W. T. Coffey, Yu.P.Kalmykov and S.V.Titov, 'Anomalous dielectric relaxation in an N-fold cosine potential' in Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 67, 6, (2003), pp 061115 - 1 - 061115 -11
Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 67 6
The fractional Klein-Kramers (Fokker-Planck) equation describing the fractal time dynamics of an assembly of fixed axis dipoles rotating in an N-fold cosine potential representing the internal field due to neighboring molecules is solved using matrix continued fractions. The result can be considered as a generalization of the solution for the normal Brownian motion in a cosine periodic potential to fractional dynamics (giving rise to anomalous diffusion) and also represents a generalization of Fröhlich’s model of relaxation over a potential barrier. The solution includes both inertial and strong internal field effects, which in combination produce a strong resonance peak (Poley absorption) at high frequencies due to librations of the dipoles in the potential, an anomalous relaxation band at low frequencies mainly arising from overbarrier relaxation, and a weaker relaxation band at midfrequencies due to the fast intrawell modes. The high-frequency behavior is controlled by the inertia of the dipole, so that the Gordon sum rule for dipolar absorption is satisfied, ensuring a return to optical transparency at very high frequencies. Application of the model to the broadband (0–THz) dielectric loss spectrum of a dilute solution of the probe dipolar molecule CH2Cl2 in glassy decalin is demonstrated.
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