on the equilibrium thermodynamic manifold, providing a unified description of thermodynamic cycles in both the $(P,V)$ and $(T,S)$ representations. The same structure yields a direct link between cycle geometry and thermodynamic response: the work generated by infinitesimal cycles is set locally by the mixed curvature $U_{SV}$ of the equilibrium energy surface, which can be expressed in terms of measurable susceptibilities. This identifies thermodynamic work as a local geometric field over state space rather than solely a global property of cyclic processes. More broadly, the framework connects classical cycle geometry to stochastic thermodynamic trajectories, providing a geometric interpretation of nonequilibrium work relations such as the Jarzynski equality.