Bose condensation at fixed dilute density
Fix a nonzero, nonnegative, radial potential $V \in C_c^\infty(\mathbb{R}^3)$. For $L > 0$ let $\Lambda_L = (\mathbb{R}/L\mathbb{Z})^3$, let $V_L(x) = \sum_{k \in \mathbb{Z}^3} V(x + Lk)$, and on the symmetric subspace of $L^2(\Lambda_L^N)$ let $$H_{N,L} = \sum_{j=1}^N -\Delta_{x_j} + \sum_{1 \le i < j \le N} V_L(x_i - x_j).$$ Let $\Psi_{N,L}$ be its normalized nonnegative ground state, $\gamma^{(1)}_{N,L}$ its one-particle reduced density matrix, normalized to have trace $N$, and $\phi_L = L^{-3/2}$ the normalized constant function on $\Lambda_L$. Lieb and Seiringer proved complete Bose–Einstein condensation of the ground state of a dilute trapped gas in the Gross–Pitaevskii limit, where $N \to \infty$ with $Na$ fixed, $a$ being the scattering length of the interaction in units of the trap size. Chong, Liang and Nam derived condensation beyond the Gross–Pitaevskii regime, for the interaction $N^{2-2\kappa}V(N^{1-\kappa}x)$ on the unit box with $0 < \kappa < 2/11$. For $0 \le \kappa < 2/3$ this interaction corresponds to the unscaled potential at density $N^{3\kappa - 2} \to 0$, and $\kappa = 2/3$ to the thermodynamic limit. Here the potential and the density stay fixed as the volume grows.
Problem. Is there $\rho_0(V) > 0$ such that every fixed $0 < \rho < \rho_0(V)$ satisfies $$\liminf_{\substack{L \to \infty,\ N \to \infty \\ N/L^3 \to \rho}} \frac{\langle \phi_L, \gamma^{(1)}_{N,L} \phi_L \rangle}{N} > 0\,?$$
Source: Stated by AIM — Open Applied Problems, after Chong, Liang and Nam (2025), §1 and §2.2, who distinguish the thermodynamic limit from the regimes they treat.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MP — Mathematical Physics
- Secondary
- SP
- Keywords
- AIM Open Applied Problems, Bose-Einstein condensation, dilute Bose gases, thermodynamic limit
References
@incollection{lieb2003,
author = {Lieb, Elliott H. and Seiringer, Robert},
title = {{B}ose--{E}instein condensation of dilute gases in traps},
booktitle = {Advances in Differential Equations and Mathematical Physics},
series = {Contemp. Math.},
volume = {327},
publisher = {Amer. Math. Soc.},
address = {Providence, RI},
year = {2003},
pages = {239--250},
doi = {10.1090/conm/327/05818},
eprint = {math-ph/0210028},
archivePrefix = {arXiv},
}
@article{chong2026,
author = {Chong, Jacky J. and Liang, Hao and Nam, Phan Th{\`a}nh},
title = {Kinetic localization via {P}oincar{\'e}-type inequalities and applications to the condensation of {B}ose gases},
journal = {J. Funct. Anal.},
volume = {291},
number = {5},
year = {2026},
pages = {111542},
doi = {10.1016/j.jfa.2026.111542},
eprint = {2510.20493},
archivePrefix = {arXiv},
}
@misc{aim-thermodynamic-bose-condensation,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Bose condensation at fixed dilute density}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/196-thermodynamic-bose-condensation.md},
note = {Entry 196 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
For each fixed bounded measurable, nonnegative, radial interaction of finite range in three dimensions that is not zero almost everywhere, we prove a positive lower bound on the constant-orbital condensate fraction that is uniform over all sufficiently small densities and all temperatures between zero and the square of the density. The interaction and density remain fixed in the thermodynamic limit.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, build/main.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00727, "Bose condensation at fixed dilute density", clopen.solutions.