[Paper Review] Relaxation Equations: Fractional Models

Ester C. F. A. Rosa, E. Capelas de Oliveira|arXiv (Cornell University)|Oct 6, 2015

Fractional Differential Equations Solutions23 references4 citations

TL;DR

This paper proposes a generalized fractional kinetic model for dielectric relaxation using Riemann-Liouville fractional derivatives of order $\gamma$ ($0 < \gamma \leq 1$), solving the resulting equations via Laplace transforms to yield solutions in terms of Mittag-Leffler functions. The key contribution is that the classical Debye, Cole-Cole, Cole-Davidson, and Havriliak-Negami relaxation functions emerge as special cases when $\gamma = 1$, unifying these models under a fractional framework with enhanced flexibility for anomalous relaxation behavior.

ABSTRACT

The relaxation functions introduced empirically by Debye, Cole-Cole, Cole-Davidson and Havriliak-Negami are, each of them, solutions to their respective kinetic equations. In this work, we propose a generalization of such equations by introducing a fractional differential operator written in terms of the Riemann-Liouville fractional derivative of order $γ$, $0 &lt; γ\leq 1$. In order to solve the generalized equations, the Laplace transform methodology is introduced and the corresponding solutions are then presented, in terms of Mittag-Leffler functions. In the case in which the derivative's order is $γ=1$, the traditional relaxation functions are recovered. Finally, we presente some 2D graphs of these function.

Motivation & Objective

• To generalize classical dielectric relaxation models by introducing a fractional derivative of order $\gamma$ ($0 < \gamma \leq 1$) in the kinetic equations.
• To derive analytical solutions of these fractional kinetic equations using the Laplace transform method.
• To express the solutions in terms of Mittag-Leffler functions, which generalize classical relaxation functions.
• To demonstrate that the classical relaxation models (Debye, Cole-Cole, Cole-Davidson, Havriliak-Negami) are recovered as special cases when $\gamma = 1$.
• To provide graphical representations of the fractional relaxation functions for different parameter values.

Proposed method

• Formulate fractional kinetic equations using the Riemann-Liouville fractional derivative of order $\gamma$ to model relaxation processes.
• Construct memory functions based on integral kernels corresponding to each classical relaxation model (Debye, Cole-Cole, Cole-Davidson, Havriliak-Negami).
• Apply the Laplace transform to the fractional kinetic equations to derive algebraic expressions for the transformed relaxation functions.
• Use known Laplace transform identities for Mittag-Leffler functions, particularly $\mathscr{L}[t^{\gamma-1}E_{\alpha,\gamma}^{\beta}(at^{\alpha})](s) = \frac{s^{\alpha\beta - \gamma}}{(s^{\alpha} - a)^{\beta}}$, to invert the solutions.
• Recover classical models by setting $\gamma = 1$ and specific parameter values (e.g., $\beta = 1$, $\alpha = 1$).
• Generate 2D plots of the fractional relaxation functions to visualize the dependence on $\gamma$, $\alpha$, $\beta$, and $\sigma$.

Experimental results

Research questions

• RQ1 Can fractional differential equations with Riemann-Liouville derivatives of order $\gamma \in (0,1]$ generalize classical dielectric relaxation models?
• RQ2 Do the solutions of these fractional equations reduce to known classical relaxation functions (Debye, Cole-Cole, Cole-Davidson, Havriliak-Negami) when $\gamma = 1$?
• RQ3 How do the Mittag-Leffler function-based solutions behave for $\gamma < 1$, and what is their monotonicity and physical interpretability?
• RQ4 What is the role of the parameters $\alpha$, $\beta$, $\gamma$, and $\sigma$ in shaping the relaxation dynamics in the fractional framework?
• RQ5 Can the complex dielectric permittivity be consistently derived from the fractional relaxation functions via the Laplace transform relation $\tilde{\varepsilon}(s) = 1 - s^{\gamma}\tilde{\varphi}(s)$?

Key findings

• The fractional kinetic equation with Riemann-Liouville derivative of order $\gamma$ generalizes classical relaxation models, with solutions expressed as Mittag-Leffler functions.
• When $\gamma = 1$, the fractional Debye model recovers the classical Debye relaxation function $\varphi(t) = e^{-t/\sigma}$.
• The fractional Cole-Cole model yields $\varphi(t)_{CCF} = \frac{t^{\gamma-1}}{\Gamma(\gamma)} - \sigma^{-\alpha}t^{\alpha + \gamma - 1}E_{\alpha,\alpha + \gamma}^{\alpha}\left(-\frac{t}{\sigma}\right)$ for $\alpha = 0.5$, with a stretched exponential decay.
• The fractional Cole-Davidson model is given by $\varphi(t)_{CDF} = \frac{t^{\gamma-1}}{\Gamma(\gamma)} - \sigma^{-\beta}t^{\beta + \gamma - 1}E_{1,\beta + \gamma}^{\beta}\left(-\frac{t}{\sigma}\right)$, recovering the classical C-D function at $\gamma = 1$.
• The fractional Havriliak-Negami model is expressed as $\varphi(t)_{HNF} = \frac{t^{\gamma-1}}{\Gamma(\gamma)} - \sigma^{-\alpha\beta}t^{\alpha\beta + \gamma - 1}E_{\alpha,\alpha\beta + \gamma}^{\beta}\left[-\left(\frac{t}{\sigma}\right)^\alpha\right]$, which reduces to the classical H-N model when $\gamma = 1$.
• Graphical analysis confirms that the fractional relaxation functions remain completely monotonic for $t > 0$ and exhibit non-exponential, power-law-like decay characteristics dependent on $\gamma$, $\alpha$, and $\beta$.