Virial expansion for a strongly correlated Fermi gas

Using a high temperature virial expansion, we present a controllable study of the thermodynamics of strongly correlated Fermi gases near the BEC-BCS crossover region. We propose a practical way to determine the expansion coefficients for both harmonically trapped and homogeneous cases, and calculate the third order coefficient $b_3(T)$ at finite temperatures $T$. At resonance, a $T$-independent coefficient $b_{3,\mathrm{\infty }}^{\mathrm{hom}}\approx -0.29095295$ is determined in free space. These results are compared with a recent thermodynamic measurement of ${}^6$Li atoms, at temperatures below the degeneracy temperature, and with Monte Carlo simulations.