Discrete charges on a two dimensional conductor

Journal of Statistical Physics

Volumn 116, Issue 5-6, 2004, Pages 1301-1358

  Berkenbuseh, Marko Kleine ^a   Claus, Isabelle ^a   Dunn, Catherine ^a   Kadanoff, Leo P ^a   Nicewicz, Maciej ^a   Venkataramani, Shankar C ^a  

^a UNIVERSITY OF CHICAGO   (United States)

Author keywords

Electrostatic equilibrium; Fekete points; Singular domains; Symmetry breaking

Indexed keywords

EID: 26844480956     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1023/B:JOSS.0000041741.27244.ac     Document Type: Article

References (40)

1. M. Abramowitz and I. A. Stegun, editors. Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover Publications Inc., New York, 1992. Reprint of the 1972 edition.
2. O. Agam, E. Bettelheim, P. Wiegmann, and A. Zabrodin, Viscous fingering and the shape of an Electronic droplet in the Quantum hall regime. Phys. Rev. Lett. 88:236801, (2002).
3. C. D. Ahlbrandt and A. C. Peterson, Discrete Hamiltonian systems, volume 16 of Kluwer Texts in the Mathematical Sciences. Kluwer Academic Publishers Group, Dordrecht, 1996. Difference equations, continued fractions, and Riccati equations.
4. V. V. Andrievskii and H.-P. Blatt, Discrepancy of signed measures and polynomial approximation. Springer Monographs in Mathematics. Springer-Verlag, New York (2002).
5. V. I. Arnold and A. Avez, editors. Ergodic Problems of Classical Mechanics. Benjamin, New York (1968).
6. M. Bowick, A. Cacciuto, D. Nelson, and al. Crystalline order on a sphere and the generalized Thomson problem. Phys. Rev. Lett. 89:185502 (2002).
7. M. Bowick, D. Nelson, and A. Travesset, Interacting topological defects on frozen topographies. Phys. Rev. B 62:8738-8751 (2000).
8. J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, volume 290 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York, third edition, 1999. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov.
9. P. Deift, T. Kriecherbauer, and K. T.-R. McLaughlin, New results on the equilibrium measure for logarithmic potentials in the presence of an external field. J. Approx. Theory, 95(3):388-475 (1998).
10. T. A. Driscoll and L. N. Trefethen, Schwarz-Christoffel Mapping. Cambridge monographs on applied and computational mathematics. Cambridge University Press, Cambridge UK, 2002.
11. T. Erber and G. M. Hockney, Complex systems: equilibrium configurations of N equal charges on a sphere (2 ≤ N ≤ 112). In Advances in chemical physics, Vol. XCVIII, Adv. Chem. Phys., XCVIII, pages 495-594. Wiley, New York (1997).
12. M. J. Feigenbaum, I. Procaccia, and B. Davidovich, Dynamics of finger formation in Laplacian growth without surface tension. J. Statist. Phys. 103(5-6):973-1007 (2001).
13. M. Fekete, über die Verteilung der wurzeln bei gewissen algebraischen gleichungeu mit ganzzahligen koeffizienten. Math. Z. 17:228-249 (1923).
14. O. Frostman, Potentiel d'équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions. Dissertation, Lunds Univ. Mat. Sem. 3:1-118 1935. Dissertation.
15. W. Gangbo and R. J. McCann, The geometry of optimal transportation. Acta Math. 177(2):113-161 (1996).
16. I. S. Gradstein and I. M. Ryshik, Summen-, Produkt- und Integraltafeln. Band 1, 2. Verlag Harri Deutsch, Thun, language edition, 1982. Translation from the Russian edited by Ludwig Boll, Based on the second German-English edition translated by Christa Berg, Lothar Berg and Martin Strauss, Incorporating the fifth Russian edition edited by Yu. V. Geronimus and M. Yu. Tseǐtlin.
17. J. M. Greene, J. Math. Phys. 20:1183 (1979).
18. M. B. Hastings and L. S. Levitov, Laplacian growth as one-dimensional turbulence. Physica A 116(1-2):244-252 (1998).
19. T. L. Hughes, A. D. Klironomos, and A. T. Dorsey, Fingering of electron droplets in nonuniform magnetic fields, preprint, September 2002.
20. L. P. Kadanoff, Scaling for a critical Kolmogorov-Arnold-Moser trajectory, Phys. Rev. Lett. 47:1641 (1981).
21. L. P. Kadanoff and S. J. Shenker, Critical behavior of a KAM surface: I. Empirical results. J. Stat. Phys. 27:631 (1982).
22. J. Korevaar, Asymptotically neutral distributions of electrons and polynomial approximation. Ann. of Math. (2) 80:403-410 (1964).
23. J. Korevaar, Fekete extreme points and related problems. In Approximation theory and function series (Budapest, 1995), volume 5 of Bolyai Soc. Math. Stud, pages 35-62. János Bolyai Math. Soc., Budapest, 1996.
24. J. Korevaar and T. Geveci. Fields due to electrons on an analytic curve. SIAM J. Math. Anal. 2:445-453 (1971).
25. I. K. Kostov, I. Krichever, M. Mineev-Weinstein, P. B. Wiegmann, and A. Zabrodin, The τ-function for analytic curves. In Random Matrix Models and Their Applications, Vol. 40 of Math. Sci. Res. Inst Publ., 285-299. Cambridge Univ. Press, Cambridge, (2001).
26. B. A. Kupershmidt, KP or mKP, Vol. 78 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2000. Noncommutative mathematics of Lagrangian, Hamiltonian, and integrable systems.
27. C. Pommerenke, Über die Faberschen Polynome schlichter Funktionen. Math. Z. 85:197-208 (1964).
28. C. Pommerenke, Über die Verteilung der Fekete-Punkte. Math. Ann. 168:111-127 (1967).
29. C. Pommerenke, Über die Verteilung der Fekete-Punkte. II. Math. Ann. 179:212-218 (1969).
30. C. Pommerenke, Boundary behaviour of conformal maps, volume 299 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1992.
31. R. T. Rockafellar, Convex Analysis. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1997. Reprint of the 1970 original, Princeton Paperbacks.
32. I. Singer, Abstract convex analysis. Canadian mathematical society series of monographs and advanced texts. John Wiley & Sons Inc., New York, 1997. With a foreword by A. M. Rubinov, A Wiley-Interscience Publication.
33. T. J. Stieltjes, Sur quelques théorèmes d'algèbre. C. R Acad Sci. Paris Ser. I Math. 100:620-622 (1885).
34. G. Szegö, Orthogonal polynomials. American mathematical society colloquium publications, Vol. 23. Revised ed. American Mathematical Society, Providence, R.I., 1959.
35. J. Thompson, Philos. Mag 7:237 (1904).
36. V. Totik, Weighted approximation with varying weight, volume 1569 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1994.
37. L. N. Trefethen, Numerical computation of the Schwarz-Christoffel transformation. STAM J. Sci. Stat. Comp. 1:82-102 (1980).
38. G. Valent and W. Van Assche, The impact of Stieltjes' work on continued fractions and orthogonal polynomials: additional material. J. Comput. Appl. Math. 65(1-3):419-447 (1995).
39. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions, Reprint of the fourth edition.
40. P. B. Wiegmann and A. Zabrodin, Conformal maps and integrable hierarchies. Comm. Math. Phys. 213(3):523-538 (2000).