Point vortex equilibria on the sphere via Brownian ratchets

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 465, Issue 2102, 2009, Pages 437-455

  Newton, Paul K ^a,b   Sakajo, Takashi ^c  

^a University of Southern California ^*  (United States)

^b UNIVERSITY OF SOUTHERN CALIFORNIA   (United States)

^c HOKKAIDO UNIVERSITY   (Japan)

Author keywords

Brownian ratchets; Point vortices on a sphere; Shannon entropy; Singular value decomposition; Thomson's problem

Indexed keywords

BROWNIAN MOVEMENT; ENTROPY; MATRIX ALGEBRA; SPHERES; VECTORS; VORTEX FLOW;

BASIS SETS; BROWNIAN RATCHETS; CONFIGURATION MATRIXES; CONVERGENCE THRESHOLDS; EQUILIBRIUM CONFIGURATIONS; FROBENIUS NORMS; GENERAL FEATURES; HAMILTONIAN ENERGIES; NULL SPACES; OPTIMAL BASES; POINT VORTICES ON A SPHERE; RANDOM WALKS; RELATIVE EQUILIBRIUMS; SHANNON ENTROPY; SINGULAR VECTORS; THOMSON'S PROBLEM; VORTEX STRENGTHS; VORTICITY VECTORS;

SINGULAR VALUE DECOMPOSITION;

EID: 58149131998     PISSN: 13645021     EISSN: 14712946     Source Type: Journal    
DOI: 10.1098/rspa.2008.0203     Document Type: Article

References (28)

1. Altschuler, E. L. & Perez-Garrido, A. 2005 Global minimum for Thomson's problem of charges on a sphere. Phys. Rev. E 71, 047 703. (doi:10.1103/PhysRevE.71.047703)
2. Altschuler, E. L. & Perez-Garrido. A. 2006 Defect free global minima in Thomson's problem of charges on a sphere. Phys. Rev. E 73, 036 108. (doi:10.1103/PhysRevE.73.036108)
3. Altschuler, E. L., Williams, T. J., Ratner, E. R., Tipton, R., Stong, R., Dowla, F, & Wooten, F. 1997 Possible global minimum lattice configurations for Thomson's problem of charges on a sphere. Phys. Rev. Lett. 78, 2681-2685. (doi:10.1103/PhysRevLett.78.2681)
4. Aref, H., Newton, P. K., Stremler, M. A., Tokieda, T. & Vainchtein, D. L. 2003 Vortex crystals. Adv. Appl. Mech. 39, 1-79.
5. Aref, H. & Vainchtein, D. L. 1998 Point vortices exhibit asymmetric equilibria. Nature 392, 769-770. (doi:10.1038/33827)
6. Bergersen, B., Boai, D. & Palffy-Muhoray, P. 1994 Equilibrium configurations of particles on a sphere: the case of logarithmic interactions. J. Phys. A Math. Gen. 27, 2579-2586. (doi:10.1088/0305-4470/27/7/ 032)
7. Brillinger, D. R. 1997 A particle migrating randomly on a sphere. J. Theor. Prob. 10, 429-443. (doi:10.1023/A:1022869817770)
8. Cabral, H. E., Meyer. K. R. & Schmidt, D. S. 2003 Stability and bifurcations for the N+1 vortex problem on the sphere. Regular Chaotic Dyn. 8, 259-282. (doi:10.1070/RD2003v008n03 ABEH000243)
9. Campbell, L. J. & Ziff, R. 1979 Vortex patterns and energies in a rotating superfluid. Phys. Rev. B 20, 1886-1902. (doi:10.1103/PhysRevB.20.1886)
10. Edmundson, J. R. 1992 The distribution of point charges on the surface of a sphere. Acta Cryst. A 48, 60-69. (doi:10.1107/ S0108767391009133)
11. Erber, T. & Hockney. G. M. 1991 Equilibrium configurations of N equal charges on a sphere. J. Phys. A Math. Gen. 24, L1369-L1377. (doi:10.1088/0305-4470/24/23/008)
12. Glasser, L. & Every, A. G. 1992 Energies and spacings of point charges on a sphere. J. Phys. A Math. Gen. 25, 2473-2482. (doi:10.1088/0305-4470/25/9/020)
13. Jamaloodeen, M. I. & Newton, P. K. 2006 The N-vortex problem on a rotating sphere: II. Heterogeneous Platonic solid equilibria. Proc. R. Soc. A 462, 3277-3299. (doi:10.1098/rspa.2006.1731)
14. Kendall, D. G. 1974 Pole-seeking Brownian motion and bird navigation. J. R. Statist. Soc. B 36, 365-417.
15. Kidambi, R. & Newton, P. K. 1998 Motion of three point vortices on a sphere. Physica D 116, 143-175. (doi:10.1016/S0167-2789(97)00236- 4)
16. Laurent-Polz, F. 2002 Point vortices on the sphere: a case of opposite vorticities. Nonlinearity 15, 143-171. (doi:10.1088/0951-7715/15/ l/307)
17. Lim, C. C. & Nebus, J. 2006 Vorticity, statistical mechanics, and Monte Carlo simulation. Springer Monographs in Mathematics. New York, NY: Springer.
18. Lim, C. C., Montaldi, J. & Roberts, M. R. 2001 Relative equilibria of point vortices on the sphere. Physica D 148, 97-135. (doi:10.1016/S0167-2789(00)00167-6)
19. Newton, P. K. 2001 The N-vortex problem: analytical techniques. Applied Mathematical Sciences, vol. 145. New York. NY: Springer.
20. Newton, P. K. & Chamoun, G. 2007 Construction of point vortex equilibria via Brownian ratchets. Proc. R. Soc. A 463, 1525-1540. (doi:10.1098/rspa.2007.1832)
21. Newton, P. K. & Chamoun, G. In press. Vortex lattice theory: a particle interaction perspective. SIAM Rev.
22. Newton, P. K. & Shokraneh. H. 2006 The N-vortex problem on a rotating sphere. I. Multifrequency configurations. Proc. R. Soc. A 462, 149-169. (doi:10.1098/rspa.2005.15C6)
23. Riemann, P. 2002 Brownian motors: noisy transport far from equilibrium. Phys. Rep. 361, 57-265. (doi:10.1016/S0370-1573(01)00081-3)
24. Roberts, P. H. & Ursell, H. D. 1960 Random walk on a sphere and on a Riemannian manifold. Phil. Trans. R. Soc. A 252, 317-356. (doi:10.1098/rsta.1960.0008)
25. Saff, E. B. & Kuijlaars, A. B. J. 1997 Distributing many points on a sphere. Math. Intell. 19, 5-11.
26. Shannon, C. E. 1948 A mathematical theory of communication. Bell. Sys. Tech. J. 27. 379-423.
27. Smale, S. 2000 Mathematical problems for the next century. In Mathematics: frontiers and perspectives, pp. 271-294. Providence RI: AMS.
28. Trefethen, L. N. & Bau III, D. 1997 Numerical linear algebra. Philadelphia, PA: SIAM Publishing.