First Law of Thermodynamics with strong coupling.*

When systems are strongly coupled, with non-negligible energetic interactions, then state changes are accompanied not only by changes in mutual information, but also by changes in interaction energy. Strong coupling can potentially involve many degrees of freedom and work exchange on the nano-scale. The distinction between work and heat may thus become difficult in these situations. Nonetheless, since all thermodynamic considerations depend on the definition of work vs. heat, this is a central issue. 

Here we discuss how the problem can be solved by a concise and simple mathematical description for classical systems in which the change in interaction energy between the two systems can be decomposed into energy flows related to state changes of each one of the systems alone. We show that the energy flows between strongly coupled systems are composed of a dissipative part plus another part that can be interpreted as work-like. 

This result allows the formulation of local first laws. Furthermore, it gives a re-formulation of local second laws [Crooks&Still, EPL(125)40005, 2019] that provides new insights on quantities used elsewhere in the literature to quantify entropy production in feedback controlled systems.