diffusion equation study of rotational brownian motion

Diffusion Equation Study of Rotational Brownian MotionAuthor(s): James McConnellSource: Proceedings of the Royal Irish Academy. Section A: Mathematical and PhysicalSciences, Vol. 77 (1977), pp. 13-30Published by: Royal Irish AcademyStable URL: http://www.jstor.org/stable/20489063 .

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[ 13 3

2.

DIFFUSION EQUATION STUDY OF ROTATIONAL BROWNIAN MOTION

By JAMES MCCONNELL, M.R.I.A.

Dublin Institute for Advanced Studies*

[Received, 25 MAY 1976. Read, 8 NOVEMBER 1976. Published, 31 MARCH 1977.]

ABSTRACT

Sack's discussion of the rotational Brownian motion of a spherical polar molecule based on a diffusion equation is simplified. The solution of the equation for a periodic external electric field is expressed in terms of elements of the reciprocal of a matrix. If the reciprocal is expanded as a series of powers of a dimensionless para meter, the results for complex polarisability agree with those of long calculations of Ford, Lewis, McConnell and Scaife, and they may be extended without difficulty to higher orders of approximation.

TABLE OF CONTENTS Page

1. INTRODUCTION ... ... ... 13

2. THE FOKKER-PLANCK EQUATION FOR THE ROTATING SPHERE ... ... ... ... 14

3. SOLUTIONS OF THE DIFFUSION EQUATION ... ... ... ... ... 18

4. RELAXATION EFFECTS IN DIELECTRICS ... ... ... ... ... 23

1. Introduction

A lengthy study of inertial effects in dielectric relaxation processes was made by Sack in two papers (1957a, 1957b). In paper I he took the disk as a model of a

dipolar molecule and in II he took as models both the rotating needle and the rotating

sphere. His investigations were based on a diffusion equation, which is a Liouville

equation supplemented by Kramers terms. Since Sack's calculations for the needle and the sphere, though condensed, are

both long and complicated, it has seemed worthwhile to seek alternative treatments

of the diffusion equation. This has been done for the needle by Ford (1975), who

employed expansions in associated Laguerre polynomials. We have found that

similar expansions produce no simplification for the spherical model. However, it

is possible to modify Sack's method so as to shorten the calculations considerably.

This is done in the present paper. In the next section we give an alternative derivation of Sack's generalised Liouville

equation and elaborate some results from II. We then express the solution of the

equation for a periodic external electric field as a series, in which the coefficients are

*The subvention granted by the Dublin Institute for Advanced Studies towards the cost of

publication of papers by members of its staff is gratefully acknowledged by the Royal Irish Academy.

PROC. RI.A., VOL. 77, SECT. A t2I



14 Proceedings of the Royal Irish Academy

elements of the reciprocal of a matrix. The reciprocal is expanded as a power series in a dimensionless parameter introduced in earlier investigations. In the final section this matrix method is applied to relaxation effects in dielectrics and, with substantially less labour, previous results for complex polarisability are extended.

2. The Fokker-Planck equation for the rotating sphere

Sack obtained his LiouviMe equation by adapting to three-dimensional rotational motion an equation of Kramers (1940, eq. (9)) for one-dimensional linear Brownian

motion. Since Kramers' derivation of his equation is open to objection, we shall

derive the diffusion equation independently putting down Langevin equations and

following standard procedures to go to the Fokker-Planck equation. To facilitate

comparison with Sack II we retain his notation expressing the components of angular velocity of the rotating sphere (v9, v., vu) in terms of the Euler angles 0, q, X by

v0=OX, v,=4 sin 0, v5=-X+# cos 0. (1)

We take moving axes, numbered 1, 2, 3 in Fig. 1, for the rotating sphere with a dipole of constant moment p along axis 2. The sphere, whose moment of inertia is I, is

subject to a frictional drag Ifi times the angular velocity, to a random driving couple

IdW dt

FiG. I



MCCONNELL-Diffusion equation study of rotational Brownian motion 15

in no preferential direction and to an electric field of intensity F in the direction 0=0. W(t) is a Wiener process. The components of angular velocity of the axes 1, 2, 3 are respectively (0, $ cos 0, $ sin 0) and the equations of motion are

I0 + I#k

sin 0 -LF sin 0-I#d + I

dtW1 dt

d dIV2 I- d +0 cos 0)=(-Ixx+@ cos 0)+I dit dt

d dW3 *si

I d( sinfO)-Icq= -If4sin0+I

We express these in terms of vo, v,, v. by (1) as

dvo -v.(v,-vv, cot 0) +fiv+ yE sin 0] dt+dW1

dv, = [v0(v7 - v# cot 0) - fv4 dt + dW3 (2)

dvz= -flv dt + dW2.

In addition we have from (1)

dO = v6ldt, do dt, dX = (v1- v# cot 0) dt. (3) sin 0

We employ (2) and (3) to establish the Fokker-Planck equation for the probability density w specified in configuration-angular velocity space by the coordinates 0, v0,

v4,, v. and the time t, the probability density being clearly independent of the angles q and X. We write

Y =0,Y2k, Y3 X, Y4=V69 )Y5 =V+, Y6 =vz (4)

and express (2) and (3) as

dy1 =y4dt, dy2 = ;no dt, dY3=(-cot 0 Y5+Y6) dt

dY4 =-[v(v2-v.* cot 6) + flv + F

i dt + dWi (5)

dy5 = [v0(v- - Vu cot 0)- fivo] dt + d W3

dy6= -vv dt + dW2.

In general the equations

m

dXi(t)=Afi(, X) dt+ E G(t, X) dW,Q) (i= 1, 2, .... d) 1=1

lead to the Fokker-Planck equation (cf. Arnold 1974, pp. 156 - 159)

aw X 8 d d 02

Ft + Yi (JXt, y) w)-Ic2 y.oy.((GG')iiw)O, (6)

PROC. R.I.A.5 VOL. 7 7, SECT. A [2AJt




where G' is the transpose of the matrix G and c2 is determined by

< Wi(t) Wj(t')> = i c min (t, t'), (7)

the brackets on the left hand side denoting ensemble average. However, in establish ing (6) it is assumed that the volume element in the y-coordinates is dy1 dy2.. . dYd,

whereas for (4) it is

sin y1 dy1 dy2 dy3 dy4 dy5 dy6.

Then on integrating by parts, with g a smootb bounded function with continuous partial derivatives and vanishing outside a bounded domain,

{dtj' dO sin 0 4 wvo= -J'dt vofg ,- (w sin 0) dO

=-J'dtvo gJsinO (T+ w cot 0) dO,

whereas

Jdt Jd 4, wv =-{dt v, g yo do.

Hence in writing down (6) with fi, GCi given by (5) we replace

8w 8w by + cot 0.

The Fokker-Planck equation is therefore

8w 8w / 8w 8w N F 8w i+?vo gp0+(vt,PcotO-vz)Kv. yg- v p- --sin. 0 -3,Bw

8w Zw 8w2 v I

8V0 (8) / (w

w 82w 2w 82w\

_ft + oy + y+ vy t-Ic 2 + 7+ 8V2}?

To obtain the equation in the form given by Sack we use the relation (Lewis, McConnell and Scaife 1976, eq. (12))

c2 kT 2/- I'

where k is the Boltzmann constant and T the absolute temperature. This relation could be derived independently by taking for w in (8) the steady state solution given below in (12). On substituting for I c2 equation (8) becomes

8w aw / 8v,

w 8w N ,uF 8w at+ vo1+ (v, cot vpjz)

- vo

- )- sin

8( kT8w 8 a kT8w 8a kTaw\ (9)

=fl-Kvow+7r} fliw $W+ +-y +

g} Tfl Vz w+-j-~ whc g ee th I e v. cmie wthw I 3vz I )v

which agrees with II eq. (3.8) combined vvith 1 (2.3).



MOCONNELL-Diffusion equation study of rotational Brownian motion 17

For future reference we repeat more explicitly Sack's transformation of equation

(9). Writing O-V01, V=-V2, Vz=V3, U=-U1, UO=U2,

Uz=U3

3 3

I V12=v2, Z u12=U2 I-1 ~1=1

we make the Fourier transformation

r+ 00+ or+ Go3

P(O, uo, ut,, uZ, 0=J jJw (O vo, v4t, v", t) exp (-i Eu, V) dvl dv2 dV3, (10) - 00 - o00

which transforms (9) to

04D @2FD a a a ii

7 + ?im j-i coto T y-O

-u.y- -D - sin ? uo 3 O(D u2kT

___- (D. -- fiZ E au, The probability density wo for a steady state is given by the Boltzmann distribution:

w = const. exp{ - [IV2 + V(Q)]/kT}, (12)

where the potential energy V obviously depends only on the Euler angle 0. From (10)

the Fourier transform ( of wo is given by

(FO(0, ue, u#, uz, t)=const. exp (-V(0)/kT) exp (-kTu2/2I). (13)

We lighten the calculations by elimination of the second exponential putting

L'(0, i0, it, u, t) = exp (kTu2/2I) 0(0, uo, u, u,t). (14) Then (11) leads to the differential equation for P

0T a2' i8 F ap _

Aj+i @ --j kTj +mFsin0Wj a a kT

- aT alp '

3 aT (15

+tau -au" I (cot it. aujuo 1J=-fl?ui (5

which is the diffusion equation with which we shall be chiefly concerned. The equilibrium solution To of the last equation is seen from (13) and (14) to be

given by 0 = const. exp (- V(0)/kT).

The V(O) due to the external field F is - piF cos 0 and (I2) becomes

1 I 3/2 puF 1v2 ALFcos 0

o 8=ni 2nrkT1'- F e 2T e

,E (16)

kT sinhu

where the constant multiplying factor has been chosen such that

r27 02X IC r+OD+ o+ oo

S X { o sif fdo0 oco f-cn 0 0 0 00 00 00




From (16) we deduce that pP cos O kTua

1 pFe kT e 2i

kT sinh kT

and from (14) that F -cos

1 pFe kT P (0, u9, u,, uz) 2 -

kTsinh kT

If now we suppose that

kT

is sufficiently small that it is permissible to take the linear approximation

uFcos 0

+ kT

for the exponential, we write

879 Po(Q ue, uuz)=l + kT co

* (17)

3. Solutions of the diffusion equation

Sack's investigations of relaxation effects in dielectrics were based on two obser vations:

a. the polarisation of the dielectric due to the external electric field is proportional to

(2n (2 Jr it

lim dXl d4J sin Ocos OT(O, u6, uis, u,, t) dO, uO o o1 0 0 0

b. the value of ' in the linear approximation for the equilibrium case is given by (17).

Since Sack was immediately interested in TF, he studied the solution of (15) rather than that of the Fokker-Planck equation (9). He confined his attention to the linear approximation and so was guided by (17) in his search for a solution of (15).

We shall likewise confine our attention to the linear approximation. For the solution of (15) we try

8n2' = + cos 0 XI (u2, t) + sin 0 uo Z2(u2, t) + (cos 0 UZ2 + sin 0 u_2u4) X3(U2, t) (18)

where XI, X2z X3 are of order

F

kT



MCCONNELL Diffusion equation study of rotational Brownian motion 19

and terms of higher order in this dimensionless quantity are neglected. This equation is equivalent to 11 (3. 14), if we put

01 = xl(u 2, t), 02=ux2(U2, t), #3=u2x3(u, t)

and note the expressions given in 1I (2.22) and (3.16) for #. 02' 03 when F=Foe't This is in fact the case with which we shall be almost entirely concerned. Sack

obtained 11 (3.14) by expanding T in u-space as a series of spherical harmonics. On substituting (18) into (15) it is found that 8X2 the left-hand side of (15) is equal

to

Cos i( +2iZ2+2iXa2 (u2 + U2)-j x 2(UO + U))

+sin 0 (a -2i I

ikT

iX3)

+ (cos 0 uz2 +sin 0 uz Xu) -X3-2i sin 0 u zX2 2i Uz2 COS 0 OX2

.20ax2ik T. ikT 6 2 ikT OS0U2 +2 2iU uCos 0 2+ sinO0uuo I2 +-cos Ou X2A I Co 054X2+'

=CosoQ0 p+2ix22U2 x2 ikcT u2x -CO 0( t1+ 2i X2 +2i TU2 -i U X2)

+ sin 0 (X2- 2i DX'ikT _ipF+F (19)

+(Cos0 V2+sin0uzu X -+j2i, 2+

i T

We introduce the dimensionless variable z by

kTu2 21 (20)

There will be no danger of confusing this z with the subscript in u. or v:. We then

deduce that (19) equals

cos 0 ( ,+2i X2 +2iz X2-2iz X2

at a x-+x >h(1

OX2ikTX,ikrTx ikTF + sin 0 + ) (21)

+(cos 0 U, 2 OX sn u";a3 _ikT DX2 ik T )




Similarly for 8n2 the right-hand side of (15)

3 O OW O -S2f2lZui-= -82fl - = -l16nr2flz

Now from (18)

8z aT=O? ax, +snU Oe@0 X2 Oau?)+(COS 0 U 2+i 0 z X)3a gn2Lcosok'+sinO oT( 0 2 ?UdlI. Oz Jr ~~~~~~~azo

X3 [2 cos 0 u z+sino (+ 8*

so that

- 16.n2z =Z -2/1 jcos Oz ax, j u.9 sin z 9X2 +IX2

+(cos 0 u.2 +sin 0 u u4)( a++3)}+ * (22)

On equating the coefficients of cos 0, is0 sin 0, cos 0 U z2+sin 0 uz u,p in (21) and (22) we obtain the equations determining Xi, X2, X3

0Xj +2i X2 +2iz X2--2iz x2=-2fiz x (23)

0X2 ikTD0X1 ikT i4sF 0X2 at2_'ka@Xt+ I xl-I -+ix =-2fl z-F-fX2 (24)

0X3 ikT 0X2 ikT 0X3 _Ik Oz?+ IkT X2 21

z 0z-2PX3 (25)

The solutions of these equations when substituted into (18) give the W(0, uo, u,A, ui2, t)

resulting in linear approximation from the field F. We now specialise to the case

where F= Fo e"'t and try the solutions

X1(U2, t)=eiwt > Ar zr, x2(u2, t)=elw >1 Br Zr, rO r0o

oD (26)

X3(u2, t)=efitt E Cr zr. r=O

Equation (23) yields

(icoQ+2r) Ar+2i(r+1)Br-2iBr,_1=0. (27) Equation (24) gives

ikT ikT i#F0 ic Brzr- T E rAr zr +

I1E Ar Z-y

+iZ C,r zr+2p z rBr 4+PE Br Z=o.




On account of the presence of

ipFo

it is necessary to distinguish between r=0 and r= 1, 2,.

ikT ikT ipFo (iw+fl)Bo+ I Ao+iCo- I A1l

- , (28)

ikT ikT (iw+2flr+f) Br+ -I Ar+iC- y(r+ 1) A,+1=O.(r 1I2*..) (29)

Lastly, from (25) ikT ikT

(io 2flr + 2f0) Cr + Br-- (r+1) Br+1 (30)

On eliminating Ar, Ar+ 1, Cr between (27), (29) and (30) we get

(iw+ 2flr +fl)B+yjkT 2

1 ( 2r+2

+ 2r+3

(c OBr+TL io + 2r B -

+\icw + 2fJr 3i1 + 2)r + 2

(r + 1)(2r + 1)

ico2SB+2pBr+l =0-.(r=1,s2,.* ) iw? +2flr?2fl

When r =0 and l =0 we cannot eliminate Ar from (27). We shall therefore assume

that co # 0, which implies that our subsequent calculations do not apply to the case of

a constant field Fo. Then for r = 0 equation (31) will be true provided that we put

ipFo

I

on the right-hand side, in accordance with (28). On introducing the dimensionless

parameters cv' and y by cc kT

cv =gp s vy~ = Ip(32)

(31) and its related equation for r = 0 may be expressed as

_2 / 2r+2 2r+3 0 (ico'F+ 2r + 1)BF+v[1+2 Br_ 1 +

ico' + 2r + 2r Br

(r + 1) (2r+ 5) IiF0 (33)

iwl+2r+2 Br+1 B '1= rO

Let us put

La, = (ico'+ 2n + 1) bnl (34)

2 ( 2n+2 2n+3 \ (n+1)(2n+5)

Mn-ico+ 2n ico6 11+\t2n+ iw +2n+2 2 ico'+ 2n+2

so that (33) becomes

(Ln I + yMn 1) Bz =

ipFo

1 ?




We write this in matrix notation

JBJ 0

(L+yM) l2 4sFO 0 (35)

Li ~~~J where the elements of L and M are given in (34).

Equation (35) shows that

Br= jy? ((L+yM)' )rO. (36)

From (27) 2i(r +1) 21

(ico'+2r)Ar=- f Br +fBL_

and then from (36)

Ar = (i1' + {2r)f1 {(r + 1)((L + yM) - )ro-((L + yM) _ o}. (37)

The second term in the bracket will vanish for r= 0 and so explicitly

2pF0 (8 Ao=.- ofl ((L+ M)-)o (38)

Similarly we find from (30) that

Cr = ( +2r +2I -(r + )((L + yM) -

)r + 1, 0 + ((L + YM)E- )rO} (39)

Equations (18) (20) (26), (36), (37), (39) give the solution in linear approximation of the differential equation (15) for T(P, UBs, up, uZ, t) when F=Fo eIt.

In studies of the rotational Brownian motion of a sphere based on a stochastic differential equation (Lewis, McConnell and Scaife 1976; Ford, Lewis and McConnell 1976) the autocorrelation function of cos 0 was expressed as a series in powers of y. We therefore try to express Ar, Br, Cr as such series taken to a finite power of y. For this

we need the sum to a finite number of terms of a power series in y for ((L+ yM) 1)Io with I= r -1, r, r + 1. We are not concemed here with the problem of the convergence of the infinite series expansion of (L+ yM) U Let us write out from (34) the matrices L-1 and M using the abbreviations

io= (0), ico'+ 1= (1),iwk'+ 2= (2), etc.: (40)

1 2 3 1.5 -00 ? o l (0) (2) (2) 0

1 2 4 5 2.7 (1 L(3) , M= (2) (2) (4) (4) . . ()

1 2 6 7 3.9 00 O50 0 .4) (4) (6-) (6)




These are infinite matrices but for our purpose we truncate them to lI+k dimensions, where k will depend on the power of y to which we wish to take the calculations.

If we neglect terms proportional to yS+ 1, . .s., we have for the truncated L

and M the approximation

(L+yM)1-(L(l +yL-1M))-1=(l +yL-M)-Y L-'

=L- _L-y1ML-1+y2(L-1M)2L- - _ +(-)sys(L- M)3L-1

The diagonal form of L greatly simplifies the calculation of ((L+yM- ') ; indeed

((L+yM)-')jo=L11O-0L-yL , M1oL- loo +y2L- 11(ML-'M)10 L- loo

(42) - + (-_)sytsL - 1 Al(ML- l0

(42)

Moreover, since Mpq vanishes unless p= q -1, q, q+ 1, it follows that ((L1' Mf),0 vanishes unless s> 1. For such values of s the first non-vanishing term on the right

hand side of (42) is proportional to Y9 and, since

((ML 1) )jo=Mj9 1_1 L 11_1, 1-1 MI-1, 1_2 ...

L-11, Mlo L-100,

the first non-vanishing term in the series for ((L+y M)-f')o is explicitly

2' yI

(iw)'+l)(ko+?2). (io'+21+I) (43)

We see from (36), (37), (38), (39), (43) that

A ; {2r ri}(

Br- f- (io:)I + 1) (Ico.) + 2; .... (co' + 2r 1) } AO p2 A).(io + ... }

ipuF0 21%r (k'2?lr*I

puF0f_2_____yr______I

r I (ico + 1) (ico? + 2) .* (icow +2r+2) 5

The continuation of these series will present no difficulty, as we shall illustrate in the next section. The solution of the diffusion equation (15) in the linear approximation as a seres in powers of y is therefore essentially completed for the field Fo eiwt No explicit solution appears to have been obtained by any other method.

4. Relaxation effects in dielectrics

We take as a model of a dielectric a system of non-interacting spherical molecules

each carrying a dipole of fixed moment p. Let N be the number of molecules per

unit volume. A dipole making an angle 0 with the direction of the electric field F




has energy p F cos 0 and the polarisation P(t) of the system is defined in configuration

angular velocity space by the integral of N p cos 0 w over the space; that is,

2Z 2n + j+ +cC +oo P(t)=Np dx do cos 0sin 0 dOJ w(0 v, v, vz, t) dvo dv4 dvz.

0 0 o0 -c - O -o

Hence

P(t)=N,i lximJ wcos 0 sin 0 exp (-iZut vl) dv1 dv2 dv3 dO d4 dx

=Ng lim {J0J ((0, uo9, uu,;, t) cos 0 sin 0 dO do dx, by (10),

Np lim fJf|(0, u6, ut, tuz, t) cos 0 sin 0 dO do dx, u b0

by (14). For the value of TP in (18)

Np 'rZ 2n pn

P()= XI (0 d t) xJ dJ Cos 20 sin 0 dO

0 0 ) (44) Np

=- zi (0,)

In the case of the field Fo e(f equation (26) gives

P(t)= 3 Ao e't

We write this P*(w) eiwt, so that

Np P*(.

N Ao. 3

For a constant field Fo, TP is given by (17) with F=F0, so the respective polarisation

PO is obtained on replacing x1(0, t) in (44) by

pFo

kT

Hence

P*(co) AokT,

PO pFo

and on using (32) and (38)

P*() 2yfi ((L + yM)0 0)oo (45) Po fro




As a direct application of this we examine the overdamped case of very large ,B. Then co' and y are both very small but

_=_=2a)TO, (46) v kT

where To is the Debye relaxation time. In the limit as both co' and y tend to zero (41)

yields

1+2y 0 O

L+yM= 0 3 0..

so that

((L+ ym)- t)Oo= 1+27

1a}

and, by (45) and (46),

P*(wo) 1 (7 (47)

PO I + io)xo

This is the well-known result for the Debye theory (Debye 1929, p. 90, eq. (62)). For the application of (45) to the general case we evaluate the terms in the series

for ((L+yM)->)oo. Previous calculations of P*(w)/Po as a series in powers of y went as far as y4-terms, and we shall now take them as far as y5-terms. Equation (45)

shows that this requires the expansion of ((L+yM)- ')o as far as y4-terms. Putting I=0 and s= 4 in (42) we evaluate

L oo-_(L-y 1 ML- ')oo +y2(L-1 ML-

I ML- ')%o

-y3(L- I ML-

I ML-

I ML- )'o% + y4(L- 1 ML- ' ML-

I ML- V ML- V)0.

According to (41) L-1 ML

L -1

(L-' ML-' ML- 1)00 = Loo -'(Moo Loo- I Moo + MO, LI L

' M,O) Loo

4 12 9 10

(0)2(1)3 (0)(1)3(2) (1)3(2)2 (1)2(2)2(3)




(L-' ML-' ML-' ML-')00=Loo -1

{Moo Loo

- 1 MO( Loo

t Moo

+Moo LOO-1 MO, L,j-' M,0+M01 L,-j'Mlo Loo0'Moo

+M01 LI,-' MA, LI,j'M,0) Loo1

8 36 54 27 (0)3(1)4 (0)2(1)4(2) (0)(1)4(2)2 (1)4(2)3

40 60 40 50 + _+ + +.....+-+

(0)(1)3(2)2(3) (1)3(2)3(3) (1)2(2)3(3)2 (1)2(2)2(3)2(4)

(LU ML MU MU- 1 ML- 1)fo=Loo - '(ML 1 ML- 1 ML 1 M)00 Loo 1.

(48) Employing the relations

(ML-' M)==0Moo Loo0 Moo+MOI LI Ij Ml

(ML-' M)o1 =Moo Loo( MO1 +MOI LI Ij Ml I

(ML-' M),0=MA0 LOO-1 M00+M1 LI,-' Mlo

(ML-' M),I=M, M L0oA MO,1+M,l LI I M1 1 + Ml 2L22-1 M21

we deduce that

(ML-' ML- 1 ML' M)OO =Loo-3 Moo4 + 3L0o-2 L, I- ' M002 Mol Mlo

+2Loo0 L,I,2 Moo MAl MO, Mt0+L00-j L112(Mo, M10)2

+L,-3 MA12 Mo0 M,0+L,7-2 L22-1 M01 M10 MA2 M21,

where Li,7' is to be interpreted as (Lii 'f) and MA as (MIk)r. We now see from (41) and (48) that

(L-U ML-1 ML-1 MU-1 M -1)

16 96 216 120 216 =_-_ + + __ + +_ (0)4(1)5 (0)3(1)5(2) (0)2(1)5(2)2 (0)2(1)4(2)2(3) (0)(1)5(2)3

360 160 + 200 81

(0)(1) (23(3 (OXI)3(2)3( )2 (0)(1)3(2)2(3)2(4) (1)5(2)4

270 340 300 160 + , + . + ..+..

(1)4(2)4(3) (1)3(2)4(3)2 (1)3(2)3(3)2(4) (1)2(2)4(3)3

400 250 280 (1)2(2)3(3)3(4) (1)2(2)2(3)3(4)2 i()2(2)2(3)2(4)2(5)

For the above calculations it is sufficient to take L and M as third order matrices. On referring back to (40) we conclude from (45) that

P i(W)

2y 2y 2 +3

Po i 2y'(iwl + ) icoQwl ++1)2 io/ iaw+2 }

2y3 5 4 12 +

tCoCo + 1)2 }(ko')2(iCo' + 1) ic'(ito' + 1)(iw' + 2)

9 + 10 } yCo, + 1)(iCot +2)2 oio + 2)2(i' +3



MCCoNNELL-Diffusion equation study of rotational Brownian motion 27

2y4 ( 8 36 iw'(iwt' + 1)2 j(i')3(iw0' + 1)2 (ico')2ito' + 1)2(io + 2)

54 40

iwj'(ito' + 1)2(iCo' +2)2 io'(io' + 1)(io' +2)2(ito' + 3)

27 60 ? . + ,

(ito' + 1)2(ito' p+2)3 (icc' + 1)(iw' + 2)3(ito' + 3)

+ ('+ 40

+ 50 )

(ia)' + 2)3(iw' + 3)2 (ito' + 2)2(ico' + 3)2(ki c+4)

+ 2y J 16 + 96 216

iCo'(io' + 1)2 (iCo')4(ico' + 1)3 (it')3(io' + 1)3(ico' +2) (io')2(ico' + 1)3(ico' +2)2

120 216 + _.+ ..

(ic)')2(ito' + 1)2(ito' + 2)2(ico' + 3) ijc'(ijc' + 1)3(ito' + 2)3

360 + 160 +

+i 12io )(C, j ic '(-iw' +1)2(ico' +2)3(ito' +3) c'(iC0' + 1)(ico' ?2)3(ico + 3)2

200 81

ito'(ito' + 1)(ico' + 2)2(ico' + 3)2(icc' + 4) (icc' + 1)3(iCo' + 2)4

270 340 + .+ -

(icQ + 1)2(ico' + 2)4(ico + 3) (ico' + 1)(ico' + 2)4(io' + 3)2

300 160 ? ~~~~~+ (iGc' + 1)(iw' + 2)3(icc' + 3)2(ioj' +4) (icc' + 2)4(ioj' + 3)3

400 250 +28+0

(icc' ? 2)3(ico' + 3)3(io' + 4) (i2cc' +2)2(iot' + 3)3(it)' + 4)2

(io' + 2)2(ito' + 280 + 439(it' + 5) } 4

The limit of the expression on the right-hand side as co'-- 0, y - 0 but with top/y finite gives the first five terms in the formal expansion of the right-hand side of the

Debye result (47). However, (49) is not true for co=0, since this case was expressly excluded in the derivation of (33).

It will not be difficult to extend (49) to the y6 and y7- approximations. On inspection we find in the y -approximation that for 1=1,2, 3, 4, 5 the number of terms

is 2' 1, and that the sum of the coefficients of the terms multiplying (-)'-Iyl is

(21+1)! 3.21-' 1!




It would appear from a comparison of the limiting value as ft -O0 of the normalised

autocorrelation function of cos 0(t) (cf. Lewis, McConnell and Scaife 1976 eq. (58); Ford, Lewis and McConnell 1976 eq. (83), (85) and Appendix) with the value of the

same function for free notation (cf. Hubbard 1972 eq. (6.18), (7.6), (7.9)), namely,

+ ( )1(21+ 1)! (kT2\

1-13. 21 -'

(21)! 1

that the above expression for the sum of the coefficients is true for all 1 and that

consequently the extension the extension of (49) to indefinitely higher powers of y will produce a divergent series.

Sack's calculations based on his equation II (3.14) led to a set of recurrence

relations which he solved by using continued fractions. He finally obtained 11 (3.19), namely,

P(*Q1) ia I + 2y + (3-{)I

PO ic' 1ic'2 2 +ioc' I + iol +2 + ico'

+ 4y + (5-134) I (50) - ly_ 4+ico'

3 + ico' 32i

Both the derivation of this result and the expansion of the continued fraction as a

power series in y involve arduous calculations. However, it has been checked by taking successive convergents that (49) and (50) agree as far as terms of order 4

A more direct comparison can be made between (49) and the result of calculations

based on a stochastic differential equation. To do this we recall briefly how P*(cO)/PO may be related to the autocorrelation function K(n(t) . n(O))>, where n(t) is a unit

vector through the centre of the spherical molecule in the direction of the dipole axis, n(O) is the value of n(t) at time zero when the relaxation process commences and the

brackets < > signify an ensemble average (cf. Scaife 1971, section 2 and further refer

ences there). When a field F(t) is operating for a long time, it will induce in a dielectric

a moment M(t), where

M(t)=J F(t-x) da(x dx, pdx,

a(t) being the response to unit field. Thus for F(t) =Fo ei't

M(t)=F0 eXt c(4c), where

a(c)={e- iwx-dax) dx, 0

the complex polarisability. For a static field Fo, a(c() = 4(O) and from our definitions of P*(co) and PO we have that

P*(Oc) ca(c@)

PO a(O) (51)




If m(t) is the moment induced by a field eo cos wt, the Kubo relation is

a(@o) = 3kT [Km2>- ito (m(t' + t) . m(t')) eiWt dI (52) 0

where the bar denotes time average. On account of the stationarity of the Brownian

motion

(m(t +t) . m(t'))=(m(t) . m(O))

Moreover, this time average is equal to the ensemble average <(m(t) . m(O))>. In the stochastic equation method one studies the rotational Brownian motion of a single spherical molecule of moment ft and for this m(t) is y n(t). Hence (52) may be

may be expressed as

o'Qo) =4T[lI -it <(n(t). n(O))> e 1 dt

from which we deduce that

a, (0) #2 3kT

We then conclude from (51) that

P*(60) X0

p = 1- ico <(n(t) . n(O))> e iof dt. (53)

0

The autocorrelation function was calculated by using an iteration solution of

dn(t) =((t ) A n(t)), (54) dt

where the angular velocity c(t) of the sphere is assumed to satisfy the stochastic equation

It d_+ IjoIdWf (55) dt dW

the term on the right-hand side being a couple due to random driving forces arising

from the environment. It was found that (Lewis, McConnell and Scaife 1976, section 2c)

J 2S {s3(s+ s2(s+f (s+2fl)

- {A27)3 ts4(s.+.3 +S3(s+flA3(S+2f

(56)

1 8 20 3

+( +

i +3...




The autocorrelation function <(n(t) . n(O))> is a power series in t that converges for all values of t, and it is also expressible as a power series in y. If we confine our

studies to a finite power of y, the Laplace transform of this power series is the sum of the Laplace transforms. Moreover, we can then legitimately obtain

0 <(n(t) . n(O))> e-X dt

by putting s=iw in the sum of the Laplace transforms. Doing this for (56) and substituting into (53) we get (49) as far as terms proportional to y3. Furtiermore, by employing an exponential operator and a graphical method calculations based on

(54) and (55) were extended so as to give to the right-hand side of (56) the additional

term (Ford, Lewis and McConnell 1976 eq. (85))

p2y4 16 72 108

(s5(s+0t s4(S+ f4(S+2fi) S3(S+fl4(s+ 2fi)2

80 100 + s (S ? f)(s?-2S23(s + 3/)2 + ?(s + f)2(s? 2p)2(s + 3f)2(S+ 4p)

54 120 +

2(S + fi)4(S +2)3 +

2(+)3(S+2f)3(S + 3/)

+ 880

s3(s+fi)3(s+21)2(s+ 3f)J

It is seen immediately that this leads to the y4-term of (49). The method of the

present paper is much less laborious than those based on the stochastic differential equation.

The above investigations were stimulated by correspondence and conversations with Professor R. A. Sack, University of Salford. I am indebted to Professors

J. T. Lewis and B. K. P. Scaife for helpful discussions.

References

Arnold, L. 1974 Stochastic Differential Equations: Theory and Applications. New York. John

Wiley and Sons.

Debye, P. 1929 Polar Molecules. New York. Dover Publications.

Ford, G. W. 1975 unpublished.

Ford, G. W., Lewis, J. T., and McConnell, J. 1976 Graphical Study of Rotational Brownian

Motion. Proc. R. Ir. Acad. 76 A (13), 117-143.

Hubbard, P. S. 1972 Rotational Brownian Motion. Phys. Rev. A 6, 2421-2433.

Kramers, H. A. 1940 Brownian Motion in a Field of Force and the Diffusion Model of Classical Reactions. Physica 7, 284-304.

Lewis, J. T., McConnell, J. and Scaife, B. K. P. 1976 Relaxation Effects in Rotational Brownian

Motion. Proc. R. Ir. Acad. 76 A (7), 43-69.

Sack, R. A. 1957a Relaxation Processes and Inertial Effects?I: Free Rotation about a Fixed

Axis. Proc. phys. Soc. B 79, 402-413, cited as I.

Sack, R. A. 1957b Relaxation Processes and Inertial Effects?II: Free Rotation in Space, Proc.

phys. Soc. B 70, 414-426, cited as n.

Scaife, B. K. P. 1971 Complex Permittivity. English Univ. Press.



CONTENTS PAGE

TOLAND (J. F.): 1. Asymptotic linearity and nonlinear eigenvalue problems-II 1

MCCONNELL (J.): - 2. Diffusion equation study of rotational Brownian motion . 13

MCGILL (P.): -

3. Constructing smooth measures on certain classes of paved sets 31

BATES (D. R.): - 4. Problems of communication with civilisations around other

stars . . . . . . . 45

LAFFEY (T. J.): - 5. Finite groups generated by a pair of unitary matrices with

normal sum . . . . . . 61

ARTHURS (A. M.): -

6. On variational principles and the hypercircle for boundary value problems . . . . . . 75

NOVERRAZ (PH.): 7. On topologies associated with Nachbin topology . . 85

HURLEY (T. C.): 8. Residual properties of groups determined by ideals . . 97

WICKSTEAD (A. W.): 9. The structure space of a Banach lattice-Il . . . 105

O'FARRELL (A. G.): -

10. Rational approximation in Lipschitz norms-I . . 113

CORRIGENDA

Page 26, line 8. For M11 read M1o.

Page 28, line 4. For notation read rotation.



diffusion equation study of rotational brownian motion

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