Bose--Einstein Condensation without an Initial Low-Energy Concentration Assumption
Abstract
We prove that semi-strong convergence to a Bose--Einstein equilibrium below the critical temperature implies finite-time Bose--Einstein condensation (BEC) for conservative isotropic measure solutions of the spatially homogeneous quantum Boltzmann equation with low-momentum scattering degeneracy exponent 0≤η<1. For every admissible initial measure with T̅/T̅c<1, there exists a conservative solution that has a positive zero-energy atom after a finite time without any assumption of initial concentration near zero energy. We also show that it persists and converges to the equilibrium condensate mass, while the full solution converges strongly. The proof introduces a method to obtain condensation from relaxation estimate to Bose--Einstein equilibrium. The relaxation estimate first gives a fixed amount of mass below a small energy level. We then use collision estimates at smaller and smaller energy scales and add their effects in a weighted sum functional with a fixed upper bound. If no atom forms at zero energy, enough mass remains in the positive-energy shells to force this sum to exceed its upper bound in finite time, which leads to condensation in finite time.
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BibTeX
@misc{luo2026boseeinstein,
title = {{Bose--Einstein Condensation without an Initial Low-Energy Concentration Assumption}},
author = {Siwei Luo and Jian-Guo Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01436},
eprint = {2609.01436},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01436},
}
RIS
TY - UNPB
TI - Bose--Einstein Condensation without an Initial Low-Energy Concentration Assumption
AU - Siwei Luo
AU - Jian-Guo Liu
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01436
UR - https://arxiv.org/abs/2609.01436
AB - We prove that semi-strong convergence to a Bose--Einstein equilibrium below the critical temperature implies finite-time Bose--Einstein condensation (BEC) for conservative isotropic measure solutions of the spatially homogeneous quantum Boltzmann equation with low-momentum scattering degeneracy exponent 0≤η<1. For every admissible initial measure with T̅/T̅_c<1, there exists a conservative solution that has a positive zero-energy atom after a finite time without any assumption of initial concentration near zero energy. We also show that it persists and converges to the equilibrium condensate mass, while the full solution converges strongly. The proof introduces a method to obtain condensation from relaxation estimate to Bose--Einstein equilibrium. The relaxation estimate first gives a fixed amount of mass below a small energy level. We then use collision estimates at smaller and smaller energy scales and add their effects in a weighted sum functional with a fixed upper bound. If no atom forms at zero energy, enough mass remains in the positive-energy shells to force this sum to exceed its upper bound in finite time, which leads to condensation in finite time.
ER -