[Paper Review] Solution landscape of Onsager functional identifies non-axisymmetric critical points.

Jianyuan Yin, Lei Zhang+1 more|arXiv (Cornell University)|Apr 20, 2021

Quantum chaos and dynamical systems27 references4 citations

TL;DR

This study maps the solution landscapes of the Onsager free-energy functional using saddle dynamics and uniform sampling, revealing non-axisymmetric critical points—such as the novel 'tennis' state and symmetries like square, hexagonal, and icosahedral—under coupled dipolar/Maier-Saupe and Onsager potentials, demonstrating a global view of phase bifurcations beyond axisymmetric solutions.

ABSTRACT

We investigate the solution landscapes of the Onsager free-energy functional with different potential kernels, including the dipolar potential, the Maier--Saupe potential, the coupled dipolar/Maier--Saupe potential, and the Onsager potential. A uniform sampling method is implemented for the discretization of the Onsager functional, and the solution landscape of the Onsager functional is constructed by using the saddle dynamics coupled with downward/upward search algorithms. We first compute the solution landscapes with the dipolar and Maier--Saupe potentials, for which all critical points are axisymmetric. For the coupled dipolar/Maier--Saupe potential, the solution landscape shows a novel non-axisymmetric critical point, named as tennis, which exists for a wide range of parameters. We further demonstrate various non-axisymmetric critical points in the Onsager functional with the Onsager potential, including square, hexagon, octahedral, cubic, quarter, icosahedral, and dodecahedral states. The solution landscape provides an efficient approach to show the global structures as well as the bifurcations of critical points, which can not only verify the previous analytic results but also propose several conjectures based on the numerical findings.

Motivation & Objective

• To explore the global structure of critical points in the Onsager free-energy functional beyond axisymmetric solutions.
• To investigate how different potential kernels—dipolar, Maier-Saupe, coupled dipolar/Maier-Saupe, and Onsager—affect the solution landscape.
• To identify and characterize non-axisymmetric critical points that emerge under specific parameter regimes.
• To validate prior analytic results and generate new conjectures through numerical solution landscape construction.
• To develop a systematic method for visualizing bifurcations and structural transitions in liquid crystal phase behavior.

Proposed method

• Employing uniform sampling to discretize the Onsager functional for numerical stability and coverage.
• Applying saddle dynamics coupled with downward/upward search algorithms to systematically explore critical points.
• Using the solution landscape framework to map all critical points, including minima, saddles, and bifurcations.
• Analyzing the functional with four distinct potential kernels: dipolar, Maier-Saupe, coupled dipolar/Maier-Saupe, and Onsager potential.
• Tracking parameter-dependent transitions and symmetry breaking via continuous monitoring of critical point evolution.
• Validating results by comparing with known analytic solutions and identifying new numerical patterns.

Experimental results

Research questions

• RQ1 What critical points emerge in the Onsager functional when using the coupled dipolar/Maier-Saupe potential, and are they axisymmetric?
• RQ2 How do different potential kernels influence the symmetry and stability of critical points in the solution landscape?
• RQ3 What non-axisymmetric critical points appear in the Onsager potential model, and under what parameter conditions do they exist?
• RQ4 Can the solution landscape framework reveal bifurcation structures and phase transitions not captured by traditional analytic methods?
• RQ5 What new conjectures about liquid crystal phase behavior can be derived from the numerical exploration of the solution landscape?

Key findings

• The solution landscape reveals a novel non-axisymmetric critical point, termed the 'tennis' state, in the coupled dipolar/Maier-Saupe potential, existing over a wide range of parameters.
• For the Onsager potential, multiple non-axisymmetric critical points are identified, including square, hexagonal, octahedral, cubic, quarter, icosahedral, and dodecahedral states.
• All critical points under the pure dipolar and Maier-Saupe potentials remain axisymmetric, confirming previous theoretical expectations.
• The solution landscape provides a comprehensive view of global structures and bifurcations, enabling the verification of prior analytic results.
• The numerical findings suggest new conjectures regarding symmetry breaking and phase transitions in liquid crystal systems.
• The method successfully captures complex structural transitions and symmetry changes that are difficult to access through conventional analytical approaches.