kramers equations

143

Advances in Molecular Relaxation and Intemction Processes, 23 (1982) 143-178 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

NmRICAL

SOLUTION

M. FERRARIO

Chemistry

OF FOKKER/PLANCK/KRAMXRS

EQUATIONS

and M. W. EVANS

Department,

University

College

Trinity

College,

of

Wales,

Aberystwyth

SY23 lNE, Wales

(Gt. Britain) and W. T. COFFEY

School of Engineering,

(Received

26 March

Dublin

2, Ireland.

1982)

ABSTRACT

Some Fokker/Planck/Kramers numerically

for autocorrelation

that uncritical neglect

effects

by Evans

of current

functions

use of these equations

of memory

discussed

equations

inherent

interest

and spectra.

It is demonstrated

should be avoided

in their make-up.

(1976) does this type of equation

are solved

because

of the

Only in the case produce

realistic

spectra,

and then only over a limited range of temperature

and viscosity.

proceed

of this type is to use

in problems

molecular

dynamics

involving

molecular

diffusion

The way to

simulation

INTRODUCTION

In a series of articles J1-5J the relation

of spectroscopic

state and related infra-red frequencies

condensed

spectroscopy

profiles phases.

been attempting

to molecular

In particular,

with dielectric

has provided

0378-4487/82/0000-0000/$02.75

we have recently

measurements

us with a sensitive

means

motions

to describe

in the liquid

the combination at microwave of evaluating

0 1982 Elsevier Scientific Publishing Company

of far

and lower probability

144 diffusion

equations

of the Fokker-Planck-Kramers

descriptions

of the motion

of a particle,

disturbances

of stochastic

origin,

to a superimposed various

potential

energy.

of molecular

infra-red)

spectrum

frequencies

We have developed for comparison

specific

doing we have assumed from intermolecular Potential

bandshapes

that the potential

leading

course complicated, each molecule

depending

but progress

In this context

a harmonic

Coffey et al. 19,

potential

and evaluated

wide range of conditions. reproducing

also the basic features

absorption,

but not in describing

frequencies

in the peak absorption Coffey

Calderwood,

theory by developing or oscillation

rotational

et al.18.g a model

(i.o), because

previously

torsional

intermolecular manner

potential

arises

oscillation, is of of

in detail by

by Reid et al. [lo] under a

? (cm max

is successful

spectroscopy, observed -1

in

the Poley

shift to higher

).

upon this aspect of the

of the FPK formalism the dynamical

In so

the use of simple functions.

the experimentally

have improved

liquids,

on the structure

that this theory

frequency

[8]

For example:

of far infra-red

as rigid entity.

pairs with

scattering.

known as itinerant

equations

libration

are those of a librator

bound and encaged with a group of neighbours diffusion

functions

as a perturbation.

to molecular

experimentally

[7].

in the FPK equation

kinds.

through

the

theory is applicable,

has been discussed

It emerges

of physics

FPK equations

appearing

in an intricate

is possible

stressing

from static to THz (or far

and neutron

of various

well interaction,

to

of a series of dipolar

to Rayleigh

interactions

and also

transform

field can be regarded

The form of the resultant

or libration.

harmonically

liquids

forms of the general

the analysis

to

autocorrelation

that linear response

with the zero-THz

and have extended

motion

appeared

These are Fourier

of dipolar

measuring

subjected

in many branches

from FPK formalism

provided

i.e. that the external

(i)

have recently

dipole reorientation.

the electromagnetic

These are

The latter may be attributed

of the FPK type of equation

In this article we compute

or molecule,

such as in Brownian

causes and several articles

importance

(FPK) type [6].

undergoing

This is an FPK system in lR4 space

145 where the intermolecular In its simplest

form this model

and also of defining because

potential

properly

the encaged molecule

neighbours.

Evans,

relationship

with

is capable

is shielded

Grigolini

the Mori continued its relation

subjected

dipole-dipole

to internal

paper we use FPK formalism

(ii)

intermolecular

Zero-THz

dynamics

potential

spectroscopy

have recently to explain nematogen

the loss features

a hydrodynamic

n-heptyl

director.

SECTION

I:

-dp=d dt

equation

(Fo-:e-my:o)

Langevin

form.

for use with an

similar

to that of Coffey

and MHz frequencies

Chaturvedi

OF KRAMERS

is written

density

the molecular

1131

assuming

and Budo

of the the presence

potential

and Shibata

of

was cosine,

[14] using

have shown how this restriction

may be lifted.

EQUATIONS in this field the

as:

+&(PP)

--

molecules

In this

to investigate

(7CB) without

-(I-l) m

P

where p is the particle

of cosinal

1151 often quoted by authors

&

of the particle

equation

biphenyl

SOLUTION

its

form.

constraints,

In a paper by Brinkman Fokker-Planck-Kramers

theory of diffusing

the i.o. equations

at microwave

Mori formalism,

NUMERICAL

interaction

by its

it is an approximation

In this work the intermolecular

but used with geometrical convolutionless

with Budo's

essentially

in detail

In this respect Evans and Price

used a Smoluchowski

4-cyano-4

of which

can be used incisively

of liquid crystals.

collisions

1111 have discussed

fraction

to generalise

the shift in imax

mean square torque

from impulsive

and Ferrario

to be harmonic.

of reproducing

the intermolecular

Coffey 1121 discussed

arbitrary

well is again assumed

2 in phase space 1R (q,p), p being

P and q its coordinate,

Eqn.

(1.1) is derived

the momentum

from the

equation:

dp = F(q) + A(t) - (I - 2); dt where V is the potential

rni = p;

energy of P.

F(q) = -dV - (I - 3) q This is the basic equation

with which

146 we shall be concerned order

but we shall change Brinkman's

to concur with that more commonly

equation

slightly

used in contemporary

in

articles.

Accordingly: W(q,p,t);

T ~ B

We shall be interested

-i

, F ~ -~V/~q

in evaluating

- (I - 4) numerically

averages

of the type

p In

the

notation

-~w=~ ~p

-~

eqn.

(I

-

1)

becomes

~-i ~p

E n=O

the He n are the Hermite We stipulate

(~)

+~

B-I

= exp ~ _ p L i~

determined. Eqns.

4),

solution may be written

W(q,p,t) where

-

-~vW-~p-~r~w

~t whose

(I

(i

~

5 ~

as: He

P

~n (q,t)

- (I - 6)

n polynomials

and the ~n are functions

that ~n = O for n
(I.6) and (I.l) with the use of Hen(X)

recurrence

to be

n = O,1,2,3...

relations

result

in

the scheme: ~n

+ n

3t

~n =

F

B-I

~n-I - c k T ) ½~*n-I - ( n

~

m

~q

where m is the mass of the particle. difference

~t

equations, ! m

+ i) C k T ) ½ ~*n+l - (I - 7) m

The set

(I.7)

~q

is one of differential

e.g. for n = O, n = 1

~q -

(I

-

8)

!

~t This

~-I

(mkT) ½

is a complicated

when the potential W(q,p,o) which

is meant

~q

structural

?q

problem which

V is at all involved.

= exp (-p2/2mkT)

Here fo(q)

is best tackled

The initial

numerically

conditions

fo(q)

chosen are:

- (I - 9)

to cover all possible

be interested. only.

m

physical

represents

contingencies

the initial

in which we shall

distribution

in q space

Clearly:

W(q,p,o)

= exp

Of particular

(-p2/2mkT)

interest

Heo~o(q,o);

to zero-THz

('"

spectroscopy

He O = I) is rotational

- (I - IO) diffusion.

147 In order to keep things consider Kramers &n

diffusion equation

+ n

Yn =

4

i-t

on a circular

and

track so that q = 8 and our Fokker-Planck

F

Pn_l - kT ' &'_l (-1 I 4Iz?!do

- (I - 11)

- (n + 1)

p-i%:)+

$",

B_1

%

ap

the Euler equations

becomes:

-dvw-w

- bw = cl

simple we shall linearise

. Here I is the scalar moment

where p = IO. corresponding

.. IO

equation

process

[q

molecule.

oscillator

of Calderwood

model [l-3,16]

governing

the stochastic

torque experienced

When F(B) = Iate the model reduces

by the librating model

(I - 12)

t>o-

= I;(t);

where B is the Wiener

m-diffusion

The Langevin

to this is

I& + F(0)

+

of inertia.

.

, which is closely related to Gordon's

et al. I9J

We suppose

to the harmonic

that static conditions

have prevailed

up to t = 0, i.e. for t < 0: 1;

+ I&

+ F(B) + pEsine = Ii(t)

To cover the itinerant be constructed, greater

detail

In general,

= J2n 0

=I 0 because

2n

librator

i.e. involving

(I - 13)

-

and Budo formalism the FPK system in IR4 must . . IR4(el e2 f31 e2 t) and this is discussed in

I

,

,

I

later. for the present J”,

eiq

elq y.

case:

W dq dp

dq/oJ al

yo

2n eiq

‘oJ

dq dp

- (I - 14)

dq

of the normalisation: 3

J~_e-~‘

When dealing
i0

Hen (x) Hem (x)dx = 4277 n! 6 m.n with

the scalar angular

> = J2nei%'0 0

The problem

d0/OJ2'Yn

is to calculate

d0;

YO(B,t)

coordinate

8 we have:

YO E YO (0,t) for any periodic

- (I - 15) F(0) of eqn.

(Z-13).

148 We have initially: W(0, I&

0) I W(0, p, 0) = exp C-p*/*IkT)

(1 + g

cos 0)e-Bv(')

so that: Y,(w)

= (1 + JJJ cos 0) exp( - fN(o)) = Yo(O, 0) kT

To simplify

eqn. (I - 11) we write:

l=m i10 Y,(O, t) = z ,F=F A;(t)e l=-cn

and use the implications

d$; + ; A: =

(IkT)

-1

FrexrO r=-‘x

of orthogonality

to write:

m

n-1(t);=__ Fr Al_r

(kT/I)'ilfAy-l(t)

+ (n + l)Ay+'(t :) 1 - (I - 16)

so that:

.

m

<.i@,

=

I

0

*n c

.

lloelodo

Ay(

1=--m

- (I - 17)

2nm i 12_ A"(t)eilOdO ml The numerator


i0

is finite only for 1 = - 1 and the denominator 2ll

,=J

Azl(t)e

only for 1 = 0:

i0 -iO,,,~*'A~(t)d, e

0

= A"1 (t)/AE(t) If we now take the special case of: F = IW: sin@ = 1‘0% (el' - emi0)/2i this leads to the recurrence

I 2w2

- (kT)'

il (A:-'(t)

- (I - 18)

relations:

n-l . n-l l(A1+1 - Al-l)

+ (n + l)Ay+'(t))

- (I - 19)

149

Numerical

Solution

This is quite

of Eqn.(I

- 19):

straightforward

if we write

eqn.

(I - 19) in the matrix - (I - 20)

J\(t) = -C A(t) = -where A(t)

=

The formal

solution

A(t) ZZ

=

is:

- (I - 21)

exp(zt)A(o) zz

and for numerical

purposes

we must restrict

If we take the left and right eigenvectors and x:

( = 1, .. ..N) respectively,

b(o) = Ccr.X., where i LZl

c x. =hiXi, __ ew (gt)zi

form:

ourselves

of the matrix

with eigenvalues

ui = (ET, A(o)) =

to N components C: =

Xi

of A. =

(i = 1, . . ...N)

Xi, we may write:

the scalar product,

then we have:

which means:

= exP(X;t)~;

,

We finally

get: A;t A(t) = Ze 1 i

This is valid

- (I - 22)

only if the matrix 2 has a complete

set of eigenvectors,

e.g. if: (I - 23) Eqn.

(I - 19) is now amenable

to solution

Next in this section we describe equations restriction

the method

of Coffey and Calderwood of harmonic

potential

using methods

of generalising

191 and Coffey force.

of diagonalisation. itinerant

et al 181 to remove

oscillator the

150 FOKKER-PLANCK

KRAMERS

EQUATION

In this case the Langevin libration

are, for a harmonic

FOR THE ITINERANT equations

OSCILLATOR

of motion

for scalar angular

itinerant

potential, .

Il;;lW +IIBl

up)

-

12qY2(t)

yt))

-

= IlWl(t) - (I - 24)

Y2w

I2

+ 12B2 G2(t) + 12WZ ( Y2(0

These reduce

to the equations

The physical

meaning

friction

-

Eqns.

the generalisation

the harmonic

_aE+ at =

B2

in the literature

[8,9].

of the term:

force between

corresponding

a_&f) + a bik - a (wS(JI~aii2

31 . (Ji2f)

+

the inner molecule

the Poley absorption

The FPK equation

a

Y2(t) and Pi(t) and of the

systems by Reid and Evans

that generally

theoretically.

B2 = o.

[lo].

81

&f)

a

for about

They have the dis-

is too sharply defined to eqn. (1.24) is [ll]

w)

=

Gl

z

and the diffusing

B2 = o have been tested out experimentally

(I - 24) with

fifty liquid and glassy advantage

191 when

Y,(O)

which represents cage.

functions

Bl and e2 are fully explained

We wish to accomplish

I&Y2W

of Coffey and Calderwood

of the angular

coefficients

- ui1(t)) = Iqi2(t)

a

(Qfp2 -

:

$l)f)

66

+ 82 kT a2f + 81 g

-a7G

I1

5%

a2f

(I - 25)

an

with initial condition:

. f($l, $2, 111, $2, o) = A exp

and is soluble

analytically.

for a distribution ;I;w

+ L

;2w

12

concept

($2

-

$1)

=

our problem

- Ji1j2)

is to solve the FPK equation

the Langevin

equations:

iz(t)

(I - 26)

. ($2

-

$1)

=

Bl(t)

I1

where B1 and B2 are Wiener oscillator

describing

(I,;; + I ;2 + 12WZ($2 22

I2

.. @l(t) + +_ jlI(t) -v' 1

However

function + Jy

- 1 2kT

processes.

these equations

As well as generalising

also describe

and generalise

the itinerant the Budo

151 theory

1121 of the molecular

rotating

dipolar

the Budo/Coffey common

groups.

If these dipoles

coefficient

of the surroundings,

while

and ~2 again arising

from Brownian

aw +

at

solved

G2 aw + $,aw -

subject

field E.

their

In eqn.

of inertia of each equally

arising

from the Brownian

sized

movement acting on uI

motion. in IR4 corresponding

the FPK equation

to the system

(I = II = I )

This is:

+ 1 (c$3

about an axis through

Al(t) and AZ(t) are the random couples

{12} has derived

xJ2

by uI and u* then in

the measuring

I is the moment

and 5 is the friction

(I - 26).

containing

$1 and $2 are the angles l.11and u2 make with e, the

of E, at any time.

Coffey

of molecules

are denoted

to the plane containing

(I. 26) subsequently

dipole

of an assembly

theory they are compelled to rotate

centre normal

direction

dynamics

(aw av

1

Xl

+ aw

av ) =

$2 aw2 VJl ajll

1

1

(a

G2

chw + $_* 6iJlW

+ c$, a2w )

a*w + (c$+ + c$,)

a2w

ah

aqlw2

to the initial

1.

(I - 27)

w

conditions:

W($l, $2, $1, $2, 0) = A exp { - @{Ii2 ($I* + 4~~) + "($2 - $1) (I - 28)

+ (VI cos $1 + u2 cos $2) E 31 In eqn.

(I - 27) C$,, C$+, C<,, and C?+, are constants

equilibrium

Boltzmann

aw+$,aw

at =L

{a

I

+Gl aJi2 (iJ*W)

distribution.

w

- 1 ( w T

%I +

32

a

s

polynomials

30,o= fg(x,n)

+ aw

q2

-%I

i-

qTy*

av ) aJil

(I - 29)

aJ,1*

(I - 29) may be manipulated

into the set of differential - (k$

&,I+

%O) an

using two dimensional

difference

equations:

;

$m n + (2IkT) -1 8" $m n_l ,

an

(I - 30)

’

% + (kT+ Ti:

=o

(m + 1) "m+l,n

ax

from the

eqn. (I - 27) becomes:

($,W) + kT (a*w + a2w )

ax {s + 4 (m + n) I

av

q*

31

Coffey has shown how eqn. Hermite

Thereby

to be determined

"m + (n + 1) _,

an

n+l + a5m-l 9n + -’a$m n-l}

ax

a17

152 where

x = ($1 + +2)/2;

W(x,n,i,&O)

n

= exp I- BI (i2 + r&1

the probability

density

and f

= ($2 - $1)/2;

function

fo(X.ri)

being given by:

= exP { - IB(x2 + n2)J mfn Hemen ((z)';,

8

is defined by:

@&)";I) $

kT

‘;; hbs).

We wish to solve eqns.

q

Details

of the Numerical

Problem

in IR2 Space

at

P(n]n,,

t) = L

F

equation

P(nlno,

PO = N'exp Witi, L = exp

(I-31)

we obtain

=

Iw,el

solution:

{ - d/kT 1;

{d/Lk’l’)

is solved in the form:

t)

Id with the equilibrium

LF

4 = Io2/2 + V(O).

the equation:

as the equilibrium = exp

P(nlnn

=

Klz(t)

with the initial

condition

and Vollmer

, t)IJN'

(I - 33)

= /QJ1cn, 0)02(n,

function may be expressed

t)an

fI

{71 obtain the differential ai?

(I - 35) operator

(kT

- 2) 4

in the equation: Al (t) 1 ?Y

Hen

34)

of the equation:

w + a-v'(o) + 8(; + a2

n=o q = -m

as:

Oi(Q, o) = W i~o(n)

au

which results

= v%'exp ( - d/2kT)

t ion and:

Qi(Q, t) is the solution

L = - k

n Co

to show that any correlation


Risken

distribu

d/2

It is possible

where

(I - 32)

as the transformation

exp I- $/2kT}

with

7 C

for $.

Solution

In this case the Kramers a

(I-30) numerically

(w) &'

exp( - d/2kT + iqe)

= 1, I = 1) (I - 36)

153 and the recursion a

relations:

Ai (t) = - n PA: (t) - iq/z

A:+'(t)

at + 67

Y

irf r A II:(t)

- iqin Ai-l(t)

(I - 37)

r=-m with

V(0)

Y

=

fr exp(iqr)

-

(I - 38)

r = -m The two-time

correlation

Ku(t) = ;

K&lo

may now be computed. function, l-5

) andqos

intensity

is related

spectrum

red, optical

These are of spectroscopic

20(t)cos

d cos@(t) d -dt x

=

is related

us0

coefficient

because autocorrelation

by Fourier light.

transformation

The drivative

Fourier

transform

a(w) (in neper cm -1 ).

= &'

= co~(lO)~~

r = - wsin0;0

to the

function

of the far infraWe have: (I - 39)

Hel(w)

exp ( - d/2kT);

(t,w,e) dud0

= (&'/2)(el"

(I - 40) + e-ll')Heg

e

-$/2kT

and Kd(t) = IQd(o,w,0)Od(t,W,B)dwd8 wahQd(o,w,B)

transformation

(t,w,O)dwdO

= J~(osl~(o,~,~)~ccs10

with Qwl,(o,~,O)

d/dtosO(t)/t=o>

loss,E_(w),by Fourier

t = o> is the direct

power absorption

interest

of the orientational

of depolarisedscattered

cos@(t)I

with Qu(o,w,O)

analogue

to the dielectric

20(o)>

Kw(t) = I@o(o,w,O)Ow

K aosl0(t)

= <(oslO(t)caslO(o)> Kd =
is the planar

which

kd= (F&-S e(t)$-~~~(t)~~~~ )

functions

(I - 41) (I - 42)

= JN'/(2i)(eiq'_e

-iqyRe

1 e-b/2kT

(I - 43)

By defining: exp i- V'kTJ

it is possible

= .I,

g2 exp(ir0)

(see Risken

and Vollmer)

to show:

Ku(t) = J_ ? A;(t)g -9 go q=rm

(I - 44)

A;(o) = 6n 1 6

(I - 45)

with

,

q90

154 By explicit

KcoslO + x

calculation

i10 + e-ilO)Hg

t) = Jni,dB Jmmdu

Z A;(t) n,q

=/

we have:

N' CI

1 Hen(w)exp J;;r il0 + e-i1O

71

a-V/kT

(e (I - 46)

(iq0) exp(w2/2A2)

C A'(t)e lqO de = 1

) exp(-V(O)/kT)

.

q q with the initial

y32

y A”W[g_q_~ q=-03q

condition:

A;(O) =

1 6n

Ka(t) =

J”#e {mm&o{-o sinBNYe-V'kTe-w2i2A2

I

3

o

(6

q,l

(I - 47)

+ &4,-l)

and

Jo N, { (,iB _ e-ie)e-V(e)/kT

Z An(t) n,q q

6 Ai(t)elqe

e iqe 1 72

I

1 de

“2i 1/(2igo) with the initial

t=_,

A:(t)

{g_l_q

- g1-,I

condition:

A;(O) = - 1 rSn 1 (6 q,l - 6q,-1) Z'

The Budo Model with Arbitrary In this case we have {a} =

(I - 49)

Potential

(f(e)) (IR3 space)

;

w1 = "l/Al

e = $1-$J2;

w2 = $,fA,

ol,w2,8

and LP = - (Alwl - A,w,)&

+ &

(Al&

- A2k 1

+ 61 (a

Wl '& 2,

awl

+ 62 (a

1

aw2

and pO

= N exp {-w$/2 - wi/2 - V(B)/kT)

i.e. I$= $12

(I - 48)

+ $12

+ V(B)/kT

)V'(e) 2

w2 '& 2, 2

gmq+d

155 The operator L = -

L becomes

(A1wl - A202)

a/a8 + l/kT(Ala/awl

2 2 + 81(1/2 + a /UW~ -

(,@I2

e 3’2

a

=

x1 The numerical

solution

+ 13~(1/2 + a2/a$

$14)

a

wi

-

x1

- A2 a/ao,)V-(0)

e-+‘2

;

T

now proceeds

=

a

- l&4); _

as

v-(e)) -*

2kT

via the expansion

of the functions:

Cn1'"2(t,~)&Pe-v(e)'2kT~nl,n2> = Y n =O, 1 n =O 2 where we use a quantum-like notation: ~i(e,W1'W2,t)

InlPn2> = (4n2nl!n2!)-1'2Henl(til)

which

exploits

immediate

the properties

Hen2(w2)exp{-

of Hermite

uf i '$1

polynomials

in the following

way:

a/awllnl,n2>

= 1 (Jn,ln, - 1, n2> - Jn, + l'jnl + 1, nz>)

a/aw21nlrn2>

= 1 (Jn2lnl,n2

wllnl,n2>

From

1

+ x

)

+ 1, nz>) + l>)

the foil owing differential

= - Ai(mlnl

1 nl, n2 + l> + 62

Al (m

A1/(2kT)(Jnl c"l nJ ;e,

t)J,-

Cnl"'(O,t)&

+ 1, n2> + GInI

I "1, n2 -1 >)

In1 -1, n2> - -1

nl+

1, nz>)V'@)

exp (-V(@)/(2kT)) n2-l> -ml

"1. n2 + l>)V'(O)

exp(-V(O)/ZkT))

- (nl 61+ n2f32)Cn1n2(8,t)JN'e -V(@)'(2kT)jn1,n2> Summations

equation.

:

- V'/(2kT)Cn1n2(0,t))&ce-V(S)'2kT

- A2/(2kT)(alnl, x

+I>

(Cn1,"2(0,t) - V'/(2kT)Cn1n2(8,t))&Pe-V(B)'2kT +

K

(0,t)

nl,nz

nl,n2

C +%,t)

nl,n2> -v@)2kTJNHI

a/at Cnln2(0,t)e x

-l> + Jn2+TI

for the coefficient

a/at $i= L $.

fCnin2

Jn,+fl

-

I n1 - 1, 132> + Jnl+l'lnl

= (Jnl

u2jnl n2> = (JnJnl,n; We obtain

-1’

over nl and n2 are understood

in the above formula.

- l,nz>

156 We obtain for the coefficient

+

A2(4n?2Cn4n2 -1(&t)

Cnl' "'(0,t):

+ Jn2 +l'

Cnl,n2 + 1

+

(Al/kT)J"1+1Cn1+1+2

(a,t)V'(o)

-

(A2/kT)Jn2

(O,t)V'(o)

-

(n1B1 + n2R2)Cn1"2(0,t)

+ l'CnlP"L+l

(0,t))

The next step is to expand Cnlyn z(O,t) over a set of functions By denoting:

Iq> = e

The properties

iq0

we have Cnl' "2(0,t)

=

of 0.

y A;lnz(t)Iq> q=--m

of Iq> are

a_ Iq> = iqlq> ao

and V'(O)Iq>

=-_ id (14 + l> - /q 2 For V(0) = ia cos 0, where 8 is a constant,

= a(eie _e-ie)/(2i);

By inserting

the equation

and

we have V'(B) = 8 sin0

v’(e)eiqe = _ $(ei(q+l)e

_e-i(q-l)e)e

"l'"2 for C (8.t) we obtain n -1,n

a/at

A n1'n2(t) \q> = -Al (Jn,(iq) Aql q

2(t)l

n +l,n (iq) Aql 2(t)lq>)

q> + v

,n +l 2 (t) q>) + (- id Al)-2 kT n +l,n n1,n2+1 1 2(t) (14 + l> - \q - l>)) - (- id A,) v (Aq x (A (t) q -2 kT n1,n2-1

+ A,(dn2(iq)A

(t)/q> + Jn2+l'(iq)

q

An1 q

"l'"2 x (jq + 1> - lq - l>)) - (nlP1 + n B )A (t) /q> 22 q "l'"2 Finally we have an equation for A (t) which q n -1,n "l'n2 (t) = -iq Al/n1 Aql 2(t) - iq AIFAql a/at A q

nl,n2-1 +iqA/n

A

n1,n2+1 (t) + iq A2w

2 2 q n +l,n 1 -A 2(t)I + i ~~[An~~"'~t) q+l The correlation

is numerically soluble: n +l,n 2(t)

functions

of spectroscopic

n +l,n (t) - izF{Aqll

- A~~;n2"(t)~ interest

- (nlPl + n2B2) <"n2(t)

are now given in terms of

n,*n, AL L(t):

K:_(t)

= = _~m_~m&j~/~B~~l$0e-V/(2kT)JN'~nl,*2,q>~

1 =

(l/go)

4

lpowg_q

Aq

%t)

157 with the initial

**lyn2(0) q

condition:

= 6 nl,ldq,06n2'0

0 q i'l(t)g_,; with i.c. Az1'n2(0) = 6n1,0 bn2,1 6q,0 KU2(t) = 1'g z * K cos18(t) = +, q" A;"(t) (g-,-l + g-q+l) 0 "l'"2 with initial condition A (0) = I, 6 nl.0 'n2,O @q , 1 + 6q,-l) q Ka(t)

=

-

iig

;[A;"(t)

- *;'l(t)l

(g-,-l - g-q+l)'

0 with ArL'n2 (0) = - ii

+6

(6

y.0

Checks on the Numerical

Evaluation:

limits of the Correlation

(a) Equation

in IR

2

Po(O,w)

y,l

",,,o) (6q91 - 6q9-1)

Normalisation

Factors

and t + -

Functions

space

The normalisation distribution

6n2,1

factors may be calculated

from the equilibrium

function:

= N exp C-w'/2 - V(e)},

(N-l = /iexp

(x*/2

(kT = 1)

- V(f3))&8w)

We set once more V(e) = &0se.

The initial

condition

on the Kramers

equation

is

p2(e(t), w(t), t.le(o), W(O), 0) = sbdt) - do)) The limiting value distribution functions,

functions because

6(8(t) - e(o))

for t-)comay easily be calculated may be replaced

by the product

there is no correlation

t + m, so: lim w2(e(t), w(t), t ;

e(o), do), 0) -tm = Po(e(t). w(t)) PO (e(o), ~(0))

(PO independent

of t).

between

because

the two-time

of the two PO equilibrium

the system at t = 0 and at

158 So we obtain:

= NJJw2exp(-02/2



=

$,(I

=

1/(2go)

where g

n

+

=

N/~COS2ee~p(-w2/2

Cos2e)exp{&ose}

Igo

+

+

de

Bessel

function

=

= N J/w2sin2eexp

+

1/(2go)Igo

+

order

(n).

ik0se)&3

=

{-w2/2 + acose} &&I

lik0sel

ae/iexp

tacose}

ae

(9, - 82)

of the Kramers

in the development

potential.

of a molecular

These

limits may be expressed

lim t+-



a.c.f.' s are generally

theory of the mesomorphic

by:

tie),

o)

d@(t)d0(o)dtS(t)Wo)

= /Po(Ol,bb~)g~(0~,~)d0~~1~Po(02,tP2)g2(0z,~2)d02d~2 /( IPo(0,~)dOd(d)2 so we obtain: = 0, because

lim t-F YE 3-F lim t"

JUPo(O,O)dOdW=

o

=

from

importance

state of matter.

/ W2( )dO(o) . .. . . du(o)



different

This fact is of paramount

= lim IWz(@(t), a(t),t; t"

x g1(o(t),qt))g2(O(o),6$(o))

lim t+=

In addition:

g2nl

The limits t + m of the orientational zero because

of integer

9,)

at at

= 1/(2go)

de&!

{kkose} Be = I,(d)

= lflcos22eexp(-u2/2



= isin28exp

acose)

821

= Jcosneexp

1/(2go)Igo

+

= 1

Beliexp{ikoSe}

In (z) being the modified

=

+ V(e))&&

= o

159 (b) Budo's Model,

Cosine Potential

In this case: P,(@l,Y,O)

= exp

+ &co,

[z$-%

. .. .

0 //exp ]

dWld%dO

so we obtain +(0)(4(o)>

= Af; = = 1/(2go)[go




=

and

&

cosOw)t=o

cosOw~t~o>

[<81(~)6)1(o)>

+

=(g + A$]

l/(ko)

lim
:E

-02(o))

sinC(o)(6al(o)-#(o))sinO(o)>


+(o)tJ~(o)>]

k 0 - 9,)

0 ) > = liar<6)2(t)ti2(0)> = 0 t*

< d$osO(t)

d$oso(t)~t=o)=

0;

= (g,/g,)'.

and Discussion

The numerical

results

are illustrated

model and the more complicated feature

+ g2n-J =
lim < cos nO(t)cosnS(o)> t"

Results

A$

is that in neither

autocorrelation

in figs.

set of equations

case is the velocity

well-defined

as t+ o.

not zero, and can be explained

(1) to (3) for both the IR‘

in 1R3 space.

An important

(or angular velocity)

This is because

the initial

slope is

as follows:

(a) lR2 Space In this case eqn. (I-2) h as no memory friction

coefficient.

autocorrelation which begins:

A direct mathematical

function

of the velocity 2

1 - 8 t + C(t ) where

(b) The Budo equations

of neutron

matrix

is that the

or angular velocity

S is the friction

equation

scattering,

apart from the delta function

consequence

(I-24) can be regarded

of the integro-differential in the context

kernel

has a Taylor

frequency

expansion

factor.

as a zeroth order approximant

develop&

by Damle et al 1171

i.e. the associated

memory

matrix

has

160

FIGURE

1

IR* model,

normalised

values of friction behaviour.

autocorrelation

coefficient

Abscissa

functions

B and potential

in reduced

and spectra

strength

time units of & [)

Solid like

d.

1 where

for various

I is the moment

of inertia. a) Angular

velocity

autocorrelation

function. 1

-----_---_----- 6 = 0.05 reduced -_-_---_ -_-

-

-

units of F (1

, d = 0.25.

= 0.05, d = 0.50 _

-

- = 0.05, d = 1.00

-

-

= 0.05, de2.00

b) and c). Abscissa

frequency

Real and imaginary

in reduced

d) rotational

frequency

velocity

the orientational

parts of the angular velocity

spectrum

units

autocorrelation

autocorrelation

function,

function.

the second derivative

of

161

(bl,

5.0 4.5 4.0 3.5 3.0 2. 5 2.0 1.5 1.0 0.5 0.0

0.8

0.6

0.4

0.2

0.0 (

-0.2

-a,

162

1.2 1.0 0.8 0. *

0.

c

0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 4.2

2. ‘ 2. * 2.0 l.0 1.6 I.‘ l.2 7.0 0.8 0.d 0. ‘ 0.2 0. 0

e)

and f).

Far infra-red

respectively. librations

dispersion

Note the multiple

and power absorption

peak structure

in the solid state.

g) h)and i).

Fourier

transform

components

of g)

j) k) and 1).

Fourier

transform

components

of j).

transform

components

of m).

m) n) and 0).

Fourier

reminiscent

coefficient of lattice

163

0.8

0.6

0.4

0.2

0. 0

2

4

6

8

-0.2

-0.4

2.0 1.8 l.d i.‘ 7.Z 1.0 0.8 0.d 0. I 0.2 0.0

5. 0 4. 5

(i)

164

0.8

0.6

0.4

0.2

0.0

1

2

3

4

5

6

-0.2

-0.4

(j)

I.8 1.6 1.4

7.2 1.0

0.8 0.6 0.1 0.2 0. c

165

-0.2

t

-0.4

(Ill

2.0 1.8 1.6 1.4 1.2 ?.O 0.8 0.6 0.‘ 0.2 0.0

4.0 3. 5

3.0 2.5 2.0 f.5 1.0 0.: o.,

0

1

2

3

4

5

6

166 FIGURE

2

B = 0.5

Intermediate

damping,

as for figure 1.

d = 0.25 d = 0.5 d = 1.0 d = 2.0

0.2

‘“____,L____-----

\ \

-0.2

\

//’

\ \

/’

\

/’

\ 4.‘

-

‘\ ‘-__*

,/’ (0)

167

0.8

-0.4

-

-

0.20

0.15 0.10 0. OS 0.00

-0. OS -0.10

-0.15

/

\

‘./’

/

168

0.8

0.6

0.4

0.2

0.0

-0.2

-0.4

(9)

0.40

-

0.35

-

0.30

-

(h)

-----________

0

2

3

4

5

*

169

0.8 0.7 0.6 0.5 0. I 0.3 0.2 0.1 0.0

0.8

0.6

0.4

0.2

0.0

-0.2

-0.1

0.22 0.20 0. 18 0. 16 0. 14 0. 12 0. 10 0.08 0.06 0.04 0.02 0.00

170

0.20 0.15

-

0.10

-

0.05

-

0.W

0.

0

---________---

I

2

3

5

4

6

e

0.6

0.4

4

0.2

0.0

00

0.5

1.0

1.5

2.0

3.0

2.5

3.5

-0.2

-0.4

(ll)

0.20 0. 18 0.16 0.14 0.12 0.10

_---------.

0.08 0.06 0.04 0.02 0.00 0

1

2

3

4

5

*

171

delta function angular

off-diagonal

velocity

correlation

1 - 62 t + 0(t2),

function

This is when the model fraction

Both the models

considered

Kramers/Fokker/Planck

here are therefore

equation

the description

of phenomena

superconduction

and second-order

equations,

but in physical

in this context vitreous

revealed

temperature

viscous

with,

phase

the mathematical terms the most

describe

and spectral

the spectral

features

peaks

theoretically.

situation

by virtue

With

Josephson

point for tunnelling,

of the Kramers

experimental

technique

of ultra-viscous

loss the spectrum three times

of frequency.

elsewhere

to be used

in laser technology.

and of low

(a, B and y) over At room temperatures

altered.

in the latter case fairly

peaks can be produced is improved

incisive

are drastically

(as described

that the

17J as the starting

for example,

spectroscopy

solutions

features

fail at high viscosities

first derived

flawed except

therefore

limitations

In terms of dielectric

molecular

fundamentally

loop analysis

a very wide range of about twelve decades the viscosity

the correct

they are based continues

literature

connected

is that of zero-THz

media 1101.

or

is also

to the version

It is surprising

on which

in much of the physical

We have already

reduces

of the velocity

form by Evans IS(l)].

in the case B2 = o for the second.

uncritically

from this model

i.e. only in the case b2 = o do we recover

initial behaviour. in Mori continued

The initial behaviour

elements.

Evans equations

satisfactorily,

1107) b ecause

the more complicated

of the fact that the theoretical

but

only two loss case B2# o the y peak may

172 FIGURE 3

case, as for figure

damped

Heavily

1.

b = 2.0

d = 0.25 d = 0.50

0.8

0.6

0.4

0.2

a0

,

-0.2

-0.4

0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0. 10

----____

0.05 0.00

0

1

2

3

1

2

3

4

5

6

4

5

6

1

d= 1.00 d= 2.00

0.0

0.6

0.1

0.2

0.0

-0.2

-0.4

0. IO 0.00

0.06 0.04 0.02 0.00 -0.02 -0.04 -0. w -0.0.9 -0.

(e)

IO

(f) 0

1

2

5

6

( h)

0.00

L.,/-

0

I

2

3

4

5

I

6

(i)

0.4

-

0. 0 0

,

2

3

4

5

6

175

0.8

0. *

0. 4

0.2

0.0

-0.2

-0.4

I

(1)

0.20 0.24 0.22 0.20 0.18 0.16 0.14 0. I2 0. IO 0.08 0.06 0.04 Y.

w<

0.05

0.45 0.40 0.35 0. 30 0.25 0. 20 0.15 0. IO 0.05

(k)

176

0.10 0.08 0.06 0.04 0.02

(n) 0. 00

0

1

1

I

2

3

I

5

2

3

4

5

02.3 0.2.5 0.21 0.2* 0.20 0. 1.5 0. 16 0. 10 0. I2 0. IO 0. 08 0. 06 0. 04 0.02 0.00

(0)

0

1

1

6

-1 but at extreme

now be broadened,

high frequencies

mathematical

flaw at t -t o manifests

transparency

which

To emphasize technique

itself in a theoretical

in detail

the shortcomings

is to use the far infra-red molecular

known as the Poley absorption of the dielectric

frequency

(rad set

infra-red

range

to static whose

return

to

is far too slow.

neper cm-l) of dipolar

version

(around 200 cr.; f the

-1

power absorption

This is sometimes

by a multiplication

by n(w) the refractive

high frequency

shape represents

(o(m)/

, but should always be considered

loss weighted

) and division

the most useful

coefficient

liquids and solutions. l-5

is the extreme

of both models

a molecular

by the angular The far

index.

limit of a spectrum dynamical

as a

evolution

extending extending

from ps onwards. In conclusion, form of potential zero-THz

therefore,

used in FPK equations

spectra when the underlying

(i.e. based on naive Fokker-Planck results

concepts

equation

by numerical

is irrelevant

mathematical

of the molecular

of the type considered

in other branches

phenomena

of physics

analysis

in the description

structure

dynamics).

of

is imperfect The Kramers

here succeeds

only because

that the

in describing

the data available

on these

do not cover a wide enough range of conditions.

Suggested (i)

we see clearly

improvements

may be listed as follows:

The Fokker-Planck-Kramers

other than a delta function.

equations

should be given a memory

This would rectify

the incorrect

function

behaviour

as

t + 0. (ii) where

Increased

these equations

dynamics (iii)

effots

simulations

Until memory

cannot be useful potential. harmonic

should be made to coordinate

are used as starting and broad-band

and inertial

in discriminating

the research

points with the results

in fields of molecular

spectroscopy.

effects

are involved

between

Only then will it be possible

properly

these equations

forms of the inter-molecular to progress

cosine forms, chosen only for reasons

from such crudities

of analytical

tractability.

as

178 Acknowledgement We thank the S.R.C. for continuing

financial

support

to MWE and MF.

REFERENCES M.W. Evans, Adv.Mol.Rel.Int.Proc., J.S. Rowlinson M.W. Evans, Reporter:

10 (1977) 203-271.

and M.W. Evans, A.Rep.Chem.Soc.A.

Dielectric

and Related

M. Davies)

Molecular

(The Chemical

M.W. Evans, A.R. Davies

(1975) p.5 ff.

Processes,

Society),

Vol. 3 (Senior

(1977) pp.1 - 44.

and G.J. Evans, Adv.Chem.Phys.,

(1980) in press

(ca. 300 pp. rev.). M.W. Evans, W.T. Coffey, Wiley-Interscience,

H.A. Kramers, H. Risken ibid.,

N.Y.,

"Dynamical

E. Nelson,

G.J. Evans and P. Grigolini, ca. 700 pp., in press,

Theories

Physica,

of the Brownian

Dynamics",

(1981).

Motion",

(1967), U.P. Princeton;

7 (1940) 284.

and H.D. Vollmer,

Z.Physik.B,

34B, 313;

35B (1979) 313;

33B, 297;

31B (1978) 209.

M.W. Evans,

Chem.Phys.Letters,

M.W. Evans and G.H. Wegdam,

39 (1976) 601;

J.Chem.Soc.

W.T. Coffey,

W.T. Coffey,

Faraday

M.W. Evans, W.T. Coffey and J.D. Pryce,

Trans.11,

and W.T. Coffey, Proc.R.Soc.A.,

10 C.J. Reid and M.W. Evans, J.Chem.Soc. (1980) in press;

Acta 35A (1979) 679;

Faraday

G.J. Evans,

74 (1978) 310;

Chem.Phys.Letters,

G.J. Evans and M.W. Evans, Mol.Phys.,

J.H. Calderwood

ibid,

"Molecular

63 (1979) 133;

38 (1979) 477.

356 (1977) 269. Trans.11,

75 (1979) 1218;

C.J. Reid, G.J. Evans and M.W. Evans,

Spectrochim.

C.J. Reid and M.W. Evans, Adv.Mol.Rel.Int.Proc.,

15

(1979) 281. 11 M.W. Evans, M. Ferrario in press;

and P. Grigolini,

ibid., Mol.Phys.,

12 W.T. Coffey,

Mol.Phys.,

15 H.C. Brinkman, 16 R. Gordon,

Adv.Magn.Res.,

17 P.S. Damle, A. Sjolander

Trans.11

(1980)

(1980) in press.

J.Chem.Soc.

and B. Shibata, Physica,

Faraday

38 (1979) 437.

13 M.W. Evans and A.H. Price, 14 S. Chaturvedi

J.Chem.Soc.

Faraday Trans.11

Z.Physik.,B,

(1980) in press.

38 (1979) 256.

22 (1956) 29. 3 (1968) 1. and K.S. Singwi, Phys.RBv.,

165 (1968) 277.