Fluctuation Analysis in Simple Fluids

14 Pages·2011·1.19 MB·English

PHYSICAL HKVIE% A VOLUME 2, NUMB ER 3 SEPTEMBEH, 1970
Fluctuation Analysis in Simple Fluids
A, Zlya Akcasu
Cekmeee Nuclear Research and Training Center, &.&.1, guava AEani, Istanbul, TNrkey
Rnd
E. Daniels
university ofMichigan, Ann arbor, Michigan 48104
(Received 21November 1969)
A classical analysis oftime correlations in simple fluids based on the generalized Langevin
equation is presented. Formulas -for the current-current correlations are developed explic-
itly in a region offrequencies ({d 103sec ') and wave numbers (k-10 cm ') whj.ch are ex-
plored in typical slow-neutron-scattering measurements. %here applicable, comparisons
are made with the results ofthe numerical calculations of Bahman in argonlike liquids, and
good agreement is generally found. The analysis is based on ahydrodynamic description of
fluids involving frequency- and wavelength-dependent transport parameters. The frequency
and wavelength dependence of shear and longitudinal viscosities are given explicitly for argon-
like liquids.
I. INTRODUCTION In this approach, the conventional hydrodynamic
equations are used to describe the linear response
The calculation of coxrelation functions in clas- of the fluid, and the correlation functions are
sical simple fluids in terms of microscopic quan- then related to the linear response by means of
tities plays an important role in statistical me- the fluctuation-dissipation theorem. Using the
chanics, both for the interpretation of scattering formulation described by Landau and Lifschitz, ~0
experiments and the evaluation of the frequency- Rytov applied the fluctuation-dissipation theorem
and wavelength-dependent transport coefficients. to distributed parameter systems and calculated,
Among these, the density-density {orVan Hove) among others, density-density correlation func-
correlation function has received the most atten- tions in an arbitrary continuous medium. This
tion becRuse of its dix'ect xelation to the cbfferen- hydrodynamic approach has been used to interpret
tial scattexing cross sections. Othex corxelation light scattering from liquids successfully by
'
functions, such as the transverse cuxrent-current Mountain. A systematic and general hydrody-
correlation functions, which are not readily acces- namic description of fluids fox the calculation of
sible in experiments, have been subject to quanti- correlations and transport coefficie"nts has been
tative investigations only recently following the developed by Kadonoff and Martin. The latter
publication of Rahman's' computer calculations in, approach has been applied by Chung and Yip to
argonlike liquids. Computer studies of correla- Rahman 8 cRlculatlons of curx'eQt-current corre-
tions using molecular dynamical calculations pro- 1Rtlons.
vide a stringent test of the validity of the various The objective of this paper is to present a clas-
classical theories introduced in correlation analy- sical analysis of coxrelations in simple fluids
sis because they only assume aknown model inter- based on the generalized Langevin equation devel-
particle potential, and involve no quantum effects. oped by Zwanzig" and Mori~6'~~ and to interpret
The classical analysis of correlations is usually quantitatively the current-current correlations
based on either akinetic or hydrodynamic de- computed by Rahman for liquid argon. This ap-
scription of fluids. The kinetic description devel- proach has several appealing features. First by
oped extensively by Welkin and his co-workers choosing the dynamical variable in the desex'iption
has been justified theox'etically for dilute gases, of the fluid as the microscopic phase density func-
and used successfully to interpret Brillouin scat- tion one obtains" an exact kinetic equation for t e
tering from gases. '6 It has also been extended to correlation function I"{v,v'; x,x',t), which re-
dense fluidsv Rnd applied to Rahman's molecular- duces by approximation to the kinetic equation de-
dynamics calculations for liquid argon with poor rived by Zwanzig' and welkin. However, by
quantitative agreement. 8 making an alternative choice of the dynamical
The hydrodynamic description of fluids has long variables to be microscopic densities in configu-
been in use in the fluctuation analysis in axbitrary ration space {e.g., mass, current, and energy
continuous media as aphenomenological theory. densities) one arrives at an exact hydrodynamic
FLUCTUAT ION ANA LYSIS IN SIMP LE FLUIDS 963
description of correlation functions in terms of theory of Brownian motion. The random-force
frequency- and wavelength-dependent transport vector f(t) is given formally by
"
parameters. The choice of the appropriate set
of dynamical variables is arbitrary. For any f(t)=e u (1—P)a (a=iLa), (2.2)
choice of these variables one obtains exact ex- where I- is the classical Liouville operator and P
pressions for the correlation functions of the is aprojection operator defined for any arbitrary
variables in the set. Different levels of approxi- phase function G(t) by
mations can be obtained for aparticular correla- '
tion function by adding new variables to the set PG(t) =(G(t)a ) ~(aalu) a (2.4)
and using the same simplifying assumption (e.g., Here a denotes the row vector wh'ich is the Her-
Markov assumption) in each case. The continued mitian conjugate of a. The (aa~) is the inverse
fraction expansion of correlation functions by of the square matrix [(a,a,)]which is the static
Mori, for example, can be obtained by using an correlation matrix. It should be noted that the
orthogonal extension of the set starting from a evolution of the force vector "f(t,) is determined by
e'"
given dynamical variable. The separation of the special propagator whereas the evo-
thermodynamic and transport parameters can be lution of the state vector a(t) is given by
given a geometric interpretation in terms of pro- =e"
a(t) a (2.6)
jections of dynamical variables on appropriate
orthogonal axes, and the extension of their defi- It is shown by Mori" that
nitions to short wavelengths where the anisotropies
(f(t)a') =o, thoy-a-ott . (2.6)
become significant can be made in a systematic
way. The square matrices y(t) (damping function) and
In this paper, we will use the configuration- n (frequency matrix) are defined by
vspesatciegat(ehydtrhoedycnuarmreicn)t-cudrersecnrtiptiocnorreolfatfilounidsfutonctiino-ns, y(t) =(f(t)f'(0)) ~(aa') ' (2. I)
'
and obtain approximate formulas for transverse in =(a'a') ~(aa') (2.6)
and longitudinal current-current correlations in '
Multiplying (2.1) (aalu) from the right-hand
the frequency and wavelength regions encountered
side, taking the thermal average of the resulting
in neutron scattering using a Markov assumption.
equation, and using (2.6) yield
The results will then be compared to Rahman's
tcaoimnptuhteerwadvaetlae~ngftohr laiqnudidfreaqrugeonnc.y Wdeepewnidllenacelsooofb- dR(t)—in ~R(t)+ ty(t —u).R(u) du=0, t~0 (2.9)
shear and longitudinal viscosities explicitly and where R(t), the normalized dynamic correlation
discuss the influence of thermal effects as a func- matrix, is defined by
tion of wavelength. The objective of this paper is
similar to that by Chung and Yip; however their
approach is based on the correlation-function for- The one-sided Fourier transform of R(t) is ob-
malism developed by Martin and Kadonoff, rath- tained from (2.9) as
er than on the projection-operator formalism by - '
Zwanzig and Mori. R(i&a)=[iu) in y+(i~)] (2.it)
where """"
II. GENERALIZED LANGEVIN EQUATION y(i&a) =lim J e y(t)dt (2.12)
Extending the projection-ope"rator technique 0
first introduced by Zwanzig, Mori' proves The projection technique enables one to find a
closed set of linear equations for the correlation
that the equation of motion of a set of dynamical
variables a&(t) can be written in the form matrix R(t) when the state variables a&(t) are
chosen as fluctuations from thermal equilibrium.
da t —tn a(t) ' y(t —u) a(u) du =f(t), t~0 . The theory, although formally exact, serves only
to transform the calculation from the direct com-
(2.1)
putation of R(t) to the computation of n and y(t).
The state vector a(t) is defined such that it has no However, the frequency matrix 0is determined
invariant contribution, e.g., from static correlations which are generally much
a(t)=A(t) -(A(t)) (2.2) easier to compute than the time-dependent corre-
lation functions, and we may use approximations
where (~ ~ ~) denotes the thermal average of the to compute the damping matrix y(t). In particu-
vector A(t). Equation (2.1)is the generalized lar, we will consider representations in which we
form of the Langevin equation ' in the stochastic can make a Markov approximation on rp(t), viz. ,
964 A. Z. AKCASU AND E. DANIE LS
(2.ls) the volume of the system, and p0 is the equilibrium
density. Here C«(k) is one of the elastic moduli
to approximate the transform of the correlation calculated by Schofield. The frequency matrix
matrix R(i~) by 0is calculated using (3.4) and (3.5), and noting
R(i~)=-[i~-iQ+W] ' (2.14) a~=ika2 as follows:
0=k
%e shall follow this formal procedure to calculate (3.6)
gC«(k)/p, 0
the current-current correlation function and its
transform by choosing the components of the state Since (1—P)a&=ik(1—P)a2-0, th—e random force
vector as the spatial Fourier transforms of the component f,(t) is identically equal to zero and the
local densities of conserved variables. damping matrix y has only one nonzero element,
(3.Ia)
%'e first calculate the cosine transform of the
transverse current correlation function as a sim- where "
ple application of the generalized Iangevin equa- y»(t) =- (1—P)as e'u (1—P)a, a,a2 '
& & & &
tion, and then compare it with Rahman' s compu- (3.7b)
tations. For this purpose, we choose the compo-
The generalized Langevin equation for the set
nents of the state vector as
[a,(t),a,(t)]becomes
a,=Z,(k),
da,(t)
(3.8a)
a,=II„(k) (3.1b) dt
where J(k) and II(k) are the mass current density da~(t) —i.k C« a~(t) + ' y»(t -u)a~(u) du =f2(t) .
and the stress tensor, respectively. They are dt
(s.sb)
defined by
The transverse current correlation function is
J&(k) = Qmv,'exp(ik x') (S.2a) defined by
R (k t) &ag(t)&g& (s.9)
2P(841e) ~X (s.2b) Its cosine transform Rr(k, &) is the transverse
current power spectral density. The latter can
(3.2c) be obtained directly from (3.8) by multiplying it
'"'" by a,*, taking the thermal averages, and using
ddVR(R) 1——eik R (3.2d) &f,(t)a,*)=-0 as ',
Rr(k, &u)=Re[i~+ (k'/po)qr(k, i&a)] (3.10)
In these definitions, x and v denote the position
and velocity of the &th particle in the system, and where qr(k, i~), "which we refer to as the trans-
the subscripts i and j refer to the Cartesian com- verse viscosity, is defined by
ponents in a coordinate system in which k is par- qr(k, i(u) -=(po/ik) (ap(iu))a,*&/& a,(i~)a,"),
allel to the z axis. With this choice of variables (3.11a)
the static c'o&rrae,laa,t*io)n m0atrix is diagonal, viz.(3,.3) qr(k, i(u) ='C«(k)/[i&up y»(k, iv)] . (3,lib)
0 (a,ag& Rahman' has computed Rr(k, ~) for various values
soidndceantdheevVeRn11fuRnbcletisonQs1 RonfdthQep Rplaert,ic1leespveecltolvceitliye,s p0reanssdio&n. fVorfeithaivnetebreinens aobflethetoLoabptalainceantraenxsafoctrmex-
so that &a&a~*&=0. Furthermore, a direct evalua- oevfethr,e tdhaempeivnagluaftuionnctioonf, thveizl.at,teyr»(uks,inig&u)(.3.'HIbo)wi-s as
tion of the diagonal terms using (3.2) yields
(P/V) &.,.,*&=p. , (s.4) pdeifrftiucrublattivaes sotlevcihnngiquthees Lsiouuchvillaes eexqpuaantisoino,ns alitnhough
. . .
(tI/V) *&=C, (k) density or interparticle potential' may be used in
& dilute systems. For sufficiently small frequen-
= n—+n'z d33Rg R 82V(1—coskz) cies, we can approximate (3.10)by replacing
y»(k,i&a) by its zero-frequency limit y»(k,i~=0).
(3.5) This approximation corresponds to a, Markov de-
where P=(1/keT)(ke =Boltzmann's constant), Vis scription of the fluid in terms of a,(t) and a,(t), in
FLUCTUATION ANALYSIS IN SIMPLE FLUIDS
which the convolution integral in (3.8b) is replaced case of longitudinal current correlation function
by because it is related directly to the neutron scat-
J duy22(t-u)a2(u):-a2(t) f du y22(u) . (3.12a) tering cross section where the large values of k
correspond to large momentum transfer to the
Ttiohne fmreaqyuebnecyexpreacntgeed itnowbheicvhalitdhecaMnarbkeovestaimssautmedp- sacpaptrtoearcinhg-zmereodifuomr.la)rgHeenvcaelu,esweofrekq,uiir.ee.&, r (k) to
by considering the next term in the expansion of &sr(k) 2k2C44(k)/p2 (3.19)
y22 (k,iv) in powers of (i~): Thus, we obtain the asymptotic behavior of +r(k)
y22(k, i~) dy228(2k,i~) ice- 0 (3.12b) fkordespmenadlelncaend olfarg&eur(kk) ffroormthe(3.in1t6e)rmaneddia(t3e.19v)a.luTeshe
of k can be obtained by interpolating it between the
It is clear that the Markov assumption ceases to zero and large-0 limits by the following formula:
be valid if y22(k, i&=0), the leading term in this
expansion, vanishes for some values of k. In such ~'r(k) =2k'C«(k)/po
acamseosreitctoumrnpsletoeutdteoscbreipmtioonre ocfonthveenfileunitd btoy giontrtoo- ~2r(0)—2k2 (C«(k) —p2/ prn )/p2 , (3.20)
ducing new variables. + 0
With these remarks, we obtain the following ap- where ko is an adjustable parameter whose choice
proximate form for Rr(k, ~): will be discussed presently.
It is interesting to compare (3.13)using the
er(k)k2C«(k)/p2 above expression for &ur(k) to the ideal-gas result
~ (u (k) [k C (k)/p —&u'] in (3.17)for zero frequency and large k, because
the Markov approximation becomes exact at ~=0.
where we have introduced
Noting that C«(k)/po- (I/Pan) [cf. Eq. (3.5)], we
&ur(k) = limy22(k, i&a) . (3.14) obtain Rr(k, 0) from (3.13)as (2P~)'/'/k whereas
(3.17)yields (vPm/2)'/'/k. The ratio is (4/m)' 2
Calculation of &ur(k) directly from (3.14) is still a =1.12. Thus (3.13)recaptures the ideal-gas re-
formidable task in dense fluids, although it is sult in the limit of small & and large k. It may
simpler than calculating y22(k, iv). Therefore, we be pointed out here that (3.10) can be approximated
choose to try to guess its 4' dependence by consid- for large frequencies by replacing y»(k, i~) by
ering the asymptotic behavior of qr(k, i&u) in the y22(k, &=0)/i&u [short-time expansion of y22(k, t)].
small-k limit, and of Rr(k, &) in the large-k limit. Since our interest lies in the small-frequency re-
We shall show in Sec. IV [cf. (4.54)] that gion we shall not dwell on this point further even
lim qr(k, i&a)=C44(0)/&ur(0) (3.15) thoWugihth yth»e(ka,idf=o0f)(i3s.2c0a)lcwuelabhaleve ebxeaecntlya.ble to ob-
k2f2) 0
tain an expression for the transverse current pow-
is equal to the conventional shear viscosity p,.
er spectral density (3.13), which contains only
Hence, the small-k limit of &or(k) is given by one adjustable parameter ko. The value of 40 de-
~r(0)=C44(0)/'/t 2 (3.16) termines the transition from the small- to large-
[note that C«(0) is G in Zwanzig's notation ']. & limits. It is expected to be in the vicinity of the
main peak of the structure function S(k), which
The large-k behavior of Rr(k, &u) may be pre- occurs at k=2 A0 ' in Rahman's computer calcula-
dicted from the transverse current spectrum ofan '
tions for liquid argon. We have chosen F0=1.5P.
ideal gas, viz. , which yielded a good fit to the computed curves
RIC(k &) (&p~/2k2)1/2 &-Bme2/2k2 (3.17) although the value of A'0 is found to be not too crit-
ical. The other constant in (3.20) is ~r(0) which
This function has a single maximum at &=0for is obtained from (3.15) as +r(0)=0.1&&10'2sec
all k. On the other hand, (3.13)attains its maxi- corresponding to avalue for the shear viscosity
mum for afixed 0 at a frequency &2=2.8X10 P at p2=1.407 g/cm2 and T=76 'K
NTm(k) =k C«(k)/Po 2~r(k)- (3.18) fpourtedliquaicdcoradrginogn. toTEhqe.v(a3lu.5e)sinofwChi«c(hk)thweerientecropmar--
for ~r(k) &2k C«(k)/p2, and at ~=0 otherwise. ticle potential is taken, following Rahman, as
[The function &ur (k) is often referred to as the &(R)=4E[(t7/R)"—((7/R)']
dispersion relation. ] The dispersion relation
(3.18-)is expected to approach that of an ideal gas with (e/ks) =120'K (ks=Boltzmann constant) and
as & ~because for large 4 the particles behave v=3.4 A. The variation of C«(k) is plotted in
as free particles. (This is more apparent in the Fig. 1.
AKC ASU AND E. DANIE LS
IOO
eo-
60—
40
I.O
.8
20
,6
IO 4
e
P .2
.08
.06
4I
kinA & .04
FIG. 1. Variation of elastic moduli with wavelength.
.02 I,IO 2I0 3I0 4,I0 5,0
0
Figures 2 and 3 show the variation of &ur(k) and k inA
qr(k, 0)=C«(k)/~r(k) with k. We observe that the
k-dependent shear viscosity decreases very rap- FIG. 3. Variation of shear viscosity with wavelength.
idly by a factor of 100in the region of k from
A,
zero to 2 and approaches zero as 1/k. This
k dependence of qr(k, 0) appears to be crucial to able a,(t) [e.g., J,(k, t)] can be obtained by solving
the behavior of Rr(k, &) for the k values in 1-4 the appropriate generalized Langevin equation. If
'.
A only the autocorrelation function is of interest, the
Figures 4 and 5 show comparison between the one-component description of the system is suffi-
calculated curves and Rahman's data. It is note- cient in principle. The correlation function in
worthy that the present model predicts well the this case is obtained by solving
cutoff wavelength in the dispersion curve, i.e., R,(t)+J,'du y,(t-u)R,(u)=0 (4.1)
&ur (k). Other features are self-explanatory.
IV. GENERALIZED HYDRODYNAMIC DESCRIPTION (note that the frequency matrix is always zero in
one-dimensional description). The damping func-
The previous application indicates that the cor- tion y,(t) involves f(t)=exp[t(1 —P)iL](1—P)a,(0),
relation function associated with a dynamical vari- where P projects a phase function onto a~(0). Al-
though (4.1)is exact, the calculation of y,(t) is as
difficult as calculating (a,(t)a,*)directly. Crude
I.6 approximations for y&, such as the Markov as-
sumption, are generally not precise enough to in-
1.4 clude even the qualitative features of the correla-
I.2 tion function, or the power spectral density asso-
ciated with it, for large values of ~and k. By in-
I.O
O troducing instead a multidimensional description
IIDA of the system, one actually extracts a great deal
O
.6 of information about the collective motion of the
system through the frequency matrix even though
4
one may still be interested only in the autocorrela-
2 tion function of a single variable. This information
~
00- I.IO 2,I0 0 3I.0 4.I0 5.0 itisonco.ntaAinperdopeinr pc,h(ot)icien oofnet-hdeimaedndsitiioonnaall vdaersicabrilpe-s
k inA in a given system can lead to a sufficiently pre-
FIG. 2. Variation ofthe transverse and longitudinal cise expression for the correlation function in a
relaxation frequencies with wavelengths. wide range of (d,0 even with crude approximations
FLUCTUATION ANALYSIS IN SIMPLE FLUIDS 967
24-
N
2.2— p(k)=mme'" * (4.3)
2.0— Ki1.3k' e=1
11,.86 e(k)= E(k)-((E(k))p„*((„k))~)p(k)~, (4.4)
11..42 o11(k)=II''(k) (Il„(k)p*(k))p(k)
+ I.p ( IkI) 3&( ))
08 -
0.6 (Il„(k)8*(k)),
0.4 (e(k)e*(k)) (4.6)
0,2
q (k)=-Q (k) —(Q1J*(k))~ (J(k)J*(k)) J(k). (4.6)
3.4—
The definitions of J(k) and II,1(k) have already been
given in (3.2a) and (3.2b), respectively. The
32,.86 1Iuantities E(k) and Q1(k) are the energy density and
I 2.4— the energy current density, respectively. They
.g 2.2— are defined as
2.0—
A~
1,6
1.4
12
1.0
0.8
0.6 4p
0.4 IC3
0.2 .g 30-
0 020.4080.810IQI31s.2ec1,4I 1,6 1.82.0 0 020.40,61&0l.81JlIQ101I3Z 14. 1.6 1,82Q C) 2.0-
FIG. 4. Transverse current-current correlation A3 IP ~ oo ~~ ~0e ~~se,ss~~~~~~„e
B&(k,&) versus &for various k. Op .41 .81 1I,2 1I6. 2I.0 2I.4 2g
K1nk.&
on the multidimensional damping function. The 5
variables J,(k) and II„(k)introduced in Sec. III
provide such adescription for the transverse cur- Oi
CVp
rent correlations. The description of the longitu- C3 .3-
dinal current correlations requires a more de- A,
tailed description of the fluid including thermal and 2-
viscosity effects as will be demonstrated in this
&I-
section. Moreover, a multidimensional descrip-
tion also allows the computation of the various 00 .4I .8I 11.2 I1 2.0 2I6 2I8 3.2
cross-correlations between the variables in the set Kink '
in terms of the same thermodynamic and transport
parameter s.
The purpose of this section is then to present a IC3 4,0-
14-dimensional description of a simple liquid, and
to compute specifically the transverse and longitu- 3,0-
p
dinal current correlations. This description in-
cludes thermal effects, and sheds light on the an- 3lt 2P
~+~ss+esyoss+
isotropies in the fluid for large-0 values. 0-
For state variables, choose C 1, iI,
a=col[p& 8,o~,—J„q1j, (4.2a) 00 8 1.2 1.6 2Q 24 2.8 3.2
Kink '
w(ih,jer=e1,J&2,a3n)danq&d aor~eivseactosirxs-cowmithponthernete vcoemctpoornewnittsh FIG. 5. (a) Maxima of a transverse current-current
i1=1,...,6,) viz. , dcuoerretolatpiorensenftuncthtieoonryvearnsduspowinatvsernepumrebseenrt'Hka.hmSaonli'ds line
V» =COI[+11& +22& +33& O13& +23& +121. (4.2b) data. (b}Frequency at which the transverse current-
current correlation function is maximum. (c) Trans-
The variables 8, 0~, and q& are defined by verse current-current correlation at zero frequency.
A. Z. AKC ASU AND E. DANIE LS
E(k) =ZN —'m, v" v +2KN V(!x —xol) e'"' ", where pp'&
e=1 (&/P=e1) y'-=&a'a"& = & 0 (gg*) 0 (4.13a)
(4.7)
0 0 &oot&
Qj(k) =Pv,. —'m, v v'+—',gN V(~x —8[) e'"'*'
e /=1 (JJ)
x(B"Pe), a' 0
and Q'=-&a t& = (4.13b)
e8 &qq')
+—'Q, "(v'+v') lx~'™, ,8!'P '(k)e'"'"'
g (4.8) The block diagonality of Q' and Q is a, consequence
[x'o and P'o(k) were defined previously in (3.2c) of the choice of the state variables as in (4.4),
and (3.2d)]. (4.5), and (4.8), which imply the following orthog-
The following usual conservation laws prevail onality relations:
:
a(am)o—nBgp=tihke'vJa,'ria(bbl)es—BJp=,ikJ&,—II,'II;,(,c)E—B,Ea=ndikQ,Q. .' &pg*) =&&; p*)=&o; g*&=&q;J,*& =o . (4.14)
at at at (49) The static correlation functions appearing in Eqs.
The tensors II;,.and o;,. are symmetric and have (4.13a)and (4.13b}will be discussed la'ter.
only six independent components as implied in The frequencymatrix (iQ)=&aa ) ~P can be writ-
ten
(4.2b). The variables o„and denote the viscous
stress tensor and the thermal enqe&rgy flux vector. = 0 &u uot& .@0~ (4.15)
The g$), defined by (4.4), will be replaced later by &g ~o 8~&.@8-~ 0
&
T(k)=g(k)/p, C„(a} (4.10) As a consequence of the conservation relations
whose average with respect to a perturbed distri- (4.9),g(0)=(1—P)u has only nine nonzero compo-
bution function yields the temperature in the con- nents:
vention'al linearized hydrodynamic description of a f(0)=col[0,0,f~,0,0,0,f&] (4.ia)
fluid. Such an identification, however, is not
needed for the present. The quantity C„(k)will be Therefore, the damping matrix g(t) is of the fol-
defined later (it will be identified as the specific lowing form: 0—
heat at constant volume). 0 0 0
The average values of p(k), E(k), and II;,(k) are q(t)= 0 g"(t) 0 @"(t) (4.iv)
zero for ktp. When k=p, we have 0 0 0 0
o y"(t) 0 g"(t)
(i/v) &p(o))=p„ (4.11a)
where y and cp are 6x6 and 3x3 square ma-
— ' ' +1-,2' trices. The off-diagonal matrices y"(t) and q"
(E(o))=-, i~ d'ft v(ft)g(ft), are 3x6 and 6x3.
(4.11b)
Substituting (4.15)and (4.17)into the generalized
—
&11,,(0)&=5,,P, Langevin equation discussed in Sec. II, we obtain
the following set of equations:
2
(gr)) p(t)=ik J(t) (4.18)
rnp Gm
(4.11c) 0
(4.19)
Here, P0 is the equilibrium pressure. We shall pp
farlowmaysp,asEs,umaned tIhI;a&t wthheesneevearvertahgeeys araerenostubzterarcot,edso g(t)+&gJ ) (JJ ) ' J(t)=ik q(t) (4.20)
thaEtigthhet sotfattheev1e4ctocormapownielnltsdenooftethefluscttautaetiovnesc.tor a o(t)-&oq'& (qqt) '~q(t)+ g q"(t-u) ~q(u)du
a.re even, and the remaining six are odd functions —&~J'& &JJ~& ' J(t)+f' qP'(t-u) g(u)du=f'(t),
of particle velocities. Hence a can be decomposed
(4.21)
into even and odd parts as
a=col[a', 0]+col[0,ao]. q(t) —ggQ~ g(t)+ y"(t-u) ~q(u)du
0
Consequently, the static correlation matrix '
-&qo'& &oo'& o(t)
P=&aat&splits into two disjoint submatrices as
t
e 0 (4.12) + J q (t-u) g(u)du=f'(t) .
0 o 0 (4.22)
FLUCTUATION ANALYSIS IN SIMPLE FLUIDS
Equations (4.18)-(4.22) describe the time evolution g(R) being the static pair correlation function. We
of the state variables p, 8,p~,J;, and q; exactly. will refer to Cz,(k) as the longitudinal isothermal
We shall approximate them by neglecting the cou- speed of sound because one can also define a trans-
pling between the viscous stress tensor and the verse isothermal speed as
esunmerpgtyionfluixn v(e4c.t2o1r)anadnd(4in.t2ro2d)u.ciTnghe tfhirestMaaprkporovxia-s- C' (k)-=&II„p*&/(pp*) =&If„p*&/(pp*), (4.30a)
g"
m-=a0tiionn(4is.2e1q)uainvdale(n4t.2t2o),seatntdingthe(osqe'c&on=d0taond C,. (k) S(k), „P d.R x R—z ddVRg(R) sinkk n
f —
zp"(t u)~ o(u)du =W'(k) g(t) (4.23) +p~r x —z dV y )1—e'
g
f —
rp"(t u) q(u)du=W'(k) q(t) (4.24)
(4.3Ob)
f where g(R,k}is the Fourier transform of the three-
where W'(k) = dt@"(t) (4.25a) particle distribution function g(R,S) with respect
f . to S, i.e.,
W'(k)= dt &"(t) (4.25b)
0 n'g(5, k)=-U—(,2; 5(5-x")e*' ' ""&
Furthermore, the following definitions and equal-
o r
ities will be introduced:
&0Sp*&/&pp+& =c',(k-)ik, (4.26a) (P,y&zz; P&y) . (4.30c)
(JT+&/&TT+& =-C',(k)p,p,(k)ik, (4.26b) Imneathne liisnoetarropifcorcleimriatngweh,erek k5a,««11,anadbeCizng= Cthre.
(I/I') d;d;)=- (po/P) 5; (4.26c) In this limit, the distinction between the longitudi-
&
nal and the transverse speeds becomes unneces-
sary. The conventional isothermal speed of
(EE ) ( g& To C~(k) sound Co(k) is defined by '6~ z'
. 4.26d
- , SP 1 &(Trli„)p+&
(P/V) (o,g+&=tk„Z,, „(k), (4.26e) ( ) sp, (4.sod)
-. 3 &pp&
(qS*)/&ee+&=zk I" (4.26f) where P is the thermodynamic part pressure' & '
Then, we obtain the following approximate descrip- P=& Trli;z (the nonthermodynamic part, by defi-
tion: nition, has no projection on p and T). Thus, the
longitudinal and transverse isothermal speeds are
p (t)=ik,J,(t), (4.27a) related to C,(k) by
J;(t)—Cz,(k)ikz[P(t) +Pz,(k)POT(t)] =ik~5z~(t), (4.27b) c',(k)=-', [c',(k)+2c',(k)] .
e(t)—p,(k)c',(k)T,p(t) =ik,q,(t), (4.27c) In the isotropic limit, we have C,(k)=Cz (k). We
o,(t)-+„„(k)~(,t)+W;.(k)o.(t)=f;(t), (4.27d) may note that only the longitudinal isothermal
qz(t) —ik I';„tI(t)+W,' q (t)=fz(t), (4.27e) fsupnecetdionisSc(okm)pwlehteenlykdiestenromtinsemdall.by the structure
where E„„is the rate of strain tensor, i.e., The quantity Pz (k) in (4.26b) can be defined
equivalently by
e „(k)=(i/2p, )(k J„+k„d) (4.28) P,(k}-=[1/p,C',(k)] (II„T*&/& TT*& (4.Sl)
We shall discuss the physical implications of the
various quantities appearing in this set. (use ik&II» T ) = (O', T*&). We refer to Pz (k) as
the longitudinal thermal- expansion coefficient for
The Cz2 (k) introduced in (4.26a) can be defined the following reasons. Similar to (4.soa) we can
also by
also define a transverse thermal-expansion coef-
Cz(k)=(II„p*)/(pp*&, (4.29a) ficient as
P,(k) -=[1/p C (k)](II„T*)/(TT*). (4.32)
Czz(k) =1/PmS(k), (4.29b)
where S(k) is the structure factor defined by (Note that (II„T ) =(II22T&.) The conventional
definition of the thermal-expansion coefficient
p0
S(k)=1+(pp/m) fd Re' ' "[g(R)—I], (4.29c) asusrea16d,e2r1iviative of the thermodynamic part of pres-
970 A. Z. AKCASU AND E. DANIE LS
1 1 &(Trrr„)r*& p*),
p,c',(k) s &re'& &Ii„p*&=5„&11,, (4.S7a)
1 BP &II(gT ) =6(;ⅈ(T ) (4.37b)
p3C30(k) s I' p. (4.ss) and using (4.36a) in (4.34b) we obtain the form of
Then, po(k) is related to pL,(k) and pr(k) by Z~„as
Pg (k)C~ (k)+2Pr(k)cr(k) ~11 ~12 ~13
( ) Cg(k)+2cr(k) ~12 ~11 ~13
~13 ~13 ~33
In the isotropic limit defined above P3(k)=P~(k) , (4.36a)
=Pr(k). Since they reduce to the derivative of the
pressure with respect to temperature as in (4.33)
in the limit of k-o, Pz(k) and Pr(k) are identified
as thermal-expansion coefficients. where Z» =Cqq-P3(cr/Cq)yr, (4.39a}
thaTthbeydSefcihnoitfiioenld.o2f' CI„t(iks) iinden(4ti.fi2e6dd)aiss tihdeentki-cdaelpetno- Z(3=C13—pp(C T/CI, }yr (4.39b)
dent specific heat at constant volume. Its expres- Zss =Css- poCI2. yr (4.39c)
sion in terms of the multiparticle distribution Z13——C13-PpCz2 yz~ (4.39d)
functions is given in the cited reference [Eq. (50)
of Ref. 21]. In (4.39), we have introduced
The symmetric tensor g„„introduced in (4.26e) y,(k) =I+[r,C'-,(k)P', (k)/C, (k)], (4.4Oa)
can be defined alternatively by
y, (k) =1+[T,C',(k)P', (k)/C, (k)], (4.40b)
Z„„=(p/V) o.o.*&, (4.S4a)
& y„(k)-=I+[TC', (k)P,(k)P,(k)/C,(k)]. (4.4Oc)
= P &Il,o"& &pII„'&
~u gv v &pp+& In the isotropic limit, yr(k), y~(k), and y»(k) be-
come equal to
(II„T'*& Til*„&
TT&*) (4.34b) y =1+(TocoP3/Cv), (4.41)
&
where C,„is the elastic-moduli tensor defined by which is the conventional ratio of the specific heats.
c,„= It is interesting to note that Z»(k) —Z»(k) =2ZM(k)
(P/v)&11„11„*&. (4.s5)
holds also for Z~„ for all values of k. In the iso-
The elements' C44, C3„and C33have been computed tropic limit we obtain
bpyartSicchleofiedlidstribuatsiofnuncftiuonncstiono.f k Oinnetercmans sohfowthethtawto- ~12 ~1st ~11 ~33& ~33 ~13 2~44 ~ (4.42)
C„„(k)has the following form: The matrix I'(k) introduced in (4.26f) has the
IS] following alternative definition:
C11 C12 C13
C12 C11 C1s 0 r„(k)-=&q,q,*&/&re*&
C13 C13 Css = ee*&'[&e@*&—(p/p. v)1&~'~;*&l'1
c(k)= C44 (4.S6a) &
C44 It can be shown that (q,q~& is diago*na)l. with the
C66- elemrents &q,[q*r,&„=r&„q3rq3,)],and (q,q, Hence
with the following relation: =diag (4.43a)
Cgg(k) —Cg3(k) =2C33(k) . (4.36b) where
r.
This relation which can be verified directly is a (k) =-&~9*&'[&e e*&-(p/p. v) l&@~ &I'],
consequence of the isotropy in a plane perpendicu- (4.43b)
lar to k. In the limit of k-0, one finds in addi-
r.
tion to (4.36b) (k) =-«0*& '[&e q*& -(p/pov)l«11*&l'],
C11—Css~ C13—C12, C33—C13=2C4 (4.S6c) (4.4Sc)
Hence, C(k) has in general only five distinct non- where &Q3J3&=(Ell„& from stationarity. We shall
zero elements, i.e., C11 C12 C13 C33 and C44. relate I' later to the thermal diffusivity tensor.
In the isotropic limit there are only two: C» and
Relaxation Frequencies
C44
Verifying We now focus our attention upon (4.27d) and
FLUCTUATION ANALYSIS IN SIMPLE FLUIDS 971
(4.27e) which are generalizations of Maxwell's ) )
& 4 48
model for the relaxation of the viscous stress ten- i
sor, and Fourier's law. They both contain the k-
dependent relaxation frequency matrices W' and which is the conventional form, 1144and (Ii12+—32Il44)
W' which are defined by (4.24) and (4.25), respec- being the shear and bulk viscosity coefficients.
tively. Although the elements of these matrices Using the form of Z „ in (4.38a) and (4.44c) we
can be obtained in principle from their definitions can obtain a form for the relaxation frequency
[cf. (4.25)], we are forced in applications to mod- matrix:
el them as explained below. [A direct evaluation
of W' and W' using (4.24) and (4.25) is equivalent +11 12 +13
+f +1
to solving Liouville's equation with the reduced 12 1 3
Liouville operator (1-P)I., and thus not yet possi- W = (df3 (df3 6033 (4.47)
ble.] (d44
We first consider W' appearing in (4.27d). The
mean of this equation reduces in steady state to +88
w„'„(o„) =z.„&6„), (4.44a) with the following restrictions:
where &o„) and &6„) are the expected values of o 13~33 ~12 ~11) ~ ( 33 12 11}
and 2„(recall that &f;) =0 in the linear approxima- 11 12 88 (4.48b}
tion'6). Since the linear relation between the vis-
cous stress tensor and the rate of the strain ten- 13 f2p &44 f f—1 2, as k 0. (4.48c)
sor defines the viscosity tensor g, according There are four independent relaxation frequencies
~
to [i.e., ~„(k), Id12(k), &u13(k), and &d44(k)] for arbi-
ⅈ;) =n;; &2 ) (m, n=1, 2, 3), (4.44b) t4ra=r0y. vAallutheosughof ki,t iasndpotswsoibl(ei.eto.,ob&tda33inandthe&ut»im) efor
we have correlation functions in terms of these relaxation
frequencies we prefer working with a simpler mod-
~yv=W (4.44c) el in which cof2= (df3=0for all k. It will be seen
later that this choice of relaxation frequencies
We shall use (4.44c) as a guide in modelling the
eliminates the coupling between the components of
components of W~'„. The form of „consistent
g&~ the stress tensor during their relaxation. Setting
with the cylindrical symmetry about k is
j3 0 and denoting (df1and (d44 by wi and (dq
~1i ~12 ~13 respectively, in (4.47) and (4.48), we find
~~1123 ~~1ff3 ~~3133 (4.45) ~z. Id2 &s ~s Idz}. (4.48)
$44 Thus, we have to model only two frequencies,
~44 Idi(k) and &us(k), which satisfy Id2(0) =~s(0).
~88 We next consider W' in (4.27e) whose mean re-
duces in steady state to
with the condition
811(k) q12(k) 2q66(k) (4.48a) W„'&(q&) =r„&2k&&8) (4.5Oa)
1
Comparing t.his equation to
which implies isotropy in the plane perpendicular
to k. In order to-ensure isotropy in all directions (q,) =D, 2k.&e), (4.50b)
in the limit of k 0we must also require
where D, is the thermal diffusivity tensor, we ob-
&
833(k} nl ( 3)-kn244 (kt) tain
Il13(k) 1112(k), (4.48b) 8"~D =I'. (4.5Oc)
II33(k)—Il„(k), as k—0. Since I' is diagonal (cf. 4.43a), we choose
Thus, the number of independent components of W' =diag(~„ Id1| Ids), (4.51)
qbIi4~it4„r}ariinsyfthivveaeluli(emis.ieto.fo,fkI),k11-an0df1.2onInl7y/1t3htwis7o)3l3i(mi.iaetn,.d,(7I4)74.4»)44afnbod)r raer-- wsohric.h Dimenpoltiiensg athdeiageolenmalentthsermofaDl bdyiff(uDszivr,ityDrt,enD-2),
we find
duces to r,
D,(k) = (k)/~,(k), (4.52a)
~&v
&OI~) =(~13+3~44) & D,(k) =r, (k)/~,(k) . (4.52b)