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Temperature and density-dependence of shear viscosity in fluids
Volume 117, number 2
PHYSICS LETTERS A
28 July 1986
T E M P E R A T U R E AND D E N S I T Y - D E P E N D E N C E O F SHEAR VISCOSITY IN FLUIDS
Fukuo YOSHIDA Research Reactor Institute, Kyoto University, Kumatori-cho, Sennan-gun, Osaka 590-04, Japan Received 6 January 1986; revised manuscript received 14 May 1986; accepted for publication 14 May 1986
The shear viscosity is shown to consist of a gas- and a solid-like contribution by using a relaxation theory for fluids. It is evaluated for liquid argon near the triple point, and found to be in good agreement with experiments. The possibility of observing a minimal shear viscosity as in the classical one-component plasma is discussed.
Recently we have explained a curious property of the shear viscosity ~/ in classical one-component plasmas (OCP) [1] by investigating the relaxation mechanism of effective fields associated with transverse collective excitations [2,3]. In the strongly coupled regime structure relaxation dominates, whereas current relaxation dominates in the weakly coupled regime. As a result of the competition between them, the reduced viscosity has a minimum at some value of the plasma parameter, where both mechanisms contribute with nearly equal weight. It seems that such a physical argument would be applicable with little change to fluids with a short-range interaction potential compared with the OCP, considering the close correspondence between the OCP and these systems' dynamical behaviour. In this letter we discuss, from the abovementioned point of view, the appearance of a minimum shear viscosity as a function of the temperature T and density n in classical fluids. For that purpose we introduce a coupling parameter /" which reduces to the plasma parameter in the OCP. It is found that for liquid argon at reduced density n * = 0 . 8 5 and temperature T * = 0 . 7 2 , which corresponds to F = 161, our result for agrees reasonably well with the experimental result. It will be shown by using an empirical formula for the self-diffusion coefficient D~ that there is a possibility of having a minimal shear viscosity at some value of F depending on T and n. Consider a fluid composed of N particles of
mass m in a volume V, interacting by means of a pairwise additive potential v(r). The transverse current correlation function Jr(k, t) defined by
Jr(k,
t)= (jr(k,
t)jt(-k))/(jt(k)jt(-k)),
N
j r ( k ) = ~] vjx exp(ikrj),
(1)
j=l
is formulated by using the memory-function method [4,5] as
Jr(k, z)=Jf(k,
z)/[l-/-/t(k,
z ) J f ( k , z)].
(2) Here Jr(k, forms of
z) and Ht(k, zO are the Laplace transJr(k, t) and a polarization function I-It(k, t), respectively. The function Jf(k, z) is also the Laplace transform of the correlation function corresponding to a non-interacting system. In eq. (1) the z-axis is taken to be in the direction of the wavevector k, and the angular bracket stands for the ensemble average. Our basic approximation is to express H t ( k , t) in the form [2,3]
1-It(k,
t ) = [ 1 - q~(t)q_Ml.f(k, t)
--[~ot(k)2/(kvv)2]L(k, t).
(3)
In the above equation q~(t) is a normalized velocity auto-correlation function, Ml.r(k, t) a normalized first-order memory function, wt(k ) a characteristic frequency for the transverse collective excitation given in eq. (8) below, v x = (kBT/m) 1/2
0375-9601/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
77
Volume 117, number 2
PHYSICS LETTERS A
the thermal velocity of particles, and Fs(k, t) is a scaled self-correlation function. Eq. (3) is obtained straightforwardly by extracting a self-part in the first-order m e m o r y function. The function Ml,f(k, t) guarantees the correct non-interacting limit, and ~ ( t ) is squared because we have adopted the decoupling approximation for the velocity correlation (Ojx(t)vjz(t)vjx(O)vjz(O)) for the j t h particle. The first term in eq. (3) vanishes at t = 0, and represents the current relaxation essentially described by q~(t). On the other hand, the second term of the structure relaxation describes how the initially constructed effective field relaxes as time increases. The function Fs(k, t) physically represents that particles make a diffusive motion, and this incoherent motion results in reducing effectively the coupling constant between particles. It is not a self-correlation function in a usual sense since the propagator is modified by a projection operator [4]. The effect of the modification would not be large at small wavenumbers due to the conservation laws as far as collective variables are concerned [6]. In the continued fraction expansion high-order memory functions are expected to decay faster, while it is generally difficult to estimate their complicated time-dependence. It would be reasonable for the self-correlation to assume that
t)= t* = a ( k ) t
t*),
= [WZE+ 2(kv-r) :~]t/2(kvT) 2,
(4)
at this stage, just as done in the OCP [2]. Taking the small wavenumber and frequency limits of the polarization function, and also taking into account that Fs(k, t) behaves as exp(-kEDd) for small k, we obtain
* i = - n m lim k-EH(k, ito)=*io+*is, k,o~--,O
no
= nkBT
*is = k.T/D~l,
dt ~ ( t ) 2,
n=
N/V,
l= l(n, T) = w2/2nc 2.
(5) (6) (7)
The Einstein frequency o~E in the above equations and 60t(k ) in eq. (3) are given by
28 July 1986
We have used the fact that ~0t(k ) = ctk for small k with c t being the transverse sound velocity. It is to be noticed that *is is inversely proportional to D s. AS OJE and ~0t(k ) are written in terms of the pair-correlation function g(r) and the second derivative of v(r) we can evaluate in principle the temperature and density-dependent correlation length l introduced above if enough information is available for g(r). The expression of *is agrees with that obtained by Eyring in the significant structure theory [7,8]. It is derived by assuming that the activation process is essential as an elementary reaction process in liquids. We have associated it with the relaxation of effective fields for collective excitations, and in this sense our result can be considered as its generalization. In addition to this solid-like contribution with *Is inversely proportional to D, we have the first term *IG. If we adopt the following form of q ( t ) [9] for simplicity, q~(t) = exp( -
t/2"r)[cos(¢t) + sin(ct)/2,,],
c 2 = ~0~ - (2~-) -2,
I.-1 = Os(o.~ E//OT) 2,
(9)
we obtain
*i~ = mn ( D s / 2 + to2~2).
(10)
The second term in the above equation was neglected in our previous work. It comes from the oscillatory behaviour of 4~(t). It is however much smaller than *is when D s is small, and also smaller than the first term of *io when D s is large. Thus it is a small correction to the whole #1. This gas-like contribution differs by a factor two from the viscosity of an ideal gas when the correction term is neglected. The contribution of F is important at high temperature or low density, in marked contrast to *is. In the OCP, the plasma parameter is defined by F = e2/akaT, and it is actually found that for F < 1 *is is less than 1%, while *io is less than 3% for F > 100 [2]. For the system considered here, just as for Lennard-Jones liquids, we introduce the coupling parameter F as
r
= (¢oEa)
2
2 loT,
(11)
w2=n fdrg(r)V2v(r), wt(k)2=n fdrg(r)V2v(rl(1-coskz). 78
(8)
#1 We note that in the OCP the first and second terms in eq. (10) have the dependence of F -1"34 and /`-0.66, respectively and ~ behaves nearly as/,0.34.
Volume 117, number 2
PHYSICS LETTERS A
28 July 1986
where a is a characteristic radius defined by a =
(3/4'rrn) 1/3. It reduces to the plasma parameter since cOE/¢~- can be identified with the plasma frequency in the OCP [1]. The quantity OT/cOE is the radius of the thermal cloud appearing in the D e b y e - W a l l e r factor of Einstein solids. Thus physically F - 1 represents the degree of the extensions of the particle's motion measured in the unit of mean interparticle distance. For liquid argon there is an empirical formula for D s obtained by Levesque and Verlet [10],
I
I
I0 20
I
I
,50 I00
I
P
Fig. 1. F-dependence of r/* with T* fixed at 1, 5 and 10 for liquid argon.
D* = 0 . 0 4 4 5 T * ( n * ) -2 + 0.1538 - 0.1490n*, (12) where Ds* = Ddr/o 2, T* = T / c and n* = n o 3, with o = 3.405 ,~, c = 120 K and r = (c/mo2) -112. Eq. (5) is rewritten in the reduced form as
7* ~7"r/mno 2= 7D;1 . + (1/2F + ~,rra./l.) X (a*)2T*/D*.
(13)
It is found that the state at T* = 0.72 and n * = 0.85 corresponds to F = 161. Using the above equation for D*, and the values of COE and c t as 7.7 x 1012 s -1 [11] and 0.79 X 105 c m / s [12], respectively, we obtain 7 * = 3.98, and thus 7 is 3.2 x 1 0 - 3 P. This estimation is very close to the result of the computer simulation, i.e., 3.65 X 10 -3 P [13]. We note that the correlation length 1" = l/o turns out to be 6.5 for this state ~,2 and 7~ is about 1% of 7*. When we take into account the property that WE and also c0t(k ) have the dependence of T1/2n, we can estimate with eq. (12) for D* the F-dependence of 7*- Fig. 1 shows the calculated results at T* = 1, 5 and 10. It is found that the minimum of 7* takes place at F = 35, 25 and 22, for, respectively, T * = I , 5 and 10. According to eq. (12), 7* diverges at a finite n* (F) since D* vanishes there. These are 1.02 (179), 1.38 (121) and 1.64 (107) for T * = 1, 5 and 10, respectively. Fig. 2 shows the F-dependence of 7" with n* fixed at 0.85, 0.6 and 0.4. For larger T*, , 2 In our previous work [2], we made some comments on I in liquid argon saying it is of the order of the interparticle distance. These should be corrected, but this does not change the main conclusion obtained there.
7~ approaches a constant and the feature of ,/* is governed by 7~. As T* decreases the behaviour of 7* depends on the value of n*. When n* is larger than n c* = 0.79, 7* shows a similar behaviour as shown in fig. 1. We note that at n * = 0 . 8 5 , 7* takes a minimum value of 3.3 at T* = 4.0. For n* smaller than no, * 7* decreases monotonously as F increases. In summary we have discussed the shear viscosity 7 of fluids in analogy with that of the OCP, by investigating the effective field-relaxation associated with the transverse collective excitations. Our key expression for 7 given by eq. (5) consists of a gas- and a solid-like contribution, corresponding to the current and structure relaxation of the effective field as given by eq. (3). The former gives a dominant effect in a weakly coupled regime, and the latter in a strongly coupled regime. This clearly means that a simple formula for 7 inversely pro-
4 ~n*=0,85
2~~n*:o.6 ~.-~n*=04 0 i i , , 20 93 I00 I" Fig. 2. F-dependence of */* with n* fixed at 0.4, 0.6 and 0.85 for liquid argon.
79
Volume 117, number 2
PHYSICS LETTERS A
portional to the self-diffusion coefficient D~ is physically not appropriate. Our expression can explicitly be written in terms of the thermal velocity v T, the Einstein frequency w E, the transverse s o u n d velocity c t, a n d D~. We have f o u n d that it is in good agreement with the experimental result for. liquid argon near the triple point, where 4~ is very small c o m p a r e d with 4~- This indicates that our formula is practically useful as it gives a simple estimate even in fluids with a short-range interaction potential. W e have further investigated the T * - a n d n * - d e p e n d e n c e of 4* by i n t r o d u c i n g a coupling p a r a m e t e r F given b y eq. (11) in analogy with the OCP. It is f o u n d that 4" has a m i n i m u m a r o u n d F = 2 0 - 4 0 as shown in fig. 1. These values are n o t very different from the value of 4" of the m i n i m u m for the OCP, in which case the two relaxation m e c h a n i s m s are almost equally effective. W e m a y therefore conclude that the property that 4* becomes m i n i m u m is a rather general one for fluids. While the formula for D* varies from system to system, our results suggest that such a property would be observable if it was systematically investigated u n d e r experimentally favourable conditions.
80
28 July 1986
The present author would like to express his sincere thanks to Professor. S. T a k e n o for valuable discussions a n d c o n t i n u e d encouragement.
References [1] [2] [3] [4] [5]
M. Baus and J.P. Hansen, Phys. Rep. 59 (1980) 1. F. Yoshida, Phys. Lett, A 113 (1985) 207. F. Yoshida, J. Phys. Soc. Japan 53 (1984) 4098. H. Mori, Prog. Theor. Phys. 33 (1965) 423. S. Takeno and F. Yoshida, Prog. Theor. Phys. 60 (1978) 1304. [6] Y. Pomeau and P. Resibois, Phys. Rep. 19 (1975) 63. [7] H. Eyring and M.S. John, Significant liquid structures (Wiley, New York, 1965) ch, 5. [8] P.A. Egelstaff, An introduction to the liquid state (Academic Press, New York, 1967) p. 153. [9] B.J. Berne, J.P. Boone and S.A. Rice, J. Chem. Phys. 45 (1966) 1086. [10] D. Levesque and L. Verlet, Phys. Rev. A 2 (1970) 2514. [11] J.R.D. Copley and S.W. Lovesey, Phys. Rep. 38 (1975) 461. [12] S. Takeno and M. Goda, Prog. Theor. Phys. 45 (1971) 331. [13] D. Levesque and L. Verlet, Phys. Rev. A 7 (1973) 1690.
PHYSICS LETTERS A
28 July 1986
T E M P E R A T U R E AND D E N S I T Y - D E P E N D E N C E O F SHEAR VISCOSITY IN FLUIDS
Fukuo YOSHIDA Research Reactor Institute, Kyoto University, Kumatori-cho, Sennan-gun, Osaka 590-04, Japan Received 6 January 1986; revised manuscript received 14 May 1986; accepted for publication 14 May 1986
The shear viscosity is shown to consist of a gas- and a solid-like contribution by using a relaxation theory for fluids. It is evaluated for liquid argon near the triple point, and found to be in good agreement with experiments. The possibility of observing a minimal shear viscosity as in the classical one-component plasma is discussed.
Recently we have explained a curious property of the shear viscosity ~/ in classical one-component plasmas (OCP) [1] by investigating the relaxation mechanism of effective fields associated with transverse collective excitations [2,3]. In the strongly coupled regime structure relaxation dominates, whereas current relaxation dominates in the weakly coupled regime. As a result of the competition between them, the reduced viscosity has a minimum at some value of the plasma parameter, where both mechanisms contribute with nearly equal weight. It seems that such a physical argument would be applicable with little change to fluids with a short-range interaction potential compared with the OCP, considering the close correspondence between the OCP and these systems' dynamical behaviour. In this letter we discuss, from the abovementioned point of view, the appearance of a minimum shear viscosity as a function of the temperature T and density n in classical fluids. For that purpose we introduce a coupling parameter /" which reduces to the plasma parameter in the OCP. It is found that for liquid argon at reduced density n * = 0 . 8 5 and temperature T * = 0 . 7 2 , which corresponds to F = 161, our result for agrees reasonably well with the experimental result. It will be shown by using an empirical formula for the self-diffusion coefficient D~ that there is a possibility of having a minimal shear viscosity at some value of F depending on T and n. Consider a fluid composed of N particles of
mass m in a volume V, interacting by means of a pairwise additive potential v(r). The transverse current correlation function Jr(k, t) defined by
Jr(k,
t)= (jr(k,
t)jt(-k))/(jt(k)jt(-k)),
N
j r ( k ) = ~] vjx exp(ikrj),
(1)
j=l
is formulated by using the memory-function method [4,5] as
Jr(k, z)=Jf(k,
z)/[l-/-/t(k,
z ) J f ( k , z)].
(2) Here Jr(k, forms of
z) and Ht(k, zO are the Laplace transJr(k, t) and a polarization function I-It(k, t), respectively. The function Jf(k, z) is also the Laplace transform of the correlation function corresponding to a non-interacting system. In eq. (1) the z-axis is taken to be in the direction of the wavevector k, and the angular bracket stands for the ensemble average. Our basic approximation is to express H t ( k , t) in the form [2,3]
1-It(k,
t ) = [ 1 - q~(t)q_Ml.f(k, t)
--[~ot(k)2/(kvv)2]L(k, t).
(3)
In the above equation q~(t) is a normalized velocity auto-correlation function, Ml.r(k, t) a normalized first-order memory function, wt(k ) a characteristic frequency for the transverse collective excitation given in eq. (8) below, v x = (kBT/m) 1/2
0375-9601/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
77
Volume 117, number 2
PHYSICS LETTERS A
the thermal velocity of particles, and Fs(k, t) is a scaled self-correlation function. Eq. (3) is obtained straightforwardly by extracting a self-part in the first-order m e m o r y function. The function Ml,f(k, t) guarantees the correct non-interacting limit, and ~ ( t ) is squared because we have adopted the decoupling approximation for the velocity correlation (Ojx(t)vjz(t)vjx(O)vjz(O)) for the j t h particle. The first term in eq. (3) vanishes at t = 0, and represents the current relaxation essentially described by q~(t). On the other hand, the second term of the structure relaxation describes how the initially constructed effective field relaxes as time increases. The function Fs(k, t) physically represents that particles make a diffusive motion, and this incoherent motion results in reducing effectively the coupling constant between particles. It is not a self-correlation function in a usual sense since the propagator is modified by a projection operator [4]. The effect of the modification would not be large at small wavenumbers due to the conservation laws as far as collective variables are concerned [6]. In the continued fraction expansion high-order memory functions are expected to decay faster, while it is generally difficult to estimate their complicated time-dependence. It would be reasonable for the self-correlation to assume that
t)= t* = a ( k ) t
t*),
= [WZE+ 2(kv-r) :~]t/2(kvT) 2,
(4)
at this stage, just as done in the OCP [2]. Taking the small wavenumber and frequency limits of the polarization function, and also taking into account that Fs(k, t) behaves as exp(-kEDd) for small k, we obtain
* i = - n m lim k-EH(k, ito)=*io+*is, k,o~--,O
no
= nkBT
*is = k.T/D~l,
dt ~ ( t ) 2,
n=
N/V,
l= l(n, T) = w2/2nc 2.
(5) (6) (7)
The Einstein frequency o~E in the above equations and 60t(k ) in eq. (3) are given by
28 July 1986
We have used the fact that ~0t(k ) = ctk for small k with c t being the transverse sound velocity. It is to be noticed that *is is inversely proportional to D s. AS OJE and ~0t(k ) are written in terms of the pair-correlation function g(r) and the second derivative of v(r) we can evaluate in principle the temperature and density-dependent correlation length l introduced above if enough information is available for g(r). The expression of *is agrees with that obtained by Eyring in the significant structure theory [7,8]. It is derived by assuming that the activation process is essential as an elementary reaction process in liquids. We have associated it with the relaxation of effective fields for collective excitations, and in this sense our result can be considered as its generalization. In addition to this solid-like contribution with *Is inversely proportional to D, we have the first term *IG. If we adopt the following form of q ( t ) [9] for simplicity, q~(t) = exp( -
t/2"r)[cos(¢t) + sin(ct)/2,,],
c 2 = ~0~ - (2~-) -2,
I.-1 = Os(o.~ E//OT) 2,
(9)
we obtain
*i~ = mn ( D s / 2 + to2~2).
(10)
The second term in the above equation was neglected in our previous work. It comes from the oscillatory behaviour of 4~(t). It is however much smaller than *is when D s is small, and also smaller than the first term of *io when D s is large. Thus it is a small correction to the whole #1. This gas-like contribution differs by a factor two from the viscosity of an ideal gas when the correction term is neglected. The contribution of F is important at high temperature or low density, in marked contrast to *is. In the OCP, the plasma parameter is defined by F = e2/akaT, and it is actually found that for F < 1 *is is less than 1%, while *io is less than 3% for F > 100 [2]. For the system considered here, just as for Lennard-Jones liquids, we introduce the coupling parameter F as
r
= (¢oEa)
2
2 loT,
(11)
w2=n fdrg(r)V2v(r), wt(k)2=n fdrg(r)V2v(rl(1-coskz). 78
(8)
#1 We note that in the OCP the first and second terms in eq. (10) have the dependence of F -1"34 and /`-0.66, respectively and ~ behaves nearly as/,0.34.
Volume 117, number 2
PHYSICS LETTERS A
28 July 1986
where a is a characteristic radius defined by a =
(3/4'rrn) 1/3. It reduces to the plasma parameter since cOE/¢~- can be identified with the plasma frequency in the OCP [1]. The quantity OT/cOE is the radius of the thermal cloud appearing in the D e b y e - W a l l e r factor of Einstein solids. Thus physically F - 1 represents the degree of the extensions of the particle's motion measured in the unit of mean interparticle distance. For liquid argon there is an empirical formula for D s obtained by Levesque and Verlet [10],
I
I
I0 20
I
I
,50 I00
I
P
Fig. 1. F-dependence of r/* with T* fixed at 1, 5 and 10 for liquid argon.
D* = 0 . 0 4 4 5 T * ( n * ) -2 + 0.1538 - 0.1490n*, (12) where Ds* = Ddr/o 2, T* = T / c and n* = n o 3, with o = 3.405 ,~, c = 120 K and r = (c/mo2) -112. Eq. (5) is rewritten in the reduced form as
7* ~7"r/mno 2= 7D;1 . + (1/2F + ~,rra./l.) X (a*)2T*/D*.
(13)
It is found that the state at T* = 0.72 and n * = 0.85 corresponds to F = 161. Using the above equation for D*, and the values of COE and c t as 7.7 x 1012 s -1 [11] and 0.79 X 105 c m / s [12], respectively, we obtain 7 * = 3.98, and thus 7 is 3.2 x 1 0 - 3 P. This estimation is very close to the result of the computer simulation, i.e., 3.65 X 10 -3 P [13]. We note that the correlation length 1" = l/o turns out to be 6.5 for this state ~,2 and 7~ is about 1% of 7*. When we take into account the property that WE and also c0t(k ) have the dependence of T1/2n, we can estimate with eq. (12) for D* the F-dependence of 7*- Fig. 1 shows the calculated results at T* = 1, 5 and 10. It is found that the minimum of 7* takes place at F = 35, 25 and 22, for, respectively, T * = I , 5 and 10. According to eq. (12), 7* diverges at a finite n* (F) since D* vanishes there. These are 1.02 (179), 1.38 (121) and 1.64 (107) for T * = 1, 5 and 10, respectively. Fig. 2 shows the F-dependence of 7" with n* fixed at 0.85, 0.6 and 0.4. For larger T*, , 2 In our previous work [2], we made some comments on I in liquid argon saying it is of the order of the interparticle distance. These should be corrected, but this does not change the main conclusion obtained there.
7~ approaches a constant and the feature of ,/* is governed by 7~. As T* decreases the behaviour of 7* depends on the value of n*. When n* is larger than n c* = 0.79, 7* shows a similar behaviour as shown in fig. 1. We note that at n * = 0 . 8 5 , 7* takes a minimum value of 3.3 at T* = 4.0. For n* smaller than no, * 7* decreases monotonously as F increases. In summary we have discussed the shear viscosity 7 of fluids in analogy with that of the OCP, by investigating the effective field-relaxation associated with the transverse collective excitations. Our key expression for 7 given by eq. (5) consists of a gas- and a solid-like contribution, corresponding to the current and structure relaxation of the effective field as given by eq. (3). The former gives a dominant effect in a weakly coupled regime, and the latter in a strongly coupled regime. This clearly means that a simple formula for 7 inversely pro-
4 ~n*=0,85
2~~n*:o.6 ~.-~n*=04 0 i i , , 20 93 I00 I" Fig. 2. F-dependence of */* with n* fixed at 0.4, 0.6 and 0.85 for liquid argon.
79
Volume 117, number 2
PHYSICS LETTERS A
portional to the self-diffusion coefficient D~ is physically not appropriate. Our expression can explicitly be written in terms of the thermal velocity v T, the Einstein frequency w E, the transverse s o u n d velocity c t, a n d D~. We have f o u n d that it is in good agreement with the experimental result for. liquid argon near the triple point, where 4~ is very small c o m p a r e d with 4~- This indicates that our formula is practically useful as it gives a simple estimate even in fluids with a short-range interaction potential. W e have further investigated the T * - a n d n * - d e p e n d e n c e of 4* by i n t r o d u c i n g a coupling p a r a m e t e r F given b y eq. (11) in analogy with the OCP. It is f o u n d that 4" has a m i n i m u m a r o u n d F = 2 0 - 4 0 as shown in fig. 1. These values are n o t very different from the value of 4" of the m i n i m u m for the OCP, in which case the two relaxation m e c h a n i s m s are almost equally effective. W e m a y therefore conclude that the property that 4* becomes m i n i m u m is a rather general one for fluids. While the formula for D* varies from system to system, our results suggest that such a property would be observable if it was systematically investigated u n d e r experimentally favourable conditions.
80
28 July 1986
The present author would like to express his sincere thanks to Professor. S. T a k e n o for valuable discussions a n d c o n t i n u e d encouragement.
References [1] [2] [3] [4] [5]
M. Baus and J.P. Hansen, Phys. Rep. 59 (1980) 1. F. Yoshida, Phys. Lett, A 113 (1985) 207. F. Yoshida, J. Phys. Soc. Japan 53 (1984) 4098. H. Mori, Prog. Theor. Phys. 33 (1965) 423. S. Takeno and F. Yoshida, Prog. Theor. Phys. 60 (1978) 1304. [6] Y. Pomeau and P. Resibois, Phys. Rep. 19 (1975) 63. [7] H. Eyring and M.S. John, Significant liquid structures (Wiley, New York, 1965) ch, 5. [8] P.A. Egelstaff, An introduction to the liquid state (Academic Press, New York, 1967) p. 153. [9] B.J. Berne, J.P. Boone and S.A. Rice, J. Chem. Phys. 45 (1966) 1086. [10] D. Levesque and L. Verlet, Phys. Rev. A 2 (1970) 2514. [11] J.R.D. Copley and S.W. Lovesey, Phys. Rep. 38 (1975) 461. [12] S. Takeno and M. Goda, Prog. Theor. Phys. 45 (1971) 331. [13] D. Levesque and L. Verlet, Phys. Rev. A 7 (1973) 1690.