Symmetries and isometries of the thermodynamic metric

Geometric approaches to nonequilibrium thermodynamics have developed into a versatile toolkit for finding optimal protocols and fundamental physical limitations in slowly-driven microscopic systems. Typically, one makes a mapping from the space thermodynamic control variables (e.g. volume or temperature) to a smooth, Riemannian manifold in which geometric lengths correspond to physical energy dissipation; geometric methods can then be applied to assist in analyzing problems for a specific model thermal system when, for example, operated as a heat engine or ratchet. In this talk, we will instead focus more directly on the geometric side of this mapping and investigate possible physical interpretations of symmetries directly encoded in thermodynamic metrics. Notably, in certain simple cases, such symmetries have important physical implications, connecting (minimally-dissipative) geodesic paths of the underlying geometric space to entropy-preserving adiabats in a thermodynamic space. We will attempt to address this question more broadly, studying the deep connections between geometry and the thermodynamics of slowly-driven nonequilibrium systems.