the role of mittag–leffler functions in anomalous relaxation

Journal of Molecular Liquids 114(2004) 27–34

0167-7322/04/$ - see front matter� 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.molliq.2004.02.003

The role of Mittag–Leffler functions in anomalous relaxation

D.S.F. Crothers *, D. Holland , Yu.P. Kalmykov , W.T. Coffeya, a b a,c

Department of Applied Mathematics and Theoretical Physics, Queen’s University Belfast, Belfast BT7 1NN, Northern Ireland, UKa

Centre d’Etudes Fondamentales, Universite de Perpignan, 52 Avenue de Villeneuve, 66860 Perpignan Cedex, Franceb ´Department of Electronic and Electrical Engineering, School of Engineering, Trinity College, Dublin 2, Irelandc

Abstract

The intertia-corrected Debye model of rotational Brownian motion of polar molecules was generalized by Coffey et al.wPhys.Rev. E, 65, 32 102(2002)x to describe fractional dynamics and anomalous rotational diffusion. The linear-response theory of thenormalized complex susceptibility was given in terms of a Laplace transform and as a function of frequency. The angular–velocity correlation function was parametrized via fractal Mittag–Leffler functions. Here we apply the latter method and complex-contour integral-representation methods to determine the original time-dependent amplitude as an inverse Laplace transform usingboth analytical and numerical approaches, as appropriate.� 2004 Elsevier B.V. All rights reserved.

Keywords: Fractal; Anomalous; Diffusion

1. Introduction and physical background

Inertial effects in anomalous dielectric relaxation wereconsidered byw1x. The linear-response theory of thenormalized complex susceptibility was given in termsof a Laplace transform and as a function of frequency.The angular–velocity correlation function was parame-trized via fractal Mittag–Leffler functions. Here, weapply the latter method and complex-contour integral-representation methods to determine the original time-dependent amplitude as an inverse Laplace transformusing both analytical and numerical approaches, asappropriate. The fractional Klein–Kramers equation foranomalous rotational diffusion is

dW ≠W ≠W ymEsinf ≠W 1ya˙s qf s D L W. (1a)0 t FP˙dt ≠t ≠f I ≠f

Here, is the probability density function˙W f,f,tŽ .(pdf). The fractional Fokker–Planck(FP) operator hasthe property

*Corresponding author. Tel.:q44-28-9033-5048; fax:q44-28-9023-9182.

E-mail address: [email protected](D.S.F. Crothers).

1yat z1ya 1yaD L Ws D0 t FP 0 t I

2w z≠ k T ≠ WB˙= fW q (1b)x |Ž . 2˙ ˙≠f I ≠fy ~

wherek is Boltzmann’s constant,T is the temperature,B

z is the viscous damping coefficient of a dipole,I is themoment of inertia of the rigid dipolem, f is theazimuthal angle of the rigid rotator,t is the time,t isthe intertrapping time-scale, identifiable with the Debyerelaxation time(zy(k T);10 s) anda is the anom-y11

B

alous exponent characterizing the fractal-time process.A weak uniform electric fieldE is suddenly switchedoff at time ts0, when anomalous diffusion ensues. Thefractal operator is given by1yaD0 t

≠1ya yaD ' D0 t 0 t≠t

where we have, suppressing dependence,f

t W f,t9 dt9Ž .1yaD W f,t s . (2)Ž .0 t | 1yaG a tyt9Ž . Ž .0

Here G is the gamma function. We may seek a

28 D.S.F. Crothers et al. / Journal of Molecular Liquids 114 (2004) 27–34

solution of Eq.(1a) as for normal diffusion, as

q` `ipf2 2˙ ˙ ˙W f,f,t sexpyh f c t H hfŽ . Ž . Ž .Ž .ep,n n8 8

psy`ns0

(3)

where we define

1yw z 2x |hs Iy 2k TŽ .By ~

so that

w z1 1qmEcosf˙2 2yh f˙W f,f,0 f he . (4)x |Ž . 3y22p k Ty ~B

Linear-response(ps1) theory requires the solutionof the differential–recurrence relation:

d i w zx |c t q 2 nq1 c t qc tŽ . Ž . Ž . Ž .1,n 1,nq1 1,ny1y ~dt 2h

nz1ya 1yasy D t c t . (5)Ž .0 t 1,nI

The usual Laplace operatorL, given by

`yst˜µ ∂L f t 'f s s e f t dt,Ž . Ž . Ž .|

0

yields

1yaµ ∂L D f tŽ .0 t

S B E0-a-11ya ya˜s f s yD f t ± C FŽ . Ž .t ts0T D G1-s-2

UsB E1Fa-2

1ya˜Ts f s C FŽ .D GV 0-sF1

where we have introduced, for subsequent convenience,s given byss2ya.Transforming Eq.(5), we obtain

2 1yaw zx |˜2tsqn g9 ts c sŽ . Ž . Ž .1,ny ~

w zx |˜ ˜qig9 2 nq1 c s qc s sc 0 (6)Ž . Ž . Ž . Ž .1,nq1 1,ny1 1,ny ~

where we have

t 2 bg9s sz ' .y yh Ik T 2Ž .B

As shown byw1x, Eq. (6) can be solved in terms ofcontinued fractions to yield the normalized complexsusceptibility, given by linear-response theory, namely

c ivŽ .1,0c v s1yiv (7)Ž .

c 0Ž .1,0

wherev is the angular frequency. In detail,

c v s1Ž .

sB ivtŽ .y

BsB ivt qŽ .2Bs1qB ivt qŽ .

3Bs2qB ivt qŽ .3q∆

(8)

and by comparison with

based on the exact recurrence relation for the regularconfluent hypergeometric functionM(A,B,C), namely

b 1ybqz M a,b,z yazM aq1,bq1,zŽ . Ž . Ž .

qb by1 M ay1,by1,z s0 (10)Ž . Ž .

we obtain

1c v s1y M 1,1qBC,B (11)Ž . Ž .ys1q itvŽ Ž . .

where

2y2sBsb ivt (12)Ž .sCs1q itv (13)Ž .

and where

yBCM 1,1qBC,B s BC B expB g BC,B (14)Ž . Ž . Ž . Ž .

with the incomplete gamma function given by

zyt by1g b,z ' e t dt. (15)Ž . |

0

29D.S.F. Crothers et al. / Journal of Molecular Liquids 114 (2004) 27–34

2. New developments

However, the question is: what is the time-evolutionof c (t) or F(t,s) given by1,0

gqi`c t h tŽ . Ž .1,0 1 ut˜F t,s h t ' s du e c u (16)Ž . Ž . Ž .1,0|c 0 2piŽ . gyi`1,0

Here, naturally enough, we haveh(t) the Heavisidestep function and the Laplace transform

`yutc u s e c t dt. (17)Ž . Ž .1,0 1,0|

0

Cancellingh(t), it follows that

˜ `c ivŽ .1,0yivts e F t,s dt (18)Ž .|c 0Ž . 01,0

where we have

c tŽ .1,0F t,s sŽ .

c 0Ž .1,0

utgqi`1 du es M 1,1qBC,B (19)Ž .| ys2pi u 1q utŽ .gyi`

with B and C given by Eqs.(12) and (13) with ivmapped tou. Can we now evaluateF(t,s)?

2.1. Analytical check

`yivt˜ivc iv siv e F t,s dt (20)Ž . Ž .1,0 |

0

where

gqi` `1 ut yuTF t,s s du e e F T,t dT (21)Ž . Ž .| |2pi gyi` 0

and settingusyqiv,

q` `1gqiv t y gqiv TŽ . Ž .s dv e e F T,s dT (22)Ž .| |2p q` 0

` q`1gqiv tyTŽ .Ž .s dT F T,s dv e (23)Ž .| |2p 0 y`

`g tyTŽ .s dT F T,s e d tyT (24)Ž . Ž .|

0

sF t,s h t (25)Ž . Ž .

sF t,s t)0´h t s1 . (26)Ž . Ž Ž . .

This completes our first check: consistency. Noticewe may takev to be Revyi´ to speed up the conver-gence of the Fourier Integral. In effect,´ is g.

2.2. Known result for ss1

Eq. (19) gives

st nw z`gqi`1 ds e bF t,1 s 1qŽ . x || 8B E2pi 1qbqbst1 Ž .gyi` ny ~ns1C Fsq

D Gt

(27)

where (.) is the Pochhammer symbol. Thus we have,n

using partial fractions,

stgqi`1 ds eF t,1 sŽ . | B E2pi 1gyi`C Fsq

D Gt

nw zS WT T

1y qyr( )U XPT T

` n V Yqs1, q/rnx |= 1q b . (28)8 8 rqbqbsty ~ns1 rs1

Using gamma functions, their reflection formula, theCauchy Residue theorem and L’Hospital’s rule, we mayˆshow that

ry1n y1Ž .1P s . (29)

qyr ry1 ! nyr !qs1, q/r Ž . Ž .

Since there are simple poles on the negative real-s

axis at and at1

ssyt

y rqbŽ . 1ss -y 1FrFn , (30)Ž .

bt t

it follows that

ytytF t,1 se M 1,1,bŽ . Ž .

ry1` n y1Ž .

n yt rqb ybtŽ .q b e (31)8 8 ry1 ! nyr !Ž . Ž .ns1 rs1

B Er 1 1C Fbt y y qD Gbt t t


n` nt t tb rB Eby y ynC Fse qe c ye (32)t t btr8 8 D Gn!ns1 rs1

n`t t tb w znB Eby y yC Fx |se qe 1ye y1 (33)t t bt8 D Gy ~n!ns1

t ytby ybexp yŽ .btse (34)t

B B EEt tC C FFsE y qbybE y (35)1,1 1,1D D GGt bt

in agreement withw2x and where we define the Mittag–Leffler function by

k` zE z ' (36)Ž .a,b 8G bqakŽ .ks0

(w3x: Ch. 18, Eq.(19)), with the understanding that

E z 'E z . (37)Ž . Ž .a a,1

A complex-contour integral representation is given by

ayb(0q)T expTŽ .1

E z s dT (38)Ž .a,b | a2pi T yzŽ .y`

which may be derived by using the Hankel transformfor the reciprocal of the gamma function. In AppendixA, we show that an exact but normally divergentasymptotic series, valid for±z±41, is given by

B E1 1C FE z (y E . (39)Ž .a,b ya,byaD Gz z

The normal usage in asymptotic expansions, such asEq. (39), is that one sums up to the smallest term, evenif this is just the first term:

1 1E z ( y . (40)Ž .a,b z G byaØzØ™` Ž .

This is an indication of the long-lasting fractal tail ofsuch a function.

2.3. Numerical calculation and physical understanding

For 0FtF2t, we take

F t,s/1'f tŽ . Ž .5

w g Cy1Ž .1 uts dx Ime M 1,1qBC,Bx Ž .|p Cy y`

zy0 Cy1Ž .utq dyRe e M 1,1qBC,B (41)|Ž .| C ~0

whereu is given, respectively, byusxyiy andusgq0

iy. This embraces an infinite rectangular contour with,typically, g;0.1 andy ;6.0, so as to include any poles0

and the branch-cut along the negative realu-axis, withany u assigned to zero on the positive real axis.Of course if then there are two poles at3ss y us2

. We use numerical quadratures for the twoB EpiC Fexp "D Gs

integrals in (41), based on five-point Lobatto andinternal subdivision to reach the required tolerance andaccuracy. Typically , ts1.123 and1ss2yas y2Revs0.1237.For tG2t, we expand theM in Eq. (19) as an infinite

sum over indexn:n` B

M 1,1qBC,B s . (42)Ž . 8 1qBCŽ .nns0

Then we set

F t,s sf qf qf qf . (43)Ž . 1 2 33 4

Taking the termns0, we have(w3x)sut

(0q) e utŽ .1 duf s1 | s2pi u 1q utŽ .y`

T s(0q)1 dT e Ts utsTŽ .| sB E2pi T ty` s C FT q

D Gt

sB B E EtC C F FsE y (44)sD D G Gt

sB E1 tC F( (45)D Gtt4t yp

The sum over 1FnFq`, using partial fractionstwice may be written as

ut` gqi`1 du e 2y2s nn ( )b utŽ .| ys8 2pi u 1q utŽ .gyi`ns1

ry1n y1Ž .

= (46)8 ry1 ! nyr !Ž . Ž .rs1

2y2s 2ysrqb ut qb utŽ . Ž .


and we have

1ys 2y2s 2ys1q ut rqb ut qb utŽ Ž . .Ž Ž . Ž . .

2ysb1 y utŽ .ryrs y . (47)ys 2ys 2y2sµ ∂1q ut rqb ut qb utŽ . Ž . Ž .

The second term in Eq.(47) gives rise tof and f33 4

of Eq. (43) while the first term creates

2y2s nŽ .n` n gqi` utŽ .b 1 dury1 utnf s C y1 eŽ .2 r | ys8 8n! 2pi u 1q utŽ .gyi`ns1 rs1

(48)

n s` B E B B E Eb t ty21ys nŽ .C F C C F Fs E y . (49)s,1y2 1ys n( )8

D G D D G Gn! t tns1

For 0-s-1, we see immediately why this is diver-gent for smallt. Also, using the representation of theE (z) given by Eq. (36), with dummy summationa,b

index k, then forss1y2 and for evenk, the gammafunction in the denominator has poles fork)2ny2. Itfollows that some care must be exercised in calling theNAG routine s14aaf (x, ifail). Regarding the secondterm in Eq.(47), we have

n` n ynŽ .rb 1f qf sb33 4 8 8n! r! 2pins1 rs1

ut 2y2s nq2ysŽ .gqi` e utŽ .du

= (50)| 2y2s 2sµ ∂u rqb ut qb utŽ . Ž .gyi`

For a givenr, f is obtained from the contributions4

of the conjugate pair of poles, which lie in the secondand third quadrants of theu-plane. Forss1y2, say, athird pole lies on the next sheet(arg(ut)s"2pi) andis inaccessible. Collapsing the contour in Eq.(50) ontothe upper and lower lips of the branch cut along thenegative realu-axis, we have, since for 0-s-1, thereare no poles on this line(unlike the ss1 case: seeSection 2.2),

n` n ynŽ .rb 1f sb33 8 8n! r! 2pins1 rs1

ut 2y2s nq2ysŽ .e utŽ .du0qŽ .= (51)| 2y2s 2ysµ ∂u rqb ut qb utŽ . Ž .y`

2ys n``bt byvt 1ys 21ys n 2n 1ysŽ . Ž .s dv e v t v| 8p n!0 ns1

n ynŽ .rIm8 r!rs1

y 2nq1spiŽ .e= (52)2ys 2y2syips y2ipsw z

x |rqb vt e qb vt eŽ . Ž .y ~

where we have setusve on the upper lip andusip

ve on the lower. Once again thev-quadrature isyip

performed using five-point Lobatto plus interval subdi-vision. Eq.(51) could be expressed in terms of gener-alizations of the Mittag–Leffler functions, namelyWright functions(w3x, §18.1, p. 211, Eq.(27)). How-ever, this would not be numerically useful since the useof the Binomial series on expansion of the denominator,leads to divergence.The conjugate pair of simple poles of the integrand

in Eq. (50) is given by z (r) and . Either is*z rŽ .` `

obtained iteratively using complex Newton–Raphson:

f zŽ .nz sz y (53)nq1 n f9 zŽ .n

where

r 2y2s 2ysf z ' qz qz . (54)Ž .b

It was observed that±Im z (r)±-6 so that f is` 5

consistent(for 0FtF2t) and we have

tz r 21ys nqsŽ . Ž .`n µ ∂e z r` n yn Ž .tŽ . `rb

f s2t Re .4 8 8 w zsx |n! r! µ ∂2ys z r q2 1ysŽ . Ž . Ž .`ns1 rs1 y ~

(55)

Clearly z (r) and are interchangeable in Eq.*z rŽ .` `

(55). In Fig. 1, we present sample zeros off(z) in thesecond quadrant of thez-plane, wherezsut. Clearlythey lie on a curve, which is very nearly a straight line.Returning to Eqs.(19) and(20), we calculate

M 1,1qBC,BŽ .Result 1s (56)ys1q itvŽ .

whereB andC are given by Eqs.(12) and(13) and

`yivtResult 2siv e F t,s dt. (57)Ž .|

0


Fig. 1. We plot the complex zerosz of f(z) as a function of Rez` `

and Imz and of r, where f(z), as a function of the integerr, is`

defined by Eq.(54). The values ofb, s andt are inset.

Table 1For specimen numerical values of´sIm v, ss1y2 and rs1.123,we tabulate results 1 and 2, corresponding to Eqs.(56) and(57)

(a) ´sy0.1 Result 1s(0.3278, 0.1581)Result 2s(0.3278, 0.1585)

(b) ´sy0.2 Result 1s(0.4284, 0.1198)Result 2s(0.4285, 0.1193)

(c) ´sy0.3 Result 1s(0.5102, 0.0945)Result 2s(0.5100, 0.0940)

(d) ´sy0.0! Result 1s(0.2158, 0.2186)Result 2s(0.2156, 0.2143)

In Table 1, we present results 1 and 2 forss1y2,ts1.123 and Revs0.1237. The Imv is given by´.Clearly the results agree correct to three significantfigures except for´sy0.0. For 0FtF2t, we useF(t,s)sf(5) and for 2tFt, F(t,s)sf qf qf qf .1 2 33 4

Because of the aymptotic property of the Mittag–Lefflerfunctions, f and f fall-off very slowly. Pragmatically,1 2

we took t ;500 000 (´syf.f) and took the firstmax

term of the asymptotic expansion(Eq. (39)) for f for1

t)30 and for f for t)15. Meanwhile bothf and f2 33 4

die-off fairly quickly. For 0FtF2t, we usedf , since5

f diverges.2

3. Conclusions

In conclusion, we find that our analysis is robust andreliable, but that the theory of Mittag–Leffler functionsrequires numerical reinforcement, including the locationof poles of the integrand.In the case ofss1y2, our main consideration in this

paper, we have

2y2sutqb ut su tqbt (58)Ž . Ž .

so that

1B EB Etqbt 21 C FC Ff qf sf 'E y1 2 12 ,1

2 D D G Gt

1 1B EB E B Et t2 21 1C FC F C F( E y (59)y ,

2 2D G D D G Gtqbt tqbt

which provides an independent check on the work.

Acknowledgments

One of us(DH) was supported by NIDEL and QUBLarmour Awards. One of us(WTC) acknowledges aQUB Distinguished Visiting Fellowship. All of usacknowledge support by USAF.

Appendix A:

A.1. Properties of Mittag–Leffler functions

ayb t(0q)1 t eE z s dt (A1)Ž .a,b | a2pi t yzy`

`(0q)1 yb t n yans dt t e z t (A2)| 82pi y` ns0

n` z) )s Hankel Smallz . (A3)Ž . Ž .8G bqanŽ .ns0

Also, we have

aybqan`(0q)B E1 1 ttC FE z s y dt e (A4)Ž .a,b | 8 nD Gz 2pi zy` ns0

yn`y1 zs Hankel (A5)Ž .8z G byayaŽ .nns0

y1) )s E 1yz Large z . (A6)Ž .Ž .ya,byaz

By the ratio test, in general(A3) converges and(A6)diverges(0-a-1).

A.2. Asymptotics of Mittag–Leffler functions

n` zE z s (A7)Ž .a 8G 1qanŽ .ns0


n` zs (A8)8anG anŽ .ns0

n` `z Ÿs c an (A9)Ž .Ÿ8 8anns0 Ÿs1

(w4x, Eq. 6.1.34) (NB cs1)

` `L n Ls1q c a z a (A10)Lq18 8

Ls0 ns1

` `LL nq1s1q c a z nq1 (A11)Ž .Lq18 8

Ls0 ns0

` `zc1 LL ns1q qz c a z nq1 (A12)Ž .Lq18 81yz Ls1 ns0

where

` `1L Ln nq1z nq1 s nq1 z (A13)Ž . Ž .8 8zns0 ns0

L` nq1B E1 d zC Fs z (A14)8D Gz dzns0

LB E B E1 d zC F C Fs z (A15)D G D Gz dz 1yz

LB Ey1 d 1C Fs z LG1 (A16)Ž .D Gz dz zy1Ž .

Ly1 ds f g t (A17)Ž Ž ..Lz dt

where we have

t tzse , lnzst, g t sey1 (A18)Ž .

and

1f g t s (A19)Ž Ž ..

g tŽ .

L Ly1 aj9m jŽ . Ž .s f g t L;a ya = g t (A20)µ ∂Ž Ž .. Ž . Ž .1 L8 8 2z ms0 js1

using Faa di Bruno’s formula(w4x, p. 823) and summing´over: (See alsow5x)

a q2a q∆qLa sL (A21)1 2 L

a qa q∆qa sm (A22)1 2 L

and where

9 mŽ .L;a ya 'S (A23)Ž .1 L L8

namely a Stirling number of the 2nd kind.However,

tjŽ .g t sesz 1FjFL (A24)Ž . Ž .L

aj mt mj t a qa q∆qaŽ . Ž .1 2 L´ g t se se sz (A25)µ ∂Ž .2js1

m` L y1 m!Ž .

Ln mmŽ .[ z nq1 sy S z (A26)Ž . Lmq18 8 zy1Ž .ns0 ms0

since

my1 m!Ž .mŽ .f g t s (A27)Ž Ž .. mq1µ ∂g tŽ .

zc1[E z s1qŽ .a 1yz

mq1` L B E1 zL mŽ .C Fq c a m! S . (A28)Lq1 L8 8D Gz 1yzLs1 ms0

However,

LLym mŽ .y1 m!S s1 (A29)Ž . L8

ms0

(w4x, p. 825, IIB)

`z y1 L[E z ( 1q c ya (A30)Ž . Ž .a Lq181yz zz™` Ls1

`z 1 1 Ÿs1q q q c ya LsŸy1 (A31)Ž . Ž .Ÿ81yz z azŸs1

1 1 1 1s q q (A32)1yz z az G yaŽ .

1 1 y1(y y ( (A33)2z zG 1ya zG 1yaŽ . Ž .

Similarly, we have


1E z ( y . (A34)Ž .a,b zG byaz™` Ž .

Appendix B:

Check on Eq.(3.12) of w6xWith

azsygt (B1)

t2kT2N Mx t s dt tyt E z (B2)Ž . Ž . Ž .a|m 0

na`t ygtŽ .2kT

s dt tyt (B3)Ž .| 8m G 1qanŽ .0 ns0

`2kT 1s 8m G 1qaŽ .nns0

n n tanq1 anq2w zyg t yg tŽ . Ž .x |= t y (B4)1qan 2qany ~ts0

`2kT 1s 8m G 1qanŽ .ns0

nanq2w zanq2 n t ygŽ .t (yg)x |= y (B5)1qan 2qany ~

n2 aw` t ygtŽ .2kT xs 8m G 2qany Ž .ns0

n2 a zt ygt 1qanŽ . Ž . |y (B6)G 3qan ~Ž .

na w x` ygt 2qayny1yaynŽ .2kT 2s t (B7)8m G 3qanŽ .ns0

2kT 2 as t E ygt (B8)Ž .a,3m

References

w1x W.T. Coffey, Yu.P. Kalmykov, S.V. Titov, Phys. Rev. E 65(2002) 032102.

w2x W.T. Coffey, A. Morita, J. Phys. D 9(1976) 47.w3x Bateman Manuscript Project. Higher Transcendental Functions,

3, McGraw-Hill Book Co, Inc, New York, 1953, Ch 18.w4x M. Abramowitz, I.A. Stegun(Eds.), Handbook of Mathemati-

cal Functions, Dover, 1970.w5x K.B. Oldham, J. Spanier, The Fractional Calculus, Academic

Press, New York and London, 1974.w6x E. Barkai, R.J. Silbey, J. Phys. Chem. B 104(2000) 3866.

the role of mittag–leffler functions in anomalous relaxation

Documents