Partitioning Quantum Work: Missing Term Revealed in the Thermodynamics of Open Quantum Systems

A key question in the thermodynamics of open quantum systems is how to partition thermodynamic quantities such as entropy, work, and internal energy between the system and its environment. We show that entropy is only well-defined under a Hilbert-space partition, which assigns half the system-environment coupling to the system and half to the environment. However, we find that quantum work is non-trivial under Hilbert-space partition, and derive a Work Sum Rule (analogous to the Friedel sum rule) that accounts for nonlocal quantum work-at-a-distance. We show how the work sum rule resolves the issue of path-dependence of the subsystem entropy that has been claimed in the literature as an argument against Hilbert-space partitioning. The thermodynamics of two classes of quasi-statically driven open quantum systems is analyzed: a subsystem of a finite system in the grand canonical ensemble, and a finite subsystem of an infinite system, focusing on a time-dependent double quantum dot and the driven resonant-level model. Closed-form expressions for all terms appearing in the First Law are derived in terms of Nonequilibrium Green's functions (NEGF). Extensions of this thermodynamic partitioning scheme to interacting systems and driving beyond the quasi-static limit are discussed.