Physica lllA (1982)255272 NorthHolland Publishing Co.
DIELECTRIC
RELAXATION STOCHASTIC
AS A MULTIPLICATIVE PROCESS
I. GENERAL THEORYt M.W. EVANS and M. FERRARIO Chemistry
Department,
University College of Wales, Aberystwyth, Dyfed, UK, SY23
INE
and P. GRIGOLINI* Institute of Molecular Biophysics, The
Florida
State
Llniuersity, Tallahassee,
Florida,
32306, USA
Received 29 July 1981 A rigorous and general approach is developed to the relaxation of molecular dipoles on the microscopic scale, embodied in the orientational timeautocorrelation function. The usual difficulties of using the stochastic Liouville equation (SLE) are bypassed by replacing the cumulant expansion with a continued fraction. This reduces to that of Sack or Gross in the appropriate limit. The autocorrelation function is formed from approximants of this continued fraction, which is ideally suited for numerical computation, and as a basis for the newly developed technique of semistochastic molecular dynamics simulation. The numerical solution automatically produces the spectral moments of interest to order of truncation, so that the number of unknowns is reduced to one at each and every stage of approximation. This concerns the rate of energy dissipation, denoted by fi, a scalar, tensor or supertensor according to the nature of the diffusion process under consideration. The new continued fraction can be used to describe spatial rotational diffusion of the asymmetric top using the appropriate FokkerPlanck diffusion operator. It is a considerable improvement therefore on a model such as the planar itinerant librator, an approximant of the Mori continued fraction.
1. Introduction
Inertial effects in relaxation processes have been treated rigorously by Sack’), who produced a continued fraction expansion of the dielectric response function. The problem has also been considered by Gross? on the basis of a generalised Liouville equation. He obtained the following expression for t This work was supported in part by the U.S. National Institute of Health Grant No. GM 23942 and Contract No. EY765052690 between the Division of Biomedical and Environmental Research of the Department of Energy and Florida State University. MWE and MF acknowledge financial support from the S.R.C. Grant No. GR/A/8398.5. *Permanent address: Istituto di Fisica, Gruppo Nazionale C.N.R., Piazza Torricelli, 2, 56100, Pisa, Italy.
0378437 1/82/oooooooO /$02.75 @ 1982 NorthHolland
di Stmttura
della Material del
M.W. EVANS et al
256
the frequency
response
of the polarization:
(1.1) This
means
considered,
that
although
a macroscopic
a single
microscopic
description
(of the
fluctuation far
mechanism
infrared
and
is
dielectric
spectrum)
in terms of the usual relaxation concepts (Sack) would involve an form an arithmetic of discrete relaxation times, whose reciprocals The leading longest relaxation time (TJ is the original one of progression.
infinity
Debye. The last decade has seen the evolution of zeroTHz frequency dielectric spectroscopy=) to the point where it has become obvious that eq. (1.1) fails qualitatively to serve as a simple description of the observable spectral features in the far infrared’) while seeming to work at lower frequencies. Many explanations have been proffered, based for example of the Liouville equation by Mori to a continued fraction. showed exactly
that an approximant of Mori’s in physical terms with the planar
continued fraction corresponded itinerant librator model of Coffey
and Calderwood’). Subsequently this approximantlmodel for its ability to reproduce zeroTHz spectra by Evans, liquid and related phases. Both
the phenomenological
approximant
on the reduction Evans’) in 1976
and the model
has been examined Reid et al.‘? in the have
several
con
ceptual weaknesses, discussed elsewhere”)), and in practical terms are hampered by the involvement of too many effectively unknown phenomenological quantities. Perhaps the most pervasive fault in both approaches has been the purely technical necessity of disposing of nonlinearities such as those present in the (Euler) equations governing dimensions. This means that itinerant
rotation libration
of the asymmetric top in 3 was considered in the context
of the asymmetric top diffusing with its permanent dipole constrained to two dimensions. This implies, of course, that the Mori approximant is also devoid of nonlinearities in the behaviour of the total angular momentum vector J and of the dipole
p. The nonlinearities
are projected
into the noise
term
of
the Mori equation”). Notwithstanding the greatly improved ability of these approximant models (and offshoots’2)) to match the complete zeroTHz profile a fresh look at the problem is needed in order (a) to cut down the use of adjustable parameters to the absolute minimum; and (b) to consider rigorously the effect of nonlinearities on rotational diffusion. The technical reasons for the failure in the far infrared of theories such as those of Sack and Gross, and lately of McConnell and coworkers”) is well known by now to be rooted in the nature of /3 in eq. (1.1). This has evolved
DIELECTRIC
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hand in hand with the theory of nonequilibrium statistical mechanics14) from a friction coefficient, invariant with time, to a correlation function of the random forces on a diffusing molecule. The new theories inject a sense of history into p which has become a memory function’5). Yet the implied generalisation of Sack and Gross has not been accomplished, partly because of the difficulty of unravelling the memory function from the linear framework of the Mori continued fraction16). Mori’s theory works in the context of additive stochastic processes, while those of Sack, Gross, McConnell et al. deal with multiplicative stochastic processes constrained by Markov’s hypothesis on /3; i.e. its statistical behaviour is taken as independent of past events. Of course at short times (or at f.i.r. frequencies) this conflicts diametrically with the selfevident fact that the “free rotation” of a molecule is governed exclusively by past events according to the fundamental laws of dynamics. Independently, the nonequilibrium statistical mechanics (classical and quantum) of a very wide range of excitation/relaxation processes has been developing at an ever accelerating pace. The stochastic concepts have been extended to describe intramolecular phenomena such as radiationless decay and fluorescence of molecules excited by transient laser pulses. Lately Grigolini and coworkers have removed’7B) some of the conceptual difficulties of linear response theory and have shown also how to remove Markov’s constraint by working with multidimensional vectors of dynamically independent variables in both quantum and classical regimes. The main aim of this paper is to describe a new approach to the evaluation of the spectrum of the variable cc, for example an electric or magnetic dipole moment. The relaxation of p is described by: dt = iLsp + n(t) x p,
(1.2)
where the Liouvillian Ls is concerned with the variable of interest (cc) and a, for example a molecular angular velocity or a Larmor frequency, is in turn driven by a Liouvillian La as follows: (1.3) Eqs. (1.2) and (1.3) form a multiplicative stochastic system developed by Kubo”), who neglects the contribution to the time evolution of a due to the interaction between Ls and LB. As described in refs. 17 to 28 it has been possible to rewrite the exact motion eq. (1.3) as an additive stochastic, but multidimensional, equation similar in structure to a matrix Langevin equation:
M.W.
258
EVANS
et al
$,4= ~A+F, where
(1.4)
A is the column
vector
(1.5)
consisting reduce
of the dynamical
to those
variables
of fi when fI = .
behaviour in fi is disregarded. One of the major benefits’7‘x)
fo, f,, . . . , f,(fl
= f,,). The dynamics of fo n = 0 any nonMarkov
= fn = 0, i.e. when
of replacing
eq. (1.3) with
(1.4), and
con
sequently of Mori’s equation’) with (1.4), is that it enables us to construct easily the FokkerPlanck equation for an. We shall denote this by & F(A, Ao ( t, 0) = DJ’(A
(1.6)
Ao 1f. 01,
where DA is the diffusion operator and P a conditional probability. Eqs. (1.2) and (1.3) represent the theory “.“,T of the stochastic equation (SLE), whose major feature consists with the diffusion operator of the stochastic suitable
lefteigenstates5.‘4)
of replacing variable a.
eq. (1.3) may be rewritten
Liouville
the Liouvillian LB If use is made of
as
$l=D,fi
(1.7)
(Dn denotes the usual Markoffian regarded as an operator in the space 19 it was demonstrated by replacing
diffusion operator), where 0 is now spanned by the eigenstates of Dn. In ref.
that SLE theory
may be generalised
to nonMarkov
n
eq. (1.7) with
(1.8) Eq. (1.2) may then be written
d’“d& dt
as (1.9)
where
(1.10)
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It is important to note that 9, is a dynamical operator which is neither hermitian nor antihermitian. Eq. (1.10) expresses the generalised SLE in operator form’?. In ref. 19 it was used to deal with a column vector with a single element only. From a purely computational point of view the mathematical approach of ref. 19 rapidly becomes intractable as elements are added to the column vector of interest (i.e. as ~1,for example, becomes multidimensional or nonMarkovian). This is because the problem is dealt with by building up the diffusion equation for the process involving both p and a. This makes the numerical diagonalisation of the diffusion operator a costly business in terms of time and storage. Nee and Zwanzig30) have studied the problem3’) of calculating the autocorrelation function of the variable p of eq. (1.2) when Ls = 0. They found that the correlation function may be described by the following nonMarkov equation: $P
* p(t)) =  2 / dsD(t  s)(~
 p(s)),
(1.11)
0
where D(t) is obtained from the inverse Laplace transform D(w) = Y(flfi(t))
of the tensor: (1.12)
in an isotropic sample of dielectric. In section 2 we generalise the equations used by Nee and Zwanzig (NZ) en route to developing a solution to (1.10) suitable for application on a computer. This reduces to NZ when p is Markovian. The new solution contains matrices which can be diagonalised in a way depending only on the size of A without the added difficulty of degrees of freedom resulting from the inclusion of a in the set of stochastic variables. In a forthcoming paper we shall apply the new theory to zeroTHz results using a fast FORTRAN algorithm capable of calculating to any order the spectral moments of @p(O)) (as defined by Gordon3’) from successive approximants of a continued fraction which is a generalisation of that of Sack or Gross. We note that the latter leave all spectral moments undefined. The theory of section 2 is in fact effective in building up a general algorithm for many fields of excitation/relaxation, including that of semistochastic computer simulation33) of molecular dynamics.
M.W.
X0
EVANS
et al
2. General theory In section cc governed fraction
1 we defined
which
In appendix
the problem
by eq. (1.9). Mori”)
of evaluating
develops
the spectrum
the solution
a continued
operator .Yo is antihermitiun.
is valid only when the dynamical
A we show that Mori’s solution
of a variable
through
can be generalised
to involve
the
use of nonhermitian Liouvillians. This seemingly trivial extension has wide reaching practical implications which include: i) the development of a zeroTHz theory with none of the disadvantages of section 1; ii) the development of a rapidly semistochastic simulations.
convergent
continued
fraction
for use in
Define the ket or state If,,) as (the state (p,(0)) is the equilibrium of DA, i.e., Da(po) = 0 and (jio/ is its left conjugate):
eigenstate
(2.1)
If”) = &%1(.n)). If we write eq. (A.47) for k = 0. we obtain
(2.2)
As shown in appendix A this is obtained with a suitable definition products. If the state If,,) is to be given by eq. (2.1) it is convenient the scalar
product
as follows.
Take
the observables
of scalar to define
(Y and p as
a = if,
(2.3)
P = NY)&%
(2.3’)
where
y is the physical
(P I a> = (1
[email protected]*(r) where
space
of the variable
p. Then
 cp(v)wo(r))(Bolq*(A)f(A)lp,,).
we(y) is the equilibrium
the scalar
product
is (2.4)
distribution.
Eq. (2.2) is an important result whose physical meaning is as follows. It is possible to replace a multiplicative stochastic process”) with an additive one provided that we introduce a memory kernel. In other words, a nonadditive stochastic process is equivalent to a nonMarkovian one. Similar results have been indicated by Mori and Fujisaka35) and by Hyne?). The major advantage of the theory in appendix A is embodied in the continued fraction, eq. (A#), which is the key to fast computation of the spectrum of variables satisfying eq. (1.2). This result is as easily applicable to EPR spectroscopy? as it is to dielectric relaxation, and is directly compar
DIELECTRIC
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261
I
able to perturbation theories such as that of ref. 29. This will be the subject of a future numerical paper. The application of the generalised SLE to relaxation involving rotational Brownian motion consists therefore of replacing eq. (1.2) with (2.2). In order to emphasise the generality of this procedure we point out the physical conditions required to obtain from eq. (2.2) the results by Nee and Zwanzig30). Firstly we have to assume that part of the total hamiltonian is zero, Ls = 0. This is always the case for dielectric relaxation. Because DAIP&W
=
0,
(2.5)
then by using eq. (
[email protected] we can write If”>= (I  P)(iI,l+
D&O)
= (1  P) iI,,(
(2.6)
We also have (POWI
(2.6’)
fl IPO(W = 0.
This is because the physical meaning of this matrix element is the average value of the molecular angular velocity at equilibrium. We can write the memory kernel of eq. (2.2) as follows: A,@+(t) =  ~~lfo>‘
~,lfl>’ x ($11izr(I  P) exp{(I  P)%r1%#0) =  U0lf0>‘cf0l%(l  P) exp{(l  P)%r13&).
(2.7)
To obtain the result of NZ we must assume e(lP’Zot
=
eDA’ (2.8)
and using eq. (2.6) A,@,(t) =  cfo~f~>~cfo~~ X a(t)
X ~bo(fi)).
(2.9)
In an isotropic specimen (&
(2.10)
= l(cL2>,
where I is the identity follows:
matrix and (. . .) denote
an average
evaluated
as
(2.11) If we apply the definition U0lf0>= l(!.&
of scalar product
given in eq. (2.4), we obtain
M.W. EVANS
262
By applying
again
the definition
et al.
of eq. (2.4) and the isotropic
assumption
we
have
= (~01 j
drcLx(r)[WMt)cL,
 ~z(a(t)F., = 
 f%(f)/.4
 fL(~h&)llPo)
(2.12)
2(/_Lf><.nn(t,,.
In order to obtain top. Consequently:
this result
we have
assumed
that the rotator
is a spherical
(2.13)
[email protected](~) = 2(00(r)), which
is the result
of Nee and Zwanzig.
Eq. (2.8) is an approximation which can only be valid when Y?, is small compared with Da. In other words, the result by NZ is correct only as p tends to the limit of Markovian behaviour. Any continued fraction based on (].I]), such as that of Quentrec and Bezotj7), is not realistic in any other limit, as has been pointed out by Evans et al.%‘), who have tested out the continued fraction expansion of eq. (1.11) with zeroTHz spectroscopy. This is fully in agreement with a recent paper by Ferrario and Evans26) who used a less efficient cumulantbased theory (appendix B). The neglect of cumulants of higher order than the second is incorrect (appendix B) when the stochastic variable n is multidimensional. The theory of appendix A and this section is unaffected by any inaccuracy of this kind and can be regarded as a general and rigorous approach to the problem of dielectric relaxation as considered originally by Sack or Gross. In particular we can avoid the use of cumulant expansion when attempting to relate (0(t>0(0)> and (am.>, for 3D diffusion a hideously complicated problem. We would also like to stress the fact that in principle we have no need even to assume
that the stochastic
variable
a is Gaussian
(a prerequisite
of cumulant method?) when we have available the relevant FokkerPlanck equation. In a subsequent paper we shall evaluate (p(t)p(O)) and (tin(t)) numerically as indicated in this section. To indicate the nature of the specific calculation we provide here explicit expressions for Ao, A: and A, as defined by eqs. (A.36) and (A.41), from which: (2.14)
A0 = 0, ’ ([email protected]:lPd + (BolRSlPoL A:=
 ceol&SzvlPo,,
 (~“l&&lPO),
~ (POl&i,bXlPd.
 &lb*bxlPd
MkJlRilPd+ (Polntlpo,,
~ (F”ld,b,lPd
~ ~Y0l.ri,fQ,,L
(~oln:lPd + (yoln8lPo)
(2.15)
DIELECTRIC
RELAXATION
 T
AS A STOCHASTIC
(Bol~~Ip,)E,(B,I6,Ipo)
PROCESS.
T
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263
(dol.ri,lpi)Ei(dil,ri.Ipo)
AI = (A:)’
(2.16) where the Ei’s and the Ipi)‘s are the eigenvalues respectively of the diffusion operator DA defined by
and the eigenvectors
(2.17)
DA(pi>= Eilpi>.
The order of magnitude of the matrix A: is that of the mean square molecular angular velocity. The matrix A, has the same order of magnitude as the rate of energy dissipation of the molecule. When (A,[ < (A,( the truncation results from assuming A: = 0. This is a “medium memory” case. When the memory is small, lAr( e)A,( is equivalent to the Markov limit. A strong dynamical memory (such as in the free rotor limit) will require the evaluations of several subsequent contributions, a straightforward numerical procedure, which automatically gives up the spectral moments A:, A:, . . . , A’,, i.e. provides sum rules to order n. Finally we would like to emphasise the relation between the generalised continued fraction, eq. (AM) and that of Kubo, Gross or Sack. Kubo”) showed that in the monodimensional case when DA can be replaced by the FokkerPlanck operator, Dn = /3 $
(A’$+
R>,
(2.18)
we have the wellknown result (~~(t))/(~2)=exp
(2.19)
,
providing the bandshape 05 I(W)=iRe
$(e8’l+pl)iwt 0
1
dt.
(2.20)
264
M.W.
Kubo
shows
that this can be replaced
EVANS
et al.
by the continued
fraction
I(w) = ;A iw+p+?Z iw + 2/3 +. . . In appendix
(2.21)
C we show that in this, the simplest
case of isotropic
rotational
diffusion, or when any of the matrices A, and Af are diagonal, eq. (A.44) reduces to eq. (2.21), which is Sack’s eq. (4.11). As far as the state of the art in this field is concerned the most significant recent reference is to the work of McConnell et al.‘3a), who rederived eq. (2.21) for planar rotational diffusion. Subsequently Ford et al.‘3b) extended the calculation for the asymmetric top to a higher order of approximation. Without a higher dimensionality (or memory) the latter theories are not suitable for experimental application4).
Acknowledgement S.R.C.
Appendix
is thanked
for financial
support.
A
We shall generalise the continued case where the equation of motion
fraction
expansion
of Mori to the general
(A.11
$A=~.A involves hermitian.
a dynamical
operator
ip,, which
is neither
hermitian
nor
anti
We shall use a quantumlike formalism where the scalar product between two observables B and C is denoted by (B ( C). The “state” IC) is the usual hermitian conjugate of the operator C. We shall build up a chain of “states”, the first of which is (A.2)
If”> = /A). According to the quantumlike formalism starter projection operator Pa as follows: PO = Ifo>(fo I MYfol.
introduced
above
we can write the
(A.3)
DIELECTRIC
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265
We can then write Ifo(t)) = Polfo(r)> + (1  Po)lfo(t)> = Ifl+W)
+ K(r)),
(A.4)
I fo(t))
(A.3
where @o(t) = (fo I h‘Uo and
(A6)
Ifh(t)) = (1  Po)lfo(r)) By using
the following
definitions:
Zr=(lP&Y&l
(A.7)
If1)= ~e,lfo>
(A8)
and from eqs. (A.l),
If&(t))= ~IlfMO)+ Ifl(t))@o(t).
$
It is convenient I_fr(r)) = From
(A.4) and (A.6) we obtain (A.9)
to define
exp{~~tWilf0)
=
91
(A. 10)
expW~tllf0>.
eq. (A.9) we obtain
IfKtN =
I Ifdt  s)M’o(s)ds.
(A. 11)
0
Eq. (A.4) may therefore
Ifo(t))
= Ifo)@o(t)
+
be written
as
I Ifdt  s)M’o(s)ds.
(A. 12)
0
In order
to extend
this result
to kth generator
state we define
Ifk) = %CIfkl)r
(A.13)
16(t)> = exPI%JW&J,
(A. 14)
where (A. 15)
,ie, = (1  Pk_,)&,. From
eqs. (A.13) and (A.15) we obtain j_fk>= (1
Pkl)(l
Pkb2)"
' (I
pO)~Olfkd
(A. 16)
M.W.
266
It is convenient
EVANS
to define a leftstate
in the following
et al
to be associated
with the rightstate
Ifk)
way:
(fkl =
(A. 17)
(1  Ph :)(I  Ph I)
and (A. 17’)
&I = Uol. The ‘projection
operator’
P,, can then be written
as (A.18)
PL = IfkXfkIfIC)%I? which,
of course,
satisfies
the indempotence
property
P: = Pk. When
(A.19)
2” is not hermitian,
ah
however,
lost. In the following
the hermitian
operators
is
property.
We shall only use eq. (A.19) and
property
we shall have
of the projection
to avoid
the use of this
PkPk’ = Pk.&.
(A.20)
The validity of eq. (A.20), in turn, depends on the fact that, by construction, the vectors Ifk) are orthogonal to the vectors 16,) eq. (A.16) according to the formula (JkIfk,) = 0, By repeating
for
kf
(A.21)
k’.
the approach
which
led us to eq. (A.12) at k’thorder
I
ds. lfdt)) = [email protected](t)+ Ifktdf  .Y))@~(s)
we obtain
(A.22)
where @k(f) = cik We focus
( fk)‘tik
(A.22)
1 fk(t)).
our attention
now on the following
equation
(A.23)
$ (f”(t)\ = (~“U)l%. Its solution
is given
by (A.24)
(JO(t)( = (JoI e ‘at = (fol e ‘O1. The vector
(Jo(t)1 is not the usual dual vector
associated
(fdt)l = &I expWP:,tl. Therefore,
of motion:
in the hermitian
&WI = U”(  t)l.
with Ifo(t)), i.e. is not (A.25)
case of ref. 8 we have (A.26)
DIELECTRIC
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261
If we define (A.27)
&(f)( = @k&k exp&>, where _

(A.28)
(A.28’) we can obtain the left equation corresponding (jo(
= &W(~IJ + j &M.f~+,U
to eqn. (A.12): (A.29)
 s)l ds,
0
where (A.30)
d(s) = (.m)lfkX.LlW’. Eqs. (A.19) and (A.20) give @‘k(t)&
Ifk)=
$k
(A.31)
\fk)@k(t),
or <_fk ) fk)‘&k(t)
@k(t)(_fk
=
(A.31’)
1 fk)’
and
e'k'i.fk+l) =
(fk(
(A.32)
tik(t)lfk+l).
For example, by expanding in series the exponential (fkl exp{~kt}(fk+l) xexp{(l
=
Pkl)'
x(1Pk)'
PO)
' ' ' (I
‘(I
p,$&(l
(1 PrJ)(l
=
(fk&?O(l
x
exp{$$}(l
=
(fk\
ex
li
PO)
"(I
Pl) PO).
tlfk+l).
' ' ' (Ipk)'
Pkb,) "
(I
pO)zO(fk)
Pkdltimes
Pkm1)'
"(1
'.(I
PO)"

pO)zO(l
pk,)
"
. (I
pO)TO]
pO)TO(fk)
~;(fkb,l~O(l
PO)'
PO)
PO)~Ot}(l
' ' (1 
x[6p,,(lx
’
(fk,lyO(l
'.(I
z;@kll"O('
x [( 1  Pk1) ’
=
=
operator:
.
"
' (lh)
Pkm,)20'
' (I
Pk,)=%I
. ’ (1  Pk,)( 1  Pk)( 1  Pk1) ’ ’ ’ (1' '(I
Pk)(lPkb1)
pO)zk\fk)
Pkb1) ' ' . (I
PO)6pOlfk)
(A.33)
268
M.W.
From
EVANS
et al.
eq. (A.14) we obtain (A.34)
& Ifk(t)) = iPkI_fk(t))= expWkrI=%lfk).
1 = Pk + (1  Pk), eq. (A.34), in turn,
By using eq. (A.13), and the property be written as follows: & I.fk(f)) = expWktHfk)k
can
(A.35)
+ expWM+,L
where (A.36)
Ak = (d I fk)‘@%(fk). Eq. (A.35) can also be written
in the following
useful
form:
$ Ifk(f)) = I.fk(f))Ak + Ifk)Gkl_fk)~‘(.!kIexpWkrM+,) + ( 1
exd~kf)(_fk
pk)
Eq. (A.32) allows
us to replace
$ k(t))
+
= k(f))Ah +
By inserting
&
ifk(t))
(1

Pk)
(A.37)
+ I>.
eq. (A.37) with
Ifk)(fkIfl)~‘(fk(t)lfk+,)
(A.38)
exp{~kf)Ifk+l).
eq. (A.29) into eq. (A.38) we obtain
=
Ifk(f)h
+
(fk)tik
I fk)’
j
@kb)Cfk+,(t

s)lfk+,)
0 +
(1

Pk)
exp&~kt)/fk+,)
1 _fk)‘@k([email protected]+,(f
x cfk+, and by using
$
Ifk(f))
1 fk+,)
ds
( 1 
+
Pk)

s)
(A.39)
exp{~~kt)lfk+,)
eq. (A.31)
=
j_fk(t))Ak
+
(1


Pk)
\fk>
1
@k(S)A:+,@k+,(f
exp{~kt}lfk+l)y

S)
ds
(A.40)
where A:+, = 
(fklfk)m’Cfk+,(fk+,h
(A.41)
DIELECTRIC
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269
By multiplying eq. (A.40) on the left by Uk I Wcikl, we obtain
$
@k(f)
=
@k(f)Ak
@k(S)A:+,@k+df


s)
ds,
(A.42)
I 0
which gives by Laplace transformation $,(z) = @k(O)(Z Ak+ A:+,&+,(Z))‘. The Laplace transform so(t), is thus given by
(A.43)
of the correlation
matrix of the variable of interest,
1
&o(z) = z&,+A:
1 1
z,+A:
zA2+A:
“’ 42,
1 z  A,, + A;&(z)
(A.44
This result is more general than that of Mori because the parameters Ai and Af are complex numbers with, in general, nonvanishing real and imaginary parts. We can easily find a generalised equation in motion for the variables If,). By taking the Laplace transform of eq. (A.22) tfk(d>
=
(fk>bk(d
+
(A.49
Ifk+I(.d)~k(z),
and inserting in eq. (A.43), we get ifk(z>>
=
(I.fk)
+
Ifk+l(z)))(z

Ak
+
&+&k+,(Z))'
b4.W
and zl&z)) 
Ifk>
=
jfk(z))&

A:@k+,(Z))
Inverse Laplace transformation
(A.46’)
fk+,(Z)).
+
gives, finally,
f $
ifk(f))
=
Ifk(f))Ak

j0
where (Pk(S)
=
A:@‘k+dS).
Ifk(S))qk(f

s)
ds
+
Ifk+l(t)),
(A.47)
M.W.
270
Appendix
et al.
B
Assuming
n(t)
the kinetic i(t)
EVANS
to be a stochastic
process,
cumulants
can be used to solve
relation = n(t)
x u(t),
(B.1)
which defines u(t), the dipole unit process. However, several difficulties nature of the rotational Nee and Zwanzig9
diffusion assumed
as a multiplicative stochastic vector, arise mainly because of the vectorial
process. a(t) to be a Gaussian
process
and came
to
the erroneous conclusion that only the first two cumulants are nonvanishing. First of all, even in the case of Markovian relaxation for n(t), the Gaussian assumption which this implies is very rough indeed becuase a nonlinear term is present generally
in the EulerLangevin possible to reduce
equation, even for the spherical top. It is not the analysis to only two cumulants as non
cummutativity occurs when dealing with the same matrix at different time?), and destroys the rules which lead to vanishing higher order cumulants in the monodimensional case. Only when the lifetime of the correlation (a(t) is very short, and approximable by a delta function 6(t), are Nee and Zwanzig correct, but this is in any case Debye relaxationJR). Consequently, even when we are looking at spherical top molecules”~~), with moment a(t)
of inertia
I and relaxation
the orientational (u(t)u(O))
where
Q(t)
correlation
time
function
l/b
for the Markovian
process
is: (B.2)
= exp{@(t)}, is an
infinite
(kUIP2). Although
the nth cumulant
the higher
order
ones,
series
when
of terms
contributes /3 is small
in the
adimensional
parameter
to only the term in (kT/IP')"and to convergence
is not
achieved
very
quickly, and it is not even clear whether the cumulant series is convergent. In fact26) in the limiting case /3 +O the cumulant approach fails to reproduce the correct free rotor limit which can be found from eq. (B.l) only by dropping the stochastic nature kinematic description
Appendix When
of the process a(t) so that eq. (B.l) of each molecule in the ensemble.
becomes
a purely
C dealing
iLsM = ia.
with the problem
of the stochastic
oscillator
we can assume (C.1)
DIELECTRIC
The stochastic
RELAXATION
AS A STOCHASTIC
PROCESS.
I
211
variable R is assumed to be driven by
cc.3 We can expand Dn on the basis set of its eigenstates: P”(R) = [(2n)‘% !]“*2“‘*H, (&)exp{
(C.3)
5,.
If we define Ifo) = IA)IPo(~~))>
(C.4)
where A is the variable obtain
of interest
with the fluctuating
frequency
Cl, we
VI>= ]I  (IA)(Al)(l~o(~t)(~o(~n)l)lit&+ fi  iDdlA)IptW)>
= ilA)lp,(~>)(p~(~>l~lPo).
cc.9
The kth order generating state is Ifk>
=
ik(A)lPk(~n))(Pkl.(l(Pkl>
f . . (PdfljPo>.
It is then easy to show that Ak =
tik Ifk)'CiklLkIfk)=iWOkp,
(C.6)
where wo is the proper frequency of the variable IA) ((A( iLs(A) = iwo). Using suitable properties of hermite polynomials34 we obtain A’, =  (6 IfM_fn, IfnI> = nAz.
(C.7)
References 1) R.A. Sack, Proc. Phys.
Sot. 70B (1957) 402.
2) E.P. Gross, J. Chem. Phys. 23 (1955) 1415. 3) M.W. Evans, Adv. Mol. Rel. Int. Proc. IO (1977) 203. 4) M.W. Evans, G.J. Evans and A.R. Davies, Adv. Chem. Phys. 44 (1980) 255. Dynamics (Wiley/Inter5) M.W. Evans, W.T. Coffey, P. Grigolini and G.J. Evans, Molecular science, in press). Adv. Mol. Rel. Int. Proc. 21 (1981) 1. 6) M.W. Evans and J. Yarwood, 7) G.J. Davies, G.J. Evans and M.W. Evans, J. Chem. Sot., Faraday Trans. II 75 (1979) 1428. 8) M.W. Evans, Chem. Phys. Letters 39 (1976) 601. and W.T. Coffey, Proc. R. Sot. A 356 (1977) 269. 9) J.H. Calderwood 10) C.J. Reid and M.W. Evans, J. Chem. Sot., Faraday Trans. II 76 (1980) 286; 75 (1980) 1213. C.J. Reid, PhD. Thesis (Univ. College of Wales, Aberystwyth, 1979); C.J. Reid, M.Sc. Thesis (Univ. College of Wales, Aberystwyth, 1977); C.J. Reid, Chem. Phys. Letters 66 (1979) 517; C.J. Reid and M.W. Evans, Mol. Phys., 40 (1980) 1357. 11) G. Wyllie, personal communication.
272
M.W.
EVANS
et al.
12) W.T. Coffey, Molec. Phys. 37 (1979) 473. 13) J.T. Lewis, J. McConnell and B.K.P. Scaife. Proc. Roy. Irish Acad. 76 (1976) 43. G.W. Ford. J.T. Lewis and J. McConnell, Proc. Roy. Irish Acad. 76A (1976) 117. 14) R. Kubo, Adv. Chem. Phys. 15 (1969) 101. 15) B.J. Berne and G.D. Harp, Adv. Chem. Phys. 17 (1970) 63. 16) H. Mori, Prog. Theor. Phys. 33 (1965) 423; 34 (1965) 399. 17) P. Grigolini, M. Ferrario and M.W. Evans. J. Chem. Sot.. Faraday Trans. 11 76 (1980) 542. 18) P. Grigolini and A. I.ami. Chem. Phys. 30 (1978) 61. 19) M. Ferrario and P. Grigolini, Chem. Phys Letters 66 (1979) 100. 20) M. Ferrario and P. Grigolini. J. Chem. Phys. 74 (1981) 235, 21) M. Ferrario and P. Grigolini, J. Math. Phys. 20 (1979) 2567, 22) M.W. Evans, M. Ferrario and P. Grigolini, Molec. Phys. 39 (1980) 1369. 23) M.W. Evans and P. Grigolini, Molec. Phys. 39 (1980) 1391. 24) M.W. Evans and M. Ferrario, Physica Al05 (1981) 31. 25) P. Grigolini, Chem. Phys. 38 (1979) 389. 26) M. Ferrario and M.W. Evans, Adv. Mol. Rel. Int. Proc. 19 (1980) 221. 27) M.W. Evans, M. Ferrario and P. Grigolini. Chem. Phys. Letters, 71 (1980) 139. 28) M.W. Evans, M. Ferrario and P. Grigolini, Z. Physik B., in press ( 1981). 29) B. Yoon, J.M. Deutch and J.H. Freed, J. Chem. Phys. 62 (1975) 4687. 30) T.W. Nee and R. Zwanzig, J. Chem. Phys. 52 (1970) 6353. 31) C.J.F. Bottcher and P. Bordewijk, Theory of Electric Polarisation (Elsevier, Amsterdam, 1978). 32) R.G. Gordon, J. Chem. Phys. 39 (1963) 2788; 40 (1964) 1973: 41 (1964) 1819; 42 (1965) 3658. 33) C.E.C.A.M. Workshop, Orsay, Sept., 1980, report, 34) R.F. Fox, Phys. Rep. 48C (1978) 181. 35) H. Mori and H. Fujisaka. Prog. Theoret. Phys. lY (1973) 764. 36) J.T. Hynes, J. Chem. Phys. 62 (1975) 2972. 37) B. Quentrec and P. Bezot, Molec. Phys. 27 (1974) 879. J. Barojas, D. L,evesque and B. Quentrec. Phys. Rev. 7A (1973) 1092. 38) P. Debye, Polar Molecules (Chem. Cat. Co., New York, 1929). 39) B. Spain and M.G. Smith. Functions of Mathematical Physics (van Nostrand/Rheinoldt, London. 1970).