Thermodynamics of Mixed Quantum-Classical Systems

We study systems composed of slow and fast degrees of freedom coupled together in thermal equilibrium. When the two subsystems have widely separated energy scales, there is an intermediate temperature regime in which the slow degrees of freedom can be treated classically while the fast ones remain quantum. Using the partial Wigner-Weyl transform, we develop a framework for understanding thermodynamics in this mixed quantum–classical regime. As a concrete example, we illustrate how a fermionic gas coupled to a piston renormalizes the thermodynamic mass of the piston and modifies its fluctuations and equipartition relation at low temperatures. The framework we discuss here is consistent with the recently developed Moving Born-Oppenheimer Approximation.