Tuning of the Dielectric Relaxation and Complex Susceptibility in a System of Polar Molecules: A Generalised Model Based on Rotational Diffusion with Resetting

Abstract

The application of the fractional calculus in the mathematical modelling of relaxation processes in complex heterogeneous media has attracted a considerable amount of interest lately. The reason for this is the successful implementation of fractional stochastic and kinetic equations in the studies of non-Debye relaxation. In this work, we consider the rotational diffusion equation with a generalised memory kernel in the context of dielectric relaxation processes in a medium composed of polar molecules. We give an overview of existing models on non-exponential relaxation and introduce an exponential resetting dynamic in the corresponding process. The autocorrelation function and complex susceptibility are analysed in detail. We show that stochastic resetting leads to a saturation of the autocorrelation function to a constant value, in contrast to the case without resetting, for which it decays to zero. The behaviour of the autocorrelation function, as well as the complex susceptibility in the presence of resetting, confirms that the dielectric relaxation dynamics can be tuned by an appropriate choice of the resetting rate. The presented results are general and flexible, and they will be of interest for the theoretical description of non-trivial relaxation dynamics in heterogeneous systems composed of polar molecules.

Anomalous and ultraslow diffusion of a particle driven by power-law-correlated and distributed-order noises

“…Studying such concepts in the frameworks developed here will significantly enlarge our current range of stochastic models for disordered systems. We also note potential generalisations with respect to subordinated GLE models such as those studied in [87,88], fractional GLE [14,31], as well as the presence of stochastic resetting [89] in the system, including resetting in the memory kernel [90,91].…”

FDTD-Based Electromagnetic Modeling of Dielectric Materials with Fractional Dispersive Response

“…In the same paper, the authors derived a new closed-form expressions for the real and imaginary parts of complex permittivities in terms of generalized hypergeometric functions. Moreover, in [79], the authors extended the conventional dielectric relaxation analysis by including stochastic resetting dynamic.…”

“…The proposed model offers more flexibility than those based on ( 8)- (11), and enables a more effective parameterization of arbitrary dispersive media properties, as well as a better fitting capability of the experimental data over broad frequency ranges. It includes the empirical four-parameter dielectric model based on viscoelastic analysis techniques [67], the fractional model illustrated in [81] as well as the mixing Debye and Cole-Cole relationship proposed in [79]. It can be also adapted to include the complex conjugate residual pairs and the Drude critical-points dispersive relationships [82,83], the two-parameter variant of the Drude model presented in [84], as well as the modified Lorentz model [85].…”

Generalized diffusion and random search processes

“…η(s) = (s + r) −α , in the absence of sink, we recover the subdiffusive search process with exponential resetting to the initial position. The corresponding diffusion equation for the PDF P r (x, t) can be written in renewal form (25) through the PDF P(x, t) in the absence of resetting mechanism (r = 0) [61], as well as in the form of the tempered fractional Fokker-Planck equation [62,63]…”