Anomalous dielectric relaxation in a double-well potential

Journal of Molecular Liquids 114 (2004) 43–49

Anomalous dielectric relaxation in a double-well potential William T. Coffeya, Yuri P. Kalmykovb,*, Sergey V. Titovc a Department of Electronic and Electrical Engineering, Trinity College, Dublin 2, Ireland Centre d’Etudes Fondamentales, Universite´ de Perpignan, 52, Avenue de Villeneuve, 66860 Perpignan Cedex, France c Institute of Radio Engineering and Electronics of the Russian Academy of Sciences, Vvedenskii Square 1, Fryazino, Moscow Region, 141190, Russian Federation b

Abstract The fractional Klein–Kramers equation describing the fractal time dynamics of an assembly of fixed axis dipoles rotating in a double-well cosine potential representing the internal field due to neighboring molecules is solved using matrix continued fractions. The solution includes both inertial and strong internal field effects which in combination produce a strong resonance peak (Poley absorption) at high frequencies due to librations of the dipoles in the potential, an anomalous relaxation band at low-frequencies mainly arising from overbarrier relaxation and a weaker relaxation band at mid frequencies due to the fast intrawell modes. The high frequency behavior is controlled by the inertia of a dipole, so that the Gordon sum rule for dipolar absorption is satisfied ensuring a return to optical transparency at very high frequencies. Application of the model to the broad-band (0-THz) dielectric loss spectrum of a dilute solution of the probe dipolar molecule CH2Cl2 in glassy decalin is demonstrated. 䊚 2004 Elsevier B.V. All rights reserved. Keywords: Fractional dynamics; Anomalous relaxation; Rotational diffusion; Klein–Kramers equation

1. Introduction The theory of the rotational Brownian motion in the presence of a potential is of fundamental importance in a number of problems involving relaxation and resonance phenomena in stochastic systems w1x. A rudimentary example is the theory of dielectric relaxation of non-interacting polar molecules due to Debye w2x which is based on the Smoluchowski equation for the rotational diffusion of the molecules. There, because interactions between dipoles are ignored, the only potential arises from the spatially uniform weak external ac field. The complex dielectric susceptibility from this theory agrees substantially with experimental data in the microwave region. The Debye theory has very recently been generalized to anomalous dielectric relaxation which (excluding inertial effects) is characterized by a nonexponential dielectric decay function w3x. In general for non-interacting dipoles the usual exponential decay function of the Debye theory is replaced by a Mittag– Leffler function which exhibits stretched exponential behavior at short times and a long time tail w4,5x. The *Corresponding author. E-mail addresses: [email protected] (Y.P. Kalmykov), [email protected] (W.T. Coffey), [email protected] (S.V. Titov).

complex dielectric susceptibility yielded by that function (Cole–Cole behavior) w6x is substantially in accord with experimental data on dielectric relaxation of amorphous polymers, glass forming liquids, etc. w5,6x. Moreover Nigmatullin and Ryabov w6x have shown how other relaxation behaviors such as the Davidson–Cole function may also be modelled using fractional calculus w6x. We emphasize that neither the Debye theory nor its various w3,6x extensions to fractional Brownian motion include the inertia or internal field effects. Inertial effects in the theory of the normal Brownian motion were studied by Rocard w7x, Gross w8x and Sack w9x. Gross and Sack w8,9x studied these effects by solving the Fokker–Planck equation (which for a separable and additive Hamiltonian is known as the Klein–Kramers equation) for the distribution function of dipolar rotators in phase space. They obtained the complex susceptibility in exact continued fraction form so predicting a return to optical transparency at high frequencies unlike the infinite integral absorption predicted by the Debye theory. Their calculations have very recently been extended to the fractional Brownian motion by Coffey et al. w10,11x ensuring a return to optical transparency at high frequencies in fractional dynamics just as in the conventional Brownian dynamics. The approach developed in

0167-7322/04/$ - see front matter 䊚 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.molliq.2004.02.005

44

W.T. Coffey et al. / Journal of Molecular Liquids 114 (2004) 43–49

w10,11x was based on the fractional Klein–Kramers equation proposed by Barkai and Silbey w12x. The solution for the complex susceptibility given in Refs. w10,11x emerges in continued fraction form in a manner entirely analogous to the conventional Brownian motion result because of a useful generalization of the integration theorem of Laplace transformation to fractional calculus w4x. As far as the inclusion of an internal field potential combined with inertial effects is concerned the problem is much more difficult than that of including inertia alone both in the normal and the fractional Brownian dynamics. Thus, referring to normal Brownian dynamics all the initial attempts to solve the problem were made in the non-inertial limit w13–16x. In particular in Ref. w16x it was shown by representing the configuration space distribution function in Fourier series how the complex susceptibility could be obtained exactly in terms of continued fractions from differential recurrence relations for the Fourier coefficients. The particular problem treated was the Brownian motion of a rotator about a fixed axis in the presence of a cos2u potential where u is the angular coordinate of the rotator. Moreover, it was demonstrated, using the final value theorem w17x for Laplace transforms, how the correlation time of the dielectric decay function could be obtained in closed form w16x. The relevance to the present problem of this potential is that it is possible to model w16x relaxation effects involving escape of dipoles over a potential barrier. We remark in passing that in the context of the present work, overbarrier relaxation due to normal difw18x. ¨ fusion has been extensively discussed by Frohlich w x ¨ Frohlich used transition state theory 19,20 and a rate equation approach originally suggested by Debye w2x so that a discrete set of orientations for the dipoles of the assembly is implicitly assumed. A continuous distribution of orientations may be treated by the use of methods based on the Klein–Kramers equation or its fractional equivalent. These diffusion equations also allow one to ¨ include explicitly in Frohlich’s model in both discrete (normal) and fractal time dynamics (i) the influence of the dissipative coupling to the heat bath on the Arrhenius (overbarrier) process and (ii) the influence of molecular librations and the fast (high-frequency) intrawell relaxation modes on the relaxation process. The Fokker–Planck equation approach described in Ref. w16x was subsequently extended to many problems in dielectric relaxation of liquid crystals and magnetic relaxation of single domain ferromagnetic particles involving rotation in space. These are comprehensively summarized in Ref. w1x. Although the non-inertial rotational motion in space in the presence of a mean field potential has been fully described, almost all the discussion concerning inertial effects and an internal field potential has been in the context of the motion of a rotator about a fixed axis in a periodic potential repre-

senting the internal field w21–23x. This problem, on expanding the phase space distribution function in the Klein–Kramers equation in a Fourier series, leads to a differential-recurrence relation in two characteristic numbers, namely the order n of the Hermite polynomials Hn(hu) in the angular velocity u and q of the circular functions eyiqu, where hsyIyŽ2kT. , I is the moment of inertia of a rotator and kT is the thermal energy. The differential-recurrence relation in two variables is a particular example of that given by Brinkman w24x in his attempt to justify the approximate Smoluchowski equation for the distribution function in configuration space from the Klein–Kramers equation for the translational Brownian motion in phase space (for a summary of the applications to dielectric relaxation see Refs. w22,23x). The first attempt to calculate the complex polarizability including inertial effects and a potential arising from the internal field was made by Reid w25x who gave numerical results in a limited number of specialized cases. Only very recently, however, has it became possible to treat the calculation of the Fourier coefficients in a systematic way for the conventional Brownian motion. The difficulty arises because when inertial effects are included the two recurring numbers n and q always give rise to a matrix recurrence relation. Matrix continued fractions are, therefore an ideal way of solving such recurrence relations. This has been accomplished in Ref. w26x where it has been shown that the linear and non-linear response of an assembly of fixed axis rotators in the presence of a strong spatially uniform external field (that is a cosu potential) may be systematically solved using the matrix continued fraction method. It is the purpose of this paper to generalize the results w10,11x by including the effect of an internal field potential (and so dielectric relaxation due to barrier crossing by dipoles) in the fractional Brownian dynamics. As in Ref. w10,11x, our approach is based on the fractional Klein–Kramers equation for the translational Brownian motion in a potential proposed by Barkai and Silbey w12x. The solution of the rotational analogue of this fractional Klein–Kramers equation will be accomplished using the matrix continued fraction method w26x and the generalized integration theorem (i.e. the properties of the inverse linear differential operator) of Laplace transformation w17,27x. These methods will also allow us to consider the mechanism underlying the high frequency (far-infrared) absorption peak in fractional dynamics. In order to simplify our presentation we shall confine ourselves to the linear response to a small a.c. applied field. 2. Recurrence relations for statistical averages for rotation about a fixed axis We illustrate by considering one of the simplest microscopic models of dielectric relaxation, namely: an

W.T. Coffey et al. / Journal of Molecular Liquids 114 (2004) 43–49

assembly of rigid dipoles each of moment m each rotating about a fixed axis through its center w8,9,23x. A dipole has moment of inertia I and is specified by the angular coordinate u so that it constitutes a system of 1 (rotational) d.f. The internal field due to molecular interactions is represented by a two-fold cosine potential. (1)

VŽu.syV0cos2u.

By supposing that the local configuration potential is uniformly distributed in a plane, we may define the averaged susceptibility x(v) according to linear response theory w28x as xŽv.s

m2N0 w x1yiv 2kT y

|

`

z

0

~

NcosDuŽt.Moeyivtdt|,

45

terms of the convolution (the Riemann–Liouville fractional integral definition) w4x 1 ya ˙ 0Dt WŽu,u,t.s GŽa.

|

t

˙ .dt9 WŽu,u,t9

0

Žtyt9.1ya

(4)

Thus, the fractional derivative is a type of memory function w4x. Moreover, a slowly-decaying power-law kernel in the Riemann–Liouville operator Eq. (4) is typical of memory effects in complex systems. We may seek a solution of the fractional Klein– Kramers equation, Eq. (3), by using the method of separation of variables in the form of the Fourier series

(2) ˙ . WŽu,u,t

where Du(t)su(t)yu(0) and N Mo means the equilibrium statistical average. The starting point in our calculation of x(v) from Eq. (2) is the fractional Klein–Kramers equation for the ˙ . in the phase space probability density function WŽu,u,t Žu,u˙ . w10,11x, which is identical to that of the onedimensional translational Brownian motion of a particle w12x, however, rotational quantities (angle u, moment of inertia I, etc.) replace translational ones (position x, mass m, etc.). We suppose that a uniform field E (having been applied to the assembly of dipoles at a time tsy ` so that equilibrium conditions prevail by the time ts0) is switched off at ts0; in addition, we suppose that the field is weak (i.e. mE
` 1 2˙2 s hpy3y2eyh u 8 2 ns0

1 fn,q(t) HnŽhu˙ .eiqu, n 2 n! qsy` (5)

where Hn(x) are the Hermite polynomials. Noting the recurrence relations and orthogonality properties of the Hermite polynomials w29x, we can obtain the recurrence relation for the functions cn,qŽt.sNHnŽhu˙ .eyiwquŽt.yuŽ0.xM0,

(6)

which is given by (see for detail Ref. w30x) hc˙ n,qŽt.q

≠W ˙ ≠W ≠W qu y2V0 sin2u ˙ ≠t ≠u I≠u

`

8

iq w cnq1,qŽt.q2ncny1,qŽt.z~ 2y x

|

qinjVwycny1,qq2Žt.ycny1,qy2Žt.z~ x

B

WE

|

2

≠ ˙ kT ≠ F. s 0D1ya t1yabC ˙ (uW)q t D ≠u I ≠u˙ 2 G

(3)

Here bsz yI, z is the damping coefficient of a dipole, t is the intertrapping time scale which we identify with the Debye relaxation time z ykT (at ambient temperatures, t is of the order 10y11 s for molecular liquids and solutions), a is the anomalous exponent or order of the fractional derivative characterizing the fractal time process. Thus the fractional dynamics emerges from the competition of Brownian motion events of average duration t interrupted by trapping events whose duration is broadly distributed w4x. Eq. (3) with anomalous exponent a such that 1F2yaF2 describes anomalous enhanced diffusion according to Barkai and Silbey w12x. The value as1 corresponds to normal diffusion. Here ≠ ya ' the operator 0D1ya in Eq. (3) is defined in t 0D t ≠t

sy0D1ya t1yanb9cn,qŽt. t

(7)

where b9sbh and jVsV0 y(kT). On using the integration theorem w17x of Laplace transformation generalized to fractional calculus w4x, viz. Lµ0D1ya fŽt.∂ t ˜ Žs.y0Dya Ts1yaf t fŽt.± ts0 Ž0-a-1. sU T 1ya ˜ Vs fŽs. Ž1Fa-2. S

, (8)

|

where f˜ Žs.sL µfŽt.∂s (7)

`

eystfŽt.dt, we have from Eq. 0

W.T. Coffey et al. / Journal of Molecular Liquids 114 (2004) 43–49

46

where the matrix continued fraction Dn(v) is defined by y zy1 DnŽv.swy2ihvIyQnyQnqDnq1Žv.Qnq1 . ~ x

(13)

|

and the matrices Qn(v), Qnq and Qny are given by QnŽv.syg92yaŽihv.1yaŽny1.I,

(14)

Qq n syi

C

B»

Fig. 1. Dielectric loss spectrum wimaginary part of the complex susceptibility x(hv)x vs. normalized frequency hv for g9s10 and jVs 3 and various values of the fractional parameter a. w y

2hsqng92yaŽhs.1yaz~c˜ n,qŽs.qiqwyc˜ nq1,qŽs.

x

x

|

z ~

w Vy ny1,qq2

q2nc˜ ny1,qŽs. qin2j c˜ |

x

F

n

n

n

n

n

UE

∆

y2

0

0

0

0

∆

∆

0

y1

0

0

0

∆

∆

0

0

0

0

0

∆ ,

∆

0

0

0

1

0

∆

∆

0

0

0

0

2

∆

DU

n

n

n

n

n

»G

Žs.

(15) Qy n sy2iŽny1.

yc˜ ny1,qy2Žs.z~ |

(9)

s2hc0,qŽ0.dn,0, ŽnG0..

Here g9styhszy2yŽIkT. is the inertial effects parameter (so that large g9 characterizes small inertial effects and vice versa, g9sy2ygS, where gSsIkTyz2 is the inertial parameter used by Sack w9x) and cn,q(0)s0 for nG1 because (10)

NHneyiquM0s0

C

B»

n

n

∆ yjV 0

n

n

y2 0

n

n

F

n UE

n

n

jV

∆ 0 0 0 ∆ jV ∆ 0 0 ∆

∆ ∆

yjV 0

∆ 0

∆

yjV 0

∆ 0

0

∆

yjV 0

∆ 0

0

0

∆

yjV 0 2 0 jV ∆

n

n

n

n

n

DU

y1 0

0 jV ∆ 0 ∆ ,

0

1 0 jV ∆∆ n

n

n

n »G

for the equilibrium Maxwell–Boltzmann distribution w1x. The complex susceptibility from Eq. (2) is then given by m2N0 w xŽv.s 1yivc˜ 0,1Živ.z~, 2kT y x

(11)

|

Eq. (9) can be solved in terms of matrix continued fractions w26,30x. This solution is given by w30x

C F CF E

Bn

c˜ 0,y2Živ.

c0,y2Ž0.

c˜ 0,y1Živ.

c0,y1Ž0.

c˜ 0,1Živ.

s2hD1Žv.

c˜ 0,2Živ.

Dn

E

c0,1Ž0. c0,2Ž0.

G

Dn

G

and I is the unit matrix of infinite dimension; the y exceptions are the matrices Qq 1 and Q2 , which are given by Qq 1 syi

3. Matrix continued fraction solution of Eq. (9)

Bn

(16)

(12)

C

B»

F

n

n

n

n

n

UE

∆

y2

0

0

0

0

∆

∆

0

y1

0

0

0

∆

∆

0

0

0

1

0

∆

∆

0

0

0

0

2

∆

DU

n

n

n

n

n

»G

,

(17)

W.T. Coffey et al. / Journal of Molecular Liquids 114 (2004) 43–49

Qy 2 sy2i

C

B»

n

n

∆ yjV 0

n

n

y2 0

n

n

n

F

n UE

0 0 0 0 ∆

y1 jV 0 0 0 ∆

∆ 0

yjV 0

∆ 0

0

yjV 0

∆ 0

0

0

yjV 1 0 jV 0 ∆

∆ 0

0

0

0

0 2 0 jV∆

n

n

n

n

n

DU

47

0 jV 0 0 ∆ ,

n

n

n »G (18)

The initial values c0,q(0) are given by c0,qŽ0.sNeyiŽqy1.uM0sdqy1,mN

ImŽjV. I0ŽjV.

,

(19)

where the In(z) are the modified Bessel functions of the first kind of order n w29x. 4. Results and discussion ˜ 1Žv. from Eqs. (12) and (13), Having calculated C we may evaluate the complex dielectric susceptibility x(v) from Eq. (11) for all values of the model parameters h, g9, jV and a w30x. The imaginary part, x0(v), of the complex susceptibility for various values of a (which in the present context pertains to anomalous diffusion in velocity space) are shown in Fig. 1 wthe calculations were carried out for m2N0 y(2kT)s1x. The shape of the dielectric spectra strongly depends on the anomalous exponent a, jV and g9. In general, three bands may appear in the dielectric loss x0(v) spectra, the corresponding dispersion regions are visible in the spectra of x9(v). One anomalous relaxation band dominates the low frequency part of the spectra and is due to the slow overbarrier relaxation of the dipoles in the ¨ double-well cosine potential as identified by Frohlich w18x. The characteristic frequency vR of this lowfrequency band strongly depends on the barrier height jV and the friction parameter g9 as well as on the anomalous exponent a. Regarding the barrier height dependence the frequency vR decreases exponentially as the barrier height jV is raised. This behavior occurs because the probability of escape of a dipole from one well to another over the potential barrier exponentially decreases with jV. The high-frequency band is is due to the fast inertial librations of the dipoles in the potential wells. This band corresponds to the THz (far-infrared) range of frequencies and is usually associated with the Poley absorption w22x. For jV41, the characteristic frequency of libra-

Fig. 2. Broad-band dielectric loss spectrum of 10% vyv solution of probe molecule CH2Cl2 in glassy decalin at 110 K. Symbols are the experimental data w31x. Curve 1 is the best fit for the anomalous diffusion in the double-well cosine potential (as1.5, jVs8, and gs 0.003); curve 2 is the best fit for the normal diffusion (as1, jVs7 and gs0.001) in the double-well cosine potential; filled circles are the non-inertial Eq. (21). Dashed line 3 is the Cole–Cole Eq. (21)

tions vL increases as ;yVoyI. The high frequency ˆ Žv. is entirely determined by (v4vL) behavior of x0 the inertia of system. Moreover, just as in the normal Brownian dynamics, the inertial effects produce a rapid ˆ Žv. at high frequencies. It can be demonfalloff of x0 strated that the fractional model under consideration satisfies the Gordon sum rule for the dipole integral absorption of rotators in a plane, viz.,

|

`

vx0Žv.dvs 0

pNom2 . 4I

(20)

It is significant that the right hand side of Eq. (20) is determined by molecular parameters only and is independent of the model parameters a, jV and z. For as1, the anomalous rotational diffusion solution coincides with that for normal rotational diffusion. Finally, it is apparent that between the low-frequency and very high-frequency bands, at some values of model parameters, a third band exists in the dielectric loss spectra (see Fig. 2). This band is due to the high frequency relaxation modes of the dipoles in the potential wells (without crossing the potential barrier) which will always exist in the spectra even in the non-inertial limit w16x. Such relaxation modes are generally termed the intrawell modes. The characteristic frequency of this band depends on the barrier height jV and the anomalous exponent a. In Fig. 2, a comparison of experimental data for 10% vyv solution of a probe molecule CH2Cl2 in glassy decalin at 110 K w31x with the theoretical dielectric loss spectrum ´0(v);(´o y´` )x0(v)y x9(0) calculated from

W.T. Coffey et al. / Journal of Molecular Liquids 114 (2004) 43–49

48

Eqs. (11) and (12) is shown. The reduced moment of inertia Ir used in the calculation is defined by Iry1s y1 Iy1 b qIc , where Ib and Ic are the principal moments of inertia about molecular axes perpendicular to the principal axis a along which the dipole moment vector is directed. For the CH2Cl2 molecule Irs0.24=10y38 g cm2 w22x. The use of the reduced moment Ir allows one to obtain the correct value for the dipolar integral absorption for two-dimensional models. The phenomenological model parameters jV, g9 and a were adjusted by using the best fit of experimental data. It is known that in order to describe the low-frequency dielectric relaxation in such organic glasses, one must consider anomalous diffusion and relaxation w22x. The highfrequency Poley absorption is also observed in molecular glasses in the far-infrared region (e.g. w22,31x). Fig. 2 ¨ indicates that our generalized Frohlich model explains qualitatively the main features of the whole broad-band (0yTHz) dielectric loss spectrum of the CH2Cl2 ydecalin solution in contrast to the normal diffusion in a periodic potential (curve 2) that cannot explain the dielectric spectra at low frequencies. One can also see in Fig. 2 that the low frequency part of the loss spectrum ˆ Žv., which may be approximated by the modified x0 Debye (Cole–Cole) equation xŽv. x9Ž0.

s

1 2ya . 1qŽivyvR.

(21)

We remark that all the above results are obtained by using the Barkai–Silbey w12x fractional form of the Klein–Kramers equation for the evolution of the probability distribution function in phase space. In that equation, the fractional derivative, or memory term, acts only on the right hand side, that is, on the diffusion or dissipative term. Thus, the form of the Liouville operator, or convective derivative is preserved wcf. the right hand side of Eq. (3)x. Thus, Eq. (3) has the conventional form of a Boltzmann equation for the single particle distribution function. The preservation of the Liouville operator is equivalent to stating that the Newtonian form of the equations of motion underlying the Klein–Kramers equation is preserved. Thus, the high frequency behavior is entirely controlled by the inertia of the system, and does not depend on the anomalous exponent. Consequently, the fundamental sum rule, Eq. (20), for the dipole integral absorption of single axis rotators is satisfied, ensuring a return to transparency at high frequencies as demanded on physical grounds. These conclusions have been verified by solving the Barkai– Silbey equation for the simple problem of the dielectric relaxation of an assembly of non-interacting dipoles following the removal of a constant field w10x. That equation is simply Eq. (3) in the absence of an external potential. We also remark that the Barkai–Sikbey equa-

tion was originally given for subdiffusion in velocity space (a-1) that corresponds to the enhanced diffusion in configuration space. However, the most interesting case is subdiffusion in configuration space, corresponding to the Cole–Cole equation wEq. (21)x in dielectric theory. This suggests extending the Barkai–Silbey equation to enhanced diffusion in velocity space, corresponding to subdiffusion in configuration space, since ss 2ya. The justification for doing this is simply that this generalization yields physically meaningful results for the broad-band spectrum of the complex susceptibility xˆ Žv., as well as yielding the Cole–Cole equation in the limit g™0. It is also apparent that the Barkai–Silbey ´ (rather than a equation must have its origin in a Levy purely fractal) time random walk, as unlike Eq (43), it does not separate into temporal and spatial parts, moreover, the exponential decay of the normal diffusion theory is not replaced by a Mittag–Leffler function, as in a fractal time random walk w32x. Such behavior is indicative of coupling between the jump length probability distribution and the waiting time probability distribution, that is, the jump length and waiting time are not independent random variables. We remark that a general characteristic of the systems we have treated is that they are non-local both in space and time, and so give rise to anomalous diffusion w32x. ¨ The generalized Frohlich model we have outlined incorporates both resonance and relaxation behavior and so may simultaneously explain both the anomalous relaxation (low-frequency) and far infrared absorption spectra of complex dipolar systems. Moreover, a third mid-frequency relaxation band may appear in the dielectric loss spectra at low temperatures due to intrawell relaxation modes. The present calculation also constitutes an example of the solution of the fractional Klein– Kramers equation for anomalous diffusion in a periodic potential and is to our knowledge the first example of such a solution. The approach which is grounded in a theorem w27x of operational calculus generalized to fractional exponents w32x and continued fraction methods clearly indicates how many existing results of the classical theory of the Brownian motion in a potential may be extended to fractional dynamics. Acknowledgments The support of this work by EOARD (contract FA8655-03-01-3027), the CNRS Enterprise Ireland– France award scheme 2003 (programme ‘Ulysses’) and the Russian Foundation for Basic Research (project no. 01-02-16050) is gratefully acknowledged. References w1x W.T. Coffey, Yu.P. Kalmykov, J.T. Waldron, The Langevin Equation, World Scientific, Singapore, 1996, Reprinted 1998.

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w19x P. Hanggi, ¨ P. Talkner, M. Borkovec, Rev. Mod. Phys. 62 (1990) 251. w20x W.T. Coffey, D.A. Garanin, D.J. McCarthy, Adv. Chem. Phys. 117 (2001) 483. w21x H. Risken, The Fokker–Planck Equation, 2nd Ed, Springer, Berlin, 1989. w22x M.W. Evans, G.J. Evans, W.T. Coffey, P. Grigolini, Molecular Dynamics and Theory of Broadband Spectroscopy, Wiley, New York, 1982. w23x W.T. Coffey, M.W. Evans, P. Grigolini, Molecular Diffusion and Spectra, Wiley, New York, 1984, Russian Edition: Mir, Moscow 1987. w24x H.C. Brinkman, Physica 22 (1956) 29. w25x C.J. Reid, Mol. Phys. 49 (1983) 331. w26x W.T. Coffey, Yu.P. Kalmykov, S.V. Titov, J. Chem. Phys. 115 (2001) 9895. w27x R. Carmichael, A Treatise on the Calculus of Operations, London: Longman, Brown, Green and Longmans, 1855. w28x B.K.P. Scaife, Principles of Dielectrics, Oxford University Press, London, 1989, revised edition, 1998. w29x M. Abramowitz, I. Stegun (Eds.), Handbook of Mathematical Functions, Dover, New York, 1964. w30x W.T. Coffey, Yu. P. Kalmykov, S.V. Titov, Phys. Rev. E 65 (2003) 061115. w31x C.J. Reid, M.W. Evans, J. Chem. Soc. Faraday Trans. 2 75 (1979) 1369. w32x R. Metzler, J. Klafter, Phys. Rep. 339 (2001) 1.