Revisiting the Curie-Von Schweidler Law for Dielectric Relaxation and Derivation of Distribution Function for Relaxation Rates as Zipf’s Power Law and Manifestation of Fractional Differential Equation for Capacitor

The classical power law relaxation, i.e. relaxation of current with inverse of power of time for a step-voltage excitation to dielectric—as popularly known as Curie-von Schweidler law is empirically derived and is observed in several relaxation experiments on various dielectrics studies since late 19 Century. This relaxation law is also regarded as “universal-law” for dielectric relaxations; and is also termed as power law. This empirical Curie-von Schewidler relaxation law is then used to derive fractional differential equations describing constituent expression for capacitor. In this paper, we give simple mathematical treatment to derive the distribution of relaxation rates of this Curie-von Schweidler law, and show that the relaxation rate follows Zipf’s power law distribution. We also show the method developed here give Zipfian power law distribution for relaxing time constants. Then we will show however mathematically correct this may be, but physical interpretation from the obtained time constants distribution are contradictory to the Zipfian rate relaxation distribution. In this paper, we develop possible explanation that as to why Zipfian distribution of relaxation rates appears for Curie-von Schweidler Law, and relate this law to time variant rate of relaxation. In this paper, we derive appearance of fractional derivative while using Zipfian power law distribution that gives notion of scale dependent relaxation rate function for Curie-von Schweidler relaxation phenomena. This paper gives analytical approach to get insight of a non-Debye relaxation and gives a new treatment to especially much used empirical Curie-von Schweidler (universal) relaxation law. *Dedicating this work to my mother Purabi Das on her 83rd birthday. How to cite this paper: Das, S. (2017) Revisiting the Curie-Von Schweidler Law for Dielectric Relaxation and Derivation of Distribution Function for Relaxation Rates as Zipf’s Power Law and Manifestation of Fractional Differential Equation for Capacitor. Journal of Modern Physics, 8, 19882012. https://doi.org/10.4236/jmp.2017.812120 Received: October 11, 2017 Accepted: November 24, 2017 Published: November 27, 2017 Copyright © 2017 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access


Introduction
The Curie-von Schweidler law relates to relaxation current in dielectric when a step DC voltage is applied and is given by ( ) , where 0 t > and the pow- er (exponent) i.e. n is called relaxation constant or decay constant, where 0 1 n < < [1] [2] [3] [4].We note that n is non-integer.This relaxation law is taken as universal law, at least for dielectric relaxations.Whereas we are used to Debye type of relaxation i.e. exponential decay law given by ( ) λ τ − = .The radioactive decay is example of ideal Debye law where the exponential decay is governed by "one-lumped" decay constant i.e. 0 λ .The Curie-von Schweidler behavior has been observed in many instances, since late 19 th Century, such as those shown in dielectric studies and experiments [3]- [10].
This power law relaxation of the non-Debye type i.e. ( ) n i t t −  has been interpreted as a many-body problem but can also be formulated as an infinite number of independent relaxing bodies meaning infinite number of time constants τ or relaxation rates λ varying from near zero to infinity [11] [12].
The observations of power law relaxation are also made in the experiments and studies with super-capacitors [13] [14] [15] [16] [17].These studies also indicate the fractional calculus is used as constituent expression to describe super-capacitors.The use of empirical power law i.e.Curie-von Schweidler Law of relaxation of current to a step input of voltage to get constituent relation with fractional derivative was proposed in [5].Apart from relaxation of current decay in dielectrics and super-capacitors, the power law type or non-Debye relaxation is observed in visco-elastic experiments strain relaxation in [18] [19] [20] [21].
In this paper, we are giving the derivation of the distribution of relaxation rates ( λ ) particularly for Curie-von Schweidler law and we observe the distribu- tion nature as Zipf's distribution [22]- [27].We try to reason out as to why this distribution of relaxation rates takes Zipfian nature.We also show that Curie-von Schweidler law has time varying rate of relaxation.This paper will not deal with the mathematics of Zipfian distribution (or power law distribution) like probability density function, cumulative probability density function, and the conditions of finding finite mean, variance or standard deviation for power law distribution.This paper describes finding the distribution function of relaxation rates (or histogram) by formulating Laplace integral, and show that the We extend this mathematical approach to get the distribution function for time constants (τ ).We observe that time constants τ are also distributed as Zipf's power law; but this observation points to a contrary physical interpretation derived from this obtained power law distribution for the relaxation rate ( λ ) distributions.Thus we can conclude that this method developed by Laplace integral approach is restricted to get only distribution of relaxation rates i.e. λ and not to get the distribution of time constants i.e. τ .Though we are discussing especially Curie-von Schweidler law, yet we will tabulate relaxation rate distributions obtained for some other relaxation functions which are obtained via this Laplace integral method.
We shall demonstrate the formation of fractional derivative in the expression relating current and voltage considering the relaxation rates as Zifian distribution; and thus forming a scale dependent power law for relaxation rates as the scale varies from zero to infinity.Though by experiments one cannot make histogram directly for the rates of relaxation for any non-Debye processes, yet this mathematical procedure that we develop helps in extracting this information from the observations relaxation function.This is new treatment, and much more research is required, across various dynamic processes.

Obtaining Fractional Derivative Directly from Curie-Von Schweidler Law for Capacitor
Practically on applying a step input voltage ( ) BB v t V = Volts at 0 t = to a ca- pacitor which is initially uncharged; we get a power-law decay of current given by empirical Curie-von Schweidler as ( ) . That we write in following way as indicated by experimental studies [5]- [10]: The parameter n K is constant.This is from observation and the evaluation of order of power-law function is 0.5 1 n < < [5] [13] [14] [15] [16] [17].Let the capacitor be excited by a step input of V BB Volts, i.e. written as ( ) ( ) ( ) where ( ) . Then taking Laplace transform of above power-law decay current (1), we obtain   [28].Then using the formula for generalization of factorial i.e. ( ) ( ) , we get the following expressions ( ) This Expression (3) i.e. ( ) G s is admittance expression in complex frequency s-domain of a capacitor.From here we write impedance expression for capacitor as following ( ) From the obtained Expressions (3) (4) i.e. ( ) ( ) ( ) and by taking inverse Laplace transform by using the identity fractional derivative operation [12] [31], we get the constituent relation for capacity as following This fractional derivative Expression (5) gives a new capacitor theory [5] and we utilize this above Formula (5) to find characteristics of super-capacitors, like the variation of n with the current excitation, and the efficiency of the energy discharged to the energy stored [13] [14] [15] [16] [17].Classically the expres- ) . with integer "one-whole" order classical derivative.Therefore Curie-von Schweidler law gives a different approach based on fractional calculus [12] [31].In experimental observations we find that capacitor has fractional order impedance [5]- [10] [13] [14] [15] [16], [17].The impedance ( ) (is obtained by writing Laplace variable s in (4) as i ;i 1 ω = − , i.e. considering steady state analysis [5] [28].This fractional impedance observed in [5]- [10] [13] [14] [15] [16] [17], has implication in dissipation [5] theory of di-electrics that we will not cover here.We state that while classical capacitor unit is in Farad, the n C the "fractional capacity" is in units of This section gives us the understanding that this Curie-von Schweidle law i.e.
the empirical law gives a relation of voltage and current of capacitor by using fractional derivative.In this paper we will show how we get the same relation (i.e. via use of fractional derivative) by considering Zipfian distribution of relaxation rates ( λ ) that we get for Curie-von Schweidler relaxation law.

About the Zipf's Power Law Distribution and Probable Hypothesis for Its Mechanism
The Zipf's law is widely referred in linguistic studies, economics studies, population studies [22]- [27].We use this for a dielectric relaxation law (i.e.Curie-von Schweidler law), which is observed as ( ) Zipf's Law is an empirical law formulated using mathematical statistics that refers to the fact that many types of data studied in the physical and social sciences can be approximated with a Zipfian distribution.This distribution is one of a family of related discrete power law probability distributions [22]- [27].
This power law distribution help to describe phenomena where large events are rare, but small ones are quite common.For example, there are few large earthquakes but many small ones.There are a few mega-cities, but many small towns.
There are few words, such as "and" and "the" that occur very frequently, but many which occur rarely.The emergence of a complex language is one of the fundamental events of human evolution, and several remarkable features suggest the presence of fundamental principles of organization.These principles seem to be common to all languages.The best known is the so-called Zipf's law, which states that the frequency of a word decays as a (universal) power law of its rank.The possible origins of this law have been controversial, and its meaningfulness is still an open question.One of the early hypotheses of Zipf of a principle of least effort for explaining the law is shown to be sound [26] [27].But still the exact mechanism how the Zipf's distribution manifests is debated.
Many of the things that we measure have a typical size or "scale".We ask ourselves why the relaxation rates λ cannot be arranged as simple "normal distri- bution".Like while we plot the height of person in X-axis and the percentage of occurrence of that particular height in Y-axis, we get a "normal distribution" peaked around mean height with a spread both ways, that is a histogram.We find that ratio of maximum height and minimum height of a person is finite (or relatively low value).For example as per Guinness book of records tallest person was having height 272 cm and shortest person was having the height of 57 cm, making this ratio 4.8.This ratio is relatively low value.We see the most adults are about 170 cm tall-there is some variations around this figure notably depending on sex, but we never measure persons having height of 1 cm or 1000 cm.
But not all things we measure are peaked around a typical value.Some may over a very large dynamic range, sometimes many orders of magnitude.For example the ratio of population of largest town to population of smallest town is about 250,000.The histogram if plotted for X-axis with population of cities and Y-axis with percentage of cities having that population; the distribution will not show the "normal-distribution".The histogram of cities & population is highly "right-skewed", meaning that while the bulk of distribution occurs for fairly small sizes-i.e.most cities have small population-there is small number of cities

Zipfian Power Law Distribution as a Result of Connected Exponential Processes-A Postulate
Having discussed the formation of a histogram as power law type, when there is very large dynamic spreads amongst the relaxation rates of a complex relaxing process we move to a probable postulate of explaining this process via exponentially distributed processes.A much more common distribution than power law is the exponential distribution.In this complex relaxation mechanism i.e.

( )
that we are discussing we consider infinite number of bodies relaxing simultaneously, in different time scales ( T ).We consider that a complex relaxation mechanism and a quantity T say survival time of a relaxing body, has exponential distribution of probability ( ) T T e a p −


. This means that a probability for a body having very large survival time (age) is very low; and vice-versa.Then ( )

T dT p
indicates the fraction of survival numbers of bodies between survival time T and T dT + .Now suppose that the real quantity that we are interested is not T but other quantity λ , say the relaxation rate of discharge which is exponentially related to T ; thus  .That implies the surviving bodies with very large time of survival (age) have a very low rate of relaxation.This also states that d dT λ = − .Then if probability distribution of λ is ( ) (effect of conservation of probability [25]).The negative sign indicates opposite movement, as T is increased from T to T dT + , then λ is decreased from λ to  The above discussion in steps (6) gives a power law distribution for relaxation rates λ's where there is combination of exponential processes.Thus we expect that in our complex relaxation process governed by Curie-von Scweidler Law ( ) n i t t − ∝ which is having infinite number of simultaneously discharging bodies will have a power law distribution for relaxation rates as a histogram ( ) m H λ λ λ −  .This we will derive subsequently.We proceed with this explanation and hypothesis.This could be one explanation in physical sense, in line with exponential distribution in the Boltzmann distribution of energies in statistical mechanics.

Complex Relaxation of Non-Debye Type Composing with Several Exponentially Relaxing Functions of Debye Type
We call the Curie-von Shweidler relaxation law ( ) .We mention that Curie-von Shweidler relaxation law is not exponential relaxation process of Debye type.
In this section we formulate the method to extract the histogram of the relaxation rates call it ( ) H λ λ , for a complex non-Debye relaxation process ( ) i t , which we assume to be composed of several Debye type exponential relaxation functions e t λ − , with λ varying from zero to infinity.The complex decay may be expressed as following with several rate constants 1 2 3 , , , , , , a a a  , where k λ is having units in sec −1 i.e. "per second", and is equal to inverse of time constant i.e.

( )
We write following composite relaxation expression as sum of several "discrete" relaxations of Debye type i.e.

( ) ( )
The coefficients k a 's in ( 7) can be positive or negative that we will elucidate in later section.In the continuum limit, we may write the above discrete expression as following integral equation Where the function i.e.

( )
H λ λ is the distribution-function of the rate of the relaxation ( λ ) of the process, or we may call it as histogram of relaxation rates.

The function ( )
H λ λ can be positive or negative that we will elucidate in later section.
While for the case with discrete set of relaxation rates i.e.
( ) From above formulation (9) we infer that if we have only one single Debye relaxation i.e. having only one rate constant say 0 λ i.e. ( ) . This is verified in the following expression In above derivation (10) we used the property of delta function [29] [30] [32] i.e.

Extraction of Rate Distribution Function by Formulating Laplace Integral
In this section we formulate Laplace integral of the complex decay given by Curie-von Shweidler relaxation law ( ) This Expression (11) is standard integral transform of a function ( ) f t from a time domain (t) to a complex frequency domain i.e.

{ }
where real part is significant in the transient response and the imaginary part of the frequency corresponds to "steady-state" response; in classical "Control Science" [28].Here ( ) F s , and we write We have in earlier Section 8 derived ( ) ( ) . Compare this with defined Laplace integral expression as follows Both expressions in (12) In the above Expression (13) x is real number larger than 0 x , where 0 x being such that ( ) i t has some form of singularity on the real line { } 0 Re t x = but is analytic in the complex plane to the right of that line, i.e. for [30].Thus in this formulation we treat time variable as complex quantity say i t x y = + in the Expression (13) of inverse Laplace Transform i.e.
( ) ( ) . Though we cannot explain presently physical meaning of concept an "imaginary time" in the expression of complex time quantity i t x y = + , yet mathematically there is no restriction in assuming time to be complex number.We thus proceed in mathematical sense to invert a function in complex time variable, by techniques of Laplace inversion.H λ λ describing the rate distribution function; mostly got from standard Laplace transform tables [28].The integral representations of ( )

Extracting the Rate Distribution Function by Performing Inverse Laplace Transform on Relaxation Function of Time Variable
H λ λ , shown in Table 1 i.e. for entries 12 to 16 is got via Berberan-Santos method [33] [34].The entry 12 is for stretched exponential decay function and entry 13 is Becquerel's compressed hyperbolic radioactive decay function; the entry 15 and 16 is for Mittag-Leffler function and the entry 14 is general power law relaxation.These integral representations of ( ) H λ λ are difficult to solve but are easy to plot via use of numerical integration techniques.
We have observed in the previous section that a Debye relaxation of ( ) ( ) ( ) If the relaxation function is of say ( ) ( ) ( ) the coefficients k a 's can have negative values as well for some type of relaxation function.

( )
H λ λ can take positive as well as negative values (8).
One interesting observation is for a relaxation function ( ) H λ λ = -a "uniform distribution", for 0 λ ≥ .All these are listed in Table 1.
The inverse Laplace transformation is usually carried out by contour integration.But the very modern technique of Berberan-Santos [33] [34] method is the analytical Laplace inversion without the usual contour integration.We describe this now briefly.
Our aim is to evaluate Laplace inverse which is given as Laplace inversion (13) integral expression i.e.
( ) ( ) Here we describe Berberan-Santos method formulas for evaluation of the Laplace inversion without going for contour integration.First is change of variable i.e. from "real time variable" to "complex time variable" as i t x y = + ; with i 1 = − .Here the real part i.e. x is constant as a vertical line calls it 0 x x = a constant.The formulas are following [33] [34] Consider a very simple case of decay function ( ) ( ) Here we say that e aλ has integral representation as

Derivation of Rate Distribution Function for Curie-Von Schweidler Law
For the Curie-von Schweidler relaxation of type function i.e. ( ) . With the known Laplace pair i.e.
, we can write the following steps Therefore above discussion suggests that for a power law type relaxation, i.e.
Curie-von Schweidler law i.e. ( ) < < , the relaxation rates λ's are having a power law distribution of type i.e.From the above discussion (18) and using our Laplace integral (8) i.e.
( ) ( ) ( ) we write for Curie-von Schweidler relaxation function the following ( ) ( ) The above Expression ( 19) is integral representation of the Using Berberan-Santos method [33] [34] we get the Laplace inversion of ( ) n i t t − = .In reality of decay functions, we can take 0 0 x = ; in complex time va- riable i.e.
as the decay function is not expected to have singularity at time From Laplace transform tables [28] (and Table 1) we have ( ) ( ) i.e. from inverse Laplace transformed of ( ) n i t t − = , therefore we write following representation ( ) From ( 21) we write the following ( ) Now by changing variable λ to t , 1 n − to n − , and rearrange above expression (22) to get Considering now u as λ we write another integral representation of n t − as follows ( ) This means that if we chose basic relaxation function as

Zipf's Distribution for Relaxation Time Constants for Curie-Von Schweidler Law-A Contradiction
Now converting to

( )
H λ λ i.e. the rate distribution function got via Laplace inversion of ( ) i t is not possible.This we demonstrate in this sec- tion.
. This is not Laplace integral.Now we do the following steps, for ( ) n i t t − = and obtained ( ) ( ) We write the two representations of n t − as following integrals ( ) Thus we have


, as we have ( )


. Now we verify the above obtained result in the subsequent discussion.
By the logic that we had constructed ( which is Laplace integral (8); we will similarly get the integral ( which is not a direct Laplace Transform formula.Following steps will convert this expression into the Laplace Transform formula, and from there we will extract ( ) Proceeding further we obtain following result ( ) Now we take different approach to verify the above (26) (27) obtained expression for ( ) H τ τ .Let us have set of relaxation functions with various time con- stants τ ranging from 0 to infinity that is { } e ,e ,e ,  , comprising of infinite number of functions, in continuum in τ .The relaxation function varies from very-very quick decay (when 0 τ ≈ ) to very-very slow decay curve (when τ ≈ ∞ ).We construct a weighted decay function as e m t τ τ − − .This shows that we The integral of (28) i.e.
By using the definition of the Gamma function [29] [32] in integral form i.e.
( ) , we write the above integral as following Earlier in (19) we have obtained ( ) ( )


is also a right-skewed distribution, where the lower time constants (faster decay) appear more than larger time constant (slower decay).This is contradiction to what we inferred for ( )  .Hence we will deal with the relaxation rate distribution function that we extracted as ( ) H λ λ from ( ) i t via our devised method of Laplace inversion.

Experimental Validation of Range of Relaxation Exponent in Curie-Von Schweidler Law
The Curie-von Schweidler empirical law of power law relaxation, i.e. ( ) n i t t − ∝ states that 0 1 n < < .This is validated via experiments on dielectric relaxations.


. In this Laponite study [35] though the exponent n was obtained on "self-discharge" curves with various charging time history-showing memory effect, the expression obtained for self-discharge decay of voltage assumes fractional capacity-that in turn assumes Curie-von Schweidler law as current relaxation function.

Time Variant Relaxation Rate for Non-Debye Relaxation & Curie-Von Schweidler Law
Any decay function ( ) i t is written as a general formulation in following way where ( ) λ ξ is the time ( ξ ) dependent rate coefficient.When the relaxation is pure exponential, one has ( ) λ ξ as constant say 0 λ described as ( ) Thus we get a Debye relaxation for a system having constant rate of relaxation.To extract ( ) ( ) ( ) ( ) < .The voltage balance equation assuming R be the total resistance of the circuit (including internal resistance of Capacitor) at where ( ) i t is the charging current flowing into the capacitor.The above integral Equation ( 38) is differentiated and is put as following, for We saw in earlier sections the relaxation rates ( λ ) distribution, for a Cu- rie-von Schweidler relaxation law, i.e. ( ) for relaxations in dielectrics.This is histogram of rates.It says that the relaxation of current is with several relaxation rates, which are distributed as discussed in Zipf's law fashion with right-skewed-histogram.Thus if we represent the equivalent relaxation rate say 1 ;0 1 q eq q λ λ < <  with λ as scale of relaxation where the scale λ varies from zero to infinity; we will not be incorrect in as- suming this.That is as we slide from a low scale λ to high scale λ the equiv- alent relaxation rate eq λ will be different at different scales of relaxation.If the index parameter i.e. 1 q = then we have single rate constant system given by eq λ λ = always at all scales of relaxation i.e.
( ) i.e. with one relaxation rate at any scale of relaxation (λ), i.e. ( ) ( ) ( ) ( ) The initial condition is given as ( ) 0 i t = for 0 t < .The above equation is having a free "scale" parameter λ varying from zero to infinity.The solution of the above is ( ) e eq t i t  q q q q q q q q x x t t q x q qx x x x t t t t Then by using definition of Gamma function i.e. ( ) q t q x q q x q q x x q x x q q q q q g t x t q x x t t q x x q x x x x t t t t t t q q q q x q x x x t t t t By changing q to n we get integral representation of n t − as following ( ) 0 , h t λ .We find that for a system where the equivalent relaxation rate is

τ
is the relaxation time constant while 0 λ denotes the relaxation rate of the process with 1 0 0 distribution thus obtained is a Zipf's power law.
number of surviving bodies having survival time between T and T dT + .Thus we write following steps: as complex process, of non-Debye type.Where a Debye type relaxation is a decay function given by function of exponential type as  the rate distribution function would be having discrete delta functions ( ( ), 1, 2, 3, k t k δ λ − =  ) at points 1 2 3 , , , λ λ λ ; which we write like following expression S. Das . This we verify with known Laplace relation i.e. distribution function is delta function at origin i.e.

1 a
= − , we have ( ) ( ) enables us to choose 0 0 x = we get following integral representation for e λ −

For<
as histogram, we infer that for Curie-von Schweidler relaxation function i.e. ( ) large number of relaxations with small λ i.e. large number of slower decay takes place, compared to fewer faster decay rates-and the histogram λ from zero to infinity.
Zipf's power law distribution.However direct taking of reciprocal of obtained inversion of

>
the decay function e t τ − .We are assuming Zipf's type distribution of τ , in form of time constant i.e. fastest decay occurs more frequent than slow decay i.e. large time constant.The time constant parameter τ let vary from 0 to infinity and construct the following integral I , i.e.
weighted average of infinite relaxation functions.We do the substitution i.e.
above we get integral representation of the power law n t − and we represent this by time constant distribution function

A.
100V step input applied to a completely discharged capacitor of 0.47 μF having metalized paper dielectric, and the current decay is recorded with time.The graphs of log-log plot i.e. t show a straight line of average slope −0.86[5]-[10].This experiment indicates a Curie-von Schweidler law, with The exponent n is in the range of 0.85 1 n < < in several di-electric relaxation experiments[5]-[10].The experiments with super-capacitors[13] [14], show range as 0.5 1 n < < .A very low value of exponent n is found in relaxation of Laponite studies averagely 0.09 n =[35].Thus in case of Laponite studies we have relaxation rate distribu- e. Debye relaxation function.We thus modify the capacitor discharge current equation, with " at a particular scale λ , i.e. we call it ( (46) actually is valid for all scale λ varying from zero to infinity.Thus on integrating this "impulse response function" ( ) variable ( λ ) from 0 to ∞ , we get the function of time and that is called "impulse response" or the Green's function ( ) g t as depicted in following deri- vation S.Das as "impulse response function" i.e.
that the Curie-von Schweidler relaxation current for dielectric excited by a step voltage that follows the relation ( ) λ λ λ −  a power law or Zipfian distribution, with scale dependent relaxation rate described as 1 n eq λ λ = .

Table 1
lists several types' relaxation functions ( ) i t and its inverse Laplace

Table 1 .
Several relaxation functions and corresponding rate distribution functions.
, as λ is varied from zero to infinity.

Scale Dependence Relaxation Rates Give Capacitors Charging Current as per Curie-Von Schweidler Law Let
a uncharged capacitor C be connected to a voltage source ( ) body relaxation where the rate is varying with time; whereas the multi-body relaxation gives simultaneous relaxations with rates distributed as Zipf's power law.S.Das13.