Histogram Distribution Function for Relaxation Rates as Zipf’s Power Law for Universal Dielectric Relaxation - A Case Study for a Complex Disordered System

This chapter gives a new treatment of analytical approach to get an insight of a non-Debye Complex-Disordered relaxation especially to the Universal Dielectric Relaxation (UDR) law. The classical power law for dielectric relaxation, is current as inverse of power of time i.e. i(t) <n <1, for a step-voltage excitation to dielectric This is UDR and popularly known as Curie-von Schweidler law This law is empirically derived and is observed in several relaxation experiments on various dielectrics studies since late 19th Century. This UDR law is a singular power law. There could be non-singular non-Debye relaxation laws too. We will briefly mention the results of a non-singular relaxation law especially by Mittag Leffler function In this chapter, we give simple mathematical treatment to derive the distribution of relaxation rates ( -in unit of per second) 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 in this chapter give Zipfian power law distribution for relaxing time constants ( ) and discuss its physical contradiction. In this chapter, we develop possible explanation that as to why Zipfian distribution of relaxation rates appears for UDR i.e. Curie-von Schweidler Law, and relate this law to time variant and scale dependent rate of relaxation. In this chapter, we derive appearance of fractional derivative while using Zipfian power law distribution for Curie-von Schweidler relaxation phenomena. We also explain the Curie-von Schweidler relaxation i(t) <n <1 as simultaneous multi-body relaxations which have a distribution/histogram for relaxation rates as right-skewed one. That is the histogram with large number of relaxations with lower value of rate (slow rates) followed with long tail of small number of relaxations with faster relaxation rates, relaxing simultaneously. The chapter gives a possible foundation for further studies in obtaining the rate relaxation distribution functions for other non-Debye type relaxation functions, and a new type of explanation regarding reasons of Zipfian distributions.