Skip to main content
SpringerLink
Log in

Scale-invariant occupancy of phase space and additivity of nonextensive entropy S q

  • Published:
Journal of Shanghai Jiaotong University (Science) Aims and scope Submit manuscript

Abstract

Phase space can be constructed for N equal and distinguishable binary subsystems which are correlated in a scale-invariant manner. In the paper, correlation coefficient and reduced probability are introduced to characterize the scale-invariant correlated binary subsystems. Probabilistic sets for the correlated binary subsystems satisfy Leibnitz triangle rule in the sense that the marginal probabilities of N-system are equal to the joint probabilities of the (N −1)-system. For entropic index q ≠ 1, nonextensive entropy S q is shown to be additive in the scale-invariant occupation of phase space.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Tsallis C. Possible generalization of Boltzmann-Gibbs statistics [J]. Journal of Statistical Physics, 1988, 52(1–2): 479–487.

    Article  MATH  MathSciNet  Google Scholar 

  2. Gell-Mann M, Tsallis C. Nonextensive entropyinterdisciplinary applications [M]. New York: Oxford University Press, 2004.

    Google Scholar 

  3. Rohlf T, Tsallis C. Long-range memory elementary 1D cellular automata: Dynamics and nonextensivity [J]. Physica A, 2007, 379(2): 465–470.

    Article  Google Scholar 

  4. Anteneodo C. Non-extensive random walks [J]. Physica A, 2005, 358(2–4): 289–298.

    Article  MathSciNet  Google Scholar 

  5. Liao G F, Ma X F, Wang L D. Individual chaos implies collective chaos for weakly mixing discrete dynamical systems [J]. Chao, Solitons and Fractals, 2007, 32(2): 604–608.

    Article  MATH  MathSciNet  Google Scholar 

  6. Latora V, Rapisarda A, Tsallis C. Fingerprints of nonextensive thermodynamics in a long-range Hamiltonian system [J]. Physcia A, 2002, 305(1–2): 129–136.

    Article  MATH  Google Scholar 

  7. Daniels K E, Beck C, Bodenschatz E. Defect turbulence and generalized statistical mechanics [J]. Physica D, 2004, 193(1–4): 208–217.

    Article  MATH  MathSciNet  Google Scholar 

  8. Baiesi M, Paczuski M, Stella A L. Intensity thresholds and the statistics of the temporal occurrence of solar flares [J]. Physics Review Letters, 2006, 96(5): 1–4.

    Article  Google Scholar 

  9. Tsallis C. Connection between scale-free networks and nonextensive statistical mechanics [J]. The European Physical Journal-Special Topics, 2008, 161(1): 175–180.

    Article  Google Scholar 

  10. Queirós S M D, Moyano L G, Desouza J, et al. A nonextensive approach to the dynamics of financial observables [J]. The European Physical Journal B, 2007, 55(2): 161–167.

    Article  MATH  MathSciNet  Google Scholar 

  11. Cortines A A G, Riera R. Non-extensive behavior of a stock market index at microscopic time scales [J]. Physica A, 2007, 377(1): 181–192.

    Article  Google Scholar 

  12. Tsallis C. Thermostatistically approaching living systems: Boltzmann-Gibbs or nonextensive statistical mechanics? [J]. Physics of Life Reviews, 2006, 3(1): 1–22.

    Article  Google Scholar 

  13. Azaele S, Pigolotti S, Banavar J R, et al. Dynamical evolution of ecosystems [J]. Nature, 2006, 444(11): 926–928.

    Article  Google Scholar 

  14. Cancho R F I. Decoding least effort and scaling in signal frequency distributions [J]. Physica A, 2005, 345(1–2): 275–284.

    Google Scholar 

  15. Sotolongo-Costa O, Posadas A. Fragmentasperity interaction model for earthquakes [J]. Physics Review Letters, 2004, 92(4): 11–14.

    Article  Google Scholar 

  16. Anastasiadis A D, Magoulas G D. Evolving stochastic learning algorithm based on Tsallis entropic index [J]. The European Physical Journal B, 2006, 50(1–2): 277–283.

    Article  Google Scholar 

  17. Asgarani S, Mirza B. Quasi-additivity of Tsallis entropies and correlated subsystems [J]. Physica A, 2007, 379(2): 513–522.

    Article  MathSciNet  Google Scholar 

  18. Piasecki R. Extended quasi-additivity of Tsallis entropies [J]. Physica A, 2006, 366(1): 221–228.

    Article  MathSciNet  Google Scholar 

  19. Tsallis C, Gell-Mann M, Sato Y. Asymptotically scale-invariant occupancy of phase space makes the entropy Sq extensive [J]. Proceedings of the National Academy of Sciences, 2005, 102(43): 377–382.

    Article  MathSciNet  Google Scholar 

  20. Tsallis C. Occupancy of phase space, extensivity of Sq, and q-generalized central limit theorem [J]. Physica A, 2006, 365(1): 7–16.

    Article  MathSciNet  Google Scholar 

  21. Tsallis C. Is the entropy Sq extensive or nonextensive? [J]. Astrophysics and Space Science, 2006, 305(3): 261–271.

    Article  MATH  MathSciNet  Google Scholar 

  22. Marsh J, Earl S. New solutions to scale-invariant phase-space occupancy for the generalized entropy Sq [J]. Physics Letters A, 2006, 349(1–4): 146–152.

    Article  Google Scholar 

  23. Moyano L G, Tsallis C, Gell-Mann M. Numerical indications of a q-generalised central limit theorem [J]. Europhysics Letters, 2006, 73(6): 813–819.

    Article  MathSciNet  Google Scholar 

  24. Marsh J A, Fuentes M A, Moyano L G, et al. Influence of global correlations on central limit theorems and entropic extensivity [J]. Physica A, 2006, 372(2): 183–202.

    Article  MathSciNet  Google Scholar 

  25. Sato Y, Tsallis C. On the extensivity of the entropy Sq for N ⩽ 3 specially correlated binary subsystems [J]. International Journal of Bifurcation and Chaos, 2006, 16(6): 1727–1738.

    Article  MATH  MathSciNet  Google Scholar 

  26. Beck C, Cohen E G D. Superstatistics [J]. Physica A, 2003, 322(1–4): 267–275.

    Article  MATH  MathSciNet  Google Scholar 

  27. Cohen E G D. Superstatistics [J]. Physica D, 2004, 193(1–4): 35–52.

    Article  MATH  MathSciNet  Google Scholar 

  28. Tsallis C, Souza A M C. Constructing a statistical mechanics for Beck-Cohen superstatistics [J]. Physical Review E, 2003, 67(2): 1–5.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Control and Simulation Center, Harbin Institute of Technology, Harbin, 150001, China

    Wei Zhao  (赵 伟) & Ye San  (伞 冶)

Authors
  1. Wei Zhao  (赵 伟)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ye San  (伞 冶)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wei Zhao  (赵 伟).

Additional information

Foundation item: the National Natural Science Foundation of China (No. 60474069)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhao, W., San, Y. Scale-invariant occupancy of phase space and additivity of nonextensive entropy S q . J. Shanghai Jiaotong Univ. (Sci.) 15, 441–446 (2010). https://doi.org/10.1007/s12204-010-1030-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12204-010-1030-2

Key words

CLC number

Search

Navigation