Critical points of the Onsager functional on a sphere

We study Onsager’s model of nematic phase transitions with orientation parameter on a sphere. We consider two interaction potentials: the antisymmetric (with respect to orientation inversion) dipolar potential and symmetric Maier-Saupe potential. We obtain a complete classification and explicit expressions of all critical points, analyze their stability, and construct the corresponding bifurcation diagrams. A theory of phase transitions in rod-like polymers and nematic liquid crystals acquired a solid mathematical background when Onsager introduced a variational model [11] relating their equilibrium states to critical points of a free energy functional. Ever since, his approach has become a standard way to describe many associated phenomena, both static and dynamic [4,7]. However, the problem of rigorous analysis and classification of all critical points of Onsager’s functional remained open, and only recently a significant progress has been achieved. A detailed study of a reduced model, where the rod orientation is assumed to lie on a circle, was accomplished in [5,2,3,9]. A full model (on a sphere) was considered in [1], where the critical points were related to the solutions of a transcendental matrix equation, and some of their properties were studied based on such representation. In this work we extend the methods employed in [5] for the reduced model (on a circle) and present explicit expressions for all critical points of the full model, analyze their stability and bifurcations. In the outcome we produce a complete solution to this remarkable model in statistical physics of polymers. Let us review the Onsager model and outline our most important results. The state of the system (generally a liquid-crystalline suspension) is described by a probability density of rod orientations ρ(s). The orientation parameter s belongs to a unit sphere S in a three-dimensional Euclidean space. Throughout the paper we routinely use alternative notations for the points on the unit sphere, e. g., s ∈ S may be represented as a unit vector in Cartesian coordinates: x = (x1, x2, x3) ∈ R, |x| = 1; or as angles, φ ∈ [0, 2π), θ ∈ [0, π] in a spherical coordinate frame. In the last case, unless the polar axis (θ = 0) and the plane φ = 0 are prescribed explicitly, they may be chosen arbitrarily. We may write the Onsager free energy functional as F [ρ] := ∫ S2 [ τρ(s) ln ρ(s) + 1 2 ρ(φ) ∫ S2 U(s, s′) ρ(s′) ds′ ] ds. (1) The first term under the integral is the entropic term, the positive parameter τ is the temperature. This term is minimized by the uniform density ρ̄(s) ≡ 1/4π and is dominant when τ is large. The second term is the interaction term, the function U(s, s′) is called