[tMcGrath]

As far as I know, a good differential-geometric understanding of nonequilibrium thermodynamics still hasn't been achieved.

The central issue is understanding how changes in control parameters (for instance concentrations of catalysts in a chemical system, or local fields in a spin system) affect the evolution of the probability distribution over states. Some work has been done in close to steady state (for instance [1,2,3]) but it's far from resolved.

This has some nice applications - designing efficient protocols for microscale devices, for instance.

[1] https://arxiv.org/abs/1603.07758 [2] https://arxiv.org/abs/1507.06269v1 [3] https://arxiv.org/abs/1201.4166