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Effective-field approximations for memory functions in simple liquids
Volume
16, number
1
CHEhlICAL
EFFECTIVE-FIELD FOR
MEMORY
de Physique
hfol&ulaire.
1972
APPROXIMATIONS
FUNCFIONS
N.M. HOANG Groupe
15 September
PHYSICS LETTERS
LIQUIDS
and L BONAMY
FacuItP
Rcceind Revised manuscript
IN SIMPLE
des Sciences,
25 BesanFon.
France
4 &lay 1972 received 16 June 1972
Various recent models for the dynamic structure factor obtained in the frame of the linear response theory and the memory fmxtion formalism are reviewed and mathematically connected. This comparison su=ests a new model. which ieads to a good ngzeement with the molecular
dynamics
1. Introduction
w490 = (P,(r)P_Jw
The dynamic structure factor is of great interest in the study of atomic motions in dense systems. The linear response theory leads to several approximate expressions for this factor. Particularly it has long been pointed out that the memory function formalism is a suitable tool for such an analysis, Some already available expressions for the dynamic structure factor are reviewed in section 2 in the frame of the linear response theory. These are re-expressed in terms of the memory function formalism in section 3. Then, in section 4, a new model is investigated and compared to the molecular dynamics data and to the previous models.
2. Linear
response
The dynamic atoms is defined S(q, w) = sr-’ where
theory
(LRT)
structure factor for a system through the equations:
and IV
p,(t) = Nml” lg exp [iq-ri(r)]
where ri(t) 6 = 1 to N) describes the coordinates of the centre of mass of the jth atom at time f. _ The linear response theory connects the function F(q, w) and the density response function x(q, w) by
[II : B-‘x(4,0) = ioh,
w>- S(q) ,
(2)
where S(q) = (pq (C)p_ (0)) = F(q, c = 0) is the usual structure factor and /3-f = kT. When an effective field is introduced at the outset in the following form [2] : yf&4.
oflV
,
a! = v-44) Gq (WI) 2
the linea_r response
theory
(3)
leads to: (4)
Re F(q, W) , where x&q, w) is the response of the system to the sum of the external potential and the effective field
63
p(q, w) = s 0
data for liquid nrgon.
dt exp(iwc)
F(q, r) ,
(3).
103
Volume 16, number (a) The tive It is mation x,,(4.
1
CHEMICAL
PHYSICS
random phase approxiimriort {simple collecmodel) weil known that the random phase approxicorresponds to the choices:
w) = x&4, w) = B[iwpt-(4. a) -
11,
where xf(q, w) is the density response function gas of non-interacting molecules, and:
(9 of a
,
S(4)Ec4. WI
G4,w>=
1 f G)[iG&,w)
(9)
11.
-
(c) The approach of Pnthak and Singwi In a recent paper, Pathak and Singwi [S] introduce another modification of relation (7). For that purpose
denominator in (4), there will still remain some residual interactions having an influence on xs,(q, w). Their approach consists in replacing the Dirac F-func-
v(q) being the Fourier transform of the interatomic potential II(~). However, the linear response theory cannot be used directly for a potential with a hard core (for which, on the other hand, IJ(CJ)does not exist). In such a case the interparticle potential is usualiy replaced by an effective potential which is approximated by [3] : = -B-laql
15 September 1972
they note that although having taken into account the collective aspects of the atomic motions through the
Q(4) = v(q) ,
Q&d
LEI-i’ERS
X~(CJ,o) in its quantum representation by gaussian functions. It may be shown that. in $e classical limit, their resulting expression for F(q, o) can be written in the form: tions which describe
(6)
where C(q) = 1 - l/S(q) is the direct correlation tion of Ornstein and Zemike. Then the expression obtained
func-
x
Il 1+
1
I-
a&)&s(q)
-1
I[
-kFf(q, Qf (4)
cd/a&))
- 1
II
(10)
for p(q, o), i.e., The coefficient (7)
.
af(q) is then calculated in order that
the fourth frequency moment of S(q, w) is exactly satisfied:
is identical
to that given by Nelkin and Rar,ganathan [4]. These authors start from the linearized Vlassov equation for the one particle distribution function and they show that the replacement of v(q) by u&q) leads to the exact value for the second frequency moment of S(q. w). Solving the eigenvalue problem of the Liouville operator of the system using a variational principle, Zwanzig
and (12)
[5] also o’ctained the same result forF(q, wj.
lb} The approach of Kerr (extended simple collective model) It has been suggested [2] that a more suitable ap-
3. Memory
proximation could be to relate xsc(q, O) and the function x&, w) which describes the actual “self-motions”
The projection operator method of Zwanzig and hlori (IO] leads to the master equation:
of an atom. The approach
%,
of Kerr [6] gives a theoret-
ical justification of this suggest&. As mentioned by Nelkin [7], the expression for F(q, w) given by Kerr can be fotind x’s,@. a) =
one finds: 104
a) = s(a)[Io
+ w&)&,
o)l-’
where fm
again by taking:
x,(0, w> E ObZ&,
functions
u> - 11;
(8)
z(q, w) =
s --m
dt exp(iot)K(q,
r) ,
,
191 03)
CHEhlICAL PHYSICS LETTERS
Volume 16, number 1
15 September
1972
K(q, t) being known as the mezrory function of F(q, 1). Mori [ 1 l] has shown that K(q, w) can be expanded
Q4,o)=&&, CQ,> ,
as a continued fraction form [ 123 (II > 2):
then the liquid phonon model (LP) of Ortoleva and Nelkin [ 151 follcws. These authors start from an approximate evolution equation for the two-particle distribution function. This equation was obtained by prcjecting the exact Liouville equation on the basis of seven functions (which describe the high-frequency phonon-like collective modes studied by Zwanzig [ 161 and by Nossal and Zwanzig [ 17]), and by keeping only linear terms with respect to these basic functions. Using notations consistent with above, their result can be rewritten in the form:
Z(q,w)
q
qq,
which can be written
in the
Al(*)
w) =
A&)
iw f .
1cdt
-
AZ(q) are determined from the 2k first derivatives FG)(q, t= 0) (i= 0,1,_..,2li), or equivalently from the 2k first moments of the spectral density
‘The quantities
S(q, 0). One finds for example:
%7,0)
(19)
S(q) & ((1.m)
=
2 1 +z(~~,w)[iwFt((7,w)
(20)
- 11
where (a) First order approximations
A first set of expressions for F(q, w) is obtained by sproximating the fist order memory function K(q, w). The simple approximation consists in taking: K(q, r) = $(4,
t! 1
(16)
where Kr(q, r) is the ideal gas-counterpart of K(q, t). The resulting expression for F(q, w) is identical to (7)
which is known as the simple collective model (SC) [4,5,
131.
It has been pointed out by Verlet (cf., ref. ] 141) that the Kerr’s approach is obtained if K(g. t) obeys the relation: K(4, t) = K&q, 0 ,
(17)
so that this approach appears as an extended simple collective model (ESC) [6]. At this point it is interesting to note that Pathak and Singwi’s approach can be recovered when K(q, t) is approximated by: K(% t) = Kf (4, a#)
1) I
(18)
+ [a&)-
11iw[G&q,
w)- l](q’/flm)-I}
-’
appears as a time dependent direct correlation function. In this form the LP model can be recovered in the LRT frame by introducing a time dependent effective field: $4, w) = -p- * F(,, cd) . Then the extended liquid phonon model (ELF) is introduced by taking: &(4, w> = &(1’
w) -
(21)
The resultins expression for F(q, ~32is identical to (20) when F&J. w) is replaced by F,(q, w) and a;(q) by: (22) where
where ar(4) is defined by eq. (11). (b) Second order approxi??lah’ons A second set of expressions for F(q, o) arises by gproximating the second order memory functions K2(4, w). When this function is taken equal to the corresponding function for the free particles:
The numerical consequences of this mode! have been studied elsewhere [ 14,15,18]. (cj TIM
order approximations
Kim and Nelkin [ 181 introduce and study the next 105
natural Q?.
approximation
which consists
in taking:
a) -
wb = Q(7’
But now, a difficulty arises due to the inaccuracy of the sixth frequency moment. which involves the threeparticle correlation function. In their calculation the quantity b:(q) = A:(q)/A&(q) is used, In fact, as an adjustable parameter_
4. A new model
The above review of the various results obtained by ‘Lhelinear response theory suggests to introduce a new approximation through the memory funcJion formalism, i.e., to modify the expression for F(q, w) [eq. (lo)] , given by Pathak and Singwi, by taking into account the Kerr model. In order to do that we classify various possible choices of the memory functions in table 1. This table suggests at least one new approximation by taking: = K&4. a,(s) 0 .
K(4,0
The same time dependence applied to the second order memory function would lead to the approximation: K,(q* 0 = K*(q,
15 September 1972
CHEMICAL PHYSICS LETTERS
Volume 16. number 1
b,(q) t)
but to date the coefficient bs(4) is not suflicientiy known to develop this approximation.
(b) Numerical results and discussion For n_umerical calculation we use the parametrization of F,(q,o) given by Levesque and Verlet [ 191. The values of the parameters for the simulated [ 201 thermodynamic state for argon considered here (p =
1.407 g/cm3 and T = 76°K) were obtained by interpolating the data o.f these authors. Fig. 1 shows the variations of the spectral function w2S(q, o)/(q2/flnt) as a function of o for different values of 4. It is inteLesting to note that, although the low-q behaviour o,f Fs(q, o) used here is very different from that of F~((I, w) used by Pathak and Singwi [8], the functions w*S(q, o)/(q2//3m) issued from both methods are very similar. This is not inconsistent since in the hydrodynamical regime (4 < 1 a-1) af(q) and a,(q) have themselves very different behaviours:
(23) Fig. 2 exhibits the variations, as a function of 9, of the position o max((l) and of the height of the maximum of o*S(q, w)/(4*//3m). The good fitting of the molecular dynamics data by the present approach Ied us to built up a general formulation to be published eisewhere [2 l] and appearing as a generalization of the continued fraction representation of Mori [ 111. Fina!ly it will be stressed that the present work gives the exact low order momenta up to the fourth Table 1
hfemory
convolution approximation (Vineyard)
0
SC 14.51 K(4. r> = Kf(4.0
ESC (Ken [61) KG?.t) = K&7, t)
0, 2
K(q, r) =
fithak and Sin@ [8] Kf (q.ofr)
KQ, 0 = k’s@. as 0
LP 1151 Kz @,0 = Kc2f(%0
EL? [ 14,151 KZ@, 0 = K2&,
0 a=1 1 (7 = a(q)
2 3
Order of the satidied moments
Self
Free
function order
Present model
Kim
0, 2,4 0
and Nelkin [ 181 K3(q, 0 = K3s(q* f)
6294.6
Volume 16, number 1
2.0 -
CHEMICAL PHYSICS LETTERS
q=1,17r
_
_
==I ..
A I. l
2.0
0
q =
2.3
q =
1.97
15 September 1972
(whereas only the zeroth and the second are satisfied in [6]). It leads to a similar fitting of the data as the ELP model [13,14]. From a theoretical point ofview, this new model introduces as does the ELP model, the irreversible behaviour of F(q, t) through the function F&q, t) which is supposed to be known elsewhere. Such a behaviour does not seem to appear in the ap preach of Pathak and Singwi [8] due to the essentially reversible character of the evolution of Ff(q. t).
h'
A-."
I.
05
1.0
15
w(d'se~)
2D
Fig. 1. Spectral function of the longitudinal current correlations w2S(q. w)/(q2/fim) ptotted as a function of the frequency. -our results, --Pathak and Singwi’s results [8], x x x Kim and Nelkin’s results [ 181,. . . molecular dynamics data [ 20].
00)
q( A-‘)
.
Fig. 2. Dispersion curve am=@) and height of the maximum of the normalized spectral function of the longitudinal current correlations as a function of the wave vector 4. -our results, --Pathak and Singwi’s results [8], x x x Kim and Nelkin’s resti*& [18],... molecular dynamics data [ 20 1.
107
Volume 16, number 1
CHEMICAL PHYSICS LETTERS
References [I] R. Kubo, Lectures in theoretical physics, Vol. 1 WileyInterscience, Xew York, 1958). [2] K.S. Singwi, K. Skold and 51.P. Tosi, Phys. Rev. Al (1970) 454. [ 31 J.K. Percus, The equilibrium theory of classical fluids (Eenjamin, New York, 1364). [4] hi. Nelkin and S. Ranganathan, Phys. Rev. 164 (1967)
222. [5] R. Zwanzig, Phys. Rev. 144 (1966) 170. [6] W.C. Kerr, Phys. Rev. 174 (1968) 316. [7] M. Nelkin, Phys. Rev. 183 (1969) 349. [S] K.N. Path& and K.S. Singwi, Phys. Rev. A2 (1970)
2427.
!S September
1972
[9 J R. Zwamig, Lectures in theoretical physics WileyInterscience, New York, 1961). [IO] H. Mori, Progr. Theoret. Phys. 33 (1965) 423. [ 111 H. hiori, Pros. Theoret. Phys. 34 (1965) 399. [12] V.F. Sears, Cm. J. Phys. 47 (1969) 199. [ 131 P. Ortolevn and hi. Nelkin, Phys. Rev. 181 (1969) 429. [ 141 J. Kurkijarvi, Ann. Acad. Sci. Fennicae, Ser. A VI 346 (1970) 1. [ 151 P. Ortoleva and hf. Nelkin, Phys. Rev. A3 (1970) 187.
[16] R. Zwanzig, Phys. Rev. 156 (1967) 190. f17] R. Nossal and R. Zwanzig, Phys. Rev. 157 (1967) 120. [ 181 K. Kim and hf. Nelkin, Phys. Rev. A4 (1971) 2065. [I91 D. Levesquc and L. Verlet, Phys. Rev. A2 (1970) 25 14. [20] A. Rahman, Neutron inelastic scAtterin& Vol. 3 (International Atomic Energy Agency, Vienna, 1968) p. 56 1. [ZlI L. Bonamy and NM Hoang, to be published.
16, number
1
CHEhlICAL
EFFECTIVE-FIELD FOR
MEMORY
de Physique
hfol&ulaire.
1972
APPROXIMATIONS
FUNCFIONS
N.M. HOANG Groupe
15 September
PHYSICS LETTERS
LIQUIDS
and L BONAMY
FacuItP
Rcceind Revised manuscript
IN SIMPLE
des Sciences,
25 BesanFon.
France
4 &lay 1972 received 16 June 1972
Various recent models for the dynamic structure factor obtained in the frame of the linear response theory and the memory fmxtion formalism are reviewed and mathematically connected. This comparison su=ests a new model. which ieads to a good ngzeement with the molecular
dynamics
1. Introduction
w490 = (P,(r)P_Jw
The dynamic structure factor is of great interest in the study of atomic motions in dense systems. The linear response theory leads to several approximate expressions for this factor. Particularly it has long been pointed out that the memory function formalism is a suitable tool for such an analysis, Some already available expressions for the dynamic structure factor are reviewed in section 2 in the frame of the linear response theory. These are re-expressed in terms of the memory function formalism in section 3. Then, in section 4, a new model is investigated and compared to the molecular dynamics data and to the previous models.
2. Linear
response
The dynamic atoms is defined S(q, w) = sr-’ where
theory
(LRT)
structure factor for a system through the equations:
and IV
p,(t) = Nml” lg exp [iq-ri(r)]
where ri(t) 6 = 1 to N) describes the coordinates of the centre of mass of the jth atom at time f. _ The linear response theory connects the function F(q, w) and the density response function x(q, w) by
[II : B-‘x(4,0) = ioh,
w>- S(q) ,
(2)
where S(q) = (pq (C)p_ (0)) = F(q, c = 0) is the usual structure factor and /3-f = kT. When an effective field is introduced at the outset in the following form [2] : yf&4.
oflV
,
a! = v-44) Gq (WI) 2
the linea_r response
theory
(3)
leads to: (4)
Re F(q, W) , where x&q, w) is the response of the system to the sum of the external potential and the effective field
63
p(q, w) = s 0
data for liquid nrgon.
dt exp(iwc)
F(q, r) ,
(3).
103
Volume 16, number (a) The tive It is mation x,,(4.
1
CHEMICAL
PHYSICS
random phase approxiimriort {simple collecmodel) weil known that the random phase approxicorresponds to the choices:
w) = x&4, w) = B[iwpt-(4. a) -
11,
where xf(q, w) is the density response function gas of non-interacting molecules, and:
(9 of a
,
S(4)Ec4. WI
G4,w>=
1 f G)[iG&,w)
(9)
11.
-
(c) The approach of Pnthak and Singwi In a recent paper, Pathak and Singwi [S] introduce another modification of relation (7). For that purpose
denominator in (4), there will still remain some residual interactions having an influence on xs,(q, w). Their approach consists in replacing the Dirac F-func-
v(q) being the Fourier transform of the interatomic potential II(~). However, the linear response theory cannot be used directly for a potential with a hard core (for which, on the other hand, IJ(CJ)does not exist). In such a case the interparticle potential is usualiy replaced by an effective potential which is approximated by [3] : = -B-laql
15 September 1972
they note that although having taken into account the collective aspects of the atomic motions through the
Q(4) = v(q) ,
Q&d
LEI-i’ERS
X~(CJ,o) in its quantum representation by gaussian functions. It may be shown that. in $e classical limit, their resulting expression for F(q, o) can be written in the form: tions which describe
(6)
where C(q) = 1 - l/S(q) is the direct correlation tion of Ornstein and Zemike. Then the expression obtained
func-
x
Il 1+
1
I-
a&)&s(q)
-1
I[
-kFf(q, Qf (4)
cd/a&))
- 1
II
(10)
for p(q, o), i.e., The coefficient (7)
.
af(q) is then calculated in order that
the fourth frequency moment of S(q, w) is exactly satisfied:
is identical
to that given by Nelkin and Rar,ganathan [4]. These authors start from the linearized Vlassov equation for the one particle distribution function and they show that the replacement of v(q) by u&q) leads to the exact value for the second frequency moment of S(q. w). Solving the eigenvalue problem of the Liouville operator of the system using a variational principle, Zwanzig
and (12)
[5] also o’ctained the same result forF(q, wj.
lb} The approach of Kerr (extended simple collective model) It has been suggested [2] that a more suitable ap-
3. Memory
proximation could be to relate xsc(q, O) and the function x&, w) which describes the actual “self-motions”
The projection operator method of Zwanzig and hlori (IO] leads to the master equation:
of an atom. The approach
%,
of Kerr [6] gives a theoret-
ical justification of this suggest&. As mentioned by Nelkin [7], the expression for F(q, w) given by Kerr can be fotind x’s,@. a) =
one finds: 104
a) = s(a)[Io
+ w&)&,
o)l-’
where fm
again by taking:
x,(0, w> E ObZ&,
functions
u> - 11;
(8)
z(q, w) =
s --m
dt exp(iot)K(q,
r) ,
,
191 03)
CHEhlICAL PHYSICS LETTERS
Volume 16, number 1
15 September
1972
K(q, t) being known as the mezrory function of F(q, 1). Mori [ 1 l] has shown that K(q, w) can be expanded
Q4,o)=&&, CQ,> ,
as a continued fraction form [ 123 (II > 2):
then the liquid phonon model (LP) of Ortoleva and Nelkin [ 151 follcws. These authors start from an approximate evolution equation for the two-particle distribution function. This equation was obtained by prcjecting the exact Liouville equation on the basis of seven functions (which describe the high-frequency phonon-like collective modes studied by Zwanzig [ 161 and by Nossal and Zwanzig [ 17]), and by keeping only linear terms with respect to these basic functions. Using notations consistent with above, their result can be rewritten in the form:
Z(q,w)
q
qq,
which can be written
in the
Al(*)
w) =
A&)
iw f .
1cdt
-
AZ(q) are determined from the 2k first derivatives FG)(q, t= 0) (i= 0,1,_..,2li), or equivalently from the 2k first moments of the spectral density
‘The quantities
S(q, 0). One finds for example:
%7,0)
(19)
S(q) & ((1.m)
=
2 1 +z(~~,w)[iwFt((7,w)
(20)
- 11
where (a) First order approximations
A first set of expressions for F(q, w) is obtained by sproximating the fist order memory function K(q, w). The simple approximation consists in taking: K(q, r) = $(4,
t! 1
(16)
where Kr(q, r) is the ideal gas-counterpart of K(q, t). The resulting expression for F(q, w) is identical to (7)
which is known as the simple collective model (SC) [4,5,
131.
It has been pointed out by Verlet (cf., ref. ] 141) that the Kerr’s approach is obtained if K(g. t) obeys the relation: K(4, t) = K&q, 0 ,
(17)
so that this approach appears as an extended simple collective model (ESC) [6]. At this point it is interesting to note that Pathak and Singwi’s approach can be recovered when K(q, t) is approximated by: K(% t) = Kf (4, a#)
1) I
(18)
+ [a&)-
11iw[G&q,
w)- l](q’/flm)-I}
-’
appears as a time dependent direct correlation function. In this form the LP model can be recovered in the LRT frame by introducing a time dependent effective field: $4, w) = -p- * F(,, cd) . Then the extended liquid phonon model (ELF) is introduced by taking: &(4, w> = &(1’
w) -
(21)
The resultins expression for F(q, ~32is identical to (20) when F&J. w) is replaced by F,(q, w) and a;(q) by: (22) where
where ar(4) is defined by eq. (11). (b) Second order approxi??lah’ons A second set of expressions for F(q, o) arises by gproximating the second order memory functions K2(4, w). When this function is taken equal to the corresponding function for the free particles:
The numerical consequences of this mode! have been studied elsewhere [ 14,15,18]. (cj TIM
order approximations
Kim and Nelkin [ 181 introduce and study the next 105
natural Q?.
approximation
which consists
in taking:
a) -
wb = Q(7’
But now, a difficulty arises due to the inaccuracy of the sixth frequency moment. which involves the threeparticle correlation function. In their calculation the quantity b:(q) = A:(q)/A&(q) is used, In fact, as an adjustable parameter_
4. A new model
The above review of the various results obtained by ‘Lhelinear response theory suggests to introduce a new approximation through the memory funcJion formalism, i.e., to modify the expression for F(q, w) [eq. (lo)] , given by Pathak and Singwi, by taking into account the Kerr model. In order to do that we classify various possible choices of the memory functions in table 1. This table suggests at least one new approximation by taking: = K&4. a,(s) 0 .
K(4,0
The same time dependence applied to the second order memory function would lead to the approximation: K,(q* 0 = K*(q,
15 September 1972
CHEMICAL PHYSICS LETTERS
Volume 16. number 1
b,(q) t)
but to date the coefficient bs(4) is not suflicientiy known to develop this approximation.
(b) Numerical results and discussion For n_umerical calculation we use the parametrization of F,(q,o) given by Levesque and Verlet [ 191. The values of the parameters for the simulated [ 201 thermodynamic state for argon considered here (p =
1.407 g/cm3 and T = 76°K) were obtained by interpolating the data o.f these authors. Fig. 1 shows the variations of the spectral function w2S(q, o)/(q2/flnt) as a function of o for different values of 4. It is inteLesting to note that, although the low-q behaviour o,f Fs(q, o) used here is very different from that of F~((I, w) used by Pathak and Singwi [8], the functions w*S(q, o)/(q2//3m) issued from both methods are very similar. This is not inconsistent since in the hydrodynamical regime (4 < 1 a-1) af(q) and a,(q) have themselves very different behaviours:
(23) Fig. 2 exhibits the variations, as a function of 9, of the position o max((l) and of the height of the maximum of o*S(q, w)/(4*//3m). The good fitting of the molecular dynamics data by the present approach Ied us to built up a general formulation to be published eisewhere [2 l] and appearing as a generalization of the continued fraction representation of Mori [ 111. Fina!ly it will be stressed that the present work gives the exact low order momenta up to the fourth Table 1
hfemory
convolution approximation (Vineyard)
0
SC 14.51 K(4. r> = Kf(4.0
ESC (Ken [61) KG?.t) = K&7, t)
0, 2
K(q, r) =
fithak and Sin@ [8] Kf (q.ofr)
KQ, 0 = k’s@. as 0
LP 1151 Kz @,0 = Kc2f(%0
EL? [ 14,151 KZ@, 0 = K2&,
0 a=1 1 (7 = a(q)
2 3
Order of the satidied moments
Self
Free
function order
Present model
Kim
0, 2,4 0
and Nelkin [ 181 K3(q, 0 = K3s(q* f)
6294.6
Volume 16, number 1
2.0 -
CHEMICAL PHYSICS LETTERS
q=1,17r
_
_
==I ..
A I. l
2.0
0
q =
2.3
q =
1.97
15 September 1972
(whereas only the zeroth and the second are satisfied in [6]). It leads to a similar fitting of the data as the ELP model [13,14]. From a theoretical point ofview, this new model introduces as does the ELP model, the irreversible behaviour of F(q, t) through the function F&q, t) which is supposed to be known elsewhere. Such a behaviour does not seem to appear in the ap preach of Pathak and Singwi [8] due to the essentially reversible character of the evolution of Ff(q. t).
h'
A-."
I.
05
1.0
15
w(d'se~)
2D
Fig. 1. Spectral function of the longitudinal current correlations w2S(q. w)/(q2/fim) ptotted as a function of the frequency. -our results, --Pathak and Singwi’s results [8], x x x Kim and Nelkin’s results [ 181,. . . molecular dynamics data [ 20].
00)
q( A-‘)
.
Fig. 2. Dispersion curve am=@) and height of the maximum of the normalized spectral function of the longitudinal current correlations as a function of the wave vector 4. -our results, --Pathak and Singwi’s results [8], x x x Kim and Nelkin’s resti*& [18],... molecular dynamics data [ 20 1.
107
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CHEMICAL PHYSICS LETTERS
References [I] R. Kubo, Lectures in theoretical physics, Vol. 1 WileyInterscience, Xew York, 1958). [2] K.S. Singwi, K. Skold and 51.P. Tosi, Phys. Rev. Al (1970) 454. [ 31 J.K. Percus, The equilibrium theory of classical fluids (Eenjamin, New York, 1364). [4] hi. Nelkin and S. Ranganathan, Phys. Rev. 164 (1967)
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