Exchange Fluctuation Theorems for Non-Markovian Baths in Quantum Collisional Model

Standard quantum exchange fluctuation theorems, such as the Jarzynski-Wójcik exchange fluctuation theorem, typically assume memoryless Markovian environments where heat flows on average from a hotter thermal bath to a colder one. However, realistic quantum open systems frequently exhibit non-Markovian dynamics due to structured environments, finite-size baths, or internal correlations. This paper investigates how exchange fluctuation theorems generalize when thermal baths possess memory effects.

The authors employ a quantum collisional model to construct non-Markovian baths microscopically. In this framework, bath memory is generated by introducing intra-bath interactions between successive auxiliary units prior to each heat-exchange event. Using the two-point measurement protocol, the authors derive exact, trajectory-dependent exchange fluctuation theorems for two distinct physical scenarios in the steady-state regime: direct bath-bath interactions and probe-mediated heat exchange.

The central result is a generalized exchange fluctuation theorem that relates forward and reverse heat-transfer probabilities through an additional memory-induced correction factor. When the intra-bath auxiliary interactions are turned off, this factor evaluates to unity, cleanly recovering the standard Markovian Jarzynski-Wójcik relation and the traditional Clausius inequality.

When non-Markovian memory is present, the correction factor modifies the exponential asymmetry between forward and reverse heat exchanges. By analyzing an illustrative model where the bath auxiliaries are qubits, the authors demonstrate quantitatively that non-Markovian environmental memory enhances the probability of observing heat transfer against the temperature gradient—meaning heat flows from the colder bath to the hotter bath with higher relative likelihood than predicted by Markovian theories.

Understanding non-equilibrium thermodynamics at the quantum scale is crucial for designing reliable quantum thermal machines, refrigerators, and quantum batteries. This work provides an exact analytical bridge between environmental memory and non-equilibrium fluctuation statistics. By establishing that bath memory can actively facilitate reverse heat-transfer fluctuations, the findings offer fundamental insights that may help control or exploit quantum fluctuations in engineered open-system devices.