Exponential bounds of the condensation for dilute Bose gases
Phan Thành Nam, Simone Rademacher
TL;DR
The paper rigorously justifies Bose-Einstein condensation for a dilute Bose gas in the Gross-Pitaevskii regime by deriving an exponential bound on the number of excitations above the condensate for low-lying eigenstates and, at low temperature, for the Gibbs state. The core technique combines a unitary transformation to an excitation space with a Bogoliubov renormalization that encodes short-distance correlations via the scattering data, enabling a positive lower bound on the excitation Hamiltonian up to controllable error terms. A Gronwall-type argument on exponential moments yields exponential tails for the excitation number, while a detailed large-deviation analysis provides precise asymptotics for fluctuations around the condensate, including a mean $\mu$ and variance $\sigma^2$ expressed through Bogoliubov parameters. The results extend to generalized scaling regimes and establish foundational probabilistic control of condensate depletion, with additional insights into the Gibbs state at low temperature. Overall, the work sharpens our understanding of fluctuations around Bose-Einstein condensation in the GP regime and provides robust tools for analyzing large deviations in interacting Bose gases.
Abstract
We consider N bosons on the unit torus $Λ= [0,1]^3$ in the Gross-Pitaevski regime where the interaction potential scales as $N^2 V (N(x -y))$. We prove that the thermal equilibrium at low temperatures exhibits the Bose-Einstein condensation in a strong sense, namely the probability of having $n$ particles outside of the condensation decays exponentially in $n$.