We consider a dilute Fermi gas in the thermodynamic limit with interaction potential scattering length $\mathfrak{ a}_0$ at temperature $T>0$. We prove the 2nd order Huang-Yang approximation for the Fermi pressure of the system, in which there is a 2nd order term carrying the positive temperature efffect.Our formula is valid up to the temperature $T<\rho^{ \frac{ 2}{ 3}+\frac{ 1}{ 6}}$, which is, by scaling, also necessary for the Huang-Yang formula to hold. Here, $T_F\sim\rho^{ \frac{ 2}{ 3}}$ is the Fermi temperature. We also establish during the course of the proof, a conjecture regarding the second order approximation of density $\rho$ by R. Seiringer \cite{ FermithermoTpositive}. Our proof uses frequency localization techniques from the analysis of nonlinear PDEs and does not involve spatial localization or Bosonization. In particular, our method covers the classical Huang-Yang formula at zero temperature.
Categories: math-ph math.AP math.MP