On the Sub-Riemannian Geometric structure of Finite-Time Thermodynamic transitions: Isoholonomic Inequalities and Bounds on Power

Sub-Riemannian geometry, which has found applications in fields like optimal control and statistical mechanics, offers a natural framework for studying systems with constrained dynamics. In the context of finite time thermodynamics, this formalism enables analyzing optimal cyclic processes from a geometric perspective. In this work, we introduce a Sub-Riemannian geometry on the Wasserstein manifold that allows us to characterize work-maximizing cycles for overdamped systems under anisotropic temperatures. Specifically, we show that quasi-static work extraction can be written as the integral of a curvature two-form, while dissipation is given by path-lengths. Consequently, the problem of minimizing dissipation over cycles of fixed work output can be cast as an isoholonomic problem on the sub-Riemannian manifold of thermodynamic states under arbitrary potentials.