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The itinerant oscillator treated and extended in terms of a mori continued fraction
Volume 48, number 2
THE ITINERANT
OSCILLATOR
CHEMICAL
PHYSICS LETTERS
TREATED
AND EXTENDED
IN TERMS OF A MORI CONTINUED
L iunc 1977
FRACTION
M.W. EVANS Chennstry Department, The University College of Wales, Aberystwyth, SY23 IN.& UK Received I6 March 1977
The angular velocity and orientatlonal autocorrelation function of B disk are cv.duatrd usmg J four-variabble hfon forrnJliwn. It is pointed out that tbc equwalcnt threevariable forrnalisni is formally tdentical to the inertiacorrected itinerant oscillator model developed recently by Coffey et al. Therefrom a sound phykxal mterprct.ltlon wn be given to the fourvariable model m terms of a disk (m the 2-D GBC) surrounded by two annuli. the outermost of which undergoes rotztiona1 diffusion
1. Introduction Tlte itinerant oscillator as a model for diffusional motion in condensed fluids has been a consistent theme in the literature over the past decade. It was first propounded by Hill [ 1] and extended by Wylhe [2] and Larkin [3] _ Lately, Coffey et al. [4] have developed a more realistic, inertia-corrected version in two dimensions. An alternative approach to the same problem IS that of generaliring the Langcvin equation so that the friction coefficient is replaced by a memory kernel. The autocorrelation functlon (acf) may then be expanded in a Mori continued fraction which has been shown [S] to converge under the right conditions. The purpose of this letter is to use SUC~L a continued fraction expansion to estimate the angular velocity acf for a disk, and then to use the reccntIy dcrivcd relations of Lewis et al. [G] to calculate the orIentationa acf therefrom [7]. It transpires that the angular velocity acf of Coffey et al. is formall” identical to “three-variable” Mori theory, and therefore a clear physical interprctation of higher order truncations suggests itself.
2. Theory Consider the Mori/Kubo generalisation [5] of the Langcvin equation for a disk undergoing itinerant torsiona oscillation in a plane. Neglecting dipole-dipole coupling [S], we have in the field-irce case, for a moment of inertia I:
CL(t)
+j
K,(t
(1)
- 7) O(T) = r(t),
0
where I r(t)
is the couple due to extraneous
random
torques,
and the factor:
t
I Ko(t
- T) co (T) d7
b
355
CHEMICAL PHYSICS LETTERS
Volume 48, number 2
is the frictional couple arising from the medium. memory function, is defined by: K&)
=
We emphasize
that r(t)
1 June 1977
is gaussian
and non-markovian.
K,,, the
- o(0)).
Solving eq. (1) @ads to the following
a-P) = (40) +mv)]l/b
in Laplace space (p):
+&-Jp)] ,
(2)
so that:
Jz-’{[p+qp)]
r(t)
-* }.
is a variate that obeys the same type cf stochastic
[p + &(P)l-’
fimctlon
differential
equation
as (I), it follows that:
+R,ti)l I-’ = ..-,
= CP +KJOVrP
where K, is the memory
(3)
of K,.
It has been shown
(4) [4] that the “three-variable
truncation”
of eqs. (4), i.e.:
K1 (0 = Kt (0) exp(-70 yields the final expression for a(o), the power absorption coefficient, formally identical to the inertia-corrected itinerant oscdlator model of Coffey et al., although these authors used an entirely different starting proposition. In this letter we propose to extend the Mori theory to the next order of truncation [9], using: K20) The physical
= K2(0)
exP(--rl
significance
C,(t)=
t).
(5)
of this IS discussed
(w(O)-w(O))
below, but first we note that eq. (2) now reads:
P b + ~2(P)l + K, (0)
Q--l
where K2(P) = K2(0)/(p + 7,)’ so that m inverting inator, viz: p4 +p3r,
1
p3 + p2 x2@) + P K&9 + K, (WI + Kg(O)r?,@> ’
+ P’ [K,,(O) + K1(0) f
the transform
in eq. (6) WC need to solve a quartic
(6) in the dcnom-
K2@)1+ mI [K&O+ K, @)I +$KOK1(0).
Writing this as: p4 + bp3 + cp2 + dp + e = 0, then It has the discriminant [lOI : A = -I$ - 27Q2, where P= bd - 4.9- c2/3, and Q = -b2e + bed/3 + 8 cc/3 - d2 - 21~~127.The equation can be reduced by the substitutionp=pl - b/4 to: p’: +qp;
+rpl
+s=o.
Now a quartic with (I, r, s real; r f 0 and A has(i) 4 distinct real roots if 4 and 4s - q2 are negative and A > 0; (ii) no real root if q and 4s - q2 arc not both negative and A > 0; (iii) 2 distinct real and 2 imaginary roots if A < 0; (iv) at least 2 equal real roots if A = 0. It is quite easy to solve the quartic numerically to any degree of accuracy (using the N.A.G. library available on most computers) given the above rules, whence it is possible to discuss the analytical forms the autocorrelation function of angular velocity will take. In case (I) we have: 386
Volume 48. number 2
CHEMICAL PHYSICS LETTERS
t suiiu :977
4 c,C0
= c n
(71
D,, cxpl-q),
where D,, are soluble in terms of the distinct roots cm by taking partial fractions. In case (ii): = Ox0 e- alt cos&t
C,(t)
+Ox, e -Wt sin Ott + Oxz eWffzt cos &t + Ox3 e-+t
sin Q,
(81
where the roots are (or, + i 0,) (cyI - i PI) (a2 + i p2)(~z - i &), and the x factors are al1 expresnble in terms of aI, Ply a2 and P2. Stmilarly in case (Iii): C,(t)=
‘x1 e-Qz’ + ‘X2 C- a2t f ‘X3
ctwajt
cos
fit
+
lx4
e-*t
Finally, in the fourth cilsc, ayl = cr2, at feast, m eq. (9). It IS possible now to calculate the orientational autocorrelation thus the far infrared/n~i~rowavc spectrum) by USCof the relation:
sin
&.
(9)
function C,(f) of the dipole unit vector & (and
derived recently by Lewis et al. [G] for the disk. Thus in case (1):
+ D, e-(Qt + D2 e-Q=’ + D, ewa3’ + D4 e;“’ 4
4
a4
4
1.
1
(11)
In case (ii):
OXI (a:- Pf>+ ----_____
(
2q3,
(g +8:f2
0
xo
)
e-““sin
Ox2 (LX;- 0;) + 2ar2p2 ox3 + __- __-___--_ -a2t \ i (e (a; +P;)2
(
Ptt
Ox2 “2 + ox3P2 --
+r (
1
a’2 +a;
Ox3(4 - 0;) - 24-9P,ox2 cospZt
_
1)
f
___
(
____.---.
fag +p2j2 2
e -“2r
-?I1
sin &r
_
-II (121
In case (iii):
387
CHEMICAL
Voiurm48,nun1bcr 2
PHYSICS
LlXlXKS
1 June 1977
cu(r)=Exp(-r[t(~+~)-(~ 2)+tcL..gY+ “1;’ +(~p21~4)’ 3 +33 (
1,
(J -_
- 0”) + 2(Y3(3 1X4\ _ (a’3 +$)” 1
1x3 (a: - 02) - 2a3p 1x4 (C-Q3t
c0.s
St
-
1)
+
(
----
emagt
(a: +fi212
)
sin fir
,
(13)
II
with a simple chnngc for the fourth case. In each of eqs. (1 I)-(13) there is no linear or t term in the Maclaurin expansion of the exponent, and at very short times, each reduces to exp(-kTt2/21), the autocorrelation function for free rotation in 2-D. At long times the form in each case is exp (--t/To) where T,-, is the equivalent of the Dcbyc relaxation time.
3. Discussion The three-vanable formahsm represented by the exponential truncation at ICI(f) may bc given physical significance by reference to the Coffey/Caldcwood model [1]. Here it is assumed that the mechanical system consisting of the vibrating central molecule and Its cage of ncighbours may be represented by an annulus which is free to rotate about a central axis pcrpendlcular to itself. ConcentrIc and co-planar with the annulus is a disk (of diameter iess than the annulus) w1uch is free to rotate about the same central axis. The disk carries a dipole plying along one of its diameters whose orientation is specified by an angle 0 (t) relative to a fixed axis determined by the applied field direcilon, while the position of a pomt on the rim of the annulus is specified by an angle $J (t) relative to the same fixed axis. The mechanical interaction between the central molecule and its neighbours is represented by a restoring torque acting on the dipole and proportIonal to the displacement 0 (t) - JI (t). In our notation, K, (0) = wi, the angdar frequency of ihe disk when the annulu=, is held stationary; K1(0) = (12/I,)u~, where I, = moment of Inertia of the annulus, I2 = that of the disk, -y = kT~“/1, ; whcro T,, is the Dcbye relaxation time. In extending the Marl series to the four-variable level by truncating cxponentmllv at K,(t) we have created the followmg equations linking r(t) to a Wiener process l12(t). f(r)
+ zK # s il
i-“(‘1+j
K,(t
- 7) r(7) dr = r”(t),
(14)
-
(1%
T) r,(r) d7. = I-#),
0
ir,(0 + Y
1
rl
(0 = mi-2to.
(16)
Now Ii(r) IS gaussian and markovlan whereas ro(‘) and r(t) are non-markovian, although still gaussian by the central limit theorem. Thus any fluctuations S r,(O) and S r(0) induce in the surrounding molecules fluctuations which after a time cause fkrrlzerones in Ii0 and K So 6 To(f) or 6 r(t) each remains partly stationary in space [ 111. On the other hand, S r,(O) flows through the ensemble, since being markovian it must not affect ‘; again, and thus S q(r) is propagated, as would bc S r(r) for the Debye sphere, where C,(t) is a single exponential with a characteristic time tD. Further, o is a characteristic operator of one or a small number of molecules - typically the nearest neighbour cagc,so that r(t) depends on the next-nearest nei&bours etc. To say that ro(t) propagates through the fluid is to assert rotational diffusion for the nearest neighbour cage (the annulus of the model above) and thus if r,(t) propagates, the next-nearest shell must be undergoing this type of motion. So we have m the four-variable theory a disk 388
Volume
48. number
2
CHEMICAL
PHYSICS
LETIERS
z Iunc
1977
surrounded by two annuli (in two dimennons), so that iQ(0) is a function of all three ~IWIICKI~S of irwtrtia and or the disk proper frequexy 00. Obviously the formalism can be extended although one cannot (by a theorem of groups) solve analytically
higher order polynomials
than the quartic
denominator
of cq_ (6).
Acknowledgement
Dr. William Coffey is thanked Fellowship.
for stimulating
conversations,
and The Kamsay Memorial
Trust for the 1976-78
References [l] N.E Hdi, Proc. Phyc. Sot. (London) 82 (1963) 723. [2] G. Wylhc. m: Dielectrrc and Related hloleculnr Proaxes, Vol. 1, Chemical Soc~cty Spcciahst Periodical Report (Chem. hoc, London, 1972) p. 21; J. Phys. C 4 (1971) 564. [3] 1-W. Larkin, J. Chcm. Sot. Faraday II 69 (1973) 1278. [4] W.T. Coffey, T. Ambrose and J.iI. Calderwood, J. Phys. D 9 (1976) L115; W.T. Coffey and J.H. C.ddcrwood. 45th Ann. Rept., Natl. Acad. SCI. Washington, to be published;Proc. Roy. Sot. (1977). to bc pubhshed. [S] Il. hlon, Progr. Thcorct. Phy~. 33 (1965) 423; M. Dupuis, Pro&r Theoret. Phys. 37 (1967) 502; hI. Tokuyama and II. Morr. Progr. Thcoret. Phys. 55 (1976) 411. [6] J.T. Lewis. J. hIcConnel1 .md B.K.P. Scaife, Proc. Roy. Irish Acad. 76A (1976) 43. [7] M.W. Evvan\, Chcm. Phys Letters 39 (1976) 601. [ 81 B.K.P. Scaife, Complex pcrmrttivity (English Universrties Press. London, 197 1) p. 35. [9] G.J. Evans and hI.W. Evans, Chcm. Phys. Letters 45 (1977) 454. [ iOl L.E. Drckson. Llemcnt~ry theory of equations (Whey. New York, 1914). [ 11] B. Qucntrcc and P. Beirot, Mol. Phys. 27 (1974) 879.
389
THE ITINERANT
OSCILLATOR
CHEMICAL
PHYSICS LETTERS
TREATED
AND EXTENDED
IN TERMS OF A MORI CONTINUED
L iunc 1977
FRACTION
M.W. EVANS Chennstry Department, The University College of Wales, Aberystwyth, SY23 IN.& UK Received I6 March 1977
The angular velocity and orientatlonal autocorrelation function of B disk are cv.duatrd usmg J four-variabble hfon forrnJliwn. It is pointed out that tbc equwalcnt threevariable forrnalisni is formally tdentical to the inertiacorrected itinerant oscillator model developed recently by Coffey et al. Therefrom a sound phykxal mterprct.ltlon wn be given to the fourvariable model m terms of a disk (m the 2-D GBC) surrounded by two annuli. the outermost of which undergoes rotztiona1 diffusion
1. Introduction Tlte itinerant oscillator as a model for diffusional motion in condensed fluids has been a consistent theme in the literature over the past decade. It was first propounded by Hill [ 1] and extended by Wylhe [2] and Larkin [3] _ Lately, Coffey et al. [4] have developed a more realistic, inertia-corrected version in two dimensions. An alternative approach to the same problem IS that of generaliring the Langcvin equation so that the friction coefficient is replaced by a memory kernel. The autocorrelation functlon (acf) may then be expanded in a Mori continued fraction which has been shown [S] to converge under the right conditions. The purpose of this letter is to use SUC~L a continued fraction expansion to estimate the angular velocity acf for a disk, and then to use the reccntIy dcrivcd relations of Lewis et al. [G] to calculate the orIentationa acf therefrom [7]. It transpires that the angular velocity acf of Coffey et al. is formall” identical to “three-variable” Mori theory, and therefore a clear physical interprctation of higher order truncations suggests itself.
2. Theory Consider the Mori/Kubo generalisation [5] of the Langcvin equation for a disk undergoing itinerant torsiona oscillation in a plane. Neglecting dipole-dipole coupling [S], we have in the field-irce case, for a moment of inertia I:
CL(t)
+j
K,(t
(1)
- 7) O(T) = r(t),
0
where I r(t)
is the couple due to extraneous
random
torques,
and the factor:
t
I Ko(t
- T) co (T) d7
b
355
CHEMICAL PHYSICS LETTERS
Volume 48, number 2
is the frictional couple arising from the medium. memory function, is defined by: K&)
=
We emphasize
that r(t)
1 June 1977
is gaussian
and non-markovian.
K,,, the
- o(0)).
Solving eq. (1) @ads to the following
a-P) = (40) +mv)]l/b
in Laplace space (p):
+&-Jp)] ,
(2)
so that:
Jz-’{[p+qp)]
r(t)
-* }.
is a variate that obeys the same type cf stochastic
[p + &(P)l-’
fimctlon
differential
equation
as (I), it follows that:
+R,ti)l I-’ = ..-,
= CP +KJOVrP
where K, is the memory
(3)
of K,.
It has been shown
(4) [4] that the “three-variable
truncation”
of eqs. (4), i.e.:
K1 (0 = Kt (0) exp(-70 yields the final expression for a(o), the power absorption coefficient, formally identical to the inertia-corrected itinerant oscdlator model of Coffey et al., although these authors used an entirely different starting proposition. In this letter we propose to extend the Mori theory to the next order of truncation [9], using: K20) The physical
= K2(0)
exP(--rl
significance
C,(t)=
t).
(5)
of this IS discussed
(w(O)-w(O))
below, but first we note that eq. (2) now reads:
P b + ~2(P)l + K, (0)
Q--l
where K2(P) = K2(0)/(p + 7,)’ so that m inverting inator, viz: p4 +p3r,
1
p3 + p2 x2@) + P K&9 + K, (WI + Kg(O)r?,@> ’
+ P’ [K,,(O) + K1(0) f
the transform
in eq. (6) WC need to solve a quartic
(6) in the dcnom-
K2@)1+ mI [K&O+ K, @)I +$KOK1(0).
Writing this as: p4 + bp3 + cp2 + dp + e = 0, then It has the discriminant [lOI : A = -I$ - 27Q2, where P= bd - 4.9- c2/3, and Q = -b2e + bed/3 + 8 cc/3 - d2 - 21~~127.The equation can be reduced by the substitutionp=pl - b/4 to: p’: +qp;
+rpl
+s=o.
Now a quartic with (I, r, s real; r f 0 and A has(i) 4 distinct real roots if 4 and 4s - q2 are negative and A > 0; (ii) no real root if q and 4s - q2 arc not both negative and A > 0; (iii) 2 distinct real and 2 imaginary roots if A < 0; (iv) at least 2 equal real roots if A = 0. It is quite easy to solve the quartic numerically to any degree of accuracy (using the N.A.G. library available on most computers) given the above rules, whence it is possible to discuss the analytical forms the autocorrelation function of angular velocity will take. In case (I) we have: 386
Volume 48. number 2
CHEMICAL PHYSICS LETTERS
t suiiu :977
4 c,C0
= c n
(71
D,, cxpl-q),
where D,, are soluble in terms of the distinct roots cm by taking partial fractions. In case (ii): = Ox0 e- alt cos&t
C,(t)
+Ox, e -Wt sin Ott + Oxz eWffzt cos &t + Ox3 e-+t
sin Q,
(81
where the roots are (or, + i 0,) (cyI - i PI) (a2 + i p2)(~z - i &), and the x factors are al1 expresnble in terms of aI, Ply a2 and P2. Stmilarly in case (Iii): C,(t)=
‘x1 e-Qz’ + ‘X2 C- a2t f ‘X3
ctwajt
cos
fit
+
lx4
e-*t
Finally, in the fourth cilsc, ayl = cr2, at feast, m eq. (9). It IS possible now to calculate the orientational autocorrelation thus the far infrared/n~i~rowavc spectrum) by USCof the relation:
sin
&.
(9)
function C,(f) of the dipole unit vector & (and
derived recently by Lewis et al. [G] for the disk. Thus in case (1):
+ D, e-(Qt + D2 e-Q=’ + D, ewa3’ + D4 e;“’ 4
4
a4
4
1.
1
(11)
In case (ii):
OXI (a:- Pf>+ ----_____
(
2q3,
(g +8:f2
0
xo
)
e-““sin
Ox2 (LX;- 0;) + 2ar2p2 ox3 + __- __-___--_ -a2t \ i (e (a; +P;)2
(
Ptt
Ox2 “2 + ox3P2 --
+r (
1
a’2 +a;
Ox3(4 - 0;) - 24-9P,ox2 cospZt
_
1)
f
___
(
____.---.
fag +p2j2 2
e -“2r
-?I1
sin &r
_
-II (121
In case (iii):
387
CHEMICAL
Voiurm48,nun1bcr 2
PHYSICS
LlXlXKS
1 June 1977
cu(r)=Exp(-r[t(~+~)-(~ 2)+tcL..gY+ “1;’ +(~p21~4)’ 3 +33 (
1,
(J -_
- 0”) + 2(Y3(3 1X4\ _ (a’3 +$)” 1
1x3 (a: - 02) - 2a3p 1x4 (C-Q3t
c0.s
St
-
1)
+
(
----
emagt
(a: +fi212
)
sin fir
,
(13)
II
with a simple chnngc for the fourth case. In each of eqs. (1 I)-(13) there is no linear or t term in the Maclaurin expansion of the exponent, and at very short times, each reduces to exp(-kTt2/21), the autocorrelation function for free rotation in 2-D. At long times the form in each case is exp (--t/To) where T,-, is the equivalent of the Dcbyc relaxation time.
3. Discussion The three-vanable formahsm represented by the exponential truncation at ICI(f) may bc given physical significance by reference to the Coffey/Caldcwood model [1]. Here it is assumed that the mechanical system consisting of the vibrating central molecule and Its cage of ncighbours may be represented by an annulus which is free to rotate about a central axis pcrpendlcular to itself. ConcentrIc and co-planar with the annulus is a disk (of diameter iess than the annulus) w1uch is free to rotate about the same central axis. The disk carries a dipole plying along one of its diameters whose orientation is specified by an angle 0 (t) relative to a fixed axis determined by the applied field direcilon, while the position of a pomt on the rim of the annulus is specified by an angle $J (t) relative to the same fixed axis. The mechanical interaction between the central molecule and its neighbours is represented by a restoring torque acting on the dipole and proportIonal to the displacement 0 (t) - JI (t). In our notation, K, (0) = wi, the angdar frequency of ihe disk when the annulu=, is held stationary; K1(0) = (12/I,)u~, where I, = moment of Inertia of the annulus, I2 = that of the disk, -y = kT~“/1, ; whcro T,, is the Dcbye relaxation time. In extending the Marl series to the four-variable level by truncating cxponentmllv at K,(t) we have created the followmg equations linking r(t) to a Wiener process l12(t). f(r)
+ zK # s il
i-“(‘1+j
K,(t
- 7) r(7) dr = r”(t),
(14)
-
(1%
T) r,(r) d7. = I-#),
0
ir,(0 + Y
1
rl
(0 = mi-2to.
(16)
Now Ii(r) IS gaussian and markovlan whereas ro(‘) and r(t) are non-markovian, although still gaussian by the central limit theorem. Thus any fluctuations S r,(O) and S r(0) induce in the surrounding molecules fluctuations which after a time cause fkrrlzerones in Ii0 and K So 6 To(f) or 6 r(t) each remains partly stationary in space [ 111. On the other hand, S r,(O) flows through the ensemble, since being markovian it must not affect ‘; again, and thus S q(r) is propagated, as would bc S r(r) for the Debye sphere, where C,(t) is a single exponential with a characteristic time tD. Further, o is a characteristic operator of one or a small number of molecules - typically the nearest neighbour cagc,so that r(t) depends on the next-nearest nei&bours etc. To say that ro(t) propagates through the fluid is to assert rotational diffusion for the nearest neighbour cage (the annulus of the model above) and thus if r,(t) propagates, the next-nearest shell must be undergoing this type of motion. So we have m the four-variable theory a disk 388
Volume
48. number
2
CHEMICAL
PHYSICS
LETIERS
z Iunc
1977
surrounded by two annuli (in two dimennons), so that iQ(0) is a function of all three ~IWIICKI~S of irwtrtia and or the disk proper frequexy 00. Obviously the formalism can be extended although one cannot (by a theorem of groups) solve analytically
higher order polynomials
than the quartic
denominator
of cq_ (6).
Acknowledgement
Dr. William Coffey is thanked Fellowship.
for stimulating
conversations,
and The Kamsay Memorial
Trust for the 1976-78
References [l] N.E Hdi, Proc. Phyc. Sot. (London) 82 (1963) 723. [2] G. Wylhc. m: Dielectrrc and Related hloleculnr Proaxes, Vol. 1, Chemical Soc~cty Spcciahst Periodical Report (Chem. hoc, London, 1972) p. 21; J. Phys. C 4 (1971) 564. [3] 1-W. Larkin, J. Chcm. Sot. Faraday II 69 (1973) 1278. [4] W.T. Coffey, T. Ambrose and J.iI. Calderwood, J. Phys. D 9 (1976) L115; W.T. Coffey and J.H. C.ddcrwood. 45th Ann. Rept., Natl. Acad. SCI. Washington, to be published;Proc. Roy. Sot. (1977). to bc pubhshed. [S] Il. hlon, Progr. Thcorct. Phy~. 33 (1965) 423; M. Dupuis, Pro&r Theoret. Phys. 37 (1967) 502; hI. Tokuyama and II. Morr. Progr. Thcoret. Phys. 55 (1976) 411. [6] J.T. Lewis. J. hIcConnel1 .md B.K.P. Scaife, Proc. Roy. Irish Acad. 76A (1976) 43. [7] M.W. Evvan\, Chcm. Phys Letters 39 (1976) 601. [ 81 B.K.P. Scaife, Complex pcrmrttivity (English Universrties Press. London, 197 1) p. 35. [9] G.J. Evans and hI.W. Evans, Chcm. Phys. Letters 45 (1977) 454. [ iOl L.E. Drckson. Llemcnt~ry theory of equations (Whey. New York, 1914). [ 11] B. Qucntrcc and P. Beirot, Mol. Phys. 27 (1974) 879.
389