Review PaperNo Access

Spatio-temporal correlations in the Manna model in one, three and five dimensions

Gary Willis

Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, United Kingdom

Search for more papers by this author

and

Gunnar Pruessner

Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, United Kingdom

E-mail Address: g.pruessner@imperial.ac.uk

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S0217979218300025Cited by:6 (Source: Crossref)

Abstract

Although the paradigm of criticality is centered around spatial correlations and their anomalous scaling, not many studies of self-organized criticality (SOC) focus on spatial correlations. Often, integrated observables, such as avalanche size and duration, are used, not least as to avoid complications due to the unavoidable lack of translational invariance. The present work is a survey of spatio-temporal correlation functions in the Manna Model of SOC, measured numerically in detail in $d$ = 1,3 and 5 dimensions and compared to theoretical results, in particular relating them to “integrated” observables such as avalanche size and duration scaling, that measure them indirectly. Contrary to the notion held by some of SOC models organizing into a critical state by re-arranging their spatial structure avalanche by avalanche, which may be expected to result in large, nontrivial, system-spanning spatial correlations in the quiescent state (between avalanches), correlations of inactive particles in the quiescent state have a small amplitude that does not and cannot increase with the system size, although they display (noisy) power law scaling over a range linear in the system size. Self-organization, however, does take place as the (one-point) density of inactive particles organizes into a particular profile that is asymptotically independent of the driving location, also demonstrated analytically in one dimension. Activity and its correlations, on the other hand, display nontrivial long-ranged spatio-temporal scaling with exponents that can be related to established results, in particular avalanche size and duration exponents. The correlation length and amplitude are set by the system size (confirmed analytically for some observables), as expected in systems displaying finite size scaling. In one dimension, we find some surprising inconsistencies of the dynamical exponent. A (spatially extended) mean field theory (MFT) is recovered, with some corrections, in five dimensions.

Keywords:

PACS: 05.65.+b, 05.70.Jk

You currently do not have access to the full text article.
Recommend the journal to your library today!

References

1. H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, New York, 1971). Google Scholar
2. U. C. Täuber, Critical Dynamics (Cambridge University Press, Cambridge, UK, 2014). Google Scholar
3. P. Bak, C. Tang and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987). Web of Science, ADS, Google Scholar
4. G. Pruessner, Self-Organised Criticality (Cambridge University Press, Cambridge, UK, 2012). Google Scholar
5. A. van der Ziel, Physica 16, 359 (1950). Web of Science, ADS, Google Scholar
6. N. W. Watkins et al., Space Sci. Rev. 198, 3 (2016). Web of Science, ADS, Google Scholar
7. H. J. Jensen, K. Christensen and H. C. Fogedby, Phys. Rev. B 40, 7425 (1989). Web of Science, ADS, Google Scholar
8. D. Stauffer and A. Aharony, Introduction to Percolation Theory (Taylor & Francis, London, UK, 1994). Google Scholar
9. S. Lübeck, Int. J. Mod. Phys. B 18, 3977 (2004). Link, Web of Science, ADS, Google Scholar
10. H.-M. Bröker and P. Grassberger, Phys. Rev. E 56, 3944 (1997). Web of Science, ADS, Google Scholar
11. K. Christensen and N. R. Moloney, Complexity and Criticality (Imperial College Press, London, UK, 2005). Link, Google Scholar
12. M. Paczuski and S. Boettcher, Phys. Rev. Lett. 77, 111 (1996). Web of Science, ADS, Google Scholar
13. G. Pruessner, Phys. Rev. E 67, 030301(R) (2003), arXiv:cond-mat/0209531. Web of Science, ADS, Google Scholar
14. P. Bak, C. Tang and K. Wiesenfeld, Phys. Rev. A 38, 364 (1988). Web of Science, ADS, Google Scholar
15. C. Tang and P. Bak, Phys. Rev. Lett. 60, 2347 (1988). Web of Science, ADS, Google Scholar
16. C. Tang and P. Bak, J. Stat. Phys. 51, 797 (1988). Web of Science, ADS, Google Scholar
17. R. T. J. McAteer et al., Space Sci. Rev. 198, 217 (2016). Web of Science, ADS, Google Scholar
18. H. N. Huynh, G. Pruessner and L. Y. Chew, J. Stat. Mech. 2011, P09024 (2011), arXiv:1106.0406. Web of Science, Google Scholar
19. H. N. Huynh and G. Pruessner, Phys. Rev. E 85, 061133 (2012), arXiv:1201.3234. Web of Science, ADS, Google Scholar
20. D. Dhar, Phys. Rev. Lett. 64, 1613 (1990). Web of Science, ADS, Google Scholar
21. D. Dhar and S. N. Majumdar, J. Phys. A, Math. Gen. 23, 4333 (1990). ADS, Google Scholar
22. S. N. Majumdar and D. Dhar, J. Phys. A, Math. Gen. 24, L357 (1991). Google Scholar
23. S. N. Majumdar and D. Dhar, Physica A 185, 129 (1992). Web of Science, ADS, Google Scholar
24. V. B. Priezzhev, J. Stat. Phys. 74, 955 (1994). Web of Science, ADS, Google Scholar
25. E. V. Ivashkevich, J. Phys. A, Math. Gen. 27, 3643 (1994). ADS, Google Scholar
26. S. Mahieu and P. Ruelle, Phys. Rev. E 64, 066130 (2001), arXiv:hep-th/0107150. Web of Science, ADS, Google Scholar
27. P. Ruelle, Phys. Lett. B 539, 172 (2002), arXiv:hep-th/0203105. Web of Science, ADS, Google Scholar
28. M. Jeng, Phys. Rev. E 71, 016140 (2005). Web of Science, ADS, Google Scholar
29. M. Jeng, Phys. Rev. E 71, 036153 (2005). Web of Science, ADS, Google Scholar
30. M. Jeng, G. Piroux and P. Ruelle, J. Stat. Mech. 2006, P10015 (2006), arXiv:cond-mat/0609284. Web of Science, Google Scholar
31. A. A. Saberi et al., Phys. Rev. E 79, 031121 (2009). Web of Science, ADS, Google Scholar
32. N. Azimi-Tafreshi et al., J. Stat. Mech. 2010, P02004 (2010), arXiv:0912.3331v2. Web of Science, Google Scholar
33. D. Dhar and R. Ramaswamy, Phys. Rev. Lett. 63, 1659 (1989). Web of Science, ADS, Google Scholar
34. G. Grinstein, in Scale Invariance, Interfaces, and Non-Equilibrium Dynamics, eds. A. McKane, M. Droz, J. Vannimenus and D. Wolf (Plenum Press, New York, NY, USA, 1995), p. 261–293, NATO Advanced Study Institute on Scale Invariance, Interfaces, and Non-Equilibrium Dynamics, Cambridge, UK, June 20–30, 1994. Google Scholar
35. R. Dickman, A. Vespignani and S. Zapperi, Phys. Rev. E 57, 5095 (1998). Web of Science, ADS, Google Scholar
36. S. Lise, J. Phys. A, Math. Gen. 35, 4641 (2002), arXiv:cond-mat/0204490. ADS, Google Scholar
37. M. Basu et al., Phys. Rev. Lett. 109, 015702 (2012). Web of Science, ADS, Google Scholar
38. S. D. da Cunha et al., J. Stat. Mech. 2014, P08003 (2014), arXiv:1405.1134. Web of Science, Google Scholar
39. R. Dickman and S. D. da Cunha, Phys. Rev. E 92, 020104 (2015). Web of Science, ADS, Google Scholar
40. P. Grassberger, D. Dhar and P. K. Mohanty, Phys. Rev. E 94, 042314 (2016). Web of Science, ADS, Google Scholar
41. P. Le Doussal and K. J. Wiese, Phys. Rev. Lett. 114, 110601 (2015). Web of Science, ADS, Google Scholar
42. G. Pruessner (2017), in preparation. Google Scholar
43. S. Lübeck and A. Hucht, J. Phys. A, Math. Gen. 35, 4853 (2002). ADS, Google Scholar
44. S. Lübeck and P. C. Heger, Phys. Rev. Lett. 90, 230601 (2003). Web of Science, ADS, Google Scholar
45. K. Christensen, H. Flyvbjerg and Z. Olami, Phys. Rev. Lett. 71, 2737 (1993). Web of Science, ADS, Google Scholar
46. H. Flyvbjerg, K. Sneppen and P. Bak, Phys. Rev. Lett. 71, 4087 (1993). Web of Science, ADS, Google Scholar
47. S. A. Janowsky and C. A. Laberge, J. Phys. A, Math. Gen. 26, L973 (1993). Google Scholar
48. S. Zapperi, K. B. Lauritsen and H. E. Stanley, Phys. Rev. Lett. 75, 4071 (1995). Web of Science, ADS, Google Scholar
49. K. B. Lauritsen, S. Zapperi and H. E. Stanley, Phys. Rev. E 54, 2483 (1996). Web of Science, ADS, Google Scholar
50. M. Vergeles, A. Maritan and J. R. Banavar, Phys. Rev. E 55, 1998 (1997). Web of Science, ADS, Google Scholar
51. A. Vespignani and S. Zapperi, Phys. Rev. Lett. 78, 4793 (1997). Web of Science, ADS, Google Scholar
52. A. Vespignani and S. Zapperi, Phys. Rev. E 57, 6345 (1998). Web of Science, ADS, Google Scholar
53. S. Lübeck, personal communication (2003). Google Scholar
54. C. B. Yang, J. Phys. A 37, L523 (2004). ADS, Google Scholar
55. T. Hwa and M. Kardar, Phys. Rev. Lett. 62, 1813 (1989). Web of Science, ADS, Google Scholar
56. M. Paczuski and K. E. Bassler (2000), eprint arXiv:cond-mat/0005340v2. Google Scholar
57. A. Chessa, E. Marinari and A. Vespignani, Phys. Rev. Lett. 80, 4217 (1998). Web of Science, ADS, Google Scholar
58. A. Barrat, A. Vespignani and S. Zapperi, Phys. Rev. Lett. 83, 1962 (1999). Web of Science, ADS, Google Scholar
59. R. Pastor-Satorras and A. Vespignani, Phys. Rev. E 62, 6195 (2000). Web of Science, ADS, Google Scholar
60. P. Le Doussal, K. J. Wiese and G. Pruessner (2016), in preparation. Google Scholar
61. D. Dhar, Physica A 263, 4 (1999), Proc. 20th IUPAP Int. Conf. Statistical Physics, Paris, France, July 20–24, 1998, arXiv:cond-mat/9808047. Google Scholar
62. S. S. Manna, J. Phys. A, Math. Gen. 24, L363 (1991). ADS, Google Scholar
63. G. Pruessner, Int. J. Mod. Phys. B 27, 1350009 (2013), arXiv:1208.2069. Link, Web of Science, ADS, Google Scholar
64. K. Christensen et al., Phys. Rev. Lett. 77, 107 (1996). Web of Science, ADS, Google Scholar
65. D. Dhar, Physica A 340, 535 (2004), arXiv:cond-mat/0309490. Web of Science, ADS, Google Scholar
66. Á. Corral, Phys. Rev. E 69, 026107 (2004), arXiv:cond-mat/0310181v1. Web of Science, ADS, Google Scholar
67. S. Lübeck, Phys. Rev. E 61, 204 (2000). Web of Science, ADS, Google Scholar
68. R. Pastor-Satorras and A. Vespignani, Eur. Phys. J. B 19, 583 (2001), arXiv:cond-mat/0101358. Web of Science, ADS, Google Scholar
69. J. A. Bonachela, Ph.D. thesis, Departmento de Electromagnetismo y Fisíca de la Materia & Institute Carlos I for Theoretical and Computational Physics, University of Granada, Granada, Spain (2008), accessed 12 Sep 2009, http://hera.ugr.es/tesisugr/17706312.pdf. Google Scholar
70. M. Matsumoto and T. Nishimura, ACM Trans. Model. Comp. Sim. 8, 3 (1998). Google Scholar
71. P. Bak and K. Sneppen, Phys. Rev. Lett. 71, 4083 (1993). Web of Science, ADS, Google Scholar
72. J. A. Bonachela and M. A. Muñoz, Physica A 384, 89 (2007), Proc. Int. Conf. Statistical Physics, Raichak and Kolkata, India, Jan 5–9, 2007. Google Scholar
73. J. A. Bonachela and M. A. Muñoz, AIP Conf. Proc. 1091, 204 (2009), Proc. Modeling and Simulation of New Materials: Tenth Granada Lectures, Granada, Spain, Sep 15–19, 2008, arXiv:0905.1827. Google Scholar
74. H. Nakanishi and K. Sneppen, Phys. Rev. E 55, 4012 (1997). Web of Science, ADS, Google Scholar
75. D. Hexner and D. Levine, Phys. Rev. Lett. 114, 110602 (2015). Web of Science, ADS, Google Scholar
76. S. Torquato and F. H. Stillinger, Phys. Rev. E 68, 041113 (2003). Web of Science, ADS, Google Scholar
77. D. Dhar et al. (2015), arXiv:1511.06088. Google Scholar
78. J.-P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic Press, London, UK, 2006). Google Scholar
79. A. Clauset, C. R. Shalizi and M. E. J. Newman, SIAM Rev. 51, 661 (2009), arXiv:0706.1062v2. Web of Science, ADS, Google Scholar
80. A. Deluca and Á. Corral, Acta Geophys. 61, 1351 (2013). Web of Science, ADS, Google Scholar
81. K. Christensen et al., Eur. Phys. J. B 62, 331 (2008). Web of Science, ADS, Google Scholar
82. P. Le Doussal and K. J. Wiese, Phys. Rev. E 88, 022106 (2013). Web of Science, ADS, Google Scholar
83. T. Thiery, P. L. Doussal and K. J. Wiese (2015), arXiv:1504.05342v1. Google Scholar
84. M. C. Kuntz and J. P. Sethna, Phys. Rev. B 62, 11699 (2000). Web of Science, ADS, Google Scholar
85. A. Baldassarri, F. Colaiori and C. Castellano, Phys. Rev. Lett. 90, 060601 (2003). Web of Science, ADS, Google Scholar
86. L. Laurson, M. J. Alava and S. Zapperi, J. Stat. Mech. 2005, L11001 (2005). Web of Science, Google Scholar
87. M. De Menech, A. L. Stella and C. Tebaldi, Phys. Rev. E 58, R2677 (1998). Web of Science, ADS, Google Scholar
88. A. Vespignani et al., Phys. Rev. Lett. 81, 5676 (1998), arXiv:cond-mat/9806249v2. Web of Science, ADS, Google Scholar
89. G. Pruessner, Phys. Rev. E 76, 061103 (2007), arXiv:0712.0979v1. Web of Science, ADS, Google Scholar
90. C. W. Gardiner, Handbook of Stochastic Methods, 2nd edn. (Springer-Verlag, Berlin, Germany, 1997). Google Scholar
91. T. W. Anderson, The Statistical Analysis of Time Series (John Wiley & Sons, New York, NY, USA, 1971). Google Scholar
92. P. Welinder, G. Pruessner and K. Christensen, New J. Phys. 9, 149 (2007). Web of Science, Google Scholar
93. S. Lübeck and P. C. Heger, Phys. Rev. E 68, 056102 (2003). Web of Science, ADS, Google Scholar
94. Z. Olami, H. J. S. Feder and K. Christensen, Phys. Rev. Lett. 68, 1244 (1992). Web of Science, ADS, Google Scholar
95. A. A. Middleton and C. Tang, Phys. Rev. Lett. 74, 742 (1995). Web of Science, ADS, Google Scholar
96. V. Frette et al., Nature 379, 49 (1996). Web of Science, ADS, Google Scholar
97. M. A. Stapleton, Ph.D. thesis, Imperial Collage London, University of London, London SW7 2AZ, UK (2007), accessed 12 May 2007, http://www.matthewstapleton.com/thesis.pdf. Google Scholar
98. A. M. Ferrenberg, D. P. Landau and K. Binder, J. Stat. Phys. 63, 867 (1991). Web of Science, ADS, Google Scholar
99. R. Dickman et al., Phys. Rev. E 64, 056104 (2001). Web of Science, ADS, Google Scholar
100. R. Dickman, T. Tomé and M. J. de Oliveira, Phys. Rev. E 66, 016111 (2002). Web of Science, ADS, Google Scholar
101. R. Dickman, Phys. Rev. E 73, 036131 (2006). Web of Science, ADS, Google Scholar
102. S. B. Lee, Phys. Rev. Lett. 110, 159601 (2013). Web of Science, ADS, Google Scholar
103. I. Jensen, J. Phys. A, Math. Gen. 32, 5233 (1999). ADS, Google Scholar
104. F. J. Wegner, Phys. Rev. B 5, 4529 (1972). Web of Science, ADS, Google Scholar
105. M. N. Barber, in Phase Transitions and Critical Phenomena, eds. C. Domb and J. L. Lebowitz, Vol. 8 (Academic Press, New York, NY, USA, 1983), p. 145–266. Google Scholar
106. A. Chua and K. Christensen (2002), arXiv:cond-mat/0203260v2. Google Scholar
107. Wolfram Research Inc., Mathematica (Wolfram Research, Inc, Champaign, IL, USA, 2014), Version 10.0.2.0. Google Scholar
108. T. Sadhu and D. Dhar, J. Stat. Phys. 134, 427 (2009). Web of Science, ADS, Google Scholar
109. D. Dhar, Physica A 270, 69 (1999), rooted in [61], arXiv:cond-mat/9902137. Google Scholar
110. G. Willis, Ph.D. thesis, Imperial College London, Department of Mathematics, London, UK (2015), accessed 6 Aug 2016, https://spiral.imperial.ac.uk/handle/10044/1/28952. Google Scholar
111. G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes, 2nd edn. (Oxford University Press, New York, NY, USA, 1992). Google Scholar
112. N. G. van Kampen, Stochastic Processes in Physics and Chemistry (Elsevier Science B. V., Amsterdam, The Netherlands, 1992), third impression 2001, enlarged and revised. Google Scholar