Breakdown of additivity of transition rates in systems connected to multiple thermal reservoirs

In stochastic thermodynamics, it is commonly assumed that for a system coupled to multiple thermal reservoirs, the transition rates between two energy levels are additive across baths. In this work, we first demonstrate through an explicit construction of two subsystems-that are parts of a single composite system, each coupled to a distinct thermal reservoir-that while each subsystem individually evolves Markovianly, their joint evolution is inherently non-Markovian. We then present a general algebraic argument showing that, even if one assumes a Markovian description with additive transition rates, the steady-state condition imposed by the master equation leads to an inconsistency. The analysis identifies the precise structural limitation that arises in describing systems simultaneously interacting with multiple baths within a Markovian framework.