Figure 1.
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 $
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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 |
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.
Keywords:
particle dynamics,
Riesz potential,
repulsion,
rectifiable curve,
uniform distribution,
porous medium equation.
Mathematics Subject Classification:
Primary: 31C20, 35K55; Secondary: 35Q70, 92D25.
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
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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. |
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. |
Figure 2.
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)
Figure 3.
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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