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Memorized relaxation with singular and non-singular memory kernels for basic relaxation of dielectric vis-à-vis Curie-von Schweidler & Kohlrausch relaxation laws

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  • We have constructed the basic dielectric relaxations evolution expression for relaxing current as convolution operation of the chosen memory kernel and rate of change of applied voltage. We have studied types of memory kernels singular, non-singular and combination of singular and non-singular (mixed) decaying functions. With these, we form constitutive equations for relaxation dynamics of dielectrics; i.e. capacitor. We observe that though mathematically we can use non-singular kernels yet this does not give presently much useful practical or physically realizable results and interpretations. We relate our observations to relaxation currents given via Curie-von Schweidler and Kohlraush laws. The Curie-von Schweidler law gives singular function power law as basic relaxation current in dielectric relaxation; whereas the Kohlraush law is Electric field relaxation in dielectric as non-Debye function taken as stretched exponential i.e. non-singular function. These two laws are used since late nineteenth century for various dielectric relaxation and characterization studies. Here we arrive at general constitutive equation for capacitor and with each type of memory kernel we give corresponding impedance function and in some cases equivalent circuit representation for the capacitor element. We classify these systems as Curie-von Shweidler type for system with singular memory kernel function and Kohlraush type for system evolved via using non-singular function or mixed functions as memory kernel. We note that use of singular memory kernel gives constituent relations and impedance functions that are experimentally verified in large number of cases of dielectric studies. Therefore, we have a question, does natural relaxation dynamics for dielectrics have a singular memory kernel, and the relaxation current function is singular in nature? Is it the singular relaxation function for capacitor dynamics with singular memory kernel remains universal law for dielectric relaxation? However, we are not questioning researchers modeling relaxation of dielectric via non-singular functions, yet we are hinting about complexity and lack of interpretability of basic constituent equation of dielectric relaxation dynamics thus obtained via considering non-singular and mixed memory kernels; perhaps due to insufficient experimental evidences presently. However, the method employed in this study is general method. This method can be used to form memorized constituent equations for other systems (say Radioactive Decay/Growth, Diffusion and Wave phenomena) from basic evolution equation, i.e. effect function as convolution of memory kernel with cause function.

    Mathematics Subject Classification: 26A33, 26E35, 44A10, 45E10, 45H05, 45E10.

    Citation:

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  • Figure 1.  Response current $ {\rm i}(t) = {\rm C}t^{-\alpha } $ to unit step voltage excitation for dielectric relaxation with singular power law memory kernel for various memory indexes: Y-Axis current $ {\rm i}(t) $ and X-Axis time $ t $

    Figure 2.  Response current $ {\rm i}(t) = {\rm C}(1+\lambda t)^{-\alpha } $ to unit step voltage excitation for dielectric relaxation with non-singular power law memory kernel for various memory indexes: Y-Axis current $ {\rm i}(t) $ and X-Axis time $ t $

    Figure 3.  Response current $ {\rm i}(t) = {\rm C}E_{\alpha } (-\lambda t^{\alpha }) $ to unit step voltage excitation for dielectric relaxation with non-singular Mittag-Leffler kernel for various memory indexes: Y-Axis current $ {\rm i}(t) $ and X-Axis time $ t $

    Figure 4.  Response current to unit step voltage excitation for dielectric relaxation with non-singular stretched exponential kernel for various memory indexes: Y-Axis current $ {\rm i}(t) $ and X-Axis time $ t $

    Figure 5.  Response current to unit step voltage excitation for dielectric relaxation with derivative of stretched exponential for various memory indexes: Y-Axis current $ {\rm i}(t) $ and X-Axis time $ t $

    Figure 6.  Response current to unit step voltage excitation for dielectric relaxation with non-singular and singular memory kernels with memory index as $ \alpha $ 0.5: Y-Axis current $ {\rm i}(t) $ and X-Axis time $ t $ : (ⅱ) - $ {\rm k}(t) = t^{-0.5} $, (ⅲ) - $ {\rm k}(t) = (1+t)^{-0.5} $, (ⅳ) - $ {\rm k}(t) = {\it E}_{0.5} (-t^{0.5} ) $, (ⅴ)- $ {\rm k}(t) = {\rm e}^{-t} $, (vi)-$ {\rm k}(t) = {\rm e}^{-t^{0.5}} $ and (ⅶ)-$ {\rm k}(t) = 0.5t^{-0.5} {\rm e}^{-{\it t}^{0.5}} $

    Table 1.  Summary of results of various singular and Non-singular Memory Kernels

    S.NoMemory kernel ${\rm k}(t)$ Type Constitutive equation of capacitor Impedance function in Laplace domain
    1Delta Function
    ${\rm C}\delta (t)$
    Singular ${\rm i}(t) = {\rm Cv}^{(1)}(t)$
    ${\rm i}(t) = {\rm C}\left({}_{0} I_{t}^{0} {\rm v}^{(1)}(t)\right)$
    ${\rm Z}(s)=\frac{1}{s{\rm C}}$
    2Power Law
    ${\rm C}{\it t}^{-\alpha}$
    $0<\alpha <1$
    Singular ${\rm i}(t) = {\rm C}_{\alpha} {\rm v}^{(\alpha)}(t)$
    ${\rm C}_{\alpha} = {\rm C}\left(\Gamma (1-\alpha)\right)$
    ${\rm i}(t) = {\rm C}_{\alpha} \left({}_{0} I_{t}^{1-\alpha} {\rm v}^{(1)}(t)\right)$
    ${\rm Z}(s)=\frac{1}{s^{\alpha} {\rm C}_{\alpha }}$
    3Non-singular Power Law
    ${\rm C}(1+\lambda t)^{-\alpha } $
    $0<\alpha <1$
    Non-Singular ${\rm i}(t)={\rm C}\sum_{n=1}^{\infty}w_n\left({}_{0}I_{t}^{n}{\rm v}^{(1)}(t)\right)$
    $w_1=1, w_2=-\alpha \lambda, $
    $w_3=(\alpha)(\alpha +1)\lambda^{2}, \ldots$
    ${\rm Z}(s)=\frac{1}{{\rm C}\left(\sum _{n=1}^{\infty }w_{n} s^{1-n}\right)}$
    4Mittag-Leffler
    ${\rm C}E_{\alpha} (-\lambda t^{\alpha})$
    $0<\alpha <1$
    Non-Singular ${\rm i}(t)={\rm C}\sum_{n=0}^{\infty}w_{n}\left({}_{0} I_{t}^{\alpha n+1} {\rm v}^{(1)} (t)\right)$
    ${w_n=(-1)^{n}\lambda^{n}}$
    ${\rm v}(t)=\frac{1}{{\rm C}} {\rm i}(t)+\frac{\lambda}{{\rm C}} \left({}_{0} I_{t}^{\alpha } {\rm i}(t)\right)$
    ${\rm Z}(s)=\frac{1}{{\rm C}\sum_{n=0}^{\infty}w_{n} s^{-\alpha n}}$
    ${\rm Z}(s)=\frac{1}{{\rm C}} +\frac{1}{\left({\frac{{\rm C}}{\lambda}}\right)s^{\alpha}}$
    5Exponential
    ${\rm Ce}^{-\lambda t}$
    $0<\alpha <1$
    Non-Singular ${\rm i}(t)={\rm C}\sum_{n=0}^{\infty}w_{n}\left({}_{0}I_{t}^{n+1}{\rm v}^{(1)}(t)\right)$
    $w_{n}=(-1)^{n}\lambda^{n}$
    ${\rm v}(t)=\frac{1}{{\rm C}}{\rm i}(t)+\frac{\lambda}{{\rm C}}\int_0^t{\rm i}(\tau){\rm d}\tau$
    ${\rm Z}(s)=\frac{1}{{\rm C}\sum_{n=0}^{\infty}w_{n} s^{-n}}$
    ${\rm Z}(s)=\frac{1}{{\rm C}}+\frac{1}{s\left({\frac{{\rm C}}{\lambda}}\right)}$
    6Stretched-Exponential
    ${\rm Ce}^{-(\lambda t)^{\alpha}}$
    $0<\alpha <1$
    Non-Singular ${\rm i}(t)={\rm C}\sum_{n=0}^{\infty}w_{n} \left({}_{0} I_{t}^{\alpha n+1}\left[{\rm v}^{(1)} ({\it t})\right]\right)$
    ${\it w}_{n} =(-1)^{n} \left({\frac{\lambda^{\alpha n} \Gamma(\alpha n+1)}{n!}}\right)$
    ${\rm Z}(s)=\frac{1}{{\rm C}\sum_{n=0}^{\infty}w_{n} s^{-\alpha n}}$
    7Derivative of
    Stretched
    Exponential
    ${\rm C}\alpha\lambda^{\alpha} t^{\alpha-1}{\rm e}^{-(\lambda {\it t})^{\alpha}}$
    $0<\alpha <1$
    Singular ${\rm i}(t)={\rm C}_{\lambda} \sum_{n=0}^{\infty}w_{n} \left({}_{0} I_{t}^{(n+1)\alpha} {\rm v}^{(1)} ({\it t})\right)$
    ${\rm C}_{\lambda}=\alpha\lambda^{\alpha} {\rm C}$
    ${\it w}_{n}=(-1)^{n} \left({\textstyle\frac{\lambda^{\alpha n} \Gamma((n+1)\alpha)}{n!}}\right)$
    ${\rm Z}(s)=$
    $\frac{1}{{\rm C}_{\lambda} \sum_{n=0}^{\infty}w_{n} s^{1-(n+1)\alpha}}$
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