Axisymmetry of equilibrium distributions for Onsager rods
For $\kappa > 0$ and probability densities $\rho \ge 0$ on $\mathbb{S}^2$ with respect to surface measure, let $$\mathcal{F}_\kappa(\rho) = \int_{\mathbb{S}^2} \rho \log \rho \, dS + \frac{\kappa}{2} \iint_{\mathbb{S}^2 \times \mathbb{S}^2} \sqrt{1 - (p \cdot q)^2}\, \rho(p)\, \rho(q)\, dS_p\, dS_q,$$ with $0 \log 0 = 0$ and $\mathcal{F}_\kappa(\rho) = +\infty$ when the entropy is infinite. This is Onsager's free energy for hard rods, at unit temperature. A global minimizer exists for every $\kappa$ by the direct method, and every critical point is smooth (Vollmer; Ball). Ball proved that a non-axisymmetric critical point with cubic symmetry exists whenever $\kappa > 2^8/\pi$, and Yin, Zhang and Zhang computed several further non-axisymmetric critical points numerically.
Problem. Is every global minimizer of $\mathcal{F}_\kappa$, for every $\kappa > 0$, axisymmetric: is there $e \in \mathbb{S}^2$ such that $\rho(Rp) = \rho(p)$ for almost every $p$ and every rotation $R$ fixing $e$?
Source: Ball (2020), closing paragraph of §3, who asks it for local or global minimizers, for general kernels and in particular for the Onsager kernel. Stated by AIM — Open Applied Problems for global minimizers and the Onsager kernel.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP, OC
- Keywords
- AIM Open Applied Problems, nematic liquid crystals, Onsager model
References
@article{ball2021onsager,
author = {Ball, J. M.},
title = {Axisymmetry of critical points for the {O}nsager functional},
journal = {Philos. Trans. Roy. Soc. A},
volume = {379},
number = {2201},
year = {2021},
pages = {20200110},
doi = {10.1098/rsta.2020.0110},
eprint = {2008.04009},
archivePrefix = {arXiv},
}
@article{vollmer2017,
author = {Vollmer, Michaela A. C.},
title = {Critical points and bifurcations of the three-dimensional {O}nsager model for liquid crystals},
journal = {Arch. Ration. Mech. Anal.},
volume = {226},
number = {2},
year = {2017},
pages = {851--922},
doi = {10.1007/s00205-017-1146-8},
}
@article{yin2022onsager,
author = {Yin, Jianyuan and Zhang, Lei and Zhang, Pingwen},
title = {Solution landscape of the {O}nsager model identifies non-axisymmetric critical points},
journal = {Phys. D},
volume = {430},
year = {2022},
pages = {133081},
doi = {10.1016/j.physd.2021.133081},
eprint = {2104.09766},
archivePrefix = {arXiv},
}
@misc{aim-onsager-axisymmetric-minimizers,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Axisymmetry of equilibrium distributions for Onsager rods}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/219-onsager-axisymmetric-minimizers.md},
note = {Entry 219 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00628, "Axisymmetry of equilibrium distributions for Onsager rods", clopen.solutions.