Advertisement

The European Physical Journal Special Topics

, Volume 145, Issue 1, pp 191–210 | Cite as

Nodal densities of planar gaussian random waves

  • M. R. DennisEmail author
Article

Abstract.

The nodal densities of gaussian random functions, modelling various physical systems including chaotic quantum eigenfunctions and optical speckle patterns, are reviewed. The nodal domains of isotropically random real and complex functions are formulated in terms of their Minkowski functionals, and their correlations and spectra are discussed. The results on the statistical densities of the zeros of the real and complex functions, and their derivatives, in two dimensions are reviewed. New results are derived on the nodal domains of the hessian determinant (gaussian curvature) of two-dimensional random surfaces.

Keywords

Correlation Function European Physical Journal Special Topic Nodal Point Nodal Density Topological Charge 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A.R. Liddle, D.H. Lyth, Cosmological Inflation and Large-scale Structure (Cambridge University Press, 2000) Google Scholar
  2. J.C. Dainty, Prog. Opt. 14, 1 (1976) Google Scholar
  3. J.W. Goodman, Speckle Phenomena in Optics (Roberts & Co, 2006) Google Scholar
  4. J.W. Goodman, Statistical Optics (Wiley, 1985) Google Scholar
  5. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995) Google Scholar
  6. U. Frisch, Turbulence (Cambridge University Press, 1995) Google Scholar
  7. K.J. Worsley, A.C. Evans, S. Marrett, P. Neelin, J. Cer. Blood Flow Met. 12, 900 (1992) Google Scholar
  8. J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 4th edn. (Oxford University Press, 2002) Google Scholar
  9. M.V. Berry, J. Phys. A: Math. Gen. 10, 2083 (1977) Google Scholar
  10. E.J. Heller, Phys. Rev. Lett. 53, 1515 (1984) Google Scholar
  11. P. O'Connor, J. Gehlen, E.J. Heller, Phys. Rev. Lett. 58, 1296 (1987) Google Scholar
  12. A. Bäcker, R. Schubert, J. Phys. A: Math. Gen. 35, 539 (2002) Google Scholar
  13. P.W. Brouwer, Phys. Rev. E 68, 046205 (2003) Google Scholar
  14. U. Kuhl, H.-J. Stöckmann, R. Weaver, J. Phys. A: Math. Gen. 38, 10433 (2005) Google Scholar
  15. J.D. Urbina, K. Richter, J. Phys. A: Math. Gen. 36, L485 (2003) Google Scholar
  16. M.L. Mehta, Random Matrices, 2nd edn. (Academic, 1991) Google Scholar
  17. S.O. Rice, Bell. Syst. Tech. J., 23, 282 (1944); 24, 46 (1945) (reprinted in N. Wax, Selected Papers on Noise and Stochastic Processes) (Dover, 1954) Google Scholar
  18. M.S. Longuet-Higgins, Proc. R. Soc. A 246, 99 (1958) Google Scholar
  19. D.E. Cartwright, M.S. Longuet-Higgins, Proc. R. Soc. A 237, 212 (1956) Google Scholar
  20. M.S. Longuet-Higgins, Phil. Trans. R. Soc. A 249, 321 (1957) Google Scholar
  21. M.S. Longuet-Higgins, Phil. Trans. R. Soc. A 250, 157 (1957) Google Scholar
  22. M.S. Longuet-Higgins, Proc. Camb. Phil. Soc. 54, 439 (1958) Google Scholar
  23. M.S. Longuet-Higgins, J. Opt. Soc. Am. 50, 845; 851 (1960) Google Scholar
  24. M.V. Berry, J.H. Hannay, J. Phys. A: Math. Gen. 10, 1809 (1977) Google Scholar
  25. R.J. Adler, The Geometry of Random Fields (Wiley, 1981) Google Scholar
  26. G. Blum, S. Gnutzmann, U. Smilansky, Phys. Rev. Lett. 88, 114101 (2002) Google Scholar
  27. E. Bogomolny, C. Schmit, Phys. Rev. Lett. 88, 114102 (2002) Google Scholar
  28. W.E. Bies, E.J. Heller, J. Phys. A: Math. Gen. 35, 5673 (2002) Google Scholar
  29. M.V. Berry, J. Phys. A: Math. Gen. 35, 3025 (2002) Google Scholar
  30. G. Foltin, arXiv nlin.CD/0302049 (2003) Google Scholar
  31. G. Foltin, S. Gnutzmann, U. Smilansky, J. Phys. A: Math. Gen. 37, 11363 (2004) Google Scholar
  32. J.P. Keating, J. Marklof, I.G. Williams, Phys. Rev. Lett. 97, 034101 (2006) Google Scholar
  33. J.F. Nye, M.V. Berry, Proc. R. Soc. A 336, 165 (1974) Google Scholar
  34. M.V. Berry, in Les Houches Session XXV - Physics of Defects, edited by R. Balian, M. Kléman, J.P. Poirier (North-Holland, 1981), pp. 453–543 Google Scholar
  35. J.F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics Publishing, 1999) Google Scholar
  36. M.V. Berry, J. Phys. A: Math. Gen. 11, 27 (1978) Google Scholar
  37. N.B. Baranova, B.Y. Zel'dovitch, A.V. Mamaev, N. Pilipetskii, V.V. Shkukov, JETP Lett. 33, 195 (1981) Google Scholar
  38. I. Freund, N. Shvartsman, Phys. Rev. A 50, 5164 (1994) Google Scholar
  39. I. Freund, Waves Rand Med. 8, 119 (1998) Google Scholar
  40. K.-F. Berggren, K.N. Pichigin, A.F. Sadreev, A. Starikov, JETP Lett. 70, 403–9 (1999) Google Scholar
  41. M.V. Berry, M.R. Dennis, Proc. R. Soc. A 456, 2059 (2000); errata 456, 3059 (2000) Google Scholar
  42. W. Wang, S.G. Hanson, Y. Miyamoto, M. Takeda, Phys. Rev. Lett. 94, 103902 (2005) Google Scholar
  43. B. Jancovici, G. Manificat, C. Pisani, J. Stat. Phys. 78, 307 (1994) Google Scholar
  44. B.I. Halperin, in Les Houches Session XXV – Physics of Defects, edited by R. Balian, M. Kléman, J.P. Poirier (North-Holland, 1981), p. 813 Google Scholar
  45. F. Liu, G.F. Mazenko, Phys. Rev. B 46, 5963 (1992) Google Scholar
  46. I. Freund, M. Wilkinson, J. Opt. Soc. Am. A 15, 2892 (1998) Google Scholar
  47. G. Foltin, J. Phys. A: Math. Gen. 36, 1729 (2003) Google Scholar
  48. M.R. Dennis, J. Phys. A: Math. Gen. 36, 6611 (2003) Google Scholar
  49. M.R. Dennis, J.H. Hannay, J. Phys. A: Math. Gen. 36, 3379 (2003) Google Scholar
  50. J.H. Hannay, J. Phys. A: Math. Gen. 31, L755 (1998) Google Scholar
  51. P. Bleher, B. Shiffman, S. Zelditch, Invent. Math. 142, 351 (2000) Google Scholar
  52. C.T. Wheeler, J. Phys. A: Math. Gen. 38, 1491 (2005) Google Scholar
  53. M.V. Berry, M.R. Dennis, Proc. R. Soc. A 457, 141 (2001) Google Scholar
  54. M.R. Dennis, Opt. Commun. 213, 201 (2002) Google Scholar
  55. M.R. Dennis, Ph.D. thesis (Bristol University, 2001) Google Scholar
  56. K. Michielsen, H. de Raedt, Phys. Rep. 347, 461 (2001) Google Scholar
  57. L.A. Santaló, Integral Geometry and Geometric Probability (Addison-Wesley, 1976) Google Scholar
  58. H. Solomon, Geometric Probability (Arrowsmith, 1978) Google Scholar
  59. M. Aigner, G.M. Ziegler, Proofs from THE BOOK, 2nd edn. (Springer, 2001) Google Scholar
  60. R. Zallen, H. Scher, Phys. Rev. B 4, 4471 (1971) Google Scholar
  61. M.B. Isichenko, Rev. Mod. Phys. 64, 961 (1992) Google Scholar
  62. A.V. Milovanov, G. Zimbardo, Phys. Rev. E 62, 250 (2000) Google Scholar
  63. I.G. Williams, Ph.D. thesis (Bristol University, 2006) Google Scholar
  64. M.R. Dennis, in Singular Optics (Optical Vortices): Fundamentals and Applications, edited by M.S. Soskin, M.V. Vasnetsov 4403, 13 (SPIE, 2001) Google Scholar
  65. A.I. Saichev, K.-F. Berggren, A.F. Sadreev, Phys. Rev. E 64, 036222 (2001) Google Scholar
  66. M. Wilkinson, J. Phys. A: Math. Gen. 37, 6763 (2004) Google Scholar
  67. F. Wolf, T. Geisel, Nature 395, 73 (1998) Google Scholar
  68. F.H. Stillinger, R. Lovett, J. Chem. Phys. 49, 1991 (1968) Google Scholar
  69. B.A. van Tiggelen, D. Anache, A. Ghysels, Europhys. Lett. 74, 999 (2006) Google Scholar
  70. P.A.M. Dirac, Proc. R. Soc. A 133, 60 (1931) Google Scholar
  71. M.R. Dennis, J. Opt. A: Pure Appl. Opt. 6, S202 (2004) Google Scholar
  72. Y.Y. Schechner, J. Shamir, J. Opt. Soc. Am. A 13, 967 (1996) Google Scholar
  73. G. Foltin, J. Phys. A: Math. Gen. 36, 4561 (2003) Google Scholar
  74. A.G. Monastra, U. Smilansky, S. Gnutzmann, J. Phys. A: Math. Gen. 36, 1845 (2003) Google Scholar
  75. A. Weinrib, B.I. Halperin, Phys. Rev. B 26, 1362 (1982) Google Scholar
  76. M.R. Dennis, J. Phys. A: Math. Gen. 34, L297 (2001) Google Scholar
  77. A. Weinrib, Phys. Rev. B 26, 1352 (1982) Google Scholar
  78. A.F. Fercher, P.F. Steeger, Opt. Act. 28, 443 (1981) Google Scholar
  79. M.V. Berry, M.R. Dennis, J. Phys. A: Math. Theor. 40, 65 (2007) Google Scholar
  80. J.D. Neuheisel, Ph.D. thesis (Johns Hopkins University, 2000) Google Scholar
  81. M.R. Dennis, J. Phys. A: Math. Gen. 38, 1653 (2005) Google Scholar
  82. J.F. Nye, J.V. Hajnal, Proc. R. Soc. A 409, 21 (1987) Google Scholar

Copyright information

© EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2007

Authors and Affiliations

  1. 1.School of Mathematics, University of SouthamptonSouthamptonUK

Personalised recommendations