In this contribution, we present for the first time, explicit expressions for time-dependent entropy production, in the classical context. Here, we pursue the understanding of the time dependence of entropy by the approach of nonlocal fractional derivatives and also with local deformed derivatives. We claim that the entropy production depends on the dynamics and the geometry of the complex system considered. Some closed expressions for the entropy growth rate were obtained. We also give a possible explanation for the generalized Pesin relations here obtained, by considering the connection between the fractal geometry of phase-space with the types of particles interactions.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
V. Latora, M. Baranger, Kolmogorov-sinai entropy rate versus physical entropy. Phys. Rev. Lett. 82(3), 520 (1999)
V. Latora, M. Baranger, A. Rapisarda, C. Tsallis, The rate of entropy increase at the edge of chaos. Phys. Lett. A 273(1–2), 97–103 (2000)
A.S. Balankin, B.E. Elizarraraz, Hydrodynamics of fractal continuum flow. Phys. Rev. E 85(2), 025302 (2012)
A.S. Balankin, B.E. Elizarraraz, Map of fluid flow in fractal porous medium into fractal continuum flow. Phys. Rev. E 85(5), 056314 (2012)
A.S. Balankin, J. Bory-Reyes, M. Shapiro, Towards a physics on fractals: differential vector calculus in three-dimensional continuum with fractal metric. Physica A 444, 345–359 (2016)
J. Weberszpil, O. Sotolongo-Costa, Structural derivative model for tissue radiation response. J. Adv. Phys. 13(4), 4779–4785 (2017)
W. Rosa, J. Weberszpil, Dual conformable derivative: Definition, simple properties and perspectives for applications. Chaos, Solitons Fractals 117, 137–141 (2018)
J. Weberszpil, M.J. Lazo, J. Helayël-Neto, On a connection between a class of q-deformed algebras and the hausdorff derivative in a medium with fractal metric. Physica A 436, 399–404 (2015)
J. Weberszpil, J.A. Helayël-Neto, Variational approach and deformed derivatives. Physica A 450, 217–227 (2016)
J. Weberszpil, W. Chen, Generalized maxwell relations in thermodynamics with metric derivatives. Entropy 19(8), 407 (2017)
J. Weberszpil, J.A. Helayël-Neto, Structural scale q-derivative and the llg equation in a scenario with fractionality. EPL (Europhysics Letters) 117(5), 50006 (2017)
J. Weberszpil, J.A. Helayel-Neto, Axiomatic local metric derivatives for low-level fractionality with mittag-leffler eigenfunctions. J. Adv. Phys. 13(3), 4751–4755 (2017)
C. Tsallis, Possible generalization of boltzmann-gibbs statistics. J. Stat. Phys. 52(1–2), 479–487 (1988)
A. Saa, R. Venegeroles, Pesin-type relation for subexponential instability. J. Stat. Mech: Theory Exp. 2012(03), P03010 (2012)
P. Nazé, R. Venegeroles, Number of first-passage times as a measurement of information for weakly chaotic systems. Phys. Rev. E 90(4), 042917 (2014)
R. Venegeroles, Quantitative universality for a class of weakly chaotic systems. J. Stat. Phys. 154(4), 988–998 (2014)
R. Venegeroles, Exact invariant measures: How the strength of measure settles the intensity of chaos. Phys. Rev. E 91(6), 062914 (2015)
P. Gaspard, X.J. Wang, Sporadicity: between periodic and chaotic dynamical behaviors. Proc. Natl. Acad. Sci. 85(13), 4591–4595 (1988)
A.P. Leopoldino, J. Weberszpil, C.F. Godinho, J.A. Helayël-Neto, Discussing the extension and applications of a variational approach with deformed derivatives. J. Math. Phys. 60(8), 083507 (2019)
X. Su, W. Xu, W. Chen, H. Yang. Fractional creep and relaxation models of viscoelastic materials via a non-newtonian time-varying viscosity: physical interpretation. Mech. Mater. p. 103222. (2019)
W. Chen, F. Wang, B. Zheng, W. Cai, Non-euclidean distance fundamental solution of hausdorff derivative partial differential equations. Eng. Anal. Bound. Elem. 84, 213–219 (2017)
K.S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations (Wiley, 1993)
K. Oldham, J. Spanier, The fractional calculus theory and applications of differentiation and integration to arbitrary order, vol. 111 (Elsevier, 1974)
E.C. Grigoletto, E.C. de Oliveira, Fractional versions of the fundamental theorem of calculus. Appl. Math. 4(07), 23 (2013)
R. Saxena, A. Mathai, H. Haubold, Unified fractional kinetic equation and a fractional diffusion equation. Astrophys. Space Sci. 290(3), 299–310 (2004)
W. Chen, Time-space fabric underlying anomalous diffusion. Chaos Solitons Fractals 28(4), 923–929 (2006)
R. Khalil, M.A. Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)
T. Abdeljawad, On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015)
D. Zhao, M. Luo, General conformable fractional derivative and its physical interpretation. Calcolo 54(3), 903–917 (2017)
W. Xu, W. Chen, Y. Liang, J. Weberszpil, A spatial structural derivative model for ultraslow diffusion. Therm. Sci. 21(1), 121–127 (2017)
A.S. Balankin, B. Mena, J. Patiño, D. Morales, Electromagnetic fields in fractal continua. Phys. Lett. A 377(10–11), 783–788 (2013)
X. Yang, Y. Liang, W. Chen, A local structural derivative pde model for ultraslow creep. Comput. Math. Appl. 76(7), 1713–1718 (2018)
X. Su, W. Chen, W. Xu, Y. Liang, Non-local structural derivative maxwell model for characterizing ultra-slow rheology in concrete. Constr. Build. Mater. 190, 342–348 (2018)
W. Chen, X. Hei, H. Sun, D. Hu, Stretched exponential stability of nonlinear hausdorff dynamical systems. Chaos, Solitons Fractals 109, 259–264 (2018)
E.P. Borges, A possible deformed algebra and calculus inspired in nonextensive thermostatistics. Physica A 340(1–3), 95–101 (2004)
V. Garcia-Morales, J. Pellicer, Microcanonical foundation of nonextensivity and generalized thermostatistics based on the fractality of the phase space. Physica A 361(1), 161–172 (2006)
O. Sotolongo-Costa, O. Sotolongo-Grau, L. Gaggero-Sager, I. Rodrıguez-Vargas, Anomalous diffusion in phase space: Relation to the entropy growth rate. Some Current Topics in Condensed Matter Physics 1–8 (2016)
V. Sithi, S. Lim, On the spectra of riemann-liouville fractional brownian motion. J. Phys. A Math. Gen. 28(11), 2995 (1995)
C. Li, D. Qian, Y. Chen, On riemann-liouville and caputo derivatives. Discret. Dyn. Nat. Soc. 2011, (2011)
H.J. Haubold, A.M. Mathai, R.K. Saxena, Mittag-leffler functions and their applications. J. Appl. Math. 2011, (2011)
F. Brouers, O. Sotolongo-Costa, Generalized fractal kinetics in complex systems (application to biophysics and biotechnology). Physica A 368(1), 165–175 (2006)
F. Brouers, The fractal (bsf) kinetics equation and its approximations. J. Mod. Phys. 5(16), 1594 (2014)
F. Baldovin, A. Robledo, Nonextensive pesin identity: Exact renormalization group analytical results for the dynamics at the edge of chaos of the logistic map. Phys. Rev. E 69(4), 045202 (2004)
O. Sotolongo-Costa, L. Gaggero-Sager, M. Mora-Ramos, A non-extensive statistical model for time-dependent multiple breakage particle-size distribution. Physica A 438, 74–80 (2015)
M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel. Progr. Fract. Differ. Appl 1(2), 1–13 (2015)
Conflicts of Interest
The authors declare that they have no conflict of interest.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Sotolongo-Costa, O., Weberszpil, J. Explicit Time-Dependent Entropy Production Expressions: Fractional and Fractal Pesin Relations. Braz J Phys (2021). https://doi.org/10.1007/s13538-021-00889-5
- Time-dependent entropy production
- Generalized Pesin relations
- Deformed derivatives
- Fractional calculus
- Fractals and multifractals
- Kolmogorov–Sinai entropy