In this talk, we derive second-order expressions for both the one- and two-particle reduced density matrices of the Gibbs state at fixed positive temperatures. We consider a translation-invariant system of N bosons in a three-dimensional torus. These bosons interact through a repulsive two-body potential with a scattering length of order 1/N in the large N limit. This analysis provides a justification of Bogoliubov's prediction regarding the fluctuations around the condensate. The talk will primarily introduce basic concepts and settings, ensuring accessibility for all attendees. This work is a joint effort with Christian Brennecke and Phan Thành Nam.
