Dimension-dependent symmetry breaking in a diffusive nonlocal interaction model

Abstract

We study the equilibrium structure and long-time behavior of a nonlinear Fokker--Planck equation with diffusion and nonlocal interactions generated by squared simplex volumes. For $1\le n<d$, the stationary states are isotropic Gaussian measures, and every solution converges to the equilibrium determined by its conserved mean. At $n=d$, the free energy fixes only the covariance determinant, allowing a continuum of anisotropic Gaussian equilibria when $d\ge 2$. We establish global well-posedness for arbitrary initial probability measures with finite second moment, identify stationary states with the global minimizers of the free energy, and prove convergence in quadratic Wasserstein distance with explicit exponential rates.