The Free Energy of Dilute Quantum Gases

We discuss bounds on the free energy of homogeneous Bose and Fermi gases at non-zero temperature. In the dilute regime, the leading order correction compared to an ideal quantum gas depends on the particle interaction only through the scattering length. Moreover, in the Bose case it depends non-trivially on temperature through the critical density for Bose-Einstein condensation. Some of the key ingredients in the proof are the use of coherent states to deal with the condensation, as well as new correlation estimates relying on the monotonicity of the relative entropy under partial traces.