On Lévy (or stable) distributions and the Williams-Watts model of dielectric relaxation

Abstract

f,~exp(_izu_[ul~)du with 0<a<2 Q~(z) = ~ ooWhen a = 2 it becomes the Gauss distribution function and when a = 1, the Cauchy distribution. When a r 2 the distribution has a long inverse power tail F(1 + a)sinl~r~ Q~(z)~,~lzl 1 §176In the regime of small a, if a[logz I << i, the distribution is mimicked by a log normal distribution. We have derived rapidly converging algorithms for the numerical calculation of Q~(z) for various a in the range 0 < a < 1. KEY WORDS: Glass; polymer; dynamical response; momentless distributions. LEVY DISTRIBUTION FUNCTIONSThe definite integral O0appears prominently in two important contemporary fields of research: (i) theory of probability, where it is called a L~vy or stable distribution function,(1-4) and (ii) theory of relaxation processes (dielectric, mechanical, and NMR), where when multiplying by ez it is referred to as the cosine transform of the Williams-Watts (5-8) function exp(-u~). Of course the special case a = 2, the Gauss or normal distribution, has been prominent in all of science for years. One finds Q2(z)= -~ ~ eiZ"e-"2du-(a~)-l/2exp(-z2/4)J--O0Integration over z from -m to oc implies that Q~(z) is normalized to unity: Qo(z)dz= ff lThe arbitrary a case was first considered by Cauchy (9) (1853) in an attempt to generalize the least-squares theory of errors. He was not aware of the fact that Q~(z) is not a proper probability distribution when a > 2 since it may, for some values of z, become negative. The use of the expression exp -[t/T] ~ as a relaxation function for dynamical processes in materials was first proposed by Kohlrausch (1866). Its reciprocal plays an important role in creep phenomenon (l~ and its Fourier transform appears in investigations of response of materials to periodic driving forces (electrical and mechanical).Closed form expressions exist for (1) for few values of a other than a = 2. When a = 1 one has the Cauchy distribution Ql(z) = (1/~r)(1 + z 2) (4) Q2/3(z ) = ( 3 )l/2z-lexp( -~7z-2)W1/2,,/6 ( 2@z2 )The value of Q~ (0)

A Statistical Approach to Relaxation in Glassy Materials

“…been also shown recently by a different method (11 ) It is interesting to explain why here in contrast to Sections 3 and 4 synnetric ((=O) LOES distributions arise.…”

Stable law densities and linear relaxation phenomena

“…In a recent paper Montroll and Bendler have presented a number of approximations to the function Qa(z) for values of a useful for polymer applications [19]. Because Q.…”

“…In particular, the data at 0 values less than 0.5 may not fit the Williams-Watts model as well as data for 0>0.5. When a satisfactory value of a is determined, using eq (18), we may find r from the relation (19) wherefm is the frequency in Hz at which the peak maximum occurs. Finally, since an estimate of the peak height will generally be available, the estimate of the parameter A can be found from (20) where relevant values of M(a) are given in table 5.…”

“…Equations (18)(19)(20) were then used to estimate a, z, and A for the last four data points. The resulting estimates are the following: Ni 00000 1--000…”

Dielectric relaxation of alkyl chains in graphite oxide and n-alkylammonium halides

“…8 and 9 show the best fitting Ngai curves to our experimental points based on the convergent expressions of Levy function 24 with various m, with the thermal background simulated by exponential extrapolation. The cases of GO/C 16 NCl of 50:24, 50:72, 50:800 mg, 10 referred to here as SIGO-1, -2 and -3, have likewise been treated in the same manner; Fig.…”