ALEX 2018 Workshop: Abstracts

Onsager reciprocity, gradient flows, and large deviations

Mark Peletier

Technische Universiteit Eindhoven (The Netherlands)

The second law states that in a thermodynamically consistent system the ’entropy’ is a Lyapunov function, a function which is monotonic along solutions of the corresponding differential equations. When the system can be written as a gradient flow of the entropy, then this statement is strengthened: not only is this functional monotonic, but it drives the dissipative part of the evolution in a precise way, mediated by a ’friction operator’.

In this talk I will go one step further. Onsager already pointed out how symmetry properties of linear friction operators arise through an upscaling procedure from a microscopic-reversibility property of the underlying system. Fluctuations figure centrally in his argument, but at that time their theory was not well developed, and more could not be said.

However, recently we have found that the connection between microscopic reversibility and macroscopic ’symmetry’ properties is not at all limited to the close-to-equilibrium, linear-friction-operator context of Onsager’s. I will describe how the large-deviation theory of fluctuations allows one to make a much more general statement, where microscopic reversibility is one-to-one coupled to ’symmetry’ at the macroscopic level - provided one generalizes the concept of symmetry in an appropriate way.

This is joint work with Michiel Renger and Alexander Mielke (both WIAS, Berlin).