Anomalous Relaxation in Dielectrics with Hilfer Fractional Derivative
DOI:
https://doi.org/10.5890/JAND.2021.09.009Abstract
We introduce a new relaxation function depending on an arbitrary parameter as a solution of a kinetic equation in the same way as the relaxation function introduced empirically by Debye, Cole-Cole, Davidson-Cole and Havriliak-Negami regarding, anomalous relaxation in dielectrics, which are recovered as particular cases. We propose a differential equation introducing a fractional operator written in terms of the Hilfer fractional derivative of order $\xi$, with $0 < \xi \leq 1$ and type $\eta$, with $0 \leq \eta \leq 1$. To discuss the solution of the fractional differential equation, the methodology of Laplace transform is required. As a by product we mention particular cases where the solution is completely monotone. %Some graphics show the behaviour of this completely monotone function. Finally, the empirical models are recovered as particular cases.References
[1] de Oliveira, E.C., Mainardi, F., and Vaz Jr., J. (2011), {Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics}, ph {The European Physical Journal}, {193}, 161-171.
[2] de Oliveira, E.C., Mainardi, F., and Vaz Jr., J. (2014), {Fractional models of anomalous relaxation based on the Kilbas and Saigo function}, ph {Meccanica}, {49}(9), 2049-2060.
[3] de Oliveira, E.C. and Tenreiro Machado, J.A. (2014), {A review of definitions for fractional derivatives and integrals}, ph {Math. Prob. Ing.}, {2014}, ID 238459.
[4] Teodoro, G.S., Machado, J.A.T., and De Oliveira, E.C. (2019), {A review of definition of fractional derivatives and others operator}, ph {J. Comput. Phys.}, {388}, 195-209.
[5] Tarasov, V.E. (2013), {No violation of the Leibniz rule. No fractional derivative}, ph {Comm. Nonlinear Sci. Num. Simulat.}, {18}, 2945-2948.
[6] Tarasov, V.E. (2018), {No nonlocality. No fractional derivative}, ph {Comm. Nonlinear Sci. Num. Simulat.}, {62}, 157-163.
[7] Akkurt, A., Yildirim, M.E., and Yildirim, H. (2017), {A new generalized fractional derivative and integral}, ph {Konuralp J. Math.}, {5}, 248-259.
[8] Caputo, M. and Fabrizio, M.A. (2015), ph{New definition of fractional derivative without singular kernel}, { Prog. Fract. Diff. Appl.}, {1}, 73-85.
[9] Losada, J. and Nieto, J.J. (2015), {Properties of a new fractional derivative without singular kernel}, {1}, 87-92.
[10] B"ohmer, R., Ngai, K., Angell, C., and Plazeck, D. (1993), {Nonexponential relaxations in strong and fragile glass formers}, ph {J. Chem. Phys.}, {99}, 4201.
[11] Hilfer, R. (2003), {On fractional relaxation}, ph {Fractals}, {11}, 251-257.
[12] Sabatier, J., Agrawal, O.P., Machado, J.A.T. (Editors) (2007), ph{Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering}, Springer, The Netherlands.
[13] Mainardi, F. (2010), ph{Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models}, Imperial College Press.
[14] Debye, P. (1929), ph{Polar Molecules}, Dover, New York.
[15] Kohlrausch, R. (1854), {Theorie des elektrischen R"uckstandes in der Leidener Flasche}, ph {Pogg. Ann. Phys. Chem.}, {91}, 179-214.
[16] Williams, G. and Watts, D.C. (1970), {Non-symmetrical dielectric relaxation behaviour arising from a simple empirical decay function}, ph {Trans. Faraday Soc.}, {66}, 80-85.
[17] Cardona, M., Chamberlin, R.V., and Marx, W. (2007), {Comment on the history of the stretched exponential function}, ph {Ann. Phys. }(Leipzig) {16}, 842-845.
[18] Cole, K.S. and Cole, R.H. (1941), {Dispersion and absorption in dielectrics I. Alternating current characteristics}, ph {J. Chem. Phys.}, {9}, 341-351.
[19] Davidson, D.W. and Cole, R.H. (1950), {Dielectric relaxation in glycerine}, (Letter to the Editor), ph {J. Chem. Phys.}, {18}, 1417. ph{Dielectric relaxation in glycerol, propylene glycol, and $n$-propanol}, J. Chem. Phys., {19}, 1484-1490 (1951).
[20] Havriliak, S. and Negami, S. (1967), {A complex plane representation of dielectric and mechanical relaxation processes in some polymers}, ph {Polymer}, {8}, 161-210.
[21] Garrappa, R. (2016), {Gr"unwald-Letnikov operators for fractional relaxation in Havriliak-Negami models}, ph {Comm. Non. Sci. Num. Simulat.}, {38}, 178-191.
[22] Bia, P., Caratelli, D., Mescia, L., Cicchetti, R., Maione, G., and Prudenzano, F. (2015), {A novel FDTD formulation based on fractional derivatives for dispersive Havriliak-Negami media}, ph {Signal Processing}, {107}, 312-318.
[23] Mainardi, F. (2018),{Fractional Calculus}: {Theorey and Applications}, ph{ Mathematica}, MDPI, Basel.
[24] Shibatov, R.T., Uchaikin, V.V., and Uchaikin, D.V. (2012), ph{Fractional wave equation for dielectric medium with Havriliak-Negami response}, in {sf Fractional Dynamics and Control}, D. Baleanu, J. A. Tenreiro Machado, A. C. J. Luo (Editors), Springer, New York.
[25] Casalini, R., Paluch, M., and Roland, C.M. (2002), {Correlation between the $alpha$-relaxation and the excess wing for polychlorinated biphenyls and glycerol}, ph {J. Thermal Anal. Calorimetry}, {69}, 947-952.
[26] Rosa, E.C.F.A. and de Oliveira, E.C. (2015), {Relaxation equations: fractional models}, ph {J. Phys. Math.}, 6, 146. doi:10.4172/2090-0902.1000146.
[27] Mainardi, F. and Garrappa, R. (2015), {On complete monotonicity of the Prabhakar function and non-Debye relaxation in dielectrics}, ph {J. Comp. Phys.}, {293}, 70-80.
[28] Garrappa, R., Mainardi, F., and Maione, G. (2016), {Models of dielectric relaxation based on completely monotone functions}, ph {Fract. Cal. $&$ Appl. Anal.}, {19}, 1105-116.
[29] Rosa, E.C.F.A. (2017), ph{Fractional Kinetic Relaxation Functions}, (in Portuguese) Doctoral thesis, Unicamp, Campinas-SP.
[30] Machado, J.A.T., Mainardi, F. and Kiryakova, V. (2015), {Fractional calculus}: {Quo vadimus}? ({Where are we going}?) (Contributions to the roundtable held in ICFDA 2014), ph {Fract. Cal. $&$ Appl. Anal.}, {18}, 495-526.
[31] Machado, J.A.T., Mainardi, F., Kiryakova, V., and Atanackovic, T. (2016), {Fractional calculus:} {Dou venons-nous}? {Que sommes-nous}? {O`u allons-nous}? (Contributions to the roundtable held in ICFDA 2016), ph {Fract. Cal. $&$ Appl. Anal.}, {19}, 1074-1104.
[32] Giusti, A. and Colombaro, I. (2018), {Prabhakar-like fractional viscoelasticity}, ph {Comm. Nonlinear Sci. Num. Simulat.}, {56}, 138-143.
[33] Chang, A., Sun, H., Zheng, C., Lu, B., Lu, C., Ma, R., and Zhang, Y. (2018), {A time fractional convection-diffusion equation to model gas transport through heterogeneous soil and gas reservoirs}, ph {Phys. A: Stat. Mech $&$ Appl.}, {502}, 356-369.
[34] Ha, S.Y. and Jung, J. (2018), {Remarks on the slow relaxation for the fractional Kuramoto model for synchronization}, ph {J. Math. Phys.}, {59}, 032702.
[35] Gorska, K., Horzela, A., Bratek, L., Dattoli, G., and Penson, K.A. (2018), {The Havriliak-Negami and its relatives}: {the response, relaxation and probability density functions}, ph {J. Phys. A: Math. $&$ Theor.}, {51}, 135202.
[36] Sun, H.G., Zhang, Y., Baleanu, D., Chen, W., and Chen, Y.Q. (2018), {A new collection of real world applications of fractional calculus in science and engineering}, ph {Comm. Nonlinear Sci. Numer. Simulat.}, {64}, 213-231.
[37] de Oliveira, E.C., Jarosz, S., and Vaz. Jr., J. (2019), {Fractional calculus via Laplace transform and its application in relaxation processes}, ph {Comm. Nonlinear Sci. Numer. Simulat.}, {69}, 58-72.
[38] Mori, H. (1965), {A continued-fractionl representation of the time correlation function}, ph {Prog. Theor. Phys.}, {30}, 399-416.
[39] Zwanzig, R. (1961), ph{Lectures in Theoretical Physics}, Interscience, New York.
[40] Hilfer, R. (2000), in ph{Applications of Fractional Calculus in Physics}, ed. R. Hilfer (World Scientific, Singapore, 2000), p.87.
[41] Novikov, V.V., Wojciechowski, K.W., Komkova, O.A., and Thiels, T. (2005), ph{Anomalous relaxation in dielectrics. Equations with fractional derivatives}, Materials Science-Poland, {23}, 977-984.
[42] Weron, K., Jurlewicz, A., and Magdziark, M. (2005), {Havriliak-Negami response in the framework of the continuous-time random walk}, ph {Acta Phys. Pol. B}, {36}, 1855-1868.
[43] Jurlewicz, A., Weron, K., and Teuerle, M. (2008), {Generalized Mittag-Leffler relaxation}: ph{Clustering-jump continuous-time random walk approach}, ph {Phys. Rev. E}, {78}, 011103.
[44] Khamzin, A.A., Nigmatullin, R.R., and Popov, I.I. (2014), {Justification of the empirical laws of the anomalous dielectric relaxation in the framework of the memory function formalism}, ph {Frac. Cal. $&$ Appl. Anal.}, {17}, 247-258.
[45] Pandey, S.C. (2017), {The Lorenzo-Hartleys function for fractional calculus and its applications pertaining to fractional order modelling of anomalous relaxation in dielectrics}, ph {Comp. Appl. Math.}, DOI. 10.1007/s40314-017-0472-7.
[46] Gorenflo, R., Kilbas, A.A., Mainardi, F., and Rogosin, S.V. (2014), ph{Mittag-Leffler Functions, Related Topics and Applications}, Springer, Heildelberg.
[47] Camargo, R.F. and de Oliveira, E.C. (2015), ph{Fractional Calculus}, (in Portuguese), Editora Livraria da F{i}sica, S~ao Paulo.
[48] Gelfand, I.M. and Shilov, G.E. (1964), ph{Generalized Functions}, Vol. 1, Academic Press, New York.
[49] Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), {Theory and Applications of Fractional Differential Equations}, ph {Elsevier}, Amsterdam.
[50] Mittag-Leffler, G.M. (1903), {Sur la nouvelle fonction $E_{alpha}(x)$}, ph {C. R. Acad. Sci.}, {137}, 554-558.
[51] Bottcher, C.J.F. and Bordewijk, P. (1978), {Theory of Electric Polarization}, ph {Elsevier Sci. Publ.}, Amsterdam.
[52] Hanyga, A. and Seredynska, A. (2008), {On a mathematical framework for the constitutive equations of anisotropic dielectric relaxation}, ph {J. Stat. Phys.}, {131}, 269-303.
[53] Havriliak Jr., S. and Negami, S. (1966), {A complex plane analysis of $alpha$-dispersion in some polymer systems}, ph {J. Polymers Sci.}, {14}, 99-117.
[54] Frohlich, H. (1958), ph{Theory of Dielectrics}, Oxford University Press, London.
[55] Manning, M.F. and Bell, M.E. (1940), {Electrical conduction and related phenomena in solid dielectrics}, ph {Rev. Mod. Phys.}, {12}, 215-257.
[56] Willians, G. (1972), {Use of the dipole correlation function in dielectric relaxation}, ph {J. Chem. Rev.}, {77}, 55-69.
[57] Boon, J.P. and Yip, S. (1980), ph{Molecular Hydrodynamic}, Dover, New York.
[58] Giraldo, R.G.R., Gomez, A.R., and de Oliveira, E.C. (2020), ph{On the complete monotonicity of fractional relaxation functions}, to be submitted for publication.
Article Metrics
Usage tracking begins September 1, 2026.