Abstract
The fractional generalizations of the relaxation equation and the wave equation in dielectrics with the response function of the Havriliak–Negami type are considered. The obtained fractional wave equation is concordant with the asymptotical equations derived by Tarasov VE (J Phys Condens Matter 20:145212, 2008) from Jonscher’s universal law. The explicit expression for the fractional operator in this equation is obtained and the Monte Carlo algorithm for calculation of actions of this operator and of the inverse one is constructed.
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References
Fröhlich H (1958) Theory of dielectrics, 2nd edn. Oxford University Press, Oxford
Debye P (1954) Polar molecules. Dover, New York
Jonscher AK (1977) The “universal” dielectric response. Nature 267:673
Ramakrishnan TV, Raj Lakshmi M (1987) Non-debye relaxation in condensed matter. World Scientific, Singapore
Cole KS, Cole RH (1941) Dispersion and absorption in dielectrics. J Chem Phys 9:341
Davidson DW, Cole RH (1951) Dielectric relaxation in glycerol, propylene glycol, and n-propanol. J Chem Phys 19:1484
Havriliak S, Negami S (1966) A complex plane analysis of α-dispersions in some polymer systems. J Polymer Sci 14:99
Novikov VV, Wojciechowski KW, Komkova OA, Thiel T (2005) Anomalous relaxation in dielectrics. Equations with fractional derivatives. Mater Sci Poland 23:977
Nigmatullin RR, Ryabov YaE (1997) Cole-Davidson dielectric relaxation as a self-similar relaxation process. Phys Solid State 39:87
Jonscher AK (1996) Universal relaxation law. Chelsea-Dielectrics Press, London
Weron K (1991) How to obtain the universal response law in the Jonscher screened hopping model for dielectric relaxation. Phys Condens Matter 3:221
Weron K, Kotulski M (1996) On the equivalence of the parallel channel and the correlated cluster relaxation models. J Stat Phys 88:1241
Nigmatullin RR (1984) To the theoretical explanation of the “universal response”. Phys Stat Sol b 123:739–745
Glöckle WG, Nonnenmacher TF (1993) Fox function representation of non-Debye relaxation processes. J Stat Phys 71:741
Jurlewicz A, Weron K (2000) Relaxation dynamics of the fastest channel in multichannel parallel relaxation mechanism. Chaos Solitons Fractals 11:303
Coffey WT, Kalmykov YuP, Titov SV (2002) Anomalous dielectric relaxation in the context of the Debye model of noninertial rotational diffusion. J Chem Phys 116:6422
Déjardin J-L (2003) Fractional dynamics and nonlinear harmonic responses in dielectric relaxation of disordered liquids. Phys Rev E 68:031108
Aydiner E (2005) Anomalous rotational relaxation: A fractional Fokker-Planck equation approach. Phys Rev E 71:046103
Weron K, Jurlewicz A, Magdziarz M (2005) Havriliak–Negami response in the framework of the continuous-time random walk. Acta Physica Polonica B 36:1855–1868
Bochner S (1949) Diffusion equation and stochastic processes. Proc Nat Acad Sci USA 35:368–370
Phillips RS (1952) On the generation of semigroups of linear operators. Pacific J Math 2:343–369
Yosida K (1980) Functional analysis. Springer, New York
Tarasov VE (2008) Universal electromagnetic waves in dielectric. J Phys Condens Matter 20:175223
Acknowledgment
The authors are grateful to the Russian Foundation for Basic Research (grant 10-01-00618) for financial support.
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Sibatov, R.T., Uchaikin, V.V., Uchaikin, D.V. (2012). Fractional Wave Equation for Dielectric Medium with Havriliak–Negami Response. In: Baleanu, D., Machado, J., Luo, A. (eds) Fractional Dynamics and Control. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0457-6_25
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DOI: https://doi.org/10.1007/978-1-4614-0457-6_25
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