I am curious about your opinion on future perspectives of non-equilibrium thermodynamics? Is it in connection with quantum systems? In financial applications? what about new possible applications (sociology, ...)?
The NET' s main contribution consists in bringing up the problem about the irreversibility of the time arrow and how it does contribute for the existence of life and order in the universe.
Cosmology and quantum field theory are two of the most conspicuous arenas for studying NET. (I have just finished a book on the subject dealing with human social systems).
You can also take a look at the NESS programme, namely Non-Equilibrium Social Science, a field not very well known by many, so far.
Is there a connection with score functions? I think it should be, especially in the statistical viewpoint at least the estimate for the location parameter, or regression setting.