Thermodynamics of fluctuations based on time-and-space averages.

We develop non-equilibrium theory by using averages in time and space as a generalized way to upscale thermodynamics in non-ergodic systems. The approach offers a classical perspective on the energy dynamics in fluctuating systems. The rate of entropy production is shown to be explicitly scale dependent when considered in this context. We show that while any stationary process can be represented as having zero entropy production, second law constraints due to the Clausius theorem are preserved due to the fact that heat and work are related based on conservation of energy. As a demonstration we consider the energy dynamics for the Carnot cycle and for Maxwell's demon. We then consider non-stationary processes, applying time-and-space averages to characterize non-ergodic effects in heterogeneous systems where energy barriers such as compositional gradients are present. We show that the derived theory can be used to understand the origins of anomalous diffusion phenomena in systems where Fick's law applies at small length scales but not at large length scales. We then characterize fluctuations in capillary-dominated systems, which are non-stationary due to the irreversibility of cooperative events.