Second Order Expansion of Gibbs State Reduced Density Matrices in the Gross-Pitaevskii Regime

We consider a translation-invariant system of $N$ bosons in ${\mathbb{T}}^3$ that interact through a repulsive two-body potential with scattering length of order $N^{-1}$ in the limit $N\to \mathrm{\infty }$. We derive second order expressions for the one- and two-particle reduced density matrix matrices of the Gibbs state at fixed positive temperatures, thus obtaining a justification of Bogoliubov’s prediction on the fluctuations around the condensate.