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A master equation for S(q,ω) leading to the hydrodynamic regime for simple fluids
CHEMICAL
Volume 21, number 3
A MASTER
EQUATION
HYDRODYNAMIC
PHYSICS LETTERS
FOR REGIME
S(q, w) LEADING FOR
SIMPLE
1 September 1973
TO THE FLUIDS
Lionel BONAMY* and Nguyen Mifi HOANG** Groupe de Pllysique hlolk-ulaire, FmulrPdes Sciences, 25030 Besnn~on. France Received 14 May 1973
A master equation relating the coherent and incoherent dynamical slructure factors S(q, w) and S,(q, w) is introduced on the basis of physical arguments suggested by previous theories. It is shown that, once S,(q, w) and the fnst six moments of S(o. w) are known, the resulting expression for S(q, w) contains the correct hydrodynamic regime as well as the free particle behaviour. The relationships thus introduced berween the transport coefficients and the molecular quantities are compared to the experimental data for liquid argon. Molecular dynamics data arc also well reproduced.
The aim of the present paper is to propose an expression for the dynamical structure factor S(q, w) which implies, for a monoatomic fluid, the following features [ 1,2] : (i) free particle regime (FPR) (large q and w) leading to a single particle diffusion peal:; (ii) hydrodynamic limit [3,4] (HL) (q + 0, w + 0, w/q = const) exhibiting both a longitudinal sound wave peak (Brillouin scattering) and a heat diffusion peak (Rzyleigh diffusion); (iii) transition from HL to FPR going through a brosdening and a subseqlient disappearance of the sound wave peaks, as is suggested by continued hydrodynamics [5] or as shown by molecular dynamics computations [4] . The memory function formalism [7,8] will p:ove to be an appropriate scheme to connect the approximations used in several existir.g theories, and in the present work. in this formalism the coherent and incoherent dynamical structure factors are written [5,6] in the form:
of the so-called memory functions M(q, t) and M,(q, t). S(q) is the static structure factor, p-l = X-, T, m is the mass of an atom, and c02(qj = pmS(q). The function Ss(q,oj describes the motion of an atom in the fluid. This function, and hence zs(q, w), will be supposed to be known from other sources [9, lo]. The two limits mentioned above for S(q, w) are easily incorporated in this memory function representation. ‘l?Le FPR is o_btained if in eq. (1) one puts S(q) = 1 and nf(q, w) =Mdq, w) (memory function for a free particle motion). On the oiher hand, it is well known that, in the HL, the Kadanoff and Martin [3] result is recovered if one takes:
S(q,w)=S(qjReCiw+c~(qjq2/[iw+fi(q,~j]
Ilri(q,w)=r*-14pq2+(r_lj~~(0)q2/[iw+yDT42]
1-l ,(la)
Ss(q,Wj=Re{iwt(q2/Pnlj/[iw+fis(q,w)]]-1,
(It)
where &:q, w) and fl?,(q, w) are the Laplace forms, say:
trans-
fi(q, w) = L {M(q) t)} = 7 dt emiw ‘&I(q, tj 0
(2) * Cha@ chcrche l * Attache cherche
470
de Recherches at the Centre National de 1; XeScientique. de Recherches at the Centre National de la ReScientifiiqw.
= L [n-ltlpq28(t)+(-pi)+o)$
exp(-wTq2tjjI
Here n is the density, 7 = CJC,, DT is the heat diffusive coastant, and qp is the longitudinal viscosity.
For intermediate 9 values the available analytical information on S(q, w), and hence on fi(q, o), are the lowest order moments. Note that in eq. (1) the zeroth and second moments ofS(q, w) are automatically satisfied whatever could be &(q, w). The limiting behaviours of $,(q, w) are [ 1 l] tidy, 0) for 4 + = and B(w) for 4 - 0. Here R(w) is the memory function for the velocity autocorrelaticn function. Therefore in the HL it is found that [2, IO] : fqq,
Cd) = X(0) + iG’(O)
I i i
q
q
0.025
..l
A
i
i i
+ O(q*) (3)
I
= L [k(O) s(t) + C.&J)] , where k-l (0) = flnzD and K’(O) = -( 1 fC/6,DnzD) define the long time behaviour of the mean square displacement of an atom (( [br(t)12) = 6Dr + C for f --f -)_ The present approach is based on a comparison of the physical contents of two previous theories linking M(q, t) and M,(q, t). The fist is the Kerr approach [ 121 which may be expressed by [ 13] : Jf(9 >t> = M&9, r> .
1 September 1973
CHEMICAL PHYSICS LETTERS
Volume 21, number 3
(4)
Physically, this approximation does not allow the appearance of Brillouin scattering in the HI, (cf. fig. 1). The second theory is the extended liquid phonon (YELP)model expressed by [ 131:
Jm 39 =qs I0 = WC4Iwq~ >O)lq4: 0 5 (5) which leads, qualitatively [14], in the HL, to Brtilouin scattering only (cf. fig. 1). It may be verified that the inability of these two approximations to describe simultaneously Rayleigh diffusion and Brillouin scattering comes from the fact that eqs. (4) and (5) mvolve the same time dependence forM(q, t) and M,(q,t), at the opposite of eqs. (2) and (3). But for intermediate 4 values eq. (4) or (5) is satisfactory since molecular dynamics tells that S(q,o) and S,(q, w) are represented then by similar and simple curves [6, 10, 151 @road and structureless peaks); it is therefore possible in this last range of 4 values to compensate for the discrepancies between S(q, w) and S,(q, w) by adjusting conveniently the lowest order moments. Now it may be expected that some functional relationship between M(q, f) and M,(q, t) better than these two previous approximations should retain their structures in some way but should modify the time dependence mentioned above. These requirements
i
1 i
I i
‘\
Y&.__, ...
0
0.1
0.2
w
,_____...=-“..‘
1
(1
O”%x-’
1.5
2.0
2.5
)
Fig. 1. The dynamical structure factor of iiquid argon in the hydrodynamic regime (state 1): _ the result of Kadanoll and Martin, eq. (2); --- the Kerr result, eqs. (11 and (41; - - - the result of the ELP model. eqs. (I) x~d (5); the present result, eqs. (l), (6) and (7).
can be cast into the following evolution equation connecting the collective aspec.ts, contained in M(q, t) M,(q, r) [and missing in Kerr’s approach, eq. (4)], with the individual aspects contained inM(q,r) M,(q, t) [and missing in the ELI’ model, eq. (5)] :
MqJ)-&(q,t)
=-Sd~~~~,r-i,mi(~.~~-~,~~(~!~~~. 0
(6)
It may be shown that the structure of this master equation already allows, through a simple ansatz for ici(q: t), to get a reasonable behaviour for S(9, w) over the whole range of 4 values. Knowing that Ms(q, t) goes as a 6(t) function when 9 -+ 0 [cf. eq. (?)I , the master equation (6) shows that the exponential term appearing in M(q, I) [cf. eq. (2)] will be obtained if N(q, t) behaves itself as a 6(r) function in this limit. A possible way to introduce this limiting behaviour is to choose:
471
CHEMICAL PHYSICS LETT’ERS
Volume 21, number 3
An analytical form for the proportionality ~~(4) may be obtained by fuling the second derivative ofM(q, t) [i.e., the shth moment S(q, w)] _Indeed from eqs. (6) and (7), one
1973
constant initial time of finds:
NO)-fis~4,0)
U21dQ2 =~s(4~o)~~5(~~o)--M(~~o)l
1 September
(8)
’
whereM(q,O) = [cl(q) - c:(q)] q2 and fi(q,O) = -rrM(q,O)/2?(4). The quantities c,(q) and ‘i-(q) are defined with the help of known molecular integrals involving the intermolecu!ar potential and the pair and triplet distribution functions g(r) and g3(r, r’) [S, 10, 161. Now it may be verified, after solving eq. (6) by a Laplace transformation, that eq. (2) is
formally recovered following relations: rD,
= u2(0)R(O)
in the HL. Moreover
:
(y-l)c$O)
one gets the
= u”(0)K2(0),
(9)
These equations may be considered as ccnnecting the various transport ccefficients to the self transport constants (D and ct) and to the molecular integrals. Note that an expression for PJ-~T~, which reduces to the first term of the rhs of eq. (lo), was previously proposed [14]. Knowing S,(q, o), or fis(q, w) [lo], and the moments of S(q, w) up to the sixth, we get, through relations (l), (6) and (7), an expression for S(q, w) which appears to describe the correct behaviour in the whcle ratige of 4 vaiues. A numerical test has been performed in the case of liquid argon in the following states: (I) I! = 1.413 g/cm3, T = 90°K and (II) II = 1.407 glcm3, T = 76°K. !: Hydrodynamic &(O,O),
limit. The constants Ms(O,O), D and C for state I were obtained by inter-
polatiilg results of simulations [lo]. In state I experimenus [ 171 give 7 = 2.00 and tile isothermal sound velocity W = 886 mlsec leading to S(0) = r/&n W2 = 0.05C; molecular dynamics computations [ 181 give S(0) = 0.04 for state II. Roughly the same value coa(0) = 1500 misec has been obtained for state II (from simulated values ofM(q,O) [15. 181) and for state I (by evaluating directly the molecular expres472 .,.
;
Fig. 2. Spectral function of the loqitudinnl current correlations of liquid argon plotted as a function of the frequency: . molecular dynamics data, ref. [ 181 ; - the present result, eqs. (I), (5) and (7). For each curve the upper number refers to the 4 value (A-‘), the following to S(q), the fhird to c,(q) (lo3 m/xc) and the fourth to 7(q) (1O-‘3 xc).
sion with a simulated g(r) [ 161). By extrapolating simulated values ofn’i(q,O) the value r(O) = 2.3 X IO-l3 set is obtained [IS] in state II, while a simulated g(r) and the superposition approximation for g3(r,r’) leads to [16] ~(0) = 1.4 X lo-13 set for state 1. In the present calculation it is to be noted that the conditions y > 1 or D, > 0 and ~a > 0 lead respectively to: ~~(0) > 7r/2B,, and ~~(0) < n/(ZBg-M,(O,O)/I z’(O)l) for k’(O) < 0, where B. = -lii,(O,O)/Ms(O,O). For state I, one finds the following inequalities: 1.266 X lo-l3 set < T(O) < 1.374 X lo-r3 sec. For a practical purpose y and W are taken equal to iheir experimental values [17] (state I), this imposes, +hrough eq. (9), the value of u2(0) [i.e., a relation between c, (0) and T(O), cf. eq. (8)] and the value of -yDT: 0.4.1 X 1O-3 cm2/sec (experimental value [17]: 1.72 X 10s3 cm2/sec). When c, (0) = 1 500 m/set is titroduwd, one finds: ~(0) = 1.325 X lo-l3 set and n-lvlp = 1 .I9 X 10e3 cm2/sec while the experimental value 3.77 X 10m3 cm2/sec is recovered if one takes c, (0) = 2 000 m/set [and T(O) = 1.295 X lo-l3 set] . Tl-ris leads to the required moments of S(q, w). The result of the subsequent calculation ofS(q, o), through eqs. (l), (6) and (7) is shown in fig. I where it may be compared with the results from eqs. (1) and (2)-
Volume 21, number 3
CHEMICAL PHYSICS LETTERS
2. Intemediate region. !n this range of q values the current correlation function w2S(q, o)/(q’/@n) deduced from eqs. (I), (6) and (7) is compared to the molecular dynamics data [18] in state II (fig. 2). The best fit of ‘Ihe curves occurs for values of the moments whkh are very close to the existing estimations [15] . The authors are greatly indebted to Professor Galatry for his continued interest and advice.
L.
References [I] F. Kohler, The liquid state, Vol. 1 (Verlag Chemie, Weinhcim, 1972). [2] B.J. Berne, in: Physical chemistry, Vo!. 8B (Academic Press, New York, 1971) pp. 539-716. [3] L.P. Kadanoff and P.C. Martin. Ann. Phys. 24 (1963) 419.
1 September
1973
Rev. &d. Phys. 38 (1966) 205. 141 R.D.hlountain, [5] V.F. Sears, Can. J. Phys. 47 (1969) 199. [61 J. Kurkijani, Ann. Acad. Sci. Fennicae, Ser. A VI 346 (1970) 1. 171 R. Zwanzig, Phys. Rev. 14.4 (1966) 170. 181 H. Xiori, Progr. Theoret. Phys. 33 (1965) 413. [91 A. Rahman, Phys. Rev. Al36 (1964) 405. 25L4. 1101 D. Levesque and L. Vcrlet, Phys. Rev. A2 (19’70) r111 G.D. Hnrp and B.J. Beme, Phys. Rev. Al (1970) 975. [121 W.C. Kerr, Phys. Rev. 174 (1968) 316. 1131 P. Ortoleva and Xi. Nelkti, Phys. Rev. A2 (1970) 165. 1141 M. Hassan and F. L.ado, J. Chem. Phys. 57 (1972) 3003. [I51 N.K. Ailawadi, A. Rahman and R. Zwmnzig, Phys. Rev. A4 (1971) 1616. it61 D. Forster, PC. Martin and S. Yip, Phys. Rev. L70 (1968) 160. D.G. Naugle, J.H. Lunsford and J.R. Singer, J. Chcm. [I71 Phys. 45 (1966) 4669. (181 A. Rahman, in: Neutron inelastic scztttering, Vol. 3 (IAEA, Vienna, 1968) pp. 56 l-572.
Volume 21, number 3
A MASTER
EQUATION
HYDRODYNAMIC
PHYSICS LETTERS
FOR REGIME
S(q, w) LEADING FOR
SIMPLE
1 September 1973
TO THE FLUIDS
Lionel BONAMY* and Nguyen Mifi HOANG** Groupe de Pllysique hlolk-ulaire, FmulrPdes Sciences, 25030 Besnn~on. France Received 14 May 1973
A master equation relating the coherent and incoherent dynamical slructure factors S(q, w) and S,(q, w) is introduced on the basis of physical arguments suggested by previous theories. It is shown that, once S,(q, w) and the fnst six moments of S(o. w) are known, the resulting expression for S(q, w) contains the correct hydrodynamic regime as well as the free particle behaviour. The relationships thus introduced berween the transport coefficients and the molecular quantities are compared to the experimental data for liquid argon. Molecular dynamics data arc also well reproduced.
The aim of the present paper is to propose an expression for the dynamical structure factor S(q, w) which implies, for a monoatomic fluid, the following features [ 1,2] : (i) free particle regime (FPR) (large q and w) leading to a single particle diffusion peal:; (ii) hydrodynamic limit [3,4] (HL) (q + 0, w + 0, w/q = const) exhibiting both a longitudinal sound wave peak (Brillouin scattering) and a heat diffusion peak (Rzyleigh diffusion); (iii) transition from HL to FPR going through a brosdening and a subseqlient disappearance of the sound wave peaks, as is suggested by continued hydrodynamics [5] or as shown by molecular dynamics computations [4] . The memory function formalism [7,8] will p:ove to be an appropriate scheme to connect the approximations used in several existir.g theories, and in the present work. in this formalism the coherent and incoherent dynamical structure factors are written [5,6] in the form:
of the so-called memory functions M(q, t) and M,(q, t). S(q) is the static structure factor, p-l = X-, T, m is the mass of an atom, and c02(qj = pmS(q). The function Ss(q,oj describes the motion of an atom in the fluid. This function, and hence zs(q, w), will be supposed to be known from other sources [9, lo]. The two limits mentioned above for S(q, w) are easily incorporated in this memory function representation. ‘l?Le FPR is o_btained if in eq. (1) one puts S(q) = 1 and nf(q, w) =Mdq, w) (memory function for a free particle motion). On the oiher hand, it is well known that, in the HL, the Kadanoff and Martin [3] result is recovered if one takes:
S(q,w)=S(qjReCiw+c~(qjq2/[iw+fi(q,~j]
Ilri(q,w)=r*-14pq2+(r_lj~~(0)q2/[iw+yDT42]
1-l ,(la)
Ss(q,Wj=Re{iwt(q2/Pnlj/[iw+fis(q,w)]]-1,
(It)
where &:q, w) and fl?,(q, w) are the Laplace forms, say:
trans-
fi(q, w) = L {M(q) t)} = 7 dt emiw ‘&I(q, tj 0
(2) * Cha@ chcrche l * Attache cherche
470
de Recherches at the Centre National de 1; XeScientique. de Recherches at the Centre National de la ReScientifiiqw.
= L [n-ltlpq28(t)+(-pi)+o)$
exp(-wTq2tjjI
Here n is the density, 7 = CJC,, DT is the heat diffusive coastant, and qp is the longitudinal viscosity.
For intermediate 9 values the available analytical information on S(q, w), and hence on fi(q, o), are the lowest order moments. Note that in eq. (1) the zeroth and second moments ofS(q, w) are automatically satisfied whatever could be &(q, w). The limiting behaviours of $,(q, w) are [ 1 l] tidy, 0) for 4 + = and B(w) for 4 - 0. Here R(w) is the memory function for the velocity autocorrelaticn function. Therefore in the HL it is found that [2, IO] : fqq,
Cd) = X(0) + iG’(O)
I i i
q
q
0.025
..l
A
i
i i
+ O(q*) (3)
I
= L [k(O) s(t) + C.&J)] , where k-l (0) = flnzD and K’(O) = -( 1 fC/6,DnzD) define the long time behaviour of the mean square displacement of an atom (( [br(t)12) = 6Dr + C for f --f -)_ The present approach is based on a comparison of the physical contents of two previous theories linking M(q, t) and M,(q, t). The fist is the Kerr approach [ 121 which may be expressed by [ 13] : Jf(9 >t> = M&9, r> .
1 September 1973
CHEMICAL PHYSICS LETTERS
Volume 21, number 3
(4)
Physically, this approximation does not allow the appearance of Brillouin scattering in the HI, (cf. fig. 1). The second theory is the extended liquid phonon (YELP)model expressed by [ 131:
Jm 39 =qs I0 = WC4Iwq~ >O)lq4: 0 5 (5) which leads, qualitatively [14], in the HL, to Brtilouin scattering only (cf. fig. 1). It may be verified that the inability of these two approximations to describe simultaneously Rayleigh diffusion and Brillouin scattering comes from the fact that eqs. (4) and (5) mvolve the same time dependence forM(q, t) and M,(q,t), at the opposite of eqs. (2) and (3). But for intermediate 4 values eq. (4) or (5) is satisfactory since molecular dynamics tells that S(q,o) and S,(q, w) are represented then by similar and simple curves [6, 10, 151 @road and structureless peaks); it is therefore possible in this last range of 4 values to compensate for the discrepancies between S(q, w) and S,(q, w) by adjusting conveniently the lowest order moments. Now it may be expected that some functional relationship between M(q, f) and M,(q, t) better than these two previous approximations should retain their structures in some way but should modify the time dependence mentioned above. These requirements
i
1 i
I i
‘\
Y&.__, ...
0
0.1
0.2
w
,_____...=-“..‘
1
(1
O”%x-’
1.5
2.0
2.5
)
Fig. 1. The dynamical structure factor of iiquid argon in the hydrodynamic regime (state 1): _ the result of Kadanoll and Martin, eq. (2); --- the Kerr result, eqs. (11 and (41; - - - the result of the ELP model. eqs. (I) x~d (5); the present result, eqs. (l), (6) and (7).
can be cast into the following evolution equation connecting the collective aspec.ts, contained in M(q, t) M,(q, r) [and missing in Kerr’s approach, eq. (4)], with the individual aspects contained inM(q,r) M,(q, t) [and missing in the ELI’ model, eq. (5)] :
MqJ)-&(q,t)
=-Sd~~~~,r-i,mi(~.~~-~,~~(~!~~~. 0
(6)
It may be shown that the structure of this master equation already allows, through a simple ansatz for ici(q: t), to get a reasonable behaviour for S(9, w) over the whole range of 4 values. Knowing that Ms(q, t) goes as a 6(t) function when 9 -+ 0 [cf. eq. (?)I , the master equation (6) shows that the exponential term appearing in M(q, I) [cf. eq. (2)] will be obtained if N(q, t) behaves itself as a 6(r) function in this limit. A possible way to introduce this limiting behaviour is to choose:
471
CHEMICAL PHYSICS LETT’ERS
Volume 21, number 3
An analytical form for the proportionality ~~(4) may be obtained by fuling the second derivative ofM(q, t) [i.e., the shth moment S(q, w)] _Indeed from eqs. (6) and (7), one
1973
constant initial time of finds:
NO)-fis~4,0)
U21dQ2 =~s(4~o)~~5(~~o)--M(~~o)l
1 September
(8)
’
whereM(q,O) = [cl(q) - c:(q)] q2 and fi(q,O) = -rrM(q,O)/2?(4). The quantities c,(q) and ‘i-(q) are defined with the help of known molecular integrals involving the intermolecu!ar potential and the pair and triplet distribution functions g(r) and g3(r, r’) [S, 10, 161. Now it may be verified, after solving eq. (6) by a Laplace transformation, that eq. (2) is
formally recovered following relations: rD,
= u2(0)R(O)
in the HL. Moreover
:
(y-l)c$O)
one gets the
= u”(0)K2(0),
(9)
These equations may be considered as ccnnecting the various transport ccefficients to the self transport constants (D and ct) and to the molecular integrals. Note that an expression for PJ-~T~, which reduces to the first term of the rhs of eq. (lo), was previously proposed [14]. Knowing S,(q, o), or fis(q, w) [lo], and the moments of S(q, w) up to the sixth, we get, through relations (l), (6) and (7), an expression for S(q, w) which appears to describe the correct behaviour in the whcle ratige of 4 vaiues. A numerical test has been performed in the case of liquid argon in the following states: (I) I! = 1.413 g/cm3, T = 90°K and (II) II = 1.407 glcm3, T = 76°K. !: Hydrodynamic &(O,O),
limit. The constants Ms(O,O), D and C for state I were obtained by inter-
polatiilg results of simulations [lo]. In state I experimenus [ 171 give 7 = 2.00 and tile isothermal sound velocity W = 886 mlsec leading to S(0) = r/&n W2 = 0.05C; molecular dynamics computations [ 181 give S(0) = 0.04 for state II. Roughly the same value coa(0) = 1500 misec has been obtained for state II (from simulated values ofM(q,O) [15. 181) and for state I (by evaluating directly the molecular expres472 .,.
;
Fig. 2. Spectral function of the loqitudinnl current correlations of liquid argon plotted as a function of the frequency: . molecular dynamics data, ref. [ 181 ; - the present result, eqs. (I), (5) and (7). For each curve the upper number refers to the 4 value (A-‘), the following to S(q), the fhird to c,(q) (lo3 m/xc) and the fourth to 7(q) (1O-‘3 xc).
sion with a simulated g(r) [ 161). By extrapolating simulated values ofn’i(q,O) the value r(O) = 2.3 X IO-l3 set is obtained [IS] in state II, while a simulated g(r) and the superposition approximation for g3(r,r’) leads to [16] ~(0) = 1.4 X lo-13 set for state 1. In the present calculation it is to be noted that the conditions y > 1 or D, > 0 and ~a > 0 lead respectively to: ~~(0) > 7r/2B,, and ~~(0) < n/(ZBg-M,(O,O)/I z’(O)l) for k’(O) < 0, where B. = -lii,(O,O)/Ms(O,O). For state I, one finds the following inequalities: 1.266 X lo-l3 set < T(O) < 1.374 X lo-r3 sec. For a practical purpose y and W are taken equal to iheir experimental values [17] (state I), this imposes, +hrough eq. (9), the value of u2(0) [i.e., a relation between c, (0) and T(O), cf. eq. (8)] and the value of -yDT: 0.4.1 X 1O-3 cm2/sec (experimental value [17]: 1.72 X 10s3 cm2/sec). When c, (0) = 1 500 m/set is titroduwd, one finds: ~(0) = 1.325 X lo-l3 set and n-lvlp = 1 .I9 X 10e3 cm2/sec while the experimental value 3.77 X 10m3 cm2/sec is recovered if one takes c, (0) = 2 000 m/set [and T(O) = 1.295 X lo-l3 set] . Tl-ris leads to the required moments of S(q, w). The result of the subsequent calculation ofS(q, o), through eqs. (l), (6) and (7) is shown in fig. I where it may be compared with the results from eqs. (1) and (2)-
Volume 21, number 3
CHEMICAL PHYSICS LETTERS
2. Intemediate region. !n this range of q values the current correlation function w2S(q, o)/(q’/@n) deduced from eqs. (I), (6) and (7) is compared to the molecular dynamics data [18] in state II (fig. 2). The best fit of ‘Ihe curves occurs for values of the moments whkh are very close to the existing estimations [15] . The authors are greatly indebted to Professor Galatry for his continued interest and advice.
L.
References [I] F. Kohler, The liquid state, Vol. 1 (Verlag Chemie, Weinhcim, 1972). [2] B.J. Berne, in: Physical chemistry, Vo!. 8B (Academic Press, New York, 1971) pp. 539-716. [3] L.P. Kadanoff and P.C. Martin. Ann. Phys. 24 (1963) 419.
1 September
1973
Rev. &d. Phys. 38 (1966) 205. 141 R.D.hlountain, [5] V.F. Sears, Can. J. Phys. 47 (1969) 199. [61 J. Kurkijani, Ann. Acad. Sci. Fennicae, Ser. A VI 346 (1970) 1. 171 R. Zwanzig, Phys. Rev. 14.4 (1966) 170. 181 H. Xiori, Progr. Theoret. Phys. 33 (1965) 413. [91 A. Rahman, Phys. Rev. Al36 (1964) 405. 25L4. 1101 D. Levesque and L. Vcrlet, Phys. Rev. A2 (19’70) r111 G.D. Hnrp and B.J. Beme, Phys. Rev. Al (1970) 975. [121 W.C. Kerr, Phys. Rev. 174 (1968) 316. 1131 P. Ortoleva and Xi. Nelkti, Phys. Rev. A2 (1970) 165. 1141 M. Hassan and F. L.ado, J. Chem. Phys. 57 (1972) 3003. [I51 N.K. Ailawadi, A. Rahman and R. Zwmnzig, Phys. Rev. A4 (1971) 1616. it61 D. Forster, PC. Martin and S. Yip, Phys. Rev. L70 (1968) 160. D.G. Naugle, J.H. Lunsford and J.R. Singer, J. Chcm. [I71 Phys. 45 (1966) 4669. (181 A. Rahman, in: Neutron inelastic scztttering, Vol. 3 (IAEA, Vienna, 1968) pp. 56 l-572.