Thermodynamic Geometry Through Phase Transitions

A common approach to quantify excess dissipation in slowly driven thermodynamic processes is through the use of a Riemannian metric on the space of control parameters, where optimal driving protocols follow geodesics. This approach, which was invented in the 70's, developed in the 80's and was revised in the 2010's,  breaks down near phase transitions as the metric diverges and geodesics may cease to exist. Using the same framework, we found that for several universality classes a the thermodynamic length across the phase transition remains finite. We demonstrate a numerical approach for computing minimal paths in such systems. We show that, in some regimes, the shortest path crosses the phase transition – even when alternative paths confined to a single phase exist.