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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 |
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.
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. |
show all references
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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