Ground states and phase transitions for an aggregation model with fast diffusion on sphere

TL;DR

This work analyzes an aggregation–diffusion free energy on the unit sphere with fast diffusion (0<m<1) and quadratic nonlocal attraction, identifying how ground states evolve with interaction strength κ for different diffusion regimes. By formulating admissible symmetric minimizers and deriving Euler–Lagrange conditions, the authors classify equilibria as fully supported densities or measure-valued states containing Dirac masses, depending on m and d. They reveal two main phase transitions: a bifurcation from the uniform distribution at κ1 to fully supported equilibria, and a second transition at κ2 where singular measures emerge, with a possible κ3 in certain ranges; numerical experiments corroborate the analytical bifurcation structure and energy ordering. The results extend Onsager-type free energy analyses to fast diffusion on spheres, illuminating when concentration persists despite diffusion and how ground states reorganize across parameter regimes. This has implications for modeling orientation phenomena in polymers and related systems on curved manifolds, and provides a framework for predicting when singular mass concentrations arise as ground states.

Abstract

We consider a free energy on the sphere that contains an entropy associated to nonlinear fast diffusion, and a nonlocal interaction energy. The two components of the free energy compete with each other, as one favours spreading and the other promotes concentration, respectively. The model is a generalization of the Onsager free energy with dipolar potential, used to study polymer orientation. We study the global energy minimizers of the energy functional, and in particular the various phase transitions that occur with respect to the strength of the nonlocal attractive interactions. In the considered regime, diffusion reduces as the density increases, for which reason the global energy minimizers can contain Dirac mass concentrations. We identify various ranges of the fast diffusion exponent and of the interaction strength, which give qualitatively different equilibria and ground states. The theoretical results are supported by numerical illustrations.

Figures (6)

• Figure 1: Plot of function $H$ defined in \ref{['eqn:H']}. (a) $1-2/d<m<1$. For this range of $m$, $H(1) = 0$. For any $\kappa>\kappa_1$, there exists a unique $\eta_\kappa >1$ such that $\kappa^{-1} = H(\eta_\kappa)$ -- see equation \ref{['eqn:kappa-eta']}. (b) $1-2/(d-1)<m<1-2/d$. In this case, $H(1) = 1/\kappa_2$, with $\kappa_2>\kappa_1$. For any $\kappa_1<\kappa<\kappa_2$, there exists a unique $\eta_\kappa>1$ that solves $\kappa^{-1} = H(\eta_\kappa)$. (c) $0<m<1-2/(d-1)$. In this case, $H(1) = 1/\kappa_2$, with $\kappa_2<\kappa_1$. For any $\kappa_2<\kappa<\kappa_1$, there exists a unique $\eta_\kappa>1$ that solves $\kappa^{-1} = H(\eta_\kappa)$. For plot (a) we used $m=0.5$ and $d =2$, for plot (b), $m=0.25$ and $d =3$, and for plot (c), $m=0.3$ and $d =3$.
• Figure 2: Plots of $f_\kappa$ and $g$ from \ref{['eqn:fg']}. (a) Case $1-\frac{2}{d-1}<m<1-\frac{2}{d}$: for any $\kappa > \kappa_2$, equation \ref{['eqn:find-alpha']} has a unique solution $\alpha_\kappa \in (0,1)$. (b) Case $0<m<1-\frac{2}{d-1}$: for $\kappa_3 <\kappa <\kappa_2$, there exist two solutions $0<\tilde{\alpha}_{\kappa}<\alpha_{\kappa}<1$ of \ref{['eqn:find-alpha']}. At $\kappa=\kappa_3$, the two solutions $\tilde{\alpha}_{\kappa}$ and $\alpha_{\kappa}$ coincide. For $\kappa>\kappa_2$, \ref{['eqn:find-alpha']} has a unique solution in the interval $(0,1)$. The numerical simulations correspond to (a) $m=0.25$ and $d=3$, and (b) $m=0.3$ and $d=5$.
• Figure 3: Case i) $1-2/d<m<1$. (a) Plot of the norm of the centre of mass of equilibria. b) Plot of the energies of equilibria for $\kappa>\kappa_1$. In both plots, blue corresponds to $\rho_{\text{uni}}$ and red to $\rho_\kappa$ -- see Proposition \ref{['prop:bif']} part a). At $\kappa=\kappa_1$, a fully supported equilibrium $\rho_\kappa$ in the form \ref{['eqn:rhok']} emerges from the uniform distribution, and then it exists for all $\kappa>\kappa_1$. The equilibrium $\rho_\kappa$ is the global energy minimizer when $\kappa>\kappa_1$ -- see Theorem \ref{['thm:global-min']} part a). The numerical simulations correspond to $m=0.5$ and $d=2$.
• Figure 4: Plot of equilibrium densities $\rho_\kappa$ from \ref{['eqn:rhok']}. (a) $m=0.5$, $d=2$. These values correspond to case i) from Proposition \ref{['prop:bif']}. The equilibrium densities $\rho_\kappa$ exist for all $\kappa>\kappa_1$. (b) $m=0.25$, $d=3$. These values correspond to case ii) from Proposition \ref{['prop:bif']}. The equilibrium densities $\rho_\kappa$ exist for $\kappa_1<\kappa<\kappa_2$ in this case (here $\kappa_2 \approx 12.4453$). For both plots, $\kappa=\kappa_1$ corresponds to the uniform distribution, and $\rho_\kappa$ concentrates around $\theta =0$ as $\kappa$ increases.
• Figure 5: Case ii) $1-2/(d-1)<m<1-2/d$. (a) Plot of the norm of the centre of mass of equilibria. (b) Plot of the energies of equilibria for $\kappa>\kappa_1$. In both plots, blue corresponds to $\rho_{\text{uni}}$ and red to either $\rho_\kappa$ (for $\kappa_1<\kappa<\kappa_2$) or $\mu_{\alpha_\kappa}$ (for $\kappa>\kappa_2$) -- see Proposition \ref{['prop:bif']}. At $\kappa=\kappa_1$, a fully supported equilibrium $\rho_\kappa$ in the form \ref{['eqn:rhok']} emerges from the uniform distribution. At $\kappa=\kappa_2$, $\rho_\kappa$ changes to a measure-valued equilibrium $\mu_{\alpha_\kappa}$ -- see \ref{['eqn:mu-alphak']} and \ref{['eqn:equil-brho']}; this transition is indicated by a black diamond. The global minimizer when $\kappa_1<\kappa<\kappa_2$ is $\rho_\kappa$, and when $\kappa>\kappa_2$, the ground state is $\mu_{\alpha_\kappa}$ -- see Theorem \ref{['thm:global-min']} part a). The numerical simulations correspond to $m=0.25$, $d=3$.

Theorems & Definitions (24)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3: Radial symmetry of global minimizers
• proof
• Remark 2.1
• Remark 2.2
• Proposition 2.1: Stability of the uniform distribution
• proof