# American Institute of Mathematical Sciences

December  2021, 41(12): 5509-5536. doi: 10.3934/dcds.2021086

## Dynamics of particles on a curve with pairwise hyper-singular repulsion

 1 Department of Mathematics, Vanderbilt University, Nashville TN 37240, USA 2 Department of Mathematics, Center for Scientific Computation and Mathematical Modeling (CSCAMM), University of Maryland, College Park MD 20742, USA 3 Department of Mathematics and Institute for Physical Science & Technology, University of Maryland, College Park MD 20742, USA

Received  October 2020 Revised  April 2021 Published  December 2021 Early access  May 2021

Fund Project: DH and ES acknowledge support, in part, by the U. S. National Science Foundation under grant DMS-1516400. RS and ET were supported by NSF grant DMS16-13911 and ONR grant N00014-1812465. ES and ET also thank the hospitality of the Laboratoire Jacques Louis Lions in Sorbonne Université during spring 2019, with support by the European Research Council advanced grant 338977 BREAD (ES) and by ERC Grant 740623 under the EU Horizon 2020 (ET)

We investigate the large time behavior of $N$ particles restricted to a smooth closed curve in $\mathbb{R}^d$ and subject to a gradient flow with respect to Euclidean hyper-singular repulsive Riesz $s$-energy with $s>1.$ We show that regardless of their initial positions, for all $N$ and time $t$ large, their normalized Riesz $s$-energy will be close to the $N$-point minimal possible energy. Furthermore, the distribution of such particles will be close to uniform with respect to arclength measure along the curve.

Citation: Douglas Hardin, Edward B. Saff, Ruiwen Shu, Eitan Tadmor. Dynamics of particles on a curve with pairwise hyper-singular repulsion. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5509-5536. doi: 10.3934/dcds.2021086
##### References:
 [1] S. V. Borodachov, Lower order terms of the discrete minimal Riesz energy on smooth closed curves, Canad. J. Math., 64 (2012), 24-43.  doi: 10.4153/CJM-2011-038-5. [2] S. V. Borodachov, D. P. Hardin and E. B. Saff, Discrete energy on rectifiable sets, Springer Monographs in Mathematics, Springer, New York, [2019], 2019, xviii+666 pp. doi: 10.1007/978-0-387-84808-2. [3] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 290. Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4757-6568-7. [4] D. P. Hardin and E. B. Saff, Discretizing manifolds via minimum energy points, Notices Amer. Math. Soc., 51 (2004), 1186-1194. [5] A. Martínez-Finkelshtein, V. Maymeskul, E. A. Rakhmanov and E. B. Saff, Asymptotics for minimal discrete Riesz energy on curves in $\mathbb{R}^d$, Canad. J. Math., 56 (2004), 529-552.  doi: 10.4153/CJM-2004-024-1. [6] K. Oelschläger, Large systems of interacting particles and the porous medium equation, J. Diff. Eq., 88 (1990), 294–346. doi: 10.1016/0022-0396(90)90101-T. [7] S. Serfaty (appendix with Mitia Duerinckx), Mean field limit for Coulomb-type flows, Duke Math. J., 169 (2020), 2887–2935. doi: 10.1215/00127094-2020-0019.

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##### References:
 [1] S. V. Borodachov, Lower order terms of the discrete minimal Riesz energy on smooth closed curves, Canad. J. Math., 64 (2012), 24-43.  doi: 10.4153/CJM-2011-038-5. [2] S. V. Borodachov, D. P. Hardin and E. B. Saff, Discrete energy on rectifiable sets, Springer Monographs in Mathematics, Springer, New York, [2019], 2019, xviii+666 pp. doi: 10.1007/978-0-387-84808-2. [3] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 290. Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4757-6568-7. [4] D. P. Hardin and E. B. Saff, Discretizing manifolds via minimum energy points, Notices Amer. Math. Soc., 51 (2004), 1186-1194. [5] A. Martínez-Finkelshtein, V. Maymeskul, E. A. Rakhmanov and E. B. Saff, Asymptotics for minimal discrete Riesz energy on curves in $\mathbb{R}^d$, Canad. J. Math., 56 (2004), 529-552.  doi: 10.4153/CJM-2004-024-1. [6] K. Oelschläger, Large systems of interacting particles and the porous medium equation, J. Diff. Eq., 88 (1990), 294–346. doi: 10.1016/0022-0396(90)90101-T. [7] S. Serfaty (appendix with Mitia Duerinckx), Mean field limit for Coulomb-type flows, Duke Math. J., 169 (2020), 2887–2935. doi: 10.1215/00127094-2020-0019.
The number $r_0$ in Lemma 4.1 is the range for which ${{\bf x}}(z)$ can be approximated by a local Taylor expansion near ${{\bf x}}(y)$ for any fixed $y$
Lemmas 5.1 and 5.2. Left: the summand in the last term of (5.2). The two terms representing the forces from $z_j$ acting on $z_{i_M}$ (red) and $z_{i_M+1}$ (blue), which decreases/increases $\delta$ respectively. Right: a local uniform distribution like $\{\tilde{z}_j\}$ makes $\frac{ \,\mathrm{d}}{ \,\mathrm{d}{t}}\delta \approx 0$ up to errors from curvature. A possible defect will release the total pushing force on $\delta$, make $\frac{ \,\mathrm{d}}{ \,\mathrm{d}{t}}\delta$ positive, and thus violate (5.11)
Proof of Theorem 2.1. Left: when (5.11) does not hold, $\delta$ is increasing very fast (i.e., $\rho_M$ is decreasing very fast). Right: when (5.11) holds, there is almost uniform distribution near $z_{i_M}$ (red parts) with average density near $\rho_M$, and the total repulsion at $z_{i_M}$ is strong (see (5.12)). The rest part has average density at most $1+\epsilon$, and Lemma 3.1 applies to give a weak total repulsion cut. The strong/weak total repulsion ((1)-(2) good contribution, $I_1$, and (3)-(4) bad contribution, $I_2$, see (6.9)) forces the green part to rotate. The parameter $r_1$ is to guarantee that (3) or (4) cannot be too short, so that the possible bad contribution from (1)-(4) or (2)-(3) (the term $I_3$) can be neglected
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