PreprintCC BY 4.0arXiv
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≤ 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≥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.
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BibTeX
@misc{kim2026dimensiondependent,
title = {{Dimension-dependent symmetry breaking in a diffusive nonlocal interaction model}},
author = {Dohyun Kim and Hansol Park and Woojoo Shim},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.16978},
eprint = {2609.16978},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16978},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Dimension-dependent symmetry breaking in a diffusive nonlocal interaction model
AU - Dohyun Kim
AU - Hansol Park
AU - Woojoo Shim
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2609.16978
UR - https://arxiv.org/abs/2609.16978
AB - 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≤ 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≥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.
ER -