Multiscale analysis of information correlations in an infinite-range, ferromagnetic Ising system

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 70, Issue 6 2, 2004, Pages

  Gheorghiu Svirschevski, S ^a   Bar Yam, Y ^a  

^a NEW ENGLAND COMPLEX SYSTEMS INSTITUTE   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; CHAOS THEORY; CORRELATION METHODS; ENTROPY; INFORMATION THEORY; MAGNETIZATION; MATHEMATICAL MODELS; ORDER DISORDER TRANSITIONS; PROBABILITY; THERMODYNAMICS;

GREEN'S EXPANSION; ISING SYSTEM; PERMUTATIONAL SYMMETRY; SHANNON ENTROPY;

FERROMAGNETIC MATERIALS;

EID: 41349101078     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.70.066115     Document Type: Article

References (42)

1. J. J. Binney, N. J. Dowrick, A. J. Fisher, and M. E. J. Newman, The Theory of Critical Phenomena (Clarendon, Oxford, 1992); D. A. Lavis and G. M. Bell, Statistical Mechanics of Lattice Systems, Vol. 2: Exact, Series and Renormalization Group Methods (Springer-Verlag, Berlin, 1999).
2. J. J. Binney, N. J. Dowrick, A. J. Fisher, and M. E. J. Newman, The Theory of Critical Phenomena (Clarendon, Oxford, 1992); D. A. Lavis and G. M. Bell, Statistical Mechanics of Lattice Systems, Vol. 2: Exact, Series and Renormalization Group Methods (Springer-Verlag, Berlin, 1999).
3. C. E. Shannon and W. Weaver, The Mathematical Theory of Communication (University of Illinois Press, Urbana, 1963).
4. D. J. Amit, Modeling Brain Function: The World of Attractor Neural Networks (Cambridge University Press, Cambridge, UK, 1989); Y. S. Abu-Mostafa, IEEE Commun. Mag. 27, 25 (1989); J. Taylor and M. Plumbley, Information Theory and Neural Networks, in Mathematical Approaches to Neural Networks, edited by J. Taylor (Elsevier Science, Amsterdam, 1993), p. 307; G. Tononi, O. Sporns, and G. Edelman, Proc. Natl. Acad. Sci. U.S.A. 91, 5033 (1994); G. Deco and D. Obradovic, An Information-Theoretic Approach to Neural Computing (Springer-Verlag, New York, 1996); O. A. Rosso, Physica A 313, 587 (2002); D. MacKay, Information Theory, Inference and Learning Algorithms (Cambridge University Press, Cambridge, UK, 2003).
5. D. J. Amit, Modeling Brain Function: The World of Attractor Neural Networks (Cambridge University Press, Cambridge, UK, 1989); Y. S. Abu-Mostafa, IEEE Commun. Mag. 27, 25 (1989); J. Taylor and M. Plumbley, Information Theory and Neural Networks, in Mathematical Approaches to Neural Networks, edited by J. Taylor (Elsevier Science, Amsterdam, 1993), p. 307; G. Tononi, O. Sporns, and G. Edelman, Proc. Natl. Acad. Sci. U.S.A. 91, 5033 (1994); G. Deco and D. Obradovic, An Information-Theoretic Approach to Neural Computing (Springer-Verlag, New York, 1996); O. A. Rosso, Physica A 313, 587 (2002); D. MacKay, Information Theory, Inference and Learning Algorithms (Cambridge University Press, Cambridge, UK, 2003).
6. D. J. Amit, Modeling Brain Function: The World of Attractor Neural Networks (Cambridge University Press, Cambridge, UK, 1989); Y. S. Abu-Mostafa, IEEE Commun. Mag. 27, 25 (1989); J. Taylor and M. Plumbley, Information Theory and Neural Networks, in Mathematical Approaches to Neural Networks, edited by J. Taylor (Elsevier Science, Amsterdam, 1993), p. 307; G. Tononi, O. Sporns, and G. Edelman, Proc. Natl. Acad. Sci. U.S.A. 91, 5033 (1994); G. Deco and D. Obradovic, An Information-Theoretic Approach to Neural Computing (Springer-Verlag, New York, 1996); O. A. Rosso, Physica A 313, 587 (2002); D. MacKay, Information Theory, Inference and Learning Algorithms (Cambridge University Press, Cambridge, UK, 2003).
7. D. J. Amit, Modeling Brain Function: The World of Attractor Neural Networks (Cambridge University Press, Cambridge, UK, 1989); Y. S. Abu-Mostafa, IEEE Commun. Mag. 27, 25 (1989); J. Taylor and M. Plumbley, Information Theory and Neural Networks, in Mathematical Approaches to Neural Networks, edited by J. Taylor (Elsevier Science, Amsterdam, 1993), p. 307; G. Tononi, O. Sporns, and G. Edelman, Proc. Natl. Acad. Sci. U.S.A. 91, 5033 (1994); G. Deco and D. Obradovic, An Information-Theoretic Approach to Neural Computing (Springer-Verlag, New York, 1996); O. A. Rosso, Physica A 313, 587 (2002); D. MacKay, Information Theory, Inference and Learning Algorithms (Cambridge University Press, Cambridge, UK, 2003).
8. D. J. Amit, Modeling Brain Function: The World of Attractor Neural Networks (Cambridge University Press, Cambridge, UK, 1989); Y. S. Abu-Mostafa, IEEE Commun. Mag. 27, 25 (1989); J. Taylor and M. Plumbley, Information Theory and Neural Networks, in Mathematical Approaches to Neural Networks, edited by J. Taylor (Elsevier Science, Amsterdam, 1993), p. 307; G. Tononi, O. Sporns, and G. Edelman, Proc. Natl. Acad. Sci. U.S.A. 91, 5033 (1994); G. Deco and D. Obradovic, An Information-Theoretic Approach to Neural Computing (Springer-Verlag, New York, 1996); O. A. Rosso, Physica A 313, 587 (2002); D. MacKay, Information Theory, Inference and Learning Algorithms (Cambridge University Press, Cambridge, UK, 2003).
9. D. J. Amit, Modeling Brain Function: The World of Attractor Neural Networks (Cambridge University Press, Cambridge, UK, 1989); Y. S. Abu-Mostafa, IEEE Commun. Mag. 27, 25 (1989); J. Taylor and M. Plumbley, Information Theory and Neural Networks, in Mathematical Approaches to Neural Networks, edited by J. Taylor (Elsevier Science, Amsterdam, 1993), p. 307; G. Tononi, O. Sporns, and G. Edelman, Proc. Natl. Acad. Sci. U.S.A. 91, 5033 (1994); G. Deco and D. Obradovic, An Information-Theoretic Approach to Neural Computing (Springer-Verlag, New York, 1996); O. A. Rosso, Physica A 313, 587 (2002); D. MacKay, Information Theory, Inference and Learning Algorithms (Cambridge University Press, Cambridge, UK, 2003).
10. D. J. Amit, Modeling Brain Function: The World of Attractor Neural Networks (Cambridge University Press, Cambridge, UK, 1989); Y. S. Abu-Mostafa, IEEE Commun. Mag. 27, 25 (1989); J. Taylor and M. Plumbley, Information Theory and Neural Networks, in Mathematical Approaches to Neural Networks, edited by J. Taylor (Elsevier Science, Amsterdam, 1993), p. 307; G. Tononi, O. Sporns, and G. Edelman, Proc. Natl. Acad. Sci. U.S.A. 91, 5033 (1994); G. Deco and D. Obradovic, An Information-Theoretic Approach to Neural Computing (Springer-Verlag, New York, 1996); O. A. Rosso, Physica A 313, 587 (2002); D. MacKay, Information Theory, Inference and Learning Algorithms (Cambridge University Press, Cambridge, UK, 2003).
11. P. Grassberger, Int. J. Theor. Phys. 25, 907 (1986); K. Lindgren and M. G. Norhdal, Complex Syst. 2, 409 (1988); Cellular Automata and Modeling of Complex Systems, Vol. 46 of Springer Proceedings in Physics, edited by P. Manneville, N. Boccara, G. I. Vichniac, and R. Bideaux (Springer, Berlin, 1990) ; Cellular Automata: Theory and Experiment, edited by H. A. Gutowitz (MIT Press, Cambridge, MA, 1991).
12. P. Grassberger, Int. J. Theor. Phys. 25, 907 (1986); K. Lindgren and M. G. Norhdal, Complex Syst. 2, 409 (1988); Cellular Automata and Modeling of Complex Systems, Vol. 46 of Springer Proceedings in Physics, edited by P. Manneville, N. Boccara, G. I. Vichniac, and R. Bideaux (Springer, Berlin, 1990) ; Cellular Automata: Theory and Experiment, edited by H. A. Gutowitz (MIT Press, Cambridge, MA, 1991).
13. P. Grassberger, Int. J. Theor. Phys. 25, 907 (1986); K. Lindgren and M. G. Norhdal, Complex Syst. 2, 409 (1988); Cellular Automata and Modeling of Complex Systems, Vol. 46 of Springer Proceedings in Physics, edited by P. Manneville, N. Boccara, G. I. Vichniac, and R. Bideaux (Springer, Berlin, 1990) ; Cellular Automata: Theory and Experiment, edited by H. A. Gutowitz (MIT Press, Cambridge, MA, 1991).
14. P. Grassberger, Int. J. Theor. Phys. 25, 907 (1986); K. Lindgren and M. G. Norhdal, Complex Syst. 2, 409 (1988); Cellular Automata and Modeling of Complex Systems, Vol. 46 of Springer Proceedings in Physics, edited by P. Manneville, N. Boccara, G. I. Vichniac, and R. Bideaux (Springer, Berlin, 1990) ; Cellular Automata: Theory and Experiment, edited by H. A. Gutowitz (MIT Press, Cambridge, MA, 1991).
15. P. Szepfalusy and G. Gyorgyi, Phys. Rev. A 33, 2852 (1986); N. B. Abraham, A. M. Albano, A. Passamante, and P. E. Rapp, Measures of Complexity and Chaos (Plenum, New York, 1989); L. Diambra and A. Plastino, Phys. Lett. A 216, 278 (1996); J. Freund, W. Ebeling, and K. Rateitschak, Phys. Rev. E 54, 5561 (1996); Y.-C. Tian, Physica D 108, 113 (1997); C. Schittenkopf, ibid. 110, 173 (1997); A. G. Bashkirov and A. V. Vityazev, Physica A 277, 136 (2000); G. Boffetta, Phys. Rep. 356, 367 (2002); A. Shabunin, V. Demidov, V. Astakhov, and V. Anishchenko, Phys. Rev. E 65, 056215 (2002); C. Beck and F. Schlogl, Thermodynamics of Chaotic Systems (Cambridge University Press, Cambridge, UK, 1993); R. Metzler, Y. Bar-Yam, and M. Kardar, Phys. Rev. E 70, 026205 (2004).
16. P. Szepfalusy and G. Gyorgyi, Phys. Rev. A 33, 2852 (1986); N. B. Abraham, A. M. Albano, A. Passamante, and P. E. Rapp, Measures of Complexity and Chaos (Plenum, New York, 1989); L. Diambra and A. Plastino, Phys. Lett. A 216, 278 (1996); J. Freund, W. Ebeling, and K. Rateitschak, Phys. Rev. E 54, 5561 (1996); Y.-C. Tian, Physica D 108, 113 (1997); C. Schittenkopf, ibid. 110, 173 (1997); A. G. Bashkirov and A. V. Vityazev, Physica A 277, 136 (2000); G. Boffetta, Phys. Rep. 356, 367 (2002); A. Shabunin, V. Demidov, V. Astakhov, and V. Anishchenko, Phys. Rev. E 65, 056215 (2002); C. Beck and F. Schlogl, Thermodynamics of Chaotic Systems (Cambridge University Press, Cambridge, UK, 1993); R. Metzler, Y. Bar-Yam, and M. Kardar, Phys. Rev. E 70, 026205 (2004).
17. P. Szepfalusy and G. Gyorgyi, Phys. Rev. A 33, 2852 (1986); N. B. Abraham, A. M. Albano, A. Passamante, and P. E. Rapp, Measures of Complexity and Chaos (Plenum, New York, 1989); L. Diambra and A. Plastino, Phys. Lett. A 216, 278 (1996); J. Freund, W. Ebeling, and K. Rateitschak, Phys. Rev. E 54, 5561 (1996); Y.-C. Tian, Physica D 108, 113 (1997); C. Schittenkopf, ibid. 110, 173 (1997); A. G. Bashkirov and A. V. Vityazev, Physica A 277, 136 (2000); G. Boffetta, Phys. Rep. 356, 367 (2002); A. Shabunin, V. Demidov, V. Astakhov, and V. Anishchenko, Phys. Rev. E 65, 056215 (2002); C. Beck and F. Schlogl, Thermodynamics of Chaotic Systems (Cambridge University Press, Cambridge, UK, 1993); R. Metzler, Y. Bar-Yam, and M. Kardar, Phys. Rev. E 70, 026205 (2004).
18. P. Szepfalusy and G. Gyorgyi, Phys. Rev. A 33, 2852 (1986); N. B. Abraham, A. M. Albano, A. Passamante, and P. E. Rapp, Measures of Complexity and Chaos (Plenum, New York, 1989); L. Diambra and A. Plastino, Phys. Lett. A 216, 278 (1996); J. Freund, W. Ebeling, and K. Rateitschak, Phys. Rev. E 54, 5561 (1996); Y.-C. Tian, Physica D 108, 113 (1997); C. Schittenkopf, ibid. 110, 173 (1997); A. G. Bashkirov and A. V. Vityazev, Physica A 277, 136 (2000); G. Boffetta, Phys. Rep. 356, 367 (2002); A. Shabunin, V. Demidov, V. Astakhov, and V. Anishchenko, Phys. Rev. E 65, 056215 (2002); C. Beck and F. Schlogl, Thermodynamics of Chaotic Systems (Cambridge University Press, Cambridge, UK, 1993); R. Metzler, Y. Bar-Yam, and M. Kardar, Phys. Rev. E 70, 026205 (2004).
19. P. Szepfalusy and G. Gyorgyi, Phys. Rev. A 33, 2852 (1986); N. B. Abraham, A. M. Albano, A. Passamante, and P. E. Rapp, Measures of Complexity and Chaos (Plenum, New York, 1989); L. Diambra and A. Plastino, Phys. Lett. A 216, 278 (1996); J. Freund, W. Ebeling, and K. Rateitschak, Phys. Rev. E 54, 5561 (1996); Y.-C. Tian, Physica D 108, 113 (1997); C. Schittenkopf, ibid. 110, 173 (1997); A. G. Bashkirov and A. V. Vityazev, Physica A 277, 136 (2000); G. Boffetta, Phys. Rep. 356, 367 (2002); A. Shabunin, V. Demidov, V. Astakhov, and V. Anishchenko, Phys. Rev. E 65, 056215 (2002); C. Beck and F. Schlogl, Thermodynamics of Chaotic Systems (Cambridge University Press, Cambridge, UK, 1993); R. Metzler, Y. Bar-Yam, and M. Kardar, Phys. Rev. E 70, 026205 (2004).
20. P. Szepfalusy and G. Gyorgyi, Phys. Rev. A 33, 2852 (1986); N. B. Abraham, A. M. Albano, A. Passamante, and P. E. Rapp, Measures of Complexity and Chaos (Plenum, New York, 1989); L. Diambra and A. Plastino, Phys. Lett. A 216, 278 (1996); J. Freund, W. Ebeling, and K. Rateitschak, Phys. Rev. E 54, 5561 (1996); Y.-C. Tian, Physica D 108, 113 (1997); C. Schittenkopf, ibid. 110, 173 (1997); A. G. Bashkirov and A. V. Vityazev, Physica A 277, 136 (2000); G. Boffetta, Phys. Rep. 356, 367 (2002); A. Shabunin, V. Demidov, V. Astakhov, and V. Anishchenko, Phys. Rev. E 65, 056215 (2002); C. Beck and F. Schlogl, Thermodynamics of Chaotic Systems (Cambridge University Press, Cambridge, UK, 1993); R. Metzler, Y. Bar-Yam, and M. Kardar, Phys. Rev. E 70, 026205 (2004).
21. P. Szepfalusy and G. Gyorgyi, Phys. Rev. A 33, 2852 (1986); N. B. Abraham, A. M. Albano, A. Passamante, and P. E. Rapp, Measures of Complexity and Chaos (Plenum, New York, 1989); L. Diambra and A. Plastino, Phys. Lett. A 216, 278 (1996); J. Freund, W. Ebeling, and K. Rateitschak, Phys. Rev. E 54, 5561 (1996); Y.-C. Tian, Physica D 108, 113 (1997); C. Schittenkopf, ibid. 110, 173 (1997); A. G. Bashkirov and A. V. Vityazev, Physica A 277, 136 (2000); G. Boffetta, Phys. Rep. 356, 367 (2002); A. Shabunin, V. Demidov, V. Astakhov, and V. Anishchenko, Phys. Rev. E 65, 056215 (2002); C. Beck and F. Schlogl, Thermodynamics of Chaotic Systems (Cambridge University Press, Cambridge, UK, 1993); R. Metzler, Y. Bar-Yam, and M. Kardar, Phys. Rev. E 70, 026205 (2004).
22. P. Szepfalusy and G. Gyorgyi, Phys. Rev. A 33, 2852 (1986); N. B. Abraham, A. M. Albano, A. Passamante, and P. E. Rapp, Measures of Complexity and Chaos (Plenum, New York, 1989); L. Diambra and A. Plastino, Phys. Lett. A 216, 278 (1996); J. Freund, W. Ebeling, and K. Rateitschak, Phys. Rev. E 54, 5561 (1996); Y.-C. Tian, Physica D 108, 113 (1997); C. Schittenkopf, ibid. 110, 173 (1997); A. G. Bashkirov and A. V. Vityazev, Physica A 277, 136 (2000); G. Boffetta, Phys. Rep. 356, 367 (2002); A. Shabunin, V. Demidov, V. Astakhov, and V. Anishchenko, Phys. Rev. E 65, 056215 (2002); C. Beck and F. Schlogl, Thermodynamics of Chaotic Systems (Cambridge University Press, Cambridge, UK, 1993); R. Metzler, Y. Bar-Yam, and M. Kardar, Phys. Rev. E 70, 026205 (2004).
23. P. Szepfalusy and G. Gyorgyi, Phys. Rev. A 33, 2852 (1986); N. B. Abraham, A. M. Albano, A. Passamante, and P. E. Rapp, Measures of Complexity and Chaos (Plenum, New York, 1989); L. Diambra and A. Plastino, Phys. Lett. A 216, 278 (1996); J. Freund, W. Ebeling, and K. Rateitschak, Phys. Rev. E 54, 5561 (1996); Y.-C. Tian, Physica D 108, 113 (1997); C. Schittenkopf, ibid. 110, 173 (1997); A. G. Bashkirov and A. V. Vityazev, Physica A 277, 136 (2000); G. Boffetta, Phys. Rep. 356, 367 (2002); A. Shabunin, V. Demidov, V. Astakhov, and V. Anishchenko, Phys. Rev. E 65, 056215 (2002); C. Beck and F. Schlogl, Thermodynamics of Chaotic Systems (Cambridge University Press, Cambridge, UK, 1993); R. Metzler, Y. Bar-Yam, and M. Kardar, Phys. Rev. E 70, 026205 (2004).
24. P. Szepfalusy and G. Gyorgyi, Phys. Rev. A 33, 2852 (1986); N. B. Abraham, A. M. Albano, A. Passamante, and P. E. Rapp, Measures of Complexity and Chaos (Plenum, New York, 1989); L. Diambra and A. Plastino, Phys. Lett. A 216, 278 (1996); J. Freund, W. Ebeling, and K. Rateitschak, Phys. Rev. E 54, 5561 (1996); Y.-C. Tian, Physica D 108, 113 (1997); C. Schittenkopf, ibid. 110, 173 (1997); A. G. Bashkirov and A. V. Vityazev, Physica A 277, 136 (2000); G. Boffetta, Phys. Rep. 356, 367 (2002); A. Shabunin, V. Demidov, V. Astakhov, and V. Anishchenko, Phys. Rev. E 65, 056215 (2002); C. Beck and F. Schlogl, Thermodynamics of Chaotic Systems (Cambridge University Press, Cambridge, UK, 1993); R. Metzler, Y. Bar-Yam, and M. Kardar, Phys. Rev. E 70, 026205 (2004).
25. P. Szepfalusy and G. Gyorgyi, Phys. Rev. A 33, 2852 (1986); N. B. Abraham, A. M. Albano, A. Passamante, and P. E. Rapp, Measures of Complexity and Chaos (Plenum, New York, 1989); L. Diambra and A. Plastino, Phys. Lett. A 216, 278 (1996); J. Freund, W. Ebeling, and K. Rateitschak, Phys. Rev. E 54, 5561 (1996); Y.-C. Tian, Physica D 108, 113 (1997); C. Schittenkopf, ibid. 110, 173 (1997); A. G. Bashkirov and A. V. Vityazev, Physica A 277, 136 (2000); G. Boffetta, Phys. Rep. 356, 367 (2002); A. Shabunin, V. Demidov, V. Astakhov, and V. Anishchenko, Phys. Rev. E 65, 056215 (2002); C. Beck and F. Schlogl, Thermodynamics of Chaotic Systems (Cambridge University Press, Cambridge, UK, 1993); R. Metzler, Y. Bar-Yam, and M. Kardar, Phys. Rev. E 70, 026205 (2004).
26. C. H. Bennett, Found. Phys. 16, 585 (1986); K. Lindgren, Phys. Rev. A 38, 4794 (1988); Complexity, Entropy and the Physics of Information, edited by W. Zurek (Addison-Wesley, Redwood City, CA, 1990); R. Wackerbauer, A. Witt, H. Atmanspacher, J. Kurths, and H. Scheingraber, Chaos, Solitons Fractals 4, 133 (1994); Y. Bar-Yam, Dynamics of Complex Systems (Addison-Wesley, Reading, MA, 1997); R. Badii and A. Politi, Complexity: Hierarchical Structure and Scaling in Physics (Cambridge University Press, Cambridge, UK, 1997); J. S. Shiner, M. Davison, and P. T. Landsberg, Phys. Rev. E 59, 1459 (1999); Yu. A. Andrienko, N. V. Brilliantov, and J. Kurths, Eur. Phys. J. B 15, 539 (2000); P. M. Binder and J. A. Plazas, Phys. Rev. E 63, 065203(R) (2001).
27. C. H. Bennett, Found. Phys. 16, 585 (1986); K. Lindgren, Phys. Rev. A 38, 4794 (1988); Complexity, Entropy and the Physics of Information, edited by W. Zurek (Addison-Wesley, Redwood City, CA, 1990); R. Wackerbauer, A. Witt, H. Atmanspacher, J. Kurths, and H. Scheingraber, Chaos, Solitons Fractals 4, 133 (1994); Y. Bar-Yam, Dynamics of Complex Systems (Addison-Wesley, Reading, MA, 1997); R. Badii and A. Politi, Complexity: Hierarchical Structure and Scaling in Physics (Cambridge University Press, Cambridge, UK, 1997); J. S. Shiner, M. Davison, and P. T. Landsberg, Phys. Rev. E 59, 1459 (1999); Yu. A. Andrienko, N. V. Brilliantov, and J. Kurths, Eur. Phys. J. B 15, 539 (2000); P. M. Binder and J. A. Plazas, Phys. Rev. E 63, 065203(R) (2001).
28. C. H. Bennett, Found. Phys. 16, 585 (1986); K. Lindgren, Phys. Rev. A 38, 4794 (1988); Complexity, Entropy and the Physics of Information, edited by W. Zurek (Addison-Wesley, Redwood City, CA, 1990); R. Wackerbauer, A. Witt, H. Atmanspacher, J. Kurths, and H. Scheingraber, Chaos, Solitons Fractals 4, 133 (1994); Y. Bar-Yam, Dynamics of Complex Systems (Addison-Wesley, Reading, MA, 1997); R. Badii and A. Politi, Complexity: Hierarchical Structure and Scaling in Physics (Cambridge University Press, Cambridge, UK, 1997); J. S. Shiner, M. Davison, and P. T. Landsberg, Phys. Rev. E 59, 1459 (1999); Yu. A. Andrienko, N. V. Brilliantov, and J. Kurths, Eur. Phys. J. B 15, 539 (2000); P. M. Binder and J. A. Plazas, Phys. Rev. E 63, 065203(R) (2001).
29. C. H. Bennett, Found. Phys. 16, 585 (1986); K. Lindgren, Phys. Rev. A 38, 4794 (1988); Complexity, Entropy and the Physics of Information, edited by W. Zurek (Addison-Wesley, Redwood City, CA, 1990); R. Wackerbauer, A. Witt, H. Atmanspacher, J. Kurths, and H. Scheingraber, Chaos, Solitons Fractals 4, 133 (1994); Y. Bar-Yam, Dynamics of Complex Systems (Addison-Wesley, Reading, MA, 1997); R. Badii and A. Politi, Complexity: Hierarchical Structure and Scaling in Physics (Cambridge University Press, Cambridge, UK, 1997); J. S. Shiner, M. Davison, and P. T. Landsberg, Phys. Rev. E 59, 1459 (1999); Yu. A. Andrienko, N. V. Brilliantov, and J. Kurths, Eur. Phys. J. B 15, 539 (2000); P. M. Binder and J. A. Plazas, Phys. Rev. E 63, 065203(R) (2001).
30. C. H. Bennett, Found. Phys. 16, 585 (1986); K. Lindgren, Phys. Rev. A 38, 4794 (1988); Complexity, Entropy and the Physics of Information, edited by W. Zurek (Addison-Wesley, Redwood City, CA, 1990); R. Wackerbauer, A. Witt, H. Atmanspacher, J. Kurths, and H. Scheingraber, Chaos, Solitons Fractals 4, 133 (1994); Y. Bar-Yam, Dynamics of Complex Systems (Addison-Wesley, Reading, MA, 1997); R. Badii and A. Politi, Complexity: Hierarchical Structure and Scaling in Physics (Cambridge University Press, Cambridge, UK, 1997); J. S. Shiner, M. Davison, and P. T. Landsberg, Phys. Rev. E 59, 1459 (1999); Yu. A. Andrienko, N. V. Brilliantov, and J. Kurths, Eur. Phys. J. B 15, 539 (2000); P. M. Binder and J. A. Plazas, Phys. Rev. E 63, 065203(R) (2001).
31. C. H. Bennett, Found. Phys. 16, 585 (1986); K. Lindgren, Phys. Rev. A 38, 4794 (1988); Complexity, Entropy and the Physics of Information, edited by W. Zurek (Addison-Wesley, Redwood City, CA, 1990); R. Wackerbauer, A. Witt, H. Atmanspacher, J. Kurths, and H. Scheingraber, Chaos, Solitons Fractals 4, 133 (1994); Y. Bar-Yam, Dynamics of Complex Systems (Addison-Wesley, Reading, MA, 1997); R. Badii and A. Politi, Complexity: Hierarchical Structure and Scaling in Physics (Cambridge University Press, Cambridge, UK, 1997); J. S. Shiner, M. Davison, and P. T. Landsberg, Phys. Rev. E 59, 1459 (1999); Yu. A. Andrienko, N. V. Brilliantov, and J. Kurths, Eur. Phys. J. B 15, 539 (2000); P. M. Binder and J. A. Plazas, Phys. Rev. E 63, 065203(R) (2001).
32. C. H. Bennett, Found. Phys. 16, 585 (1986); K. Lindgren, Phys. Rev. A 38, 4794 (1988); Complexity, Entropy and the Physics of Information, edited by W. Zurek (Addison-Wesley, Redwood City, CA, 1990); R. Wackerbauer, A. Witt, H. Atmanspacher, J. Kurths, and H. Scheingraber, Chaos, Solitons Fractals 4, 133 (1994); Y. Bar-Yam, Dynamics of Complex Systems (Addison-Wesley, Reading, MA, 1997); R. Badii and A. Politi, Complexity: Hierarchical Structure and Scaling in Physics (Cambridge University Press, Cambridge, UK, 1997); J. S. Shiner, M. Davison, and P. T. Landsberg, Phys. Rev. E 59, 1459 (1999); Yu. A. Andrienko, N. V. Brilliantov, and J. Kurths, Eur. Phys. J. B 15, 539 (2000); P. M. Binder and J. A. Plazas, Phys. Rev. E 63, 065203(R) (2001).
33. C. H. Bennett, Found. Phys. 16, 585 (1986); K. Lindgren, Phys. Rev. A 38, 4794 (1988); Complexity, Entropy and the Physics of Information, edited by W. Zurek (Addison-Wesley, Redwood City, CA, 1990); R. Wackerbauer, A. Witt, H. Atmanspacher, J. Kurths, and H. Scheingraber, Chaos, Solitons Fractals 4, 133 (1994); Y. Bar-Yam, Dynamics of Complex Systems (Addison-Wesley, Reading, MA, 1997); R. Badii and A. Politi, Complexity: Hierarchical Structure and Scaling in Physics (Cambridge University Press, Cambridge, UK, 1997); J. S. Shiner, M. Davison, and P. T. Landsberg, Phys. Rev. E 59, 1459 (1999); Yu. A. Andrienko, N. V. Brilliantov, and J. Kurths, Eur. Phys. J. B 15, 539 (2000); P. M. Binder and J. A. Plazas, Phys. Rev. E 63, 065203(R) (2001).
34. C. H. Bennett, Found. Phys. 16, 585 (1986); K. Lindgren, Phys. Rev. A 38, 4794 (1988); Complexity, Entropy and the Physics of Information, edited by W. Zurek (Addison-Wesley, Redwood City, CA, 1990); R. Wackerbauer, A. Witt, H. Atmanspacher, J. Kurths, and H. Scheingraber, Chaos, Solitons Fractals 4, 133 (1994); Y. Bar-Yam, Dynamics of Complex Systems (Addison-Wesley, Reading, MA, 1997); R. Badii and A. Politi, Complexity: Hierarchical Structure and Scaling in Physics (Cambridge University Press, Cambridge, UK, 1997); J. S. Shiner, M. Davison, and P. T. Landsberg, Phys. Rev. E 59, 1459 (1999); Yu. A. Andrienko, N. V. Brilliantov, and J. Kurths, Eur. Phys. J. B 15, 539 (2000); P. M. Binder and J. A. Plazas, Phys. Rev. E 63, 065203(R) (2001).
35. J. P. Crutchfield and D. P. Feldman, Phys. Rev. E 55, R1239 (1997). They describe the effect of ordering on information in a one-dimensional Ising model.
36. Y. Bar-Yam, Complexity 9, 37 (2004); 9, 15 (2004).
37. Y. Bar-Yam, Complexity 9, 37 (2004); 9, 15 (2004).
38. Y. Bar-Yam, Adv. Complex Syst. 7, 47 (2004).
39. D. R. Wolf, Ph. D. dissertation, University of Texas, Austin, Texas, 1996.
40. H. S. Green, The Molecular Theory of Fluids (North-Holland, Amsterdam, 1952).
41. S. Gheorghiu-Svirschevski and Y. Bar-Yam (unpublished).
42. Handbook of Mathematical Functions, edited by M. Abramowitz and I. Stegun (Dover, New York, 1970).