Recorded 03 April 2025. Andreas Deuchert of Virginia Polytechnic Institute and State University presents "The Gibbs state of the mean-field Bose gas" at IPAM's Optimal Transport for Density Operators: Theory and Numerics Workshop. Abstract: We consider the homogeneous mean-field Bose gas at temperatures proportional to the critical temperature of its Bose--Einstein condensation phase transition. Our main result is a trace-norm approximation of the grand canonical Gibbs state in terms of a reference state, which is given by a convex combination of products of coherent states and Gibbs states associated with certain temperature-dependent Bogoliubov Hamiltonians. The convex combination is expressed as an integral over a Gibbs distribution of a one-mode Φ4-theory describing the condensate. We interpret this result as a justification of Bogoliubov theory at positive temperature. Further results derived from the norm approximation include various limiting distributions for the number of particles in the condensate, as well as precise formulas for the one- and two-particle density matrices of the Gibbs state. Key ingredients of our proof, which are of independent interest, include two novel abstract correlation inequalities. The proof of one of them is based on an application of an infinite-dimensional version of Stahl's theorem. This is joint work with Phan Thành Nam and Marcin Napiórkowski. Learn more online at: https://www.ipam.ucla.edu/programs/workshops/workshop-i-optimal-transport-for-density-operators-theory-and-numerics/?tab=schedule