Dynamics of screened particles towards equi-spaced ground states
Lucia De Luca, Michael Goldman, Marcello Ponsiglione
TL;DR
The paper analyzes gradient-flow dynamics of Λ-periodic, screened one-dimensional particle systems driven by negative fractional seminorm energies ${\mathscr E}^{s}$ and their mollified versions ${\mathscr E}_\varepsilon^{s}$. It proves that equi-spaced configurations with unit lattice are global ground states for all admissible $s$ and demonstrates exponential convergence of the gradient flows to these ground states, with a rate scaling like $\Lambda^{-2s}$. In the subcritical regime, collisions are avoided and convergence is guaranteed; in the supercritical regime, energies may blow up as $\varepsilon \to 0$ but gradients remain bounded, yielding convergence to a renormalized energy flow ${\mathscr W}_0^{s}$. The critical case $s=\tfrac{1}{2}$ links to renormalized energies from Ginzburg–Landau vortices, highlighting a deep connection between one-dimensional Riesz-type interactions and periodic dislocation models. Overall, the work establishes a robust picture: convexity-driven minimization yields equi-spaced ground states, and gradient flows robustly reach these states, with a clear renormalization perspective as regularization vanishes.
Abstract
This paper deals with the dynamics - driven by the gradient flow of negative fractional seminorms - of empirical measures towards equi-spaced ground states. Specifically, we consider periodic empirical measures $μ$ on the real line that are screened by the Lebesgue measure, i.e., with $μ-d x$ having zero average. To each of these measures $μ$ we associate a {(periodic)} function $u$ satisfying $u'= d x - μ$. For $s\in (0,\frac 12)$ we introduce energy functionals $\mathcal E^s(μ)$ that can be understood as the density of the $s$-Gagliardo seminorm of $u$ per unit length. Since for $s\ge \frac 12$, the $s$-Gagliardo seminorms are infinite on functions with jumps, some regularization procedure is needed: For $s\in[\frac 12,1)$ we define $\mathcal E_\e^s(μ):= \mathcal E^s(μ_\e)$, where $μ_\varepsilon$ is obtained by mollifying $μ$ on scale $\varepsilon$. We prove that the minimizers of $\mathcal E^s$ and $\mathcal E_\varepsilon^s$ are the equi-spaced configurations of particles with lattice spacing equal to one. Then, we prove the exponential convergence of the corresponding gradient flows to the equi-spaced steady states. Finally, although for $s\in[\frac 12 ,1)$ the energy functionals $\mathcal E_\varepsilon^s$ blow up as $\varepsilon\to 0$, their gradients are uniformly bounded (with respect to $\varepsilon$), so that the corresponding trajectories converge, as $\varepsilon\to 0$, to the gradient flow solution of a suitable renormalized energy.