Abstract
This paper, as a complement to the work of Montroll and Bendler, is concerned with the Lévy-stable distributions and their connection to the dielectric response of dipolar materials in the frequency domain. The necessary and sufficient condition for this connection is found. The presented probabilistic analysis is based on the mathematically correct representation of the meaning of the relaxation function of a system of dipoles and shows why the same form of a distribution of relaxation rates, namely, the completely asymmetric Lévy-stable distribution, should apply in all different relaxing systems. This is in contrast to the traditional definition of the relaxation function, expressed as a weighted average of exponential relaxation functions, which does not explain the universality of the dielectric relaxation law. It also follows from the present considerations that not only is the imaginary part χ″(ω) of the dielectric susceptibility directly related to the Lévy-stable distribution (as was found by Montroll and Bendler), but so is the real partχ′(ω). As a consequence the relationχ″(ω)/χ′(ω)=cot(nπ/2) forω>ω p and 0<n<1, implied by experimental results, is obtained.
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References
E. W. Montroll and J. T. Bendler,J. Stat. Phys. 34:129 (1984).
K. L. Ngai,Comm. Solid State Phys. 9:127 (1979);9:141 (1980).
A. K. Jonscher,Dielectric Relaxation in Solids (Chelsea Dielectrics, London, 1983).
A. Blumen,Nuovo Cimento B 63:50 (1981).
L. A. Dissado and R. M. Hill,Proc. Roy. Soc. A 390:131 (1983).
R. G. Palmer, D. Stein, E. S. Abrahams, and P. W. Anderson,Phys. Rev. Lett. 53:958 (1984).
M. F. Shlesinger and E. W. Montroll,Proc. Natl. Acad. Sci. USA 81:1280 (1984).
M. F. Shlesinger,J. Stat. Phys. 36:639 (1984).
B. Huberman and M. Kerszberg,J. Phys. A 18:L331 (1985).
J. Klafter and A. Blumen,Chem. Phys. Lett. 119:377 (1985).
J. T. Bendler and M. F. Shlesinger,Macromolecules 18:591 (1985).
J. Klafter and M. F. Shlesinger,Proc. Natl. Acad. Sci. USA 83:848 (1986).
M. R. de la Fuente, M. A. Perez Jubindo, and M. J. Tello,Phys. Rev. B 37:2094 (1986).
K. Weron,Acta Phys. Pol. A 70:529 (1986).
L. A. Dissado and R. M. Hill,Chem. Phys. 111:193 (1987).
K. L. Ngai, A. K. Rajagopal, and S. Teitler,J. Chem. Phys. 88:5086 (1988).
J. Klafter, A. Blumen, G. Zumofen, and M. F. Shlesinger,Physica A 168:637 (1990).
K. Weron,J. Phys. Condens. Matter 3:9151 (1991).
K. Weron,J. Phys. Condens. Matter 4:10507 (1992).
K. Weron and A. Jurlewicz,J. Phys. A Math. Gen. 26:395 (1993).
A. K. Jonscher,Nature 267:673 (1977).
R. M. Hill,Phys. Stat. Sol. 103b:319 (1981).
C. J. F. Böttcher and P. Bordewijk,Theory of Electronic Polarisation, Vol. 2 (Elsevier, Amsterdam, 1978).
W. Rudin,Real and Complex Analysis (McGraw-Hill, New York, 1974).
C. W. Gardiner,Handbook of Stochastic Methods (Springer-Verlag, New York, 1983).
W. Feller,An Introduction to Probability and Its Applications, Vol. 2 (Wiley, New York, 1966).
V. M. Zolotariev,One-Dimensional Stable Distributions (American Mathematical Society, Providence, Rhode Island, 1986).
A. Janicki and A. Weron,Simulation and Chaotic Behavior of α-Stable Stochastic Processes (Marcel Dekker, New York, 1993).
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Jurlewicz, A., Weron, K. A relationship between asymmetric Lévy-stable distributions and the dielectric susceptibility. J Stat Phys 73, 69–81 (1993). https://doi.org/10.1007/BF01052751
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DOI: https://doi.org/10.1007/BF01052751