Local energy assignment for two interacting quantum thermal reservoirs

Understanding how to assign internal energy, heat, and work in quantum systems beyond weak coupling remains a central problem in quantum thermodynamics, particularly as the difference between competing definitions becomes increasingly relevant. We identify two common sets of definitions for first-law quantities that are used to describe the thermodynamics of quantum systems coupled to thermal environments. Both are conceptually non-symmetric, treating one part of the bipartition (the "system") differently from the other (the "bath"). We analyze these in a setting where such roles are not easily assigned - two large (but finite) sets of thermal harmonic oscillators interacting with each other. We further compare them with a third set of definitions based on a local, conceptually symmetric open-system approach ("minimal dissipation") and discuss their quantitative and structural differences. In particular, we observe that all three sets of definitions differ substantially even when the two subsystems are weakly coupled and far detuned, and that the minimal dissipation approach features distinct work peaks that increase with the coupling strength.