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Stress-strain fluctuations in non-linear hydrodynamics
Physica I12A (1982) 505-513 North-Holland Publishing Co.
STRESS-STRAIN FLUCTUATIONS IN NON-LINEAR HYDRODYNAMICS Ronald Forrest FOX
School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA Received 30 September 1981
Two different formulations of hydrodynamic fluctuations in the non-linear regime have been proposed. One of these involves a stress-strain fluctuation term of multiplicative type, and consequently this formulation was criticized by Fox three years ago. Recently, Saarloos, Bedeaux and Mazur have proposed such a formulation again, and have criticized Fox's original criticism. In the present paper, it is again shown that multiplicative stress-strain fluctuations lead to unphysical divergences, and that Saarloos, Bedeaux and Mazur have failed to observe that a stochastic quantity, ~, may have zero mean, (~) = 0, but a divergent second moment ((~)2) = oo. In addition, they have failed to note the limitations of time reversal invariance arguments in macroscopic physics, and they have overlooked important distinctions regarding transverse and longitudinal modes in hydrodynamics. These issues are elucidated in this paper.
1. Introduction
In the non-linear regime, two different formulations of hydrodynamic fluctuations have dominated the literature during the last decade. One of these is based upon master equation analysis, or upon schemes which are essentially equivalent to master equation analysisL2'3'4'5'6). This aporoach leads to two sets of equations. One set is the non-linear set of equations for the averaged hydrodynamical quantities 6) ~t ~ + V. ( ~ ) = 0, ~(~tL+(,.V)a~)=-
(1) ~-~-/5, ax~ ~'
~C(~ , +(a. V),)=-V.~I-P,,~D~,
(2) (3)
in which p is the mass density, u is the velocity field, T is the temperature field, C is the heat capacity per unit mass, D ~ is the strain tensor, defined by 1 /Ou,~
Ou~\
__+~._.e_ oo~ -- ~ ~0x~ 0x~)'
(4)
P~0 is the stress tensor, and q is the heat flux field. These equations become 0378-4371/82/0000-0000/$02.75 ~) 1982 North-Holland
506
RONALD FORREST FOX
autonomous when the constitutive relations are introduced: P ~ = 8 ~ P - 2"0(D~ - ½D~8~) - ~D~8o~,
(5)
in which p is the pressure and rl and ~ are the shear and bulk viscosities respectively; q, = - K ~
0
T,
(6)
in which K is the heat conductivity. The overbars in eqs. (1)-(3) denote averaged quantities. The fluctuations satisfy a second set of equations which are linear, with time and space-dependent coefficients, which arise from the solutions to eqs. (1)-(3)6): 0 ~- Ap + (~i • V)Ap + A p V • ti + tSV. Au + (Au • V)t5 = 0,
0
-
~5C
Ox. p + 2 r l
0@~
2 0 + 0 ~,, A D ° ¢ + ( ~ - ~ ) ~ ~ x . AD'~ Ox~
(7)
(8)
o
AT + ~SC(ti • V)~T + ¢SC(Au • V)T + 3 p C - ~ T + A o C ( f i • V ) T
= KV23T +
0
- B3TV
• fi - B T g
• Au + 4rl~)~3D~
.
g~ +/),~S,~,
+ 2(~ - ~ ) D ~ , ~ D o o
(9)
in which /~p, Au, and AT denote the fluctuations, B =- ~p/c~T, S,,~ is the stress tensor fluctuation and ~o is the heat flux fluctuation. The term of importance in this paper is the stress-strain fluctuation term in eq. (9), the /),eS,~ term. Notice especially that /3,~ is determined by the solutions to eqs. (1)-(3). Therefore, this term is an additive 6) fluctuation. The other formulation 7'8'9) for non-linear hydrodynamic fluctuations has the form of eqs. (1)-(3) without the overbars and with P,~o replaced by P ~ + S,~¢ while q is replaced by q + ~: 0 0~ p + V • (Ou) = 0,
(0
(10)
)
0
0 ~-u.+(u'V)u~ =-~-x (P~+S~o),
(11)
FLUCTUATIONS IN NON-LINEAR HYDRODYNAMICS
507
The crucial stress-strain fluctuation term is now So~D~ which is of multiplicative type6). Fox 6'~°'11) has argued against this term on the grounds that it leads to unphysical divergences. However, his argument was restricted to the linear regime around equilibrium where he asserted that the P,~D,~ term in (12) could be ignored because it is of quadratic order in D~, when (5) is used. Saarloos, Bedeaux and Mazur 9) have argued that keeping S ~ D ~ without P,,~D,~ is wrong because both terms are in fact of the same order as can be shown by a time reversal invariance argument. The total stress tensor is P ~ + S~, which is even under time reversal invariance when viewed microscopically. D~e, however, is odd under time reversal invariance because it is linear in the velocity. Therefore ((V,~ + S~)D~)eq = 0.
(13)
They show in addition, that for the incompressible fluid case, an explicit calculation leads to (P~D~)eq = - ~ ,
(14)
while (S~D,~)eq = +0%
(15)
in which the infinities are identical divergent integrals, except for sign. Eq. (15) is precisely Fox's earlier observation, but with (14), it appears to lose its impact entirely! The time reversal argument, for equilibrium, and the explicit computation for the incompressible fluid case, are without error in this author's view. Fox's original argument is definitely flawed. Nevertheless, the time reversal invariance argument is limited to equilibrium only, as discussed below, and, far more importantly, the incompressible fluid case can be used to compute additional physical quantities which diverge after all! In short, Saarloos, Bedeaux and Mazur have found that ( ~ ) = 0 , where ~--(P~ + S~o)D~o, but have failed to observe that (~2)= w, and that this has dramatic physical consequences. In particular, it is shown below that for the incompressible fluid case, in equilibrium, the temperature field autocorrelation becomes k2 (T(k, to)T(k', to'))¢q = 2ksT~qK(2zr)4~(k + k')~(to + to') to2 2 C~ + K2k 4 Peq
+ 4k~T2q.O25(k + k,)~(to + to,) to ~~eq 2 C ~+ K 2 k 4
× [21'-14
f d31f dO 0 02 e zq 1+ 1~214
,~21k - ll4 (to -- 0) Peq + 1"/21k - II4 (1 - 3 cos 2 a + 4 C O S 4 Or)J,
in which cos a =-1" ( k - l ) / l l k - l
(16)
I. The first part of this expression is the
508
RONALD FORREST FOX
usual 6) result for equilibrium hydrodynamic fluctuations and arises from the first two terms on the right-hand side of (12). It leads to the space-time formula
(T(r, t)T(r', t'))
knT~q[ p~qC ,3~2 [ o~qClr_r,I] peqC \4~rKIt- t'l]
exp
-
4Kit_
t'[ J'
(17)
which is unquestioned experimentally. The additional contribution to (16) is divergent and arises from the stress-strain terms in (12). It exhibits the simple fact that while (~) = 0, (~2) = ~, where 2 --- (P~ + S~)D~,~.
2. Time reversal invariance
The application of time reversal invariance arguments to macroscopic fluctuation phenomena requires care. This is a consequence of two quite different perspectives which are not always perceived. The basis for the distinction has to do with the difference between the Green-Kubo~2'13), or generalized Langevin equation~4), correlation function approach, and the Onsager ~5) phenomenological approach. Both the G r e e n - K u b o and the generalized Langevin equation approaches construct quantities from microscopic phase space variables. The momenta have negative signature under the time reversal transformation while the coordinates have positive signature. Consequently, quantities such as the stress tensor~Z), which are even in the momenta, are even under time reversal. Correlation functions also have time reversal signatures, in equilibriumS6). However, the time reversal assessment requires determination of the distribution function over which averaging is performed. For equilibrium, the distribution is the canonical distribution which depends solely upon the Hamiltonian of the microscopic system and is even under time reversal. Thus, in a correlation formula, the signatures of the phase space variables being correlated determine the signature of the correlation~6). Away from equilibrium, the distribution function must be determined by the dynamical equations, and need not possess an even signature. When there is dissipation at the macroscopic level, such as in hydrodynamics, then the even signature is lost, away from equilibrium. The Onsager perspective is macroscopic to begin with. The Onsager variables are not microscopic quantities, but macroscopic variables. Onsager also looked at their time reversal signature when establishing the reciprocal laws. However, his view is very different from the microscopic view as can be best exhibited by the example of the Brownian motion of a harmonic oscillator:
FLUCTUATIONS IN NON-LINEAR HYDRODYNAMICS d
m ~
q = p,
509 (18)
d
(19)
d--[ p = - m t ° 2 q - a p .
This may look like a microscopic dynamics, but it is not. The momentum, p, is n o t odd under time reversal, nor is it even! Neither is q even or odd! This observation was the impetus behind Fox and Uhlenbeck's tT) generalization of Onsager's theory. Hydrodynamics is just like the Brownian harmonic oscillator. Its variables, macroscopically, are neither even or odd under time reversal. This is dramatically exhibited by the constitutive relation for the stress tensor: P~
= p S ~ - 2• ( D ~ - ] D r r ~ )
- ~D~8,~,
(20)
in which p is even and D~ is odd from the microscopic, phase space point of view. We already observed, along with Saarloos, Bedeaux and Mazur that the total stress tensor, P ~ + S~, must be even12). However, here we see that the "systematic" part, eq. (20), is of mixed character. This shows how dissipation (viscosity) at the macroscopic level, changes time reversal character in variables. In summary, while the time reversal argument proposed by Saarloos, Bedeaux and Mazur is valid for equilibrium, it is not applicable away from equilibrium in hydrodynamics.
3. Incompressible fluids Fox's original criticism of equations (10)-(12) 6'1°'H) stemmed from Putterman's enticing theory 7) of 1/f noise in which he used the multiplicative stress-strain fluctuation, S~oD~o, of eq. (12). The theory was applicable only to compressible fluids because the longitudinal mode coupling between the velocity and temperature fields was the basis of the theory. To bolster their arguments, Saarloos, Bedeaux and Mazur 9) chose the incompressible fluid because it uncouples the temperature and velocity fields, thereby rendering the mathematical problem far more tractable. As will be seen, this eliminates intrinsic non-linear dependence on ~ in the compressible case, leaving only linear dependence on ; in the incompressible case. For this reason, Saarloos, Bedeaux and Mazur missed important consequences of their theory which render it unphysical. The incompressible version of eqs. (10)-(12) is V. u = 0,
(21)
510
RONALD FORREST FOX
peq
~ t/,~ - -
0X,,
p - --
P"~ - ~
~'~'
(22)
0
peqC -or T = K V : T - V • ~ - P.¢D.~ - S.~D,,~,
(23)
in which the non-linear streaming contributions have been d r o p p e d for the near equilibrium analysisg). In addition, near equilibrium we know the correlation properties of both ~ and S,,I~:
(S~(r, t)Su~(r', t')) = 2kBTeqrlS(r - r')8(t - t')[8..Se~, + 8.~8~. - i8~138~,,], (24)
(~.(r, t)g.(r', t')) =
2k~T~qK6(r
-
r')8(t
-
t')8,,.,
(25)
(S,~(r, t)~,~(r', t')) = 0.
(26)
Solutions for the infinite fluid are easily achieved by Fourier t r a n s f o r m a t i o n ,
f(k,~o)= f d3r f dt e'k'~ '~'f(r,t).
(27)
The solutions are
l
//~(k, w) = i09Peq + "ok2
(8,~ - k~k~) ik~S~(k, ~o),
r ( k , o~) = i ~ P e q + Kk 2
~
d3k '
(28)
dw'
× (P.~(k - k', to - to') + S,,~(k -- k', to - w'))k'ut~(k', to')l.
(29)
F r o m (5), (21), and (22), we get
p(k, w) = -kk-~_ S~(k, o9),
(30)
P~(k, ~o) = pS~ + i'0 (k~u.(k, w) + k.u~(k, co)).
(3l)
B e c a u s e of (21), the p8.~ term in (31) c a n n o t contribute to (29). Using (28) in (31), we get (ignoring the p&~ term) iWPeq + ~k 2
- 2k~k~ ~ ) S~,(k, ~o)J, l,,ut~(l, O) =
1 iOpeq +
- l,l~. ~ il~S,~.(l, 0). "ql2 I,~ (St~, \ 1/
(32) (33)
FLUCTUATIONS IN NON-LINEAR HYDRODYNAMR
511
Our first calculation will be a verification of the claim of Saarloos, Bedeaux and Mazur 9) that for t = t' ((P~ + Saj3)D./3>eq = 0.
(34)
In Fourier transformed variables, this requires for t = t'
l fd3kfd3k, fdtofdto, e-'kr-',r+'o,,+'o,'
(27r)
× ((P~(k, to) + S~(k, to))k'u~(k', to'))eq = 0.
(35)
The proof follows from (32) and (33), using (24): 1
e - i k . (r-r')+ito(t-t')
[
4kBTeq~2k4+
kaTeq~ k2 t; ,~ +.r/k2)) to 2peq T I_t_ 2 - 2IJ't°/leqK L4
X I -- to 2pe2q+ r/2k4
(36)
The first term arises from P ~ whereas the second arises from S~. For t = t', the w-integration simplifies; the term odd in to integrates to zero, and the claim is proved. Note, however, that for t - t'---z• 0, the to-integral is of the form o~
f
tosinto~" d t o ~ =
{ ~e-"', _ e-%
"r>0, r<0,
(37)
which implies a discontinuity at r = 0. We agree with Saarloos, Bedeaux and Mazur 9) that the correct result is obtained from ~(r ~ 0 ÷ + ~"~ 0-). Our second calculation shows that we are dealing with a case of (~)---0 while (~2) = oo. Consider (T(k, to)T(k', to')), the Fourier transform of the temperature field autocorrelation function. This is also directly computable from (32) and (33). The quantity of interest is
(T(k, to)T(k', o~')) = 2kBT~qK(27r)4,5(k + k')8(to + to') 1
k2 2 C 2 + K2k 4 tO2Peq
1
+ itopeqC + gk2ito'peqC + gk'2((?-(?-----zTr)Tr))2 f d31f dO f dSl'f dO' × ((P~(k - l, to - O) + S ~ ( k - l, to - O))l~u~(i, O)(P,,(k'- l', to' - 0') ' ' ' O9). + S,~,( k ' - r , t o ' - O))l,~u,(l, ~
(38)
Eqs. (24)-(26), in Fourier transformed form, have already been used to get (38). The first of the two terms on the right-hand side of (38) arises from the contribution to (29). It is the usual equilibrium result6). When (32) and (33) are
512
RONALD FORREST FOX
substituted into the second term in (38), fourth order averages result. They have the form
- l, ~o - 0 ) ~ ,
(l', 0 ' ) > ( ~ , , ( I ,
O)~(k'-
i', ,o'- 0')>
+ (S;v(k - 1, o~ - O)Sc.r,(k'- i', o~'- O'))(S,~(l, O)S,,~.(l', 0')),
(39)
in which the Gaussian property of the stress tensor fluctuations is explicitly indicated. The first term on the right-hand side of (39) is very much like the calculation in (35)-(36). It leads to terms containing f
.2120pea dO 2 2 ~1214=0. 0 Peq+
(40)
The second term on the right-hand side of (39) is similar, and also yields a net result of zero. The third term, however, does not vanish. A long computation finally yields 4 (T(k, o~)T(k', o~'))eq = 2kBT~qK(27r) 6(k + k')6(oJ + w')
+4k~T~qr128(k +k')8(oo + w') o~20e~C2+ K2k4 X [214--14
dO 0 ,_Peq 2 4- 7)214
(1 - 3 cos~" a + 4 cos4 a)]
"o2]k - 114 (,o - 0)~t,~ + ~ l k
d31
, 2 ,.-,~ K2k 4 (t) ~Peql~ ~ +
(41)
- ll"
where cos a = - I . ( k - l ) / l [ k - l [ . As was already pointed out in the introduction, the leading term is the well known equilibrium result, whereas the additional term is manifestly divergent! This additional term has its origin in the multiplicative stress-strain fluctuations, S,~D,~, in (23). For this reason, this type of fluctuation leads to unphysical results.
4. Concluding remarks
Saarloos, Bedeaux and Mazur °) have commented that the reason their calculation of (34) came out as oo- ~ = 0 was "(even though) the integrals are divergent if no wave-vector cut-off, necessary to tame divergencies inherent to a continuum description, is introduced". This is certainly wrong. No cut-off is required in the usual equilibrium theory, even thought it is a continuum theory. Many experimentally compatible results 6) have been calculated in the continuum limit which is not inherently divergent. The divergence here is a
FLUCTUATIONS IN NON-LINEAR HYDRODYNAMICS
513
result of the muitiplicative character of the S~,D~ term. Indeed, the theory proposed in eqs. (1)-(9) is not divergent in the continuum limit, and requires no cut-off in wave vectors. Although Fox's original criticism 6'~°'lt) of such theories is flawed in the manner demonstrated by Saarloos, Bedeaux and Mazurg), the criticism still stands on the basis of the present analysis which is far more compelling. It should not be overlooked, however, that the earlier analysis was for the compressible case, which is much more difficult mathematically.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17)
N.G. van Kampen, Can. J. Phys. 39 (1961) 551. R. Kubo, K. Matsuo and K. Kitahara, J. Stat. Phys. 9 (1973) 51. J. Keizer, Phys. Fluids 21 (1978) 198. M.H. Ernst and E.G.D. Cohen, J. Stat. Phys. 25 (1981) 153. C. van den Broeck and L. Brenig. Phys. Rev. A. 21 (1980) 1039. R.F. Fox, Phys. Rep. C 48 (1978) 179. S.J. Putterman, Phys. Rev. Letts. 39 (1977) 585. K.T. Mashiyama and H. Mori, J. Stat. Phys. 18 (1978) 385. W. van Saarloos, D. Bedeaux and P. Mazur, Physica ll0A (1982) 147. R.F. Fox, Supp. Prog. Theor. Phys. 64 (1978) 425. R.F. Fox, J. Math. Phys. 19 (1978) 1993. D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions (Benjamin/Cummings, New York, 1975). S.W. Lovesey, Condensed Matter Physics (Benjamin/Cummings, New York, 1980). H. Mori, Prog. Theor. Phys. 33 (1965) 423. L. Onsager, Phys. Rev. 37 (1931) 405; 38 (1931) 2265; 91 (1953) 1505, 1512. B.J. Berne and R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976). R.F. Fox and G.E. Uhlenbeck, Phys. Fluids B (1970) 1893, 2881.
STRESS-STRAIN FLUCTUATIONS IN NON-LINEAR HYDRODYNAMICS Ronald Forrest FOX
School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA Received 30 September 1981
Two different formulations of hydrodynamic fluctuations in the non-linear regime have been proposed. One of these involves a stress-strain fluctuation term of multiplicative type, and consequently this formulation was criticized by Fox three years ago. Recently, Saarloos, Bedeaux and Mazur have proposed such a formulation again, and have criticized Fox's original criticism. In the present paper, it is again shown that multiplicative stress-strain fluctuations lead to unphysical divergences, and that Saarloos, Bedeaux and Mazur have failed to observe that a stochastic quantity, ~, may have zero mean, (~) = 0, but a divergent second moment ((~)2) = oo. In addition, they have failed to note the limitations of time reversal invariance arguments in macroscopic physics, and they have overlooked important distinctions regarding transverse and longitudinal modes in hydrodynamics. These issues are elucidated in this paper.
1. Introduction
In the non-linear regime, two different formulations of hydrodynamic fluctuations have dominated the literature during the last decade. One of these is based upon master equation analysis, or upon schemes which are essentially equivalent to master equation analysisL2'3'4'5'6). This aporoach leads to two sets of equations. One set is the non-linear set of equations for the averaged hydrodynamical quantities 6) ~t ~ + V. ( ~ ) = 0, ~(~tL+(,.V)a~)=-
(1) ~-~-/5, ax~ ~'
~C(~ , +(a. V),)=-V.~I-P,,~D~,
(2) (3)
in which p is the mass density, u is the velocity field, T is the temperature field, C is the heat capacity per unit mass, D ~ is the strain tensor, defined by 1 /Ou,~
Ou~\
__+~._.e_ oo~ -- ~ ~0x~ 0x~)'
(4)
P~0 is the stress tensor, and q is the heat flux field. These equations become 0378-4371/82/0000-0000/$02.75 ~) 1982 North-Holland
506
RONALD FORREST FOX
autonomous when the constitutive relations are introduced: P ~ = 8 ~ P - 2"0(D~ - ½D~8~) - ~D~8o~,
(5)
in which p is the pressure and rl and ~ are the shear and bulk viscosities respectively; q, = - K ~
0
T,
(6)
in which K is the heat conductivity. The overbars in eqs. (1)-(3) denote averaged quantities. The fluctuations satisfy a second set of equations which are linear, with time and space-dependent coefficients, which arise from the solutions to eqs. (1)-(3)6): 0 ~- Ap + (~i • V)Ap + A p V • ti + tSV. Au + (Au • V)t5 = 0,
0
-
~5C
Ox. p + 2 r l
0@~
2 0 + 0 ~,, A D ° ¢ + ( ~ - ~ ) ~ ~ x . AD'~ Ox~
(7)
(8)
o
AT + ~SC(ti • V)~T + ¢SC(Au • V)T + 3 p C - ~ T + A o C ( f i • V ) T
= KV23T +
0
- B3TV
• fi - B T g
• Au + 4rl~)~3D~
.
g~ +/),~S,~,
+ 2(~ - ~ ) D ~ , ~ D o o
(9)
in which /~p, Au, and AT denote the fluctuations, B =- ~p/c~T, S,,~ is the stress tensor fluctuation and ~o is the heat flux fluctuation. The term of importance in this paper is the stress-strain fluctuation term in eq. (9), the /),eS,~ term. Notice especially that /3,~ is determined by the solutions to eqs. (1)-(3). Therefore, this term is an additive 6) fluctuation. The other formulation 7'8'9) for non-linear hydrodynamic fluctuations has the form of eqs. (1)-(3) without the overbars and with P,~o replaced by P ~ + S,~¢ while q is replaced by q + ~: 0 0~ p + V • (Ou) = 0,
(0
(10)
)
0
0 ~-u.+(u'V)u~ =-~-x (P~+S~o),
(11)
FLUCTUATIONS IN NON-LINEAR HYDRODYNAMICS
507
The crucial stress-strain fluctuation term is now So~D~ which is of multiplicative type6). Fox 6'~°'11) has argued against this term on the grounds that it leads to unphysical divergences. However, his argument was restricted to the linear regime around equilibrium where he asserted that the P,~D,~ term in (12) could be ignored because it is of quadratic order in D~, when (5) is used. Saarloos, Bedeaux and Mazur 9) have argued that keeping S ~ D ~ without P,,~D,~ is wrong because both terms are in fact of the same order as can be shown by a time reversal invariance argument. The total stress tensor is P ~ + S~, which is even under time reversal invariance when viewed microscopically. D~e, however, is odd under time reversal invariance because it is linear in the velocity. Therefore ((V,~ + S~)D~)eq = 0.
(13)
They show in addition, that for the incompressible fluid case, an explicit calculation leads to (P~D~)eq = - ~ ,
(14)
while (S~D,~)eq = +0%
(15)
in which the infinities are identical divergent integrals, except for sign. Eq. (15) is precisely Fox's earlier observation, but with (14), it appears to lose its impact entirely! The time reversal argument, for equilibrium, and the explicit computation for the incompressible fluid case, are without error in this author's view. Fox's original argument is definitely flawed. Nevertheless, the time reversal invariance argument is limited to equilibrium only, as discussed below, and, far more importantly, the incompressible fluid case can be used to compute additional physical quantities which diverge after all! In short, Saarloos, Bedeaux and Mazur have found that ( ~ ) = 0 , where ~--(P~ + S~o)D~o, but have failed to observe that (~2)= w, and that this has dramatic physical consequences. In particular, it is shown below that for the incompressible fluid case, in equilibrium, the temperature field autocorrelation becomes k2 (T(k, to)T(k', to'))¢q = 2ksT~qK(2zr)4~(k + k')~(to + to') to2 2 C~ + K2k 4 Peq
+ 4k~T2q.O25(k + k,)~(to + to,) to ~~eq 2 C ~+ K 2 k 4
× [21'-14
f d31f dO 0 02 e zq 1+ 1~214
,~21k - ll4 (to -- 0) Peq + 1"/21k - II4 (1 - 3 cos 2 a + 4 C O S 4 Or)J,
in which cos a =-1" ( k - l ) / l l k - l
(16)
I. The first part of this expression is the
508
RONALD FORREST FOX
usual 6) result for equilibrium hydrodynamic fluctuations and arises from the first two terms on the right-hand side of (12). It leads to the space-time formula
(T(r, t)T(r', t'))
knT~q[ p~qC ,3~2 [ o~qClr_r,I] peqC \4~rKIt- t'l]
exp
-
4Kit_
t'[ J'
(17)
which is unquestioned experimentally. The additional contribution to (16) is divergent and arises from the stress-strain terms in (12). It exhibits the simple fact that while (~) = 0, (~2) = ~, where 2 --- (P~ + S~)D~,~.
2. Time reversal invariance
The application of time reversal invariance arguments to macroscopic fluctuation phenomena requires care. This is a consequence of two quite different perspectives which are not always perceived. The basis for the distinction has to do with the difference between the Green-Kubo~2'13), or generalized Langevin equation~4), correlation function approach, and the Onsager ~5) phenomenological approach. Both the G r e e n - K u b o and the generalized Langevin equation approaches construct quantities from microscopic phase space variables. The momenta have negative signature under the time reversal transformation while the coordinates have positive signature. Consequently, quantities such as the stress tensor~Z), which are even in the momenta, are even under time reversal. Correlation functions also have time reversal signatures, in equilibriumS6). However, the time reversal assessment requires determination of the distribution function over which averaging is performed. For equilibrium, the distribution is the canonical distribution which depends solely upon the Hamiltonian of the microscopic system and is even under time reversal. Thus, in a correlation formula, the signatures of the phase space variables being correlated determine the signature of the correlation~6). Away from equilibrium, the distribution function must be determined by the dynamical equations, and need not possess an even signature. When there is dissipation at the macroscopic level, such as in hydrodynamics, then the even signature is lost, away from equilibrium. The Onsager perspective is macroscopic to begin with. The Onsager variables are not microscopic quantities, but macroscopic variables. Onsager also looked at their time reversal signature when establishing the reciprocal laws. However, his view is very different from the microscopic view as can be best exhibited by the example of the Brownian motion of a harmonic oscillator:
FLUCTUATIONS IN NON-LINEAR HYDRODYNAMICS d
m ~
q = p,
509 (18)
d
(19)
d--[ p = - m t ° 2 q - a p .
This may look like a microscopic dynamics, but it is not. The momentum, p, is n o t odd under time reversal, nor is it even! Neither is q even or odd! This observation was the impetus behind Fox and Uhlenbeck's tT) generalization of Onsager's theory. Hydrodynamics is just like the Brownian harmonic oscillator. Its variables, macroscopically, are neither even or odd under time reversal. This is dramatically exhibited by the constitutive relation for the stress tensor: P~
= p S ~ - 2• ( D ~ - ] D r r ~ )
- ~D~8,~,
(20)
in which p is even and D~ is odd from the microscopic, phase space point of view. We already observed, along with Saarloos, Bedeaux and Mazur that the total stress tensor, P ~ + S~, must be even12). However, here we see that the "systematic" part, eq. (20), is of mixed character. This shows how dissipation (viscosity) at the macroscopic level, changes time reversal character in variables. In summary, while the time reversal argument proposed by Saarloos, Bedeaux and Mazur is valid for equilibrium, it is not applicable away from equilibrium in hydrodynamics.
3. Incompressible fluids Fox's original criticism of equations (10)-(12) 6'1°'H) stemmed from Putterman's enticing theory 7) of 1/f noise in which he used the multiplicative stress-strain fluctuation, S~oD~o, of eq. (12). The theory was applicable only to compressible fluids because the longitudinal mode coupling between the velocity and temperature fields was the basis of the theory. To bolster their arguments, Saarloos, Bedeaux and Mazur 9) chose the incompressible fluid because it uncouples the temperature and velocity fields, thereby rendering the mathematical problem far more tractable. As will be seen, this eliminates intrinsic non-linear dependence on ~ in the compressible case, leaving only linear dependence on ; in the incompressible case. For this reason, Saarloos, Bedeaux and Mazur missed important consequences of their theory which render it unphysical. The incompressible version of eqs. (10)-(12) is V. u = 0,
(21)
510
RONALD FORREST FOX
peq
~ t/,~ - -
0X,,
p - --
P"~ - ~
~'~'
(22)
0
peqC -or T = K V : T - V • ~ - P.¢D.~ - S.~D,,~,
(23)
in which the non-linear streaming contributions have been d r o p p e d for the near equilibrium analysisg). In addition, near equilibrium we know the correlation properties of both ~ and S,,I~:
(S~(r, t)Su~(r', t')) = 2kBTeqrlS(r - r')8(t - t')[8..Se~, + 8.~8~. - i8~138~,,], (24)
(~.(r, t)g.(r', t')) =
2k~T~qK6(r
-
r')8(t
-
t')8,,.,
(25)
(S,~(r, t)~,~(r', t')) = 0.
(26)
Solutions for the infinite fluid are easily achieved by Fourier t r a n s f o r m a t i o n ,
f(k,~o)= f d3r f dt e'k'~ '~'f(r,t).
(27)
The solutions are
l
//~(k, w) = i09Peq + "ok2
(8,~ - k~k~) ik~S~(k, ~o),
r ( k , o~) = i ~ P e q + Kk 2
~
d3k '
(28)
dw'
× (P.~(k - k', to - to') + S,,~(k -- k', to - w'))k'ut~(k', to')l.
(29)
F r o m (5), (21), and (22), we get
p(k, w) = -kk-~_ S~(k, o9),
(30)
P~(k, ~o) = pS~ + i'0 (k~u.(k, w) + k.u~(k, co)).
(3l)
B e c a u s e of (21), the p8.~ term in (31) c a n n o t contribute to (29). Using (28) in (31), we get (ignoring the p&~ term) iWPeq + ~k 2
- 2k~k~ ~ ) S~,(k, ~o)J, l,,ut~(l, O) =
1 iOpeq +
- l,l~. ~ il~S,~.(l, 0). "ql2 I,~ (St~, \ 1/
(32) (33)
FLUCTUATIONS IN NON-LINEAR HYDRODYNAMR
511
Our first calculation will be a verification of the claim of Saarloos, Bedeaux and Mazur 9) that for t = t' ((P~ + Saj3)D./3>eq = 0.
(34)
In Fourier transformed variables, this requires for t = t'
l fd3kfd3k, fdtofdto, e-'kr-',r+'o,,+'o,'
(27r)
× ((P~(k, to) + S~(k, to))k'u~(k', to'))eq = 0.
(35)
The proof follows from (32) and (33), using (24): 1
e - i k . (r-r')+ito(t-t')
[
4kBTeq~2k4+
kaTeq~ k2 t; ,~ +.r/k2)) to 2peq T I_t_ 2 - 2IJ't°/leqK L4
X I -- to 2pe2q+ r/2k4
(36)
The first term arises from P ~ whereas the second arises from S~. For t = t', the w-integration simplifies; the term odd in to integrates to zero, and the claim is proved. Note, however, that for t - t'---z• 0, the to-integral is of the form o~
f
tosinto~" d t o ~ =
{ ~e-"', _ e-%
"r>0, r<0,
(37)
which implies a discontinuity at r = 0. We agree with Saarloos, Bedeaux and Mazur 9) that the correct result is obtained from ~(r ~ 0 ÷ + ~"~ 0-). Our second calculation shows that we are dealing with a case of (~)---0 while (~2) = oo. Consider (T(k, to)T(k', to')), the Fourier transform of the temperature field autocorrelation function. This is also directly computable from (32) and (33). The quantity of interest is
(T(k, to)T(k', o~')) = 2kBT~qK(27r)4,5(k + k')8(to + to') 1
k2 2 C 2 + K2k 4 tO2Peq
1
+ itopeqC + gk2ito'peqC + gk'2((?-(?-----zTr)Tr))2 f d31f dO f dSl'f dO' × ((P~(k - l, to - O) + S ~ ( k - l, to - O))l~u~(i, O)(P,,(k'- l', to' - 0') ' ' ' O9). + S,~,( k ' - r , t o ' - O))l,~u,(l, ~
(38)
Eqs. (24)-(26), in Fourier transformed form, have already been used to get (38). The first of the two terms on the right-hand side of (38) arises from the contribution to (29). It is the usual equilibrium result6). When (32) and (33) are
512
RONALD FORREST FOX
substituted into the second term in (38), fourth order averages result. They have the form
- l, ~o - 0 ) ~ ,
(l', 0 ' ) > ( ~ , , ( I ,
O)~(k'-
i', ,o'- 0')>
+ (S;v(k - 1, o~ - O)Sc.r,(k'- i', o~'- O'))(S,~(l, O)S,,~.(l', 0')),
(39)
in which the Gaussian property of the stress tensor fluctuations is explicitly indicated. The first term on the right-hand side of (39) is very much like the calculation in (35)-(36). It leads to terms containing f
.2120pea dO 2 2 ~1214=0. 0 Peq+
(40)
The second term on the right-hand side of (39) is similar, and also yields a net result of zero. The third term, however, does not vanish. A long computation finally yields 4 (T(k, o~)T(k', o~'))eq = 2kBT~qK(27r) 6(k + k')6(oJ + w')
+4k~T~qr128(k +k')8(oo + w') o~20e~C2+ K2k4 X [214--14
dO 0 ,_Peq 2 4- 7)214
(1 - 3 cos~" a + 4 cos4 a)]
"o2]k - 114 (,o - 0)~t,~ + ~ l k
d31
, 2 ,.-,~ K2k 4 (t) ~Peql~ ~ +
(41)
- ll"
where cos a = - I . ( k - l ) / l [ k - l [ . As was already pointed out in the introduction, the leading term is the well known equilibrium result, whereas the additional term is manifestly divergent! This additional term has its origin in the multiplicative stress-strain fluctuations, S,~D,~, in (23). For this reason, this type of fluctuation leads to unphysical results.
4. Concluding remarks
Saarloos, Bedeaux and Mazur °) have commented that the reason their calculation of (34) came out as oo- ~ = 0 was "(even though) the integrals are divergent if no wave-vector cut-off, necessary to tame divergencies inherent to a continuum description, is introduced". This is certainly wrong. No cut-off is required in the usual equilibrium theory, even thought it is a continuum theory. Many experimentally compatible results 6) have been calculated in the continuum limit which is not inherently divergent. The divergence here is a
FLUCTUATIONS IN NON-LINEAR HYDRODYNAMICS
513
result of the muitiplicative character of the S~,D~ term. Indeed, the theory proposed in eqs. (1)-(9) is not divergent in the continuum limit, and requires no cut-off in wave vectors. Although Fox's original criticism 6'~°'lt) of such theories is flawed in the manner demonstrated by Saarloos, Bedeaux and Mazurg), the criticism still stands on the basis of the present analysis which is far more compelling. It should not be overlooked, however, that the earlier analysis was for the compressible case, which is much more difficult mathematically.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17)
N.G. van Kampen, Can. J. Phys. 39 (1961) 551. R. Kubo, K. Matsuo and K. Kitahara, J. Stat. Phys. 9 (1973) 51. J. Keizer, Phys. Fluids 21 (1978) 198. M.H. Ernst and E.G.D. Cohen, J. Stat. Phys. 25 (1981) 153. C. van den Broeck and L. Brenig. Phys. Rev. A. 21 (1980) 1039. R.F. Fox, Phys. Rep. C 48 (1978) 179. S.J. Putterman, Phys. Rev. Letts. 39 (1977) 585. K.T. Mashiyama and H. Mori, J. Stat. Phys. 18 (1978) 385. W. van Saarloos, D. Bedeaux and P. Mazur, Physica ll0A (1982) 147. R.F. Fox, Supp. Prog. Theor. Phys. 64 (1978) 425. R.F. Fox, J. Math. Phys. 19 (1978) 1993. D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions (Benjamin/Cummings, New York, 1975). S.W. Lovesey, Condensed Matter Physics (Benjamin/Cummings, New York, 1980). H. Mori, Prog. Theor. Phys. 33 (1965) 423. L. Onsager, Phys. Rev. 37 (1931) 405; 38 (1931) 2265; 91 (1953) 1505, 1512. B.J. Berne and R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976). R.F. Fox and G.E. Uhlenbeck, Phys. Fluids B (1970) 1893, 2881.