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An alternative stochastic formulation of the itinerant oscillator model
Physica 102A (1980) 547-553 @ North-Holland
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AN ALTERNATIVE STOCHASTIC FORMULATION OF THE ITINERANT OSCILLATOR MODEL A. BELLEMANS Department
of Physics,
Universite’ Libre de Bruxelles,
C.P. 223, 1050 Brussels,
Belgium
Received 6 February 1980
The itinerant oscillator model was introduced by Hill in order to account for the high frequency dielectric absorption. In contrast to the Langevin formulation which has been considered by various authors, we here present an analytic treatment of that model, involving discrete collisions and based on two simple stochastic assumptions.
1. Introduction The so called itinerant oscillator model was introduced by Hill’) in an attempt to explain the shape of the dielectric spectrum in the far infrared region. It supposes that a molecule executes torsional oscillations around a particular axis determined by its cage of first neighbours, which itself undergoes rotational brownian motion. In the last few years, this model has been studied by many authors and its ability to reproduce experimental results investigated in great detail*-‘). Until now it seems however to have been developed essentially within the context of the (generalized) Langevin equation. Our aim here is to present an alternative treatment, involving stochastic collisions, which was inspired to us by earlier works on dielectric relaxation, dealing with inertial effects, by Gordon*) and Brat?. Like most authors we shall consider a two-dimensional version of the model i.e. a central disk and a concentric annulus which are both allowed to rotate around their common centre with respective moments of inertia Z, and Z2. Let the angle cpl define the orientation of an electric dipole ZLattached at the center of the disk, with respect to some fixed axis; similarly let cpzspecify a particular direction, rigidly bound to the annulus, towards which the dipole is attracted by a restoring torque k(~r - cpr). Instantaneous collisions occur randomly at the outer side of the annulus with frequency l/7, producing discrete changes in &. Our final goal is to obtain the dipole autocorrelation function c(r) = (cc@) - PW)/P2 = +mrcp,w - mm, or, equivalently,
its Laplace transform. 547
(1)
548
A. BELLEMANS
2. Derivation
of the dipole autocorrelation
function
Let us rewrite (1) into the more convenient C(T) = (co&$(t)
form
- 4(O) + a[NO) - wm)~
(2)
with the following definitions: ff = &II,
I = I, + 12,
4=(1-ah+w29
~=cpz-cp1,
It follows from the model that, between two successive ..
collisions,
..
cp=o, ~=-c.o4$,
(3)
where 0; = k/I’,
I’ = I, I*/( I, + I?).
We shall start from a particular situation where the annulus undergoes n successive collisions at times tl < t2 < . * * < t., during the interval (0, t). Then, by averaging cosM(t) - NO) + a[JI(O) - w)l~
(4)
over velocities at the various collisions and over collision times, according to stochastic rules defined below, we shall progressively build the autocorrelation function C(t). We first note that, on account of (3), changes of 4 and $ during the interval (t,, t) are given by * W) = 4” + J-Mt- t”), 4(t) = (cl”cos wo(t - t,) + (rn/wO)sin wo(t - t,),
(5)
where 4,,, 4” are the angular positions at the instant of the n-th collision and fl,,, r, are the corresponding velocities, immediately after that collision. Substituting (5) into (4), we obtain cosM” - 4(O) + a[JI(O) - $” cos wo(t - &)I + L?,(t - t,) - ~,(cdoo) sin oo(t - t,)}.
(6)
We now assume that the values of R and r after a collision are totally decorrelated from the values before that collision and that they obey the Boltzmann distribution (assumption 1) i.e. P(fl, I’) - exp[- (IO* + I’r2)/2kT]. Averaging (6) accordingly
over R, and r,,, we find, after some manipulations,
STOCHASTIC FORMULATION OF OSCILLATOR MODEL
cos{A - @) + dq40) - II. cos woo - CJ~ x exp[-(&T/21)(? - t,)* - (kT/21’)(a/oo)2 sin’ odt - t.11.
549
(7)
Consider next the interval (t,_,, t,) and repeat the same operations i.e. compute changes of 4 and $ during that interval, substitute them into the first factor Of (7) and average over a”_,, r._,. Carrying the same processes over all intervals including the first one (0, tJ, we obtain c0s{(Yl&(0)[1 - cos wotl . . . cos wo(l - &)I} X exp(-
(kT/2l)[t:
x exp{-
(Kr/21’)(culo0)2[1 - cos2 wotl . . . cos2 wo(t - t,)]}.
Performing
+ . * . + (t - t,)*]} (8)
a last average over 1,4(o)by means of the equilibrium distribution
P(,cI) - exp(- I’w$+b2/2kT) we get as final expression G(t; tl, . . . t,) = exp{-
(1 - a)[t: +. . . + (t - t,)2]/2}
x exp{-y[l
- cos ootl . . . cos oo(t - t,)]},
(9)
where for brevity we used (Il/kT)“2 as time unit and put y = (Y/W:. The next problem is to weight G(t ; t,, . . . tn) in relation to the mean collision frequency l/r. We shall assume that collision times are totally decorrelated (assumption 2). Then the probability for having no collisions during a time interval s is equal to exp(-s/7) and the probability density for a collision to occur at the end of such an interval is T-’ exp(-s/r); hence the probability density is 7 -n e-f/’
(LO)
for observing n collisions during the time interval (0, t). The total contribution of such situations to C(t) is obtained by multiplying (9) by (10) and integrating over the collision times inside (0, t) i.e. C”(t) = 7-” jdf. 0
1 dt,-, . . . 0
r
dfl e-“‘G(t;t,,.
. . t,).
(11)
0
At this point it is convenient
to expand (9) in powers of y
G(t ; t,, . . . , t,) = exp{- (1 - cu)[t: +. . . +(t - tnj2]} X
eey8
5
[COSmot,.. . COS b&t -
and to work with the Laplace transform
Cdl”
(12)
550
A.BELLEMANS P
c,,,(p)= 1 C,(t)
emp’dt.
(13)
0
This last integral factorizes into a product of n + 1 independent integrals, respectively associated to the subintervals delineated by the n collisions. We find
en’,(p) = eey 8
5
Y,(p
+T-‘)n+‘/P,
(14)
where m Y,(r) = I
e-(I-a)r*/2 cos" wet e-" dt.
(15)
0
The Laplace transform of C(t) itself is ultimately obtained by summing C,(p) from n equal 0 to w, i.e.
(16) Before closing this section known integral”)
let us note that Y,,,(z) is related
X(l) = i e-t2/2 eep dt.
to the well
(17)
0
Indeed, by rewriting (cos oot)m in (15) as a sum of cosines, it is easily shown that Y,(z) = (1 - a)-“*2-m 8 (;)x&i;O$‘R2r)).
(18)
3. Discussion The complex permittivity E(O) = e’- ie” is related to the Fourier transform of C(t)“); in particular, for a dilute solution of polar molecules, one has m E(O) - 1 = [E(O)- 1] \ dt e-‘“’ (-dC/dt) 0
= 1 -i oC(iw).
It follows that
(19)
STOCHASTIC
FORMULATION
OF OSCILLATOR
MODEL
551
E’(O) - 1 = [e(O) - l][l + &“(io)],
(20)
E”(W) = [e(O) - l]oP(iw),
(21)
while the absorption
coefficient
itself”) is given by
p(w) = WE”(O)/C 5 [e(O) - l]wW(iw)/c.
(22)
Three molecular parameters are involved in f?(iw) (besides the average angular frequency (I,/kT)“2 which was equated to one): (i) the collision frequency r-’ (ii) the oscillatory frequency w. and (iii) the ratio (Y= 12/(1r+ 12). For dense systems, 7-l (in reduced unit) is much larger than one and this should also be the case for oo, though perhaps to a lesser extent; (Y itself should be close to one as the moment of inertia of the annulus is obviously high compared to that of the central molecule. Hence the parameter y(= a/o;) is expected to be small and, at least for the discussion, it seems justified to limit (16) to the first power in y. As T-’ is much larger than one, X(c) may be replaced by its asymptotic formula”) X( 5) = c-1( 1 - 5-2 + . . .)
(23)
and, after some rearrangements, 1 - k&o)
one finds
(1 + ior) -’ = (1 - r)(l + iwr) 1 + iw (1 - cy)r I [ + y[ 1 + 2(io/ro@ - (02/o@]-‘.
(24)
The first term corresponds to a Debye relaxation, with a characteristic time (I- (Y)-‘7-I much larger than one, compounded with inertial terms which insure that p(o) goes to zero at high frequencies. The second term clearly describes a resonant absorption at frequency o = oo. For comparison, if we let a go to zero, we recover the model of Gordon*) where a bare central molecule is subject to the impact of its neighbours; the two-dimensional version of this model gives13) 1 - iwQi0)
= (1 + iorr)
c
1+ io
(1 + io7J2 -’ 7, , I
with Debye relaxation time 7-l. Note that the mean collision time rl, refering to a bare molecule, should be approximately equal to 27 as its cross section is nearly half that of the annulus. Let us adopt the following plausible values: 7 -I=
10,
00=5,
(Y=0.9
and compute c(io) from eq. (16). The so-called Cole-Cole plot and the absorption coefficient p(o) are shown on figs. 1 and 2, respectively. The
552
Fig. I. Cole-Cole plot of [I -i&iw)]. dashed line: 7;’ = 5, a = 0.
A. BELLEMANS
Plain line: 7-l = 10, o0 = 5, (Y= 0.9 (itinerant oscillator);
Fig. 2. Plot of &‘(iw) vs. log,, w. Plain line: T-’ = IO, w0 = 5, Q = 0.9 (itinerant dashed line: 7:’ = 5, a = 0.
oscillator);
profound modifications, introduced by the itinerant oscillator model with respect the conventional one, can be apireciated from the corresponding curves, plotted on these same two figures, for a bare molecule (7;’ = 5, a! = 0). Note that the present calculations can be readily extended to three dimensions, the results being very similar to those reported here.
STOCHASTIC
FORMULATION
OF OSCILLATOR
MODEL
553
References I) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)
N.E. Hill, Proc. Phys. Sot. 82 (1%3) 723. G. Wyllie. J. Phys. C4 (1971) 564. J.T. Lewis, J. McConnel and B.P.K. Scaife, Proc. R. Ir. Acad. Sect. A76 (1976) 43. W.T. Coffey, T. Ambrose and J.H. Calderwood, J. Phys. D9 (1976) Lll5. G.J. Evans, M.W. Evans and C.J. Reid, Adv. Mol. Relaxation Inter. Processes I2 (1978) 301; J. Chem. Sot. Faraday Trans. 2 74 (1978) 343. C.J. Reid, G.J. Evans, M.W. Evans and W.T. Coffey, Chem. Phys. Lett. 56 (1978) 529. W.T. Coffey, G.J. Evans, M.W. Evans and G.H. Wegdam, J. Chem. Sot. Faraday Trans. 2 74 (1978) 310. R.G. Gordon, I. Chem. Phys. 44 (1966) 1831. C. Brot, J. de Physique 2.8 (1967) 789. B.D. Fried and S.D. Conte, The Plasma Dispersion Function (Academic Press, New York, l%l). See e.g. E. Fatuzzo and P.R. Mason, Proc. Phys. Sot. (London) 90 (1967) 741. G.W. Chantry, Submillimeter Spectroscopy (Academic Press, London, 1971). E. Kestemont and A. Bellemans, J. Comp. Phys. 7 (1971) 515.
Publishing Co.
AN ALTERNATIVE STOCHASTIC FORMULATION OF THE ITINERANT OSCILLATOR MODEL A. BELLEMANS Department
of Physics,
Universite’ Libre de Bruxelles,
C.P. 223, 1050 Brussels,
Belgium
Received 6 February 1980
The itinerant oscillator model was introduced by Hill in order to account for the high frequency dielectric absorption. In contrast to the Langevin formulation which has been considered by various authors, we here present an analytic treatment of that model, involving discrete collisions and based on two simple stochastic assumptions.
1. Introduction The so called itinerant oscillator model was introduced by Hill’) in an attempt to explain the shape of the dielectric spectrum in the far infrared region. It supposes that a molecule executes torsional oscillations around a particular axis determined by its cage of first neighbours, which itself undergoes rotational brownian motion. In the last few years, this model has been studied by many authors and its ability to reproduce experimental results investigated in great detail*-‘). Until now it seems however to have been developed essentially within the context of the (generalized) Langevin equation. Our aim here is to present an alternative treatment, involving stochastic collisions, which was inspired to us by earlier works on dielectric relaxation, dealing with inertial effects, by Gordon*) and Brat?. Like most authors we shall consider a two-dimensional version of the model i.e. a central disk and a concentric annulus which are both allowed to rotate around their common centre with respective moments of inertia Z, and Z2. Let the angle cpl define the orientation of an electric dipole ZLattached at the center of the disk, with respect to some fixed axis; similarly let cpzspecify a particular direction, rigidly bound to the annulus, towards which the dipole is attracted by a restoring torque k(~r - cpr). Instantaneous collisions occur randomly at the outer side of the annulus with frequency l/7, producing discrete changes in &. Our final goal is to obtain the dipole autocorrelation function c(r) = (cc@) - PW)/P2 = +mrcp,w - mm, or, equivalently,
its Laplace transform. 547
(1)
548
A. BELLEMANS
2. Derivation
of the dipole autocorrelation
function
Let us rewrite (1) into the more convenient C(T) = (co&$(t)
form
- 4(O) + a[NO) - wm)~
(2)
with the following definitions: ff = &II,
I = I, + 12,
4=(1-ah+w29
~=cpz-cp1,
It follows from the model that, between two successive ..
collisions,
..
cp=o, ~=-c.o4$,
(3)
where 0; = k/I’,
I’ = I, I*/( I, + I?).
We shall start from a particular situation where the annulus undergoes n successive collisions at times tl < t2 < . * * < t., during the interval (0, t). Then, by averaging cosM(t) - NO) + a[JI(O) - w)l~
(4)
over velocities at the various collisions and over collision times, according to stochastic rules defined below, we shall progressively build the autocorrelation function C(t). We first note that, on account of (3), changes of 4 and $ during the interval (t,, t) are given by * W) = 4” + J-Mt- t”), 4(t) = (cl”cos wo(t - t,) + (rn/wO)sin wo(t - t,),
(5)
where 4,,, 4” are the angular positions at the instant of the n-th collision and fl,,, r, are the corresponding velocities, immediately after that collision. Substituting (5) into (4), we obtain cosM” - 4(O) + a[JI(O) - $” cos wo(t - &)I + L?,(t - t,) - ~,(cdoo) sin oo(t - t,)}.
(6)
We now assume that the values of R and r after a collision are totally decorrelated from the values before that collision and that they obey the Boltzmann distribution (assumption 1) i.e. P(fl, I’) - exp[- (IO* + I’r2)/2kT]. Averaging (6) accordingly
over R, and r,,, we find, after some manipulations,
STOCHASTIC FORMULATION OF OSCILLATOR MODEL
cos{A - @) + dq40) - II. cos woo - CJ~ x exp[-(&T/21)(? - t,)* - (kT/21’)(a/oo)2 sin’ odt - t.11.
549
(7)
Consider next the interval (t,_,, t,) and repeat the same operations i.e. compute changes of 4 and $ during that interval, substitute them into the first factor Of (7) and average over a”_,, r._,. Carrying the same processes over all intervals including the first one (0, tJ, we obtain c0s{(Yl&(0)[1 - cos wotl . . . cos wo(l - &)I} X exp(-
(kT/2l)[t:
x exp{-
(Kr/21’)(culo0)2[1 - cos2 wotl . . . cos2 wo(t - t,)]}.
Performing
+ . * . + (t - t,)*]} (8)
a last average over 1,4(o)by means of the equilibrium distribution
P(,cI) - exp(- I’w$+b2/2kT) we get as final expression G(t; tl, . . . t,) = exp{-
(1 - a)[t: +. . . + (t - t,)2]/2}
x exp{-y[l
- cos ootl . . . cos oo(t - t,)]},
(9)
where for brevity we used (Il/kT)“2 as time unit and put y = (Y/W:. The next problem is to weight G(t ; t,, . . . tn) in relation to the mean collision frequency l/r. We shall assume that collision times are totally decorrelated (assumption 2). Then the probability for having no collisions during a time interval s is equal to exp(-s/7) and the probability density for a collision to occur at the end of such an interval is T-’ exp(-s/r); hence the probability density is 7 -n e-f/’
(LO)
for observing n collisions during the time interval (0, t). The total contribution of such situations to C(t) is obtained by multiplying (9) by (10) and integrating over the collision times inside (0, t) i.e. C”(t) = 7-” jdf. 0
1 dt,-, . . . 0
r
dfl e-“‘G(t;t,,.
. . t,).
(11)
0
At this point it is convenient
to expand (9) in powers of y
G(t ; t,, . . . , t,) = exp{- (1 - cu)[t: +. . . +(t - tnj2]} X
eey8
5
[COSmot,.. . COS b&t -
and to work with the Laplace transform
Cdl”
(12)
550
A.BELLEMANS P
c,,,(p)= 1 C,(t)
emp’dt.
(13)
0
This last integral factorizes into a product of n + 1 independent integrals, respectively associated to the subintervals delineated by the n collisions. We find
en’,(p) = eey 8
5
Y,(p
+T-‘)n+‘/P,
(14)
where m Y,(r) = I
e-(I-a)r*/2 cos" wet e-" dt.
(15)
0
The Laplace transform of C(t) itself is ultimately obtained by summing C,(p) from n equal 0 to w, i.e.
(16) Before closing this section known integral”)
let us note that Y,,,(z) is related
X(l) = i e-t2/2 eep dt.
to the well
(17)
0
Indeed, by rewriting (cos oot)m in (15) as a sum of cosines, it is easily shown that Y,(z) = (1 - a)-“*2-m 8 (;)x&i;O$‘R2r)).
(18)
3. Discussion The complex permittivity E(O) = e’- ie” is related to the Fourier transform of C(t)“); in particular, for a dilute solution of polar molecules, one has m E(O) - 1 = [E(O)- 1] \ dt e-‘“’ (-dC/dt) 0
= 1 -i oC(iw).
It follows that
(19)
STOCHASTIC
FORMULATION
OF OSCILLATOR
MODEL
551
E’(O) - 1 = [e(O) - l][l + &“(io)],
(20)
E”(W) = [e(O) - l]oP(iw),
(21)
while the absorption
coefficient
itself”) is given by
p(w) = WE”(O)/C 5 [e(O) - l]wW(iw)/c.
(22)
Three molecular parameters are involved in f?(iw) (besides the average angular frequency (I,/kT)“2 which was equated to one): (i) the collision frequency r-’ (ii) the oscillatory frequency w. and (iii) the ratio (Y= 12/(1r+ 12). For dense systems, 7-l (in reduced unit) is much larger than one and this should also be the case for oo, though perhaps to a lesser extent; (Y itself should be close to one as the moment of inertia of the annulus is obviously high compared to that of the central molecule. Hence the parameter y(= a/o;) is expected to be small and, at least for the discussion, it seems justified to limit (16) to the first power in y. As T-’ is much larger than one, X(c) may be replaced by its asymptotic formula”) X( 5) = c-1( 1 - 5-2 + . . .)
(23)
and, after some rearrangements, 1 - k&o)
one finds
(1 + ior) -’ = (1 - r)(l + iwr) 1 + iw (1 - cy)r I [ + y[ 1 + 2(io/ro@ - (02/o@]-‘.
(24)
The first term corresponds to a Debye relaxation, with a characteristic time (I- (Y)-‘7-I much larger than one, compounded with inertial terms which insure that p(o) goes to zero at high frequencies. The second term clearly describes a resonant absorption at frequency o = oo. For comparison, if we let a go to zero, we recover the model of Gordon*) where a bare central molecule is subject to the impact of its neighbours; the two-dimensional version of this model gives13) 1 - iwQi0)
= (1 + iorr)
c
1+ io
(1 + io7J2 -’ 7, , I
with Debye relaxation time 7-l. Note that the mean collision time rl, refering to a bare molecule, should be approximately equal to 27 as its cross section is nearly half that of the annulus. Let us adopt the following plausible values: 7 -I=
10,
00=5,
(Y=0.9
and compute c(io) from eq. (16). The so-called Cole-Cole plot and the absorption coefficient p(o) are shown on figs. 1 and 2, respectively. The
552
Fig. I. Cole-Cole plot of [I -i&iw)]. dashed line: 7;’ = 5, a = 0.
A. BELLEMANS
Plain line: 7-l = 10, o0 = 5, (Y= 0.9 (itinerant oscillator);
Fig. 2. Plot of &‘(iw) vs. log,, w. Plain line: T-’ = IO, w0 = 5, Q = 0.9 (itinerant dashed line: 7:’ = 5, a = 0.
oscillator);
profound modifications, introduced by the itinerant oscillator model with respect the conventional one, can be apireciated from the corresponding curves, plotted on these same two figures, for a bare molecule (7;’ = 5, a! = 0). Note that the present calculations can be readily extended to three dimensions, the results being very similar to those reported here.
STOCHASTIC
FORMULATION
OF OSCILLATOR
MODEL
553
References I) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)
N.E. Hill, Proc. Phys. Sot. 82 (1%3) 723. G. Wyllie. J. Phys. C4 (1971) 564. J.T. Lewis, J. McConnel and B.P.K. Scaife, Proc. R. Ir. Acad. Sect. A76 (1976) 43. W.T. Coffey, T. Ambrose and J.H. Calderwood, J. Phys. D9 (1976) Lll5. G.J. Evans, M.W. Evans and C.J. Reid, Adv. Mol. Relaxation Inter. Processes I2 (1978) 301; J. Chem. Sot. Faraday Trans. 2 74 (1978) 343. C.J. Reid, G.J. Evans, M.W. Evans and W.T. Coffey, Chem. Phys. Lett. 56 (1978) 529. W.T. Coffey, G.J. Evans, M.W. Evans and G.H. Wegdam, J. Chem. Sot. Faraday Trans. 2 74 (1978) 310. R.G. Gordon, I. Chem. Phys. 44 (1966) 1831. C. Brot, J. de Physique 2.8 (1967) 789. B.D. Fried and S.D. Conte, The Plasma Dispersion Function (Academic Press, New York, l%l). See e.g. E. Fatuzzo and P.R. Mason, Proc. Phys. Sot. (London) 90 (1967) 741. G.W. Chantry, Submillimeter Spectroscopy (Academic Press, London, 1971). E. Kestemont and A. Bellemans, J. Comp. Phys. 7 (1971) 515.