Abstract
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 19th 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.
Highlights
The Curie-von Schweidler law relates to relaxation current in dielectric when a step DC voltage is applied and is given by i (t ) t−n, where t > 0 and the power i.e. n is called relaxation constant or decay constant, where0 < n < 1 [1] [2] [3] [4]
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
The empirical law that is Curie-von Schweidler law, which is a type of non-Debye relaxation, states when dielectric is stressed with a constant voltage, gives relaxation current as i (t ) t−n, 0 < n < 1
Summary
The Curie-von Schweidler law relates to relaxation current in dielectric when a step DC voltage is applied and is given by i (t ) t−n , where t > 0 and the power (exponent) i.e. n is called relaxation constant or decay constant, where. One reason that this non-Debye relaxation (explained in subsequent sections) of Curie-von Schweidler Law ( i (t ) t−n ) in dielectric is having infinite spread of relaxation rates of λ’s- forming a Zipfian power law. 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. We expect that in our complex relaxation process governed by Curie-von Scweidler Law i (t ) ∝ t−n which is having infinite number of simultaneously discharging bodies will have a power law distribution for relaxation rates as a histogram.
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