Abstract
Addition and subtraction of observed values can be computed under the obvious and implicit assumption that the scale unit of measurement should be the same for all arguments, which is valid even for any nonlinear systems. This paper starts with the distinction between exponential and non-exponential family in the sense of the scale unit of measurement. In the simplest nonlinear model dy∕dx = yq, it is shown how typical effects such as rescaling and shift emerge in the nonlinear systems and affect observed data. Based on the present results, the two representations, namely the q-exponential and the q-logarithm ones, are proposed. The former is for rescaling, the latter for unified understanding with a fixed scale unit. As applications of these representations, the corresponding entropy and the general probability expression for unified understanding with a fixed scale unit are presented. For the theoretical study of nonlinear systems, q-logarithm representation is shown to have significant advantages over q-exponential representation.
Similar content being viewed by others
References
R.C. Tolman,The Principles of Statistical Mechanics (Dover, 2010)
W. Greiner,Thermodynamics and Statistical Mechanics (Springer, 2010)
T.M. Cover, J.A. Thomas,Elements of information theory, 2nd edn (Wiley, 2006)
F. Nielsen, V. Garcia, https://arXiv:0911.4863 (2009)
C. Tsallis, Physica D 193, 3 (2004)
L. Nivanen, A. Le Mehautem, Q.A. Wang, Rep. Math. Phys. 52, 437 (2003)
E.P. Borges, Physica A 340, 95 (2004)
H. Suyari, Physica A, 368, 63 (2006)
C. Tsallis,Introduction to Nonextensive Statistical Mechanics: Approaching a ComplexWorld (Springer, 2009)
A.M. Scarfone, Entropy 15, 624 (2013)
C. Tsallis, J. Stat. Phys. 52, 479 (1988)
A. Einstein, Ann. Phys. 33, 1275 (1910)
H. Suyari, M. Tsukada, IEEE Trans. Inform. Theory 51, 753 (2005)
H. Matsuzoe, A.M. Scarfone, T. Wada, LNCS 11712, 279 (2019)
S. Thurner, R. Hanel, P. Klimek,Introduction to the Theory of Complex Systems (Oxford Univ. Press, 2018)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Suyari, H., Matsuzoe, H. & Scarfone, A.M. Advantages of q-logarithm representation over q-exponential representation from the sense of scale and shift on nonlinear systems. Eur. Phys. J. Spec. Top. 229, 773–785 (2020). https://doi.org/10.1140/epjst/e2020-900196-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjst/e2020-900196-x