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

TL;DR

This paper derives second order approximations for the reduced density matrices of a bosonic Gibbs state in the Gross-Pitaevskii regime, confirming Bogoliubov's predictions on condensate fluctuations at positive temperatures.

Contribution

It provides a rigorous derivation of second order expansions for reduced density matrices in the Gross-Pitaevskii limit, validating Bogoliubov's theory for fluctuations.

Findings

Second order expressions for reduced density matrices derived.

Confirmation of Bogoliubov's predictions on condensate fluctuations.

Rigorous justification at positive temperatures.

Abstract

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 $N\to \infty$. 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.