Predicting pattern formation in particle interactions

Mathematical Models and Methods in Applied Sciences

Volumn 22, Issue SUPPL.1, 2012, Pages

  Von Brecht, James H ^a   Uminsky, David ^a   Kolokolnikov, Theodore ^b   Bertozzi, Andrea L ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b DALHOUSIE UNIVERSITY   (Canada)

Author keywords

Nonlocal differential equations; particle self assembly; pattern formation

Indexed keywords

D DIMENSIONS; DYNAMICAL SYSTEMS APPROACH; LARGE SYSTEM; LINEAR THEORY; NONLOCAL; NONLOCAL DIFFERENTIAL EQUATIONS; PATTERN FORMATION; SPACE-FILLING; ZERO PATTERNS;

DIFFERENTIAL EQUATIONS; DYNAMICAL SYSTEMS; GROUND STATE; SPHERES; STATISTICAL MECHANICS;

SELF ASSEMBLY;

EID: 84857874283     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202511400021     Document Type: Article

References (47)

1. I. Giardina, G. Parisi, R. Santagati, F. Stefanini, A. Cavagna, A. Cimarelli and R. Tavarone, From empirical data to inter-individual interactions: Unveiling the rules of collective animal behavior, Math. Models Methods Appl. Sci. 20 (2010) 1491-1510.
2. E. L. Altschuler, T. J. Williams, E. R. Ratner, R. Tipton, R. Stong, F. Dowla and F. Wooten, Possible global minimum lattice configurations for Thomson's problem of charges on a sphere, Phys. Rev. Lett. 78 (1997) 2681-2685. (Pubitemid 127655285)
3. M. Ballerini, N. Cabbibo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Natl. Acad. Sci. 105 (2008) 1232-1237. (Pubitemid 351397036)
4. A. M. Barry, G. R. Hall and C. E. Wayne, Relative equilibria of the (1 + N)-vortex Problem, J. Nonlinear Sci., to appear.
5. N. Bellomo, H. Berestycki, F. Brezzi and J.-P. Nadal, Mathematics and complexity in life and human sciences, Math. Models Methods Appl. Sci. 20(Suppl.) (2010) 1391-1395.
6. A. J. Bernoff and C. M. Topaz, A primer of swarm equilibria, SIAM J. Appl. Dyn. Syst. 10 (2011) 212-250.
7. M. P. Brenner, P. Constantin, L. P. Kadanoff, A. Schenkel and S. C. Venkataramani, Diffusion, attraction and collapse, Nonlinearity 12 (1999) 1071.
8. J. L. Burchnall and A. Lakin, The theorems of Saalschütz and Dougall, Quart. J. Math. 2 (1950) 161-164.
9. S. Camazine, J.-L. Deneubourg, N. R. Franks, J. Sneyd, G. Theraulaz and E. Bonabeau, Self-Organization in Biological Systems (Princeton Univ. Press, 2003).
10. Y.-L. Chuang, M. R. D'Orsogna, D. Marthaler, A. L. Bertozzi and L. Chayes, State transitions and the continuum limit for a 2D interacting, self-propelled particle system, Phys. D 232 (2007) 33-47. (Pubitemid 47021228)
11. Y.-L. Chuang, Y. R. Huang, M. R. D'Orsogna and A. L. Bertozzi, Multi-vehicle flocking: Scalability of cooperative control algorithms using pairwise potentials, in 2007 IEEE International Conference on Robotics and Automation (IEEE, 2007), pp. 2292-2299. (Pubitemid 350140519)
12. H. Cohn and A. Kumar, Universally optimal distribution of points on spheres, J. Amer. Math. Soc. 20 (2007) 99-148.
13. H. Cohn and A. Kumar, Algorithmic design of self-assembling structures, Proc. Natl. Acad. Sci. 106 (2009) 9570-9575.
14. P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci. 18 (2008) 1193-1217.
15. A. M. Delprato, A. Samadani, A. Kudrolli and L. S. Tsimring, Swarming ring patterns in bacterial colonies exposed to ultraviolet radiation, Phys. Rev. Lett. 87 (2001) 158102.
16. M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi and L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Phys. Rev. Lett. 96 (2006) 104302.
17. L. Edelstein-Keshet, J. Watmough and D. Grunbaum, Do travelling band solutions describe cohesive swarms? An investigation for migratory locusts, J. Math. Biol. 36 (1998) 515-549. (Pubitemid 128735361)
18. S. N. Fejer, T. R. James, J. Hernandez-Rojas and D. J. Wales, Energy landscapes for shells assembled from pentagonal and hexagonal pyramids, Phys. Chem. Chem. Phys. 11 (2009) 2098-2104.
19. G. Gasper and W. Trebels, A Riemann-Lebesgue lemma for Jacobi expansions, in Mathematical Analysis, Wavelets, and Signal Processing: An International Conference on Mathematical Analysis and Signal Processes, Contemporary Mathematics, Vol. 90, eds. A. I. Zayed, M. E. H. Ismail, M. Z. Nashed and A. F. Gholeb (Amer. Math. Soc., 1995).
20. N. R. Franks, I. D. Couzin, J. Krauss and S. A. Levin, Effective leadership and decision-making in animal groups on the move, Nature 433 (2005) 513-516. (Pubitemid 40204306)
21. M. I. Jamaloodeen and P. K. Newton, The N-vortex problem on a rotating sphere. II. heterogeneous platonic solid equilibria, Proc. R. Soc. A 462 (2008) 3277-3299. (Pubitemid 44671025)
22. S. A. Kaufmann, The Origins of Order: Self-Organization and Selection in Evolution (Oxford Univ. Press, 1933).
23. E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol. 30 (1971) 225-234.
24. T. Kolokolnikov, H. Sun, D. Uminsky and A. L. Bertozzi, Stability of ring patterns arising from two-dimensional particle interactions, Phys. Rev. E 84 (2011) 015203.
25. R. Krasny, A study of singularity formation in a vortex sheet by the point-vortex approximation, J. Fluid Mech. 167 (1986) 65-93. (Pubitemid 16603670)
26. A. J. Leverentz, C. M. Topaz and A. J. Bernoff, Asymptotic dynamics of attractiverepulsive swarms, SIAM J. Appl. Dyn. Syst. 8 (2009) 880-908.
27. R. Lukemana, Y.-X. Lib and L. Edelstein-Keshet, Inferring individual rules from collective behavior, Proc. Natl. Acad. Sci. 10 (2010) 12576-12580.
28. A. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow (Cambridge Univ. Press, 2002).
29. A. Mogilner, L. Edelstein-Keshet, L. Bent and A. Spiros, Mutual interactions, potentials, and individual distance in a social aggregation, J. Math. Biol. 47 (2003) 353-389. (Pubitemid 38261040)
30. P. K. Newton and T. Sakajo, The N-vortex problem on a rotating sphere. III. Ring configurations coupled to a background field, Proc. R. Soc. A 463 (2007) 961-977.
31. P. K. Newton and T. Sakajo, Point vortex equilibria and optimal packings of circles on a sphere, Proc. R. Soc. A 467 (2011) 1468-1490.
32. J. K. Parrish and L. Edelstein-Keshet, Complexity, pattern, and evolutionary tradeoffs in animal aggregation, Science 284 (1999) 99-101. (Pubitemid 29282054)
33. A. Pérez-Garrido, M. J. W. Dodgson and M. A. Moore, Influence of dislocations in Thomson's problem, Phys. Rev. B 56 (1997) 3640-3643.
34. I. Prigogine, Order Out of Chaos (Bantam, 1984).
35. M. Rechtsman, F. Stillinger and S. Torquato, Designed interaction potentials via inverse methods for self-assembly, Phys. Rev. E 73 (2006) 011406.
36. M. C. Rechtsman, F. H. Stillinger and S. Torquato, Optimized interactions for targeted self-assembly: Application to a honeycomb lattice, Phys. Rev. Lett. 95 (2005) 228301.
37. R. T. Seeley, Spherical harmonics, Amer. Math. Monthly 73 (1966) 115-121.
38. L. J. Slater, Generalized Hypergeometric Functions (Cambridge Univ. Press, 1966).
39. H. Sun, D. Uminsky and A. L. Bertozzi, A generalized Birkhoff-Rott equation for 2D active scalar problems, submitted, 2010.
40. G. Szegö, Orthogonal Polynomials, 4th edn. (Amer. Math. Soc., 1975).
41. C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math. 65 (2004) 152-174.
42. S. Torquato, Inverse optimization techniques for targeted self-assembly, Soft Matter 5 (2009) 1157-1173.
43. L. Tsimring, H. Levine, I. Aranson, E. Ben-Jacob, I. Cohen, O. Shochet and W. N. Reynolds, Aggregation patterns in stressed bacteria, Phys. Rev. Lett. 75 (1995) 1859-1862.
44. J. H. von Brecht and D. Uminsky, On soccer balls and linearized inverse statistical mechanics, submitted.
45. D. J. Wales, Energy landscapes of clusters bound by short-ranged potentials, Chem. Eur. J. Chem. Phys. 11 (2010) 2491-2494.
46. D. J. Wales, H. McKay and E. L. Altschuler, Defect motifs for spherical topologies, Phys. Rev. B 79 (2009) 224115.
47. W. Yang, A. L. Bertozzi and X. Wang, Stability of a second order consensus algorithm with time delay, in 2008 IEEE Conference on Decision and Control (IEEE, 2008), pp. 2926-2931.