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Mean-field theory of hard sphere glasses and jamming

Giorgio Parisi* and Francesco Zamponi†

Giorgio Parisi*

  • Dipartimento di Fisica, INFM-CNR SMC, INFN, Università di Roma “La Sapienza,” Piazzale Aldo Moro 2, 00185 Roma, Italy

Francesco Zamponi†

  • Laboratoire de Physique Théorique, École Normale Supérieure, 24 Rue Lhomond, 75231 Paris Cedex 05, France

  • *giorgio.parisi@roma1.infn.it
  • †http://www.lpt.ens.fr/~zamponi
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Rev. Mod. Phys. 82, 789 – Published 16 March, 2010

DOI: https://doi.org/10.1103/RevModPhys.82.789

Abstract

Hard spheres are ubiquitous in condensed matter: they have been used as models for liquids, crystals, colloidal systems, granular systems, and powders. Packings of hard spheres are of even wider interest as they are related to important problems in information theory, such as digitalization of signals, error correcting codes, and optimization problems. In three dimensions the densest packing of identical hard spheres has been proven to be the fcc lattice, and it is conjectured that the closest packing is ordered (a regular lattice, e.g., a crystal) in low enough dimension. Still, amorphous packings have attracted much interest because for polydisperse colloids and granular materials the crystalline state is not obtained in experiments for kinetic reasons. A theory of amorphous packings, and more generally glassy states, of hard spheres is reviewed here, that is based on the replica method: this theory gives predictions on the structure and thermodynamics of these states. In dimensions between two and six these predictions can be successfully compared with numerical simulations. The limit of large dimension is also discussed where an exact solution is possible. Some of the results presented here were published, but others are original: in particular, an improved discussion of the large dimension limit and new results on the correlation function and the contact force distribution in three dimensions. The main assumptions that are beyond the theory presented are clarified and, in particular, the relation between static computation and the dynamical procedures used to construct amorphous packings. There remain many weak points in the theory that should be better investigated.

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References (201)

  1. Abate, A., and D. Durian, 2006, Phys. Rev. E 74, 031308.
  2. Abramowitz, H., and I. Segun, 1965, Handbook of Mathematical Functions (Dover, New York).
  3. Alder, B. J., and T. E. Wainwright, 1957, J. Chem. Phys. 27, 1208.
  4. Alexander, S., 1998, Phys. Rep. 296, 65.
  5. Angelani, L., and G. Foffi, 2007, J. Phys.: Condens. Matter 19, 256207.
  6. Anikeenko, A. V., and N. N. Medvedev, 2007, Phys. Rev. Lett. 98, 235504.
  7. Anikeenko, A. V., N. N. Medvedev, and T. Aste, 2008, Phys. Rev. E 77, 031101.
  8. Aste, T., 2005, J. Phys.: Condens. Matter 17, S2361.
  9. Aste, T., and A. Coniglio, 2004, Europhys. Lett. 67, 165.
  10. Aste, T., M. Saadatfar, and T. J. Senden, 2005, Phys. Rev. E 71, 061302.
  11. Ball, K., 1992, Int. Math. Res. Notices 1992, 217.
  12. Barrat, A., S. Franz, and G. Parisi, 1997, J. Phys. A 30, 5593.
  13. Barthel, W., A. K. Hartmann, and M. Weigt, 2003, Phys. Rev. E 67, 066104.
  14. Baxter, R. J., 1968, J. Chem. Phys. 49, 2770.
  15. Bengtzelius, U., W. Gotze, and A. Sjolander, 1984, J. Phys. C 17, 5915.
  16. Bennett, C. H., 1972, J. Appl. Phys. 43, 2727.
  17. Bernal, J., and J. Mason, 1960, Nature (London) 188, 910.
  18. Bernal, J., J. Mason, and K. Knight, 1962, Nature (London) 194, 957.
  19. Berryman, J. G., 1983, Phys. Rev. A 27, 1053.
  20. Berthier, L., and T. A. Witten, 2009a, EPL 86, 10001.
  21. Berthier, L., and T. A. Witten, 2009b, Phys. Rev. E 80, 021502.
  22. Biazzo, I., F. Caltagirone, G. Parisi, and F. Zamponi, 2009, Phys. Rev. Lett. 102, 195701.
  23. Binder, K., and W. Kob, 2005, Glassy Materials and Disordered Solids: An Introduction to their Statistical Mechanics (World Scientific, Singapore).
  24. Biroli, G., J. Bouchaud, A. Cavagna, T. Grigera, and P. Verrocchio, 2008, Nat. Phys. 4, 771.
  25. Biroli, G., and M. Mézard, 2001, Phys. Rev. Lett. 88, 025501.
  26. Biroli, G., and R. Monasson, 2000, Europhys. Lett. 50, 155.
  27. Bishop, M., and P. A. Whitlock, 2005, J. Chem. Phys. 123, 014507.
  28. Bouchaud, J. P., and G. Biroli, 2004, J. Chem. Phys. 121, 7347.
  29. Brambilla, G., D. E. Masri, M. Pierno, L. Berthier, L. Cipelletti, G. Petekidis, and A. B. Schofield, 2009, Phys. Rev. Lett. 102, 085703.
  30. Brito, C., and M. Wyart, 2006, Europhys. Lett. 76, 149.
  31. Brumer, Y., and D. Reichman, 2004, J. Phys. Chem. B 108, 6832.
  32. Cammarota, C., A. Cavagna, G. Gradenigo, T. Grigera, and P. Verrocchio, 2009, J. Stat. Mech. 2009, L12002.
  33. Cardenas, M., S. Franz, and G. Parisi, 1998, J. Phys. A 31, L163.
  34. Cardenas, M., S. Franz, and G. Parisi, 1999, J. Chem. Phys. 110, 1726.
  35. Castellani, T., and A. Cavagna, 2005, J. Stat. Mech.: Theory Exp. 2005, P05012.
  36. Cavagna, A., 2009, Phys. Rep. 476, 51.
  37. Cavagna, A., A. Attanasi, and J. Lorenzana, 2005, Phys. Rev. Lett. 95, 115702.
  38. Cavagna, A., T. S. Grigera, and P. Verrocchio, 2007, Phys. Rev. Lett. 98, 187801.
  39. Chaudhuri, P., L. Berthier, and S. Sastry, 2009, e-print arXiv:0910.0364.
  40. Chaudhuri, P., S. Karmakar, C. Dasgupta, H. R. Krishnamurthy, and A. K. Sood, 2005, Phys. Rev. Lett. 95, 248301.
  41. Clarke, A. S., and H. Jónsson, 1993, Phys. Rev. E 47, 3975.
  42. Coluzzi, B., M. Mézard, G. Parisi, and P. Verrocchio, 1999, J. Chem. Phys. 111, 9039.
  43. Conway, J. H., and N. J. A. Sloane, 1993, Sphere Packings, Lattices and Groups (Spriger-Verlag, New York).
  44. Cugliandolo, L. F., and J. Kurchan, 1993, Phys. Rev. Lett. 71, 173.
  45. Daniels, K. E., and R. P. Behringer, 2006, J. Stat. Mech.: Theory Exp. 2006, P07018.
  46. Dasgupta, C., and O. T. Valls, 1999, Phys. Rev. E 59, 3123.
  47. Dauchot, O., G. Marty, and G. Biroli, 2005, Phys. Rev. Lett. 95, 265701.
  48. Debenedetti, P. G., and F. H. Stillinger, 2001, Nature (London) 410, 259.
  49. De Dominicis, C., and P. C. Martin, 1964, J. Math. Phys. 5, 31.
  50. Dodds, J., 1975, Nature (London) 256, 187.
  51. Dodds, J., 1980, J. Colloid Interface Sci. 77, 317.
  52. Donev, A., I. Cisse, D. Sachs, E. Variano, F. Stillinger, R. Connelly, S. Torquato, and P. Chaikin, 2004, Science 303, 990.
  53. Donev, A., R. Connelly, F. H. Stillinger, and S. Torquato, 2007, Phys. Rev. E 75, 051304.
  54. Donev, A., F. H. Stillinger, and S. Torquato, 2006, Phys. Rev. Lett. 96, 225502.
  55. Donev, A., S. Torquato, and F. H. Stillinger, 2005a, Phys. Rev. E 71, 011105.
  56. Donev, A., S. Torquato, and F. H. Stillinger, 2005b, Phys. Rev. Lett. 95, 090604.
  57. Donth, E., 2001, The Glass Transition: Relaxation Dynamics in Liquids and Disordered Materials (Springer, New York).
  58. Dress, C., and W. Krauth, 1995, J. Phys. A 28, L597.
  59. Ediger, M. D., C. A. Angell, and S. R. Nagel, 1996, J. Phys. Chem. 100, 13200.
  60. Edwards, S., and R. Oakeshott, 1989, Physica A 157, 1080.
  61. Edwards, S. F., 1998, Physica A 249, 226.
  62. Finney, J. L., 1970, Proc. R. Soc. London, Ser. A 319, 479.
  63. Foffi, G., W. Götze, F. Sciortino, P. Tartaglia, and T. Voigtmann, 2003, Phys. Rev. Lett. 91, 085701.
  64. Foffi, G., W. Götze, F. Sciortino, P. Tartaglia, and T. Voigtmann, 2004, Phys. Rev. E 69, 011505.
  65. Franz, S., 2005, J. Stat. Mech.: Theory Exp. 2005, P04001.
  66. Franz, S., and G. Parisi, 1997, Phys. Rev. Lett. 79, 2486.
  67. Franz, S., and F. Tria, 2006, J. Stat. Phys. 122, 313.
  68. Frisch, H. L., and J. K. Percus, 1999, Phys. Rev. E 60, 2942.
  69. Gao, G.-J., J. Blawzdziewicz, and C. S. O’Hern, 2006, Phys. Rev. E 74, 061304.
  70. Goldbart, P., N. Goldenfeld, and D. Sherrington, 2005, Eds., Stealing the Gold–A Celebration of the Pioneering Physics of Sam Edwards (Oxford University Press, Oxford).
  71. Götze, W., 1999, J. Phys.: Condens. Matter 11, A1.
  72. Götze, W., and T. Voigtmann, 2003, Phys. Rev. E 67, 021502.
  73. Grigera, T., and G. Parisi, 2001, Phys. Rev. E 63, 045102.
  74. Gross, D. J., and M. Mézard, 1984, Nucl. Phys. B 240, 431.
  75. Hales, T. C., 2005, Ann. Math. 162, 1065.
  76. Hall, R. W., and P. G. Wolynes, 2003, Phys. Rev. Lett. 90, 085505.
  77. Hall, R. W., and P. G. Wolynes, 2008, J. Phys. Chem. B 112, 301.
  78. Hansen, J.-P., and I. R. McDonald, 1986, Theory of Simple Liquids (Academic, London).
  79. Hatano, T., M. Otsuki, and S. Sasa, 2007, J. Phys. Soc. Jpn. 76, 023001.
  80. Henderson, D., 1975, Mol. Phys. 30, 971.
  81. Hermes, M., and M. Dijkstra, 2010, EPL 89, 38005.
  82. Jerkins, M., M. Schröter, H. L. Swinney, T. J. Senden, M. Saadatfar, and T. Aste, 2008, Phys. Rev. Lett. 101, 018301.
  83. Jodrey, W. S., and E. M. Tory, 1985, Phys. Rev. A 32, 2347.
  84. Kabatiansky, G. A., and V. I. Levensthein, 1978, Probl. Inf. Transm. 14, 1.
  85. Kamien, R. D., and A. J. Liu, 2007, Phys. Rev. Lett. 99, 155501.
  86. Kansal, A. R., S. Torquato, and F. H. Stillinger, 2002, Phys. Rev. E 66, 041109.
  87. Kauzmann, W., 1948, Chem. Rev. (Washington, D.C.) 43, 219.
  88. Kim, K., and T. Munakata, 2003, Phys. Rev. E 68, 021502.
  89. Kirkpatrick, T. R., and D. Thirumalai, 1987, Phys. Rev. Lett. 58, 2091.
  90. Kirkpatrick, T. R., D. Thirumalai, and P. G. Wolynes, 1989, Phys. Rev. A 40, 1045.
  91. Kirkpatrick, T. R., and P. G. Wolynes, 1987a, Phys. Rev. A 35, 3072.
  92. Kirkpatrick, T. R., and P. G. Wolynes, 1987b, Phys. Rev. B 36, 8552.
  93. Krauth, W., 2006, Statistical Mechanics: Algorithms and Computations (Oxford University Press, New York).
  94. Krivelevich, M., S. Litsyn, and A. Vardy, 2004, Int. Math. Res. Notices 2004, 2271.
  95. Krzakala, F., and J. Kurchan, 2007, Phys. Rev. E 76, 021122.
  96. Krzakala, F., A. Montanari, F. Ricci-Tersenghi, G. Semerjian, and L. Zdeborova, 2007, Proc. Natl. Acad. Sci. U.S.A. 104, 10318.
  97. Krzakala, F., M. Tarzia, and L. Zdeborová, 2008, Phys. Rev. Lett. 101, 165702.
  98. Lechenault, F., O. Dauchot, G. Biroli, and J. Bouchaud, 2008, EPL 83, 46003.
  99. Leuzzi, L., and T. Nieuwenhuizen, 2007, Thermodynamics of the Glassy State (Taylor & Francis, London).
  100. Liu, A., and S. Nagel, 2001, Jamming and Rheology: Constrained Dynamics on Microscopic and Macroscopic Scales (Taylor & Francis, London).
  101. Lochmann, K., A. Anikeenko, A. Elsner, N. Medvedev, and D. Stoyan, 2006, Eur. Phys. J. B 53, 67.
  102. Lubachevsky, B. D., and F. H. Stillinger, 1990, J. Stat. Phys. 60, 561.
  103. Luding, S., 2001, Phys. Rev. E 63, 042201.
  104. Majmudar, T. S., M. Sperl, S. Luding, and R. P. Behringer, 2007, Phys. Rev. Lett. 98, 058001.
  105. Mari, R., F. Krzakala, and J. Kurchan, 2009, Phys. Rev. Lett. 103, 025701.
  106. Mason, G., and W. Clark, 1965, Nature (London) 207, 512.
  107. Mason, G., and W. Clark, 1966, Nature (London) 211, 957.
  108. Matheson, A. J., 1974, J. Phys. C 7, 2569.
  109. Mézard, M., 1999, Physica A 265, 352.
  110. Mézard, M., and G. Parisi, 1996, e-print arXiv:cond-mat/9602002.
  111. Mézard, M., and G. Parisi, 1999a, J. Chem. Phys. 111, 1076.
  112. Mézard, M., and G. Parisi, 1999b, Phys. Rev. Lett. 82, 747.
  113. Mézard, M., and G. Parisi, 2000, J. Phys.: Condens. Matter 12, 6655.
  114. Mézard, M., G. Parisi, and M. A. Virasoro, 1987, Spin Glass Theory and Beyond (World Scientific, Singapore).
  115. Miller, M. A., and D. Frenkel, 2004a, J. Chem. Phys. 121, 535.
  116. Miller, M. A., and D. Frenkel, 2004b, J. Phys.: Condens. Matter 16, S4901.
  117. Möller, J., I. Gutzow, and J. Schmelzer, 2006, J. Chem. Phys. 125, 094505.
  118. Monasson, R., 1995, Phys. Rev. Lett. 75, 2847.
  119. Montanari, A., and F. Ricci-Tersenghi, 2004, Phys. Rev. B 70, 134406.
  120. Montanari, A., and G. Semerjian, 2006, J. Stat. Phys. 125, 23.
  121. Morita, T., and K. Hiroike, 1961, Prog. Theor. Phys. 25, 537.
  122. O’Hern, C. S., S. A. Langer, A. J. Liu, and S. R. Nagel, 2002, Phys. Rev. Lett. 88, 075507.
  123. O’Hern, C. S., L. E. Silbert, A. J. Liu, and S. R. Nagel, 2003, Phys. Rev. E 68, 011306.
  124. Olsson, P., and S. Teitel, 2007, Phys. Rev. Lett. 99, 178001.
  125. Onoda, G. Y., and E. G. Liniger, 1990, Phys. Rev. Lett. 64, 2727.
  126. Osada, H., 1998, Probab. Theory Relat. Fields 112, 53.
  127. Ouchiyama, N., and T. Tanaka, 1981, Ind. Eng. Chem. Fundam. 20, 66.
  128. Parisi, G., 1988, Statistical Field Theory (Addison-Wesley, Reading, MA).
  129. Parisi, G., 2008, J. Stat. Phys. 132, 207.
  130. Parisi, G., and F. Slanina, 2000, Phys. Rev. E 62, 6554.
  131. Parisi, G., and F. Zamponi, 2005, J. Chem. Phys. 123, 144501.
  132. Parisi, G., and F. Zamponi, 2006a, J. Stat. Mech.: Theory Exp. 2006, P03017.
  133. Parisi, G., and F. Zamponi, 2006b, J. Stat. Phys. 123, 1145.
  134. Phan, S.-E., W. B. Russel, Z. Cheng, J. Zhu, P. M. Chaikin, J. H. Dunsmuir, and R. H. Ottewill, 1996, Phys. Rev. E 54, 6633.
  135. Pica Ciamarra, M., M. Nicodemi, and A. Coniglio, 2007, Phys. Rev. E 75, 021303.
  136. Pica Ciamarra, M., M. Tarzia, A. de Candia, and A. Coniglio, 2003, Phys. Rev. E 67, 057105.
  137. Pinson, D., R. P. Zou, A. B. Yu, P. Zulli, and M. J. McCarthy, 1998, J. Phys. D 31, 457.
  138. Powell, M. J., 1979, Phys. Rev. B 20, 4194.
  139. Pusey, P., and W. Van Megen, 1986, Nature (London) 320, 340.
  140. Richard, P., L. Oger, J.-P. Troadec, and A. Gervois, 1999, Phys. Rev. E 60, 4551.
  141. Rintoul, M. D., and S. Torquato, 1996, J. Chem. Phys. 105, 9258.
  142. Rivoire, O., G. Biroli, O. Martin, and M. Mézard, 2004, Eur. Phys. J. B 37, 55.
  143. Robles, M., and M. L. de Haro, 2003, Europhys. Lett. 62, 56.
  144. Rogers, C. A., 1964, Packing and Covering (Cambridge University Press, Cambridge).
  145. Ruelle, D., 1977, private communication.
  146. Santen, L., and W. Krauth, 2000, Nature (London) 405, 550.
  147. Santen, L., and W. Krauth, 2001, e-print arXiv:cond-mat/0107459.
  148. Schreck, C. F., and C. S. O’Hern, 2008, unpublished.
  149. Schröter, M., D. I. Goldman, and H. L. Swinney, 2005, Phys. Rev. E 71, 030301.
  150. Scott, G., 1962, Nature (London) 194, 956.
  151. Scott, G. D., and D. M. Kilgour, 1969, Br. J. Appl. Phys. 2, 863.
  152. Sellitto, M., G. Biroli, and C. Toninelli, 2005, Europhys. Lett. 69, 496.
  153. Semerjian, G., 2008, J. Stat. Phys. 130, 251.
  154. Semerjian, G., and R. Monasson, 2003, Phys. Rev. E 67, 066103.
  155. Shundyak, K., M. van Hecke, and W. van Saarloos, 2007, Phys. Rev. E 75, 010301.
  156. Silbert, L., A. Liu, and S. Nagel, 2005, Phys. Rev. Lett. 95, 098301.
  157. Silbert, L. E., A. J. Liu, and S. R. Nagel, 2006, Phys. Rev. E 73, 041304.
  158. Singh, Y., J. P. Stoessel, and P. G. Wolynes, 1985, Phys. Rev. Lett. 54, 1059.
  159. Skoge, M., A. Donev, F. H. Stillinger, and S. Torquato, 2006, Phys. Rev. E 74, 041127.
  160. Sloane, N. J. A., 2007, http://www.research.att.com/~njas/
  161. Snoeijer, J. H., T. J. H. Vlugt, W. G. Ellenbroek, M. van Hecke, and J. M. J. van Leeuwen, 2004, Phys. Rev. E 70, 061306.
  162. Snoeijer, J. H., T. J. H. Vlugt, M. van Hecke, and W. van Saarloos, 2004, Phys. Rev. Lett. 92, 054302.
  163. Somfai, E., M. van Hecke, W. G. Ellenbroek, K. Shundyak, and W. van Saarloos, 2007, Phys. Rev. E 75, 020301.
  164. Song, C., P. Wang, and H. Makse, 2008, Nature (London) 453, 629.
  165. Song, Y., E. A. Mason, and R. M. Stratt, 1989, J. Phys. Chem. 93, 6916.
  166. Speedy, R. J., 1994, J. Chem. Phys. 100, 6684.
  167. Speedy, R. J., 1998, Mol. Phys. 95, 169.
  168. Stevenson, J. D., A. M. Walczak, R. W. Hall, and P. G. Wolynes, 2008, J. Chem. Phys. 129, 194505.
  169. Stillinger, F. H., and T. A. Weber, 1982, Phys. Rev. A 25, 978.
  170. Stillinger, F. H., and T. A. Weber, 1985, Phys. Rev. B 31, 5262.
  171. Stoessel, J. P., and P. G. Wolynes, 1984, J. Chem. Phys. 80, 4502.
  172. Talagrand, M., 2003, Spin Glasses: A Challenge for Mathematicians—Cavity and Mean Field Models (Springer, New York).
  173. Talbot, J., G. Tarjus, P. R. V. Tassel, and P. Viot, 2000, Colloids Surf., A 165, 287.
  174. Tarzia, M., 2007, J. Stat. Mech.: Theory Exp. 2007, P01010.
  175. Tarzia, M., A. de Candia, A. Fierro, M. Nicodemi, and A. Coniglio, 2004, Europhys. Lett. 66, 531.
  176. Tejero, C. F., and M. Baus, 1993, Phys. Rev. E 48, 3793.
  177. Thouless, D., P. Anderson, and R. Palmer, 1977, Philos. Mag. 35, 593.
  178. Torquato, S., 1995, Phys. Rev. Lett. 74, 2156.
  179. Torquato, S., 2002, Random Heterogeneous Materials: Microstructure and Macroscopic Properties (Springer-Verlag, New York).
  180. Torquato, S., and F. H. Stillinger, 2006a, Phys. Rev. E 73, 031106.
  181. Torquato, S., and F. H. Stillinger, 2006b, Exp. Math. 15, 307.
  182. Torquato, S., T. M. Truskett, and P. G. Debenedetti, 2000, Phys. Rev. Lett. 84, 2064.
  183. Torquato, S., O. U. Uche, and F. H. Stillinger, 2006, Phys. Rev. E 74, 061308.
  184. van Meel, J. A., D. Frenkel, and P. Charbonneau, 2009, Phys. Rev. E 79, 030201.
  185. van Megen, W., and S. M. Underwood, 1993, Phys. Rev. Lett. 70, 2766.
  186. van Megen, W., S. M. Underwood, and P. N. Pusey, 1991, Phys. Rev. Lett. 67, 1586.
  187. Velenich, A., A. Parola, and L. Reatto, 2006, Phys. Rev. E 74, 021410.
  188. Wertheim, M. S., 1963, Phys. Rev. Lett. 10, 321.
  189. Wood, W. W., and J. D. Jacobson, 1957, J. Chem. Phys. 27, 1207.
  190. Woodcock, L. V., and C. A. Angell, 1981, Phys. Rev. Lett. 47, 1129.
  191. Wyart, M., 2005, Ann. Phys. 30, 1.
  192. Wyart, M., S. Nagel, and T. Witten, 2005, Europhys. Lett. 72, 486.
  193. Wyart, M., L. Silbert, S. Nagel, and T. Witten, 2005, Phys. Rev. E 72, 051306.
  194. Xia, X., and P. G. Wolynes, 2001a, Proc. Natl. Acad. Sci. U.S.A. 97, 2990.
  195. Xia, X., and P. G. Wolynes, 2001b, Phys. Rev. Lett. 86, 5526.
  196. Xu, N., J. Blawzdziewicz, and C. S. O’Hern, 2005, Phys. Rev. E 71, 061306.
  197. Yerazunis, S., S. Cornell, and B. Wintner, 1965, Nature (London) 207, 835.
  198. Yoshidome, T., A. Yoshimori, and T. Odagaki, 2007, Phys. Rev. E 76, 021506.
  199. Zamponi, F., 2007, Philos. Mag. B 87, 485.
  200. Zdeborová, L., and F. Krzakala, 2007, Phys. Rev. E 76, 031131.
  201. Zeravcic, Z., W. van Saarloos, and D. R. Nelson, 2008, EPL 83, 44001.

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Rev. Mod. Phys. 82, 789– Published 16 March, 2010

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