Description
We consider a Bose gas of N interacting particles trapped in the unit torus [0,1]^3 at zero temperature. At such a low temperature, Bose gases of a large number of particles undergo a phase transition called Bose-Einstein condensation: the majority of the particles condense into the same quantum state called the condensate.
To give a mathematical formulation of Bose-Einstein condensation, we consider a one-particle observable O acting on L^2([0,1]^3) and measure it on each of the N bosons simultaneously. This measurement yields an empirical measure \nu_N, which is expected to be close to the classical probability measure \nu_{\varphi} associated with the condensate wave function.
We confirm this expectation by proving a law of large numbers saying that the observed random empirical measures converge to the condensate measure \nu_{\varphi} in probability in the 1-Wasserstein distance as the number of particles N tends to infinity. Furthermore, we characterize the fluctuations of the empirical measures by an appropriate central limit theorem.
Joint work with Lorenzo Portinale and Simone Rademacher [1].
[1] L. Portinale, S. Rademacher, D. Virosztek. Limit theorems for empirical measures of interacting quantum systems in Wasserstein space. Adv. Math. 495 (2026), 110972.