Thermodynamically consistent collisional master equation in a low-density gas with internal structure

Quantum thermodynamics with open systems is often based on the quantum optical weak-coupling master equation or on operational repeated interaction models, whereas early works on thermalisation and on decoherence theory were mostly concerned with the kinetics of gas collisions. Here we formulate a master equation for the dynamics of a quantum system under inelastic scattering with a dilute thermal gas in three dimensions, comprised of ancilla particles that also possess internal degrees of freedom. We show thermodynamic consistency when the gas is at thermal equilibrium, irrespective of whether or not the ancillas are in resonance with the system. In contrast, when the internal and the motional state of the gas are thermalised to different temperatures, the gas acts not as two distinct heat baths, but as a structured non-equilibrium reservoir that can generate useful energy through uncontrolled collisions.