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

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 19 th 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.

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

“…i(t) = C α v (α) (t), and fractional impedance Z(s) = (s α C α ) −1 . The power-law is observed in various systems as in [40,4,50,45,12]. We do not observe the capacitor memorizing its charging history for ideal text book capacitor, i.e.…”

“…We call this as fractional capacitor-appears in real life observations. The entire experiments and analysis on dielectric studies as done in [29,44,28,15,11,31,32,16,18,17,48,10,8,46,30,6,25,47,12,14], confirm that this constituent equation, i(t) = C α v (α) (t) and fractional order impedance, Z(s) = (s α C α ) −1 is valid. The basis of dielectric relaxation current is Curie-von Schweidler law i.e.…”

“…The empirically and experimentally derived law called 'universal dielectric relaxation law' also called as Curie-von Schweidler law, is observed since late 19 th century; [29,44,30,28,12,14]. This is classical power law for current decay i.e.…”

“…singular function, is also called Universal Dielectric relaxation law, observed since late 19 th century [29,44,30,28,12,14]. This is termed as power-law [40,4,50,45,47,12].…”

Formation of desiccation crack patterns in electric fields: a review

“…It is known that a real or leaky capacitor as distinct from an ideal capacitor after charging for a time t charge , discharges in a characteristic manner as follows [31,32]. The voltage across it remains ∼ V for a time ∼ t charge and after that falls off as a power law.…”