  Controlling thermodynamic cycles to minimize the dissipated heat is a
longstanding goal in thermodynamics, and more recently, a central challenge in
stochastic thermodynamics for nanoscale systems. Here, we introduce a
theoretical and computational framework for optimizing nonequilibrium control
protocols that can transform a system between two distributions in a minimally
dissipative fashion. These protocols optimally transport a system along paths
through the space of probability distributions that minimize the dissipative
cost of a transformation. Furthermore, we show that the thermodynamic metric --
determined via a linear response approach -- can be directly derived from the
same objective function that is optimized in the optimal transport problem,
thus providing a unified perspective on thermodynamic geometries. We
investigate this unified geometric framework in two model systems and observe
that our procedure for optimizing control protocols is robust beyond linear
response.

Statistical Mechanics