- Email: [email protected]
On the microscopic derivation of balance equations in hydrodynamics
?nvs, ca
80A (1975) 585o
ON THE MICROSCOPIC DERIVATION-"' OF BALANCE EQUATIONS IN IIYDRODYNAMICS H,J, KREUZER
7?}eoreNc
W,, derive., r~ew tensor hierarchy of h.~c.~I,-.a,,,:,),~.,,, ha{ante eqtm~io'~> s[ar[mg from die quaemm-mecbav, ica~ n~any-body problem. A nov. balm~ce eqt~a~ion for ti~e kinetic press-~ re in a fluid is der{v~d and the connection with v!~:oe)as'dc ~hcory i> discussed.
1, Introductioa Ti~e p h e n o m e n o ! o g i c a ! a p p r o a c h ~.o ~,o.~?equilibrim~ ~~crn~odyx-~au~cs o f pi°oces .., ses ( N E T P ) ttsualiy starts \;ith a sel o f ba!ance e q u a t i o n s for the ; e l e v a m thcrr~'<> d y n a m i c quantities ~-%. F o r m a i l y 'ae car~ v, ri~e i>r :,m ex~e~si\e variable 2: d -
-
-
dt
v(t) = PKi
+
O{Fi,
f*}
where P[.F] d e n o t e s the p r o d u c t i o n terms wi{hi~: ti~e v o l u m e V o f ~he ,,,:s,:em a~d ,/~[F] gives the flux o f F t h r o u g h the surfitce of V, i~tt-oduci~.~g F densities b,
r = -g-i"
dr,
where 9(x) is the mass density, we find tocai balance equation, s
~t
+ divj~, = ~r[.F],
(3)
whez'e j~ is tl~e F c u r r e n t a n d a [ F ] describes local siuks arid sources o f F. C h o o s i ~ g f = 1 we recover in eq. (3) the contimfity equafio)~ for the mass density O: Laking * Work supported in part by the Nationai Research Com~cil of Canada. 585
• •!
;i, ¸
~g6
~ H+L
KREUZER
ti~rfthe velocity tield ~ we get the momentum-balance equation, i.e., the equation of nlofion of the gvstem. With f =: ½.v~ we get the kinetic-energy balance, f := u gives the internal-energy balance, etc. It is obvious that such balance equations are only useful ff we specify the source terms a[Fj according to the problem nnder study, This leads in the momentum balance to the introduction of generalized thermodyitamic fk~rces. Forsystems close to equilibrium it is further assumed that the therme,dynamic fluxes are linear in the forces, the phenomenological coefficients satis:fyi~g Onsager's reciprocity relationsS,6). l ~ get a better understanding of such macroscopic balance equations various derival ioJ~s from the microscopic theory of a many-body system have been presented startiiJg in the classical case from Lie " ~uwl,~ " ~'~'s equation• and quantum-mechanically fron~ the Liouville-Von Neumann equation for the density matrixV-~'). Such microscopic derivations can also lead to new macroscopic femures as has been shown recently~), starting tiom a many-body system with nonlocai interactions we find addilionat source terms in the macroscopic balance equations that can be imerpreted as local fluctuations. Our program for this paper is to start with a. many-body hamittonian in secondquantized form, derive a systematic hierarchy of in~nitely many balance equations for microscopic density, current, energy, and higher-rank tensor operators. Taking slfitabie thermodynamic averages this hierarchy wil! be truncated to yield the wellknown macroscopic balance equations of hydrodyllamics allowing us a term by term comparison of microscopic and macroscop'c quantities. Moreover we will be able to derive a balance equation for the pressure tensor. The significance of higherral~k tensor operators will be discussed.
2o Operator balance equations We start with a hamih_onia~~ i~; second-quantized form 7" + V + U~
(4)
w it t -.(IV/4m) .! ox ~"
[~p'(x~ /17~ (X) +. Aoq7 (x) "~;(x)],
U = ~ dxU (x) "~:'{x.,.~,~.v~,
(5)
v .--- S dx' dx"V(x', x") ~fl(x') ~p(x') ~,~(x") ,<;(x"). Writing V in this i~rm we have absorbed single-particle contributions that result from proper normal ordering> into U~ The field operators ~ (x, t) are subject to
BALANCE EQL AJIONS IN HYDROD~'NAM[CS
587
Heisenberg's e q u a t i o n o f ~ o t i o n
il/a,v' = [,p, ~t].
((9
We assume for si,~plicity that g (x. t) satisfies Bose-Einsteh~ statistics. The non~ local o p e r a t o r S. . .(x', . x") = ,t,,(x ~ ") 7,(x'),
~7)
satisfies the e q u a t i o n ~h c,.,.9 ( x . x ' ) + (h2/,'n) d • '.;¢~2(x o x")
where (~ = ~' 4-. ?"•., ? ..... ~ (~' - g"~. with i?' and d" beirlg the >~.,~.~.~,~ . s with respect, to the variables x' a n d x', respectively. We also have Wz(x',x")
=
•
-
/
5 d v. [ V ( x . y )
t
,
'
e
_V(x . } ) l { . ( y , y ) ( ; ., x . x~ " }' , ~
-
,}
. (~}1
the cut"Iy brackets indicating s y m m e t r i z e d product% Le., {AB} .... .4B + BA. Starti~ag from eq. (8) we can derive a hierarchy o f o p e r a t o r equatior~s by expal?di~lg the nonlocal ope.rator o_~(.~x ,' x ' ) in terms o f the re!ative com~',.Jmate ,,~ := x" - - x ' a r o u n d the cen;.er-of-mass coordi~3ate x = ~-(x' + x"), i.e.. write ,; (x'. x ' ) = ~ (l/n!)
{ ~ • &',;*e) ~'¢9 ( . r ' , ) * . c - . , . ' ,
i:*~ the
"local"
}irnil
x'
= x
= x
we: t:i,~d
the COl~tinuil_y eq~m',ion tbr the o p e r a t o r ~;(x) = [; (at. x} ti v,L~ (x) + ~,h°,,m) 8 .](x) ==: 0 ,
(t0)
where j(x), . . . . i lime_, o. ":e/2 (x', x ' ) is the current operator. C"o!!ecth~g *,.eun>~" " proportionat to # we get lhe e q u a t i o n o f m o t i o n # vddx) . . . . .
- (h~/rm c, 7~a (x) [ G U ( x ) ]. , ; (.x ) . + ..[dv.
[GV(x. .
r)] t.d3' ~' ) ,.,(x)~.
(t 1)
where the kinetic-energy tensor o p e r a t o r is give~ by 7P~,(x)= lime_, o (&'i:G) (a/r?~) ~ (x', x") a n d ~.~ = ~?/&x'~f r o m n o w on. Terms ~ i t h ~': yield ~ ~?,~,
(x)
+
(If/m)
•
,~;/:~, ~ " (x) = [ c" ~ / (x)]j,(x) ~ " + [?, U (x)]j~(x)
- j" dy [G v(x, y)i {&v)y,(x)} -
j~ d y t~.', " J~ (a., "•
39] [,j(y)A(x)},
(l 2)
588
H.J. KREUZER
where ~'~al(x) = --i lira4., o (~/~ff~)(d/O~a) (f~/(?~) 3 (x', x") will in turn couple in its balance equation to yet higher tensor operators namety
- [/~,eke,U(x)] 6(x) + j d3y [0,V(x~ y)l {.dO,)T~,(x)}
+ S daY [OkV(xy)] {~0') T,t(x)} + ~ d3y [~.,V(x,y)l {~(.!') L~(x)}.
(13)
It is obvious that this hierarchy of operator balance equations continues in principle ad infinitum. As we go to higher-rank tensor operators, more microscopic information is exposed through the higher-order derivatives, the complete hierarchy being equivalent to the equation of motion, eq. (8), for the nontocal operator m
'2 (x', x").
3. Macroscopic balance equations In order to define macroscopic quantities like mass density, mass current, energies, etc., we have to be able to define statistical averages oi" the corresponding operators defined in the previous section. To this end we have to know the time evolution of the statistical operator (density matrix) ~ which is only available for simple models (see e.g. reL 11). On the other hand, proceeding phenomenoIogically we can assume the existence of local equilibrium in the system under study and make an Ansatz for the formally defined stat:istical averages. It is this second approach that has been tb!towed in the past and that will also be used in this paper. Followil~g Fr~Shlich~:~) we define ~ {x. ~'" ~:, = ~r (x, ~) e ~;~':°~''b
(/4)
from which, we gel; the macroscopic mass density' ~(x) = m,,x(x, O) and ti~e macroscopic velocity v~:(x) = .h. . . .lira ..
z (ix,
(i5)
It should be remarked that, according to Galasiewiczg), Bogoliubov int:roduced macroscopic velocities at the operator level through a translatio~ e t"
It6)
BALANCE
EQUATIONS
tN HYDRODYNAMICS
589
We can readily generalize Bogo[iubov's m e t h o d by transfi)rming fl'om a fluid at rest to a fluid with a n o n c o n s t a n t macroscopic veh::city field ~:~). Thi.~ i.,~accomplished by defining a unitary transl\ormation U(x)
+/,(x)-+ ,?(x) ~r(x), by ih
- -~?8x
W(x)
=
my (x) W(x}.. .
(i 8)
with the i'\+rma[ solutioa
++.'~x)=
exr
f/ ~d+#/O ~x rCx'), d x i \
'~-'e
~t9)
J
and W(xo) = I. With a similar trap.s(ormatJor~ o~~.e can aiso h~troduce an angular velocily fie!d ~o describe vordces and reIated p h e n o m e n a in turbulence. Usiag eqs. (17)-(I9) we can transform the tensor opera~ors introduced in ti~,e !ast section ',o a m o v i n g fluid. We ,. £et [a ~up~+~.,~pt ~ ' "~,",' (0) refers to a fluid at rest] .otx) . . . =. . V. ?. ( x ) ' ' + '
j(x)
,=
"
'"
' == L'~'" : >I.X~"
j'°'(x) + (re:l,) (kx} vCv),
L,(x) = .}~~?>~x)- (,,+./~),+,~;(x)<,c,-),.:~c,O-- i,,,/,~~,.:,,.i?"+ n.f;%.
F.~,( x'~
~:o)
f.;,~ b~:) + im;iz)
4,~O~(x) +
(m ~,') -,o,~.
and. similar reductions "or highe>ral~k tensors. T o define mac~oscopic quamifie.~ we have to take stafisticaI averages o f the !oca! operators i~ eqs. {.20). This a',eraging procedure ~ecessitates the a s s u m p t i o n of ioca] equilibrium in fi~e fluid, i.e., a r o e n d each point (x~ t) i,~ space and tim.e we assm,a~:: a macroscopic but sti[~ " s m a l l " volume and dine interv~:q over wMci~ averaging is do~'~e wifi~ a n equiiibri um de~ssity matrix, Also to set mac~'oscopic qua.ntilies i'ro:m lhe <4'-.... -~" ~,~o,~. ..... eqs. (20) ,;;e have to multiply an ,xh-rauk te~?sor by h~L because according to dae correspondence 1;-~nciple it is (ili ~ ~:¢:) !:hat correspot~ds Io a macroscopic (classicai)quanti%, (momentum)~ We thus get the macroscopic mas~; deasity ~'x) :.: m -(~(x)),
(2t)
the macroscopic c~rrent j ( x } = h ( j ( x £ > .= !.?(x} v{x)+
(22)
i
)
590
H, J. K R E U Z E K
the macroscopic kinetic energy-stre:~s tensor T,:,(x)
~
-(/?/2m) ( ~ , ( x ) ) = ~,,.Tm)~x).4. ~}~'(x)t:d.'~
(23)
and a macroscopic energy-flux tensor
laity(X)
....
(l}a/2rt't 2) <&~t(X))
= F~(x)
+ v , T ~ ' ( x ) + vkT[iO" (X) + v d]q2~' (x) + -}~4x)r~r~t,~°
(24)
With these expressions we get f r e m eq, (t0) dae macroscopic continuity equation
Go(x) + aev = o.
~25)
Eq, (11) yields the macroscopic equation ol motion or m o m e n t u m balance equation
6 (t)vk) + 26 [T~°'(x) + ~-,,?
(26)
involving the density-density or ~wo-body correlaticm functim~, Usir~g ihe macroscopic equation D { 1 d ~? ~ c~(x) =- • %, ................~,(x) ....... t:!s)° Dt &v~ m &v~
(2'7'~
we can define the stress tensor %~ for ~hict: ~ e find from eq. ~27} by d~e standard expansion of the density-density corre!atio~.,v. ~~) -
F(,") rF] .['~(r)d3~........ c~'~::. F
~..0:;:>}
F
,~r~ ' contains visco~s teems, [ ] ( r ), is t:'~c radiai pair corr:~a~mt~ ' ~' " hmction. We i\~llow here the notation o f ref. 12 wI~ere also a derivation of the Nax-ier-Stokes equations Dora eq. (aTe, "~ ' c a l l be fo.u~d. {n a,~ i~c~,,~r, ie~ ..... fl~lid ~ ,~ = .
i ~.~-
i"r ?V(r) *
O ~ "
¢7Y
This justifies calling T~,t(x) the kinetic part c,f the ene~:g>~::~:ress revisor, To get a balance equation fo': the e~ergy o f the system we {:u~veessemia}~y three differem approaches, We can stall fiom tl~e eq!mtio~ o f m o d o ~ t26} o r f r o m O~,e
BALANCE EQUATIONS iN HYI)RODYNAMI'CS
591
h a m i t t o ~ i a a deasity or *.ko:n eq. (~2) for ~h¢ ki~e~icr ~he ki~e~ic eaergy we caa me~{ply the equa~{o~ of too,ion (26) with ~:l~.eve!o~:ity ~, and ge~ after so:me a!gebr~,
....... ( t':~} ev~ ~:, U .... v~ 5 J3' {7 {:](r} {;{y>~ 71>~-::~~" (x~ y},
(3())
4 i}ee'~e~,> -~ e,,~ e'~7" -~. e', ( 721'" - ~~ (.!::,e- c,,~'~}
:< {*'~{x} [:~I'{:r, ,~ } + e::ix} ,~',,{,{Xo Y}i-
{30a)
use of" eqs. {23)= We ge~
.... v~e 2H,~ i' '.. 4:,
Le~ as {:if:'s~ mves~iga~.e a~ :idea} gas~ ~!-Ee~
where P is ~:~e (scalar! !~ydros~.~.t;c press~.~'~e. T~.t~s ~aki~g ~hc ~race m : % {32} we '~}gd a c,o,!lir~u,ity equat}o~ ~:or &."~a:me!y
C~ t,* + 0~ {ic~ *~) ::,> '~},
{ 34)
592
H.L KREUZER
t h e off-dlagoaa lerms in e% (32) yield a relation (35)
¢oniirming that in an ideal fluid the rate-of-strain tensor Akt vanishes. In a real tl~fid~ h-owever, z~herate-of-strain tensor will cow,tribute to the pressure balance, so will ~ocaliy generated pressure fl,)ws through the term Gt;)~{:'(x). Moreover, the imeractions wilk through correlatior,.s, introduce local sinks and sources for the kir~el.h; part of the pressure. To understand the nature of these sinks and sources let us expat, d the de~sity.-.current correlations for small deviations from equilibrium, !Be expa~sion parameters being the deviations from the equilibrium density i}~,(x. t) :,.-:9 (x, t) ..... ,?~;~and the local velocity field v(x). Neglecting density variat i o ' r ~ we ca~l write .:, . . . . . . . ,.o~...,,, , ('X) } /" ,ig'{Y)Jf {x~;;, ::,: U'~ ,. (x + r ) . , ,.(0)
....
" r~Z', (r) + . . .&9 ...... vtZ'o (r) + cv~ . . . r , ~r'; (r) -4..... ,
(06)
wfwr¢ X~(r) a!~e unspecified functions of the dis*.ance r = tx .... y[. This expansion is ce,rt thfly r~ot complele bu! sufficient for out'purposes. Inserting (36) into (32)we .2 . . . .
dX~ ........ 2G,4~., .... 13, div ,,,3kf -., ( Tpl,O' at,, + T[g' Ovk "~ OX~ OXt / ". . . . .
7
(37)
~
where
l
~. h c ...........7 G
f
;)v ,_,
r 3 dr .......... ~ , ( r ) ,
2 hf
BI. = . . . . . . . . 3 2,,,,
ev_
r 3 d r - 2 - . 1'2(r) • (38) "
Comparir~g eq. (37) with phenomenotogica~ equations for the viscoelastic pressure ~enso:r~4), we carl ider/,:ify G as the shear modulus and B~ as a relaxatior~, bulk modu!us which is equal to .B~ = (~:/.q) G where ~ is the bulk viscosity and ~ the shear viscosity in a certain approximation~4). Thas the sinks for the kinetic wessure are due to the energy dissipation during the viscous deformation (change in vo!:ume)in the fluid. Again we find that ideal or incompressible fluids will satisfy a continuity equation for the pressure without any sinks. As in similar derivations of ~:heNavierStokes equation ~') the expansion (36) allows one to establish the structure of the balance equation (37) but makes no statement on the size or even the sign of the phenomenological coefficients. It should be obvious that our procedm-e opens up a new microscopic approach to visceelastic theory.
593
BALANCE EQUATIONS iN t:,{YDRODYNAM.[CS I
g
Let ,as return to o u r discussion of energy balance eqamtios~es. Defining the hamib. tonian energy density
k(X) .......(h?..m) ''
" "$):(x) + u(x) ?(x) .-~.,~ drY(x, .......~,),.;:x~.: =,..,:( .~
'~.... '.,?
w e ca..~ derive the balance equatiop~ for the Ioml energy (again assumii?g J>::.~ropy of the fluid)
~, <'~.;(x)'>
=
O,T ° (x) + c,c~:v ~ h('
(l
-.
'
., dy r :x, :,) ...... J ~
O?~ov, I Ofx.
Ira)
(~
:;.~7 < i~,(x) ,.(-,~',+. ........ ?. ,
. . . .
<:;;(:r~ ; (.oY ',J[," :.::c,.) :/ }o
:~ (-+0:
/
,b'
Similarly for the iater~a.[ er;< rgy density :.:(:,:)
.t:. ~=v u¢"x) == <.,/ZCt?> " " " ..... ' - ". , . --
(4i)
U(x-),::,. m ' V ~,,
we get 3_,Oz:
::=
..... d i v
(?uv)
l
......
div
,
,(.7(.v),.,] .......
/
k2
\
t'?l
! v
(,
/
:~;~ (vf ~'°' (x))
.... ,....
I
4...... ,/,(x) ~.' VU ,x: q...... div {~; j" d y g ( x , y) <2U(x)'j(y)Y) i17
h
tti
f d~,V,x...j,)f.-,~-q~:{x) ::0,:)> .-~, ,
}
,'(d;Ix)ik(y)),
& i'h
I
h j"
i~ V(x,y)
This compfetes our set of" !'lydrodynamicat balance equations. T o summarize the main points in our derivatioa, let us recall that for ail isotropic :fluid the merarch of tensor balance equatio as can be terminated wit h the s~-',. econd-rank"- kinetic el~ergy:~ r:~o) ..,tess . . . . tensor due to the fact that ~Ji = 0. F r o m the balance equation for the #. Lg: kinetic energy-stress tensor we were able to extract a balance equation f,:~r the kinetic pressure in a viscoela~,;tic m e d i u m . Tracing back the derivation of eqs° (3 l) arid (32) it can be seen that ~ ae anisotropJc m e d i u m the coupling to higher-rank tensors will remain and aploroximate truncation techniques have to be invoked. T h e detailed ~nterpretatior. o f fhe higher-rank tensor quantities, however, remains rather vague at this stage.
594
rL j. K REUZER
ACk~w~dgemen~:
lwouldlikeiolhankK~N
~ " ~ " s in this akamura for checking some of the calcu~atmm
wor'k~
Nefe~em:es ~) R oHaase~ Thermodynamics of Irreversible Processes (Addison~Wesley, New York, 1969). 2~ ,% R de Groot afld: P., M~xzur, Non4?,qt6~ibrh.tm Thermodynamics (North~Holland Publ, Co., 3) t+ Prig~'~gin,e~introduclion to Thermodynamics of h'reversible Processes (Springfield, 1955). 4) J, Mei×ner and H. G. P,eik, Handbuch der Physik II [/2 (Springer, Berlin, 1955), 5} LOr~sager, Phys, Eev, 37 (193:1) 405. f;~) H,O.G~Casifffir, Rev. rood, Phys, 17 (1945) 343, '7) H.SoGreem MoLvcular Theory of Fluids (Norlh-Holland Publ. Co., Amsterdam, 1952). g) H. F;r43Mich: Rivista det Nuovo Cimento 3 (1973) 490. 9) A.M. Galasiewicz, St~perconductiviW and Quantum Fluids (Pergamon, New York, 1970). iO;t H, J. g ret~zer aod K, Nakanmra, Physica 73 (t974.t 600. t 1) ll.. Jo Kreuzer and K. Nakamura, Physica 78 (1974) 131. 12) l-loF~6Mich, Physica 37 (t967) 2t5, :t3) Y oTakahashS, private communicalion~ i4) P,l:{gcB~aff, An In|roduclion tc, lhe L.iquid Stale (Acadet~Mc Press, New York. 1967),
80A (1975) 585o
ON THE MICROSCOPIC DERIVATION-"' OF BALANCE EQUATIONS IN IIYDRODYNAMICS H,J, KREUZER
7?}eoreNc
W,, derive., r~ew tensor hierarchy of h.~c.~I,-.a,,,:,),~.,,, ha{ante eqtm~io'~> s[ar[mg from die quaemm-mecbav, ica~ n~any-body problem. A nov. balm~ce eqt~a~ion for ti~e kinetic press-~ re in a fluid is der{v~d and the connection with v!~:oe)as'dc ~hcory i> discussed.
1, Introductioa Ti~e p h e n o m e n o ! o g i c a ! a p p r o a c h ~.o ~,o.~?equilibrim~ ~~crn~odyx-~au~cs o f pi°oces .., ses ( N E T P ) ttsualiy starts \;ith a sel o f ba!ance e q u a t i o n s for the ; e l e v a m thcrr~'<> d y n a m i c quantities ~-%. F o r m a i l y 'ae car~ v, ri~e i>r :,m ex~e~si\e variable 2: d -
-
-
dt
v(t) = PKi
+
O{Fi,
f*}
where P[.F] d e n o t e s the p r o d u c t i o n terms wi{hi~: ti~e v o l u m e V o f ~he ,,,:s,:em a~d ,/~[F] gives the flux o f F t h r o u g h the surfitce of V, i~tt-oduci~.~g F densities b,
r = -g-i"
dr,
where 9(x) is the mass density, we find tocai balance equation, s
~t
+ divj~, = ~r[.F],
(3)
whez'e j~ is tl~e F c u r r e n t a n d a [ F ] describes local siuks arid sources o f F. C h o o s i ~ g f = 1 we recover in eq. (3) the contimfity equafio)~ for the mass density O: Laking * Work supported in part by the Nationai Research Com~cil of Canada. 585
• •!
;i, ¸
~g6
~ H+L
KREUZER
ti~rfthe velocity tield ~ we get the momentum-balance equation, i.e., the equation of nlofion of the gvstem. With f =: ½.v~ we get the kinetic-energy balance, f := u gives the internal-energy balance, etc. It is obvious that such balance equations are only useful ff we specify the source terms a[Fj according to the problem nnder study, This leads in the momentum balance to the introduction of generalized thermodyitamic fk~rces. Forsystems close to equilibrium it is further assumed that the therme,dynamic fluxes are linear in the forces, the phenomenological coefficients satis:fyi~g Onsager's reciprocity relationsS,6). l ~ get a better understanding of such macroscopic balance equations various derival ioJ~s from the microscopic theory of a many-body system have been presented startiiJg in the classical case from Lie " ~uwl,~ " ~'~'s equation• and quantum-mechanically fron~ the Liouville-Von Neumann equation for the density matrixV-~'). Such microscopic derivations can also lead to new macroscopic femures as has been shown recently~), starting tiom a many-body system with nonlocai interactions we find addilionat source terms in the macroscopic balance equations that can be imerpreted as local fluctuations. Our program for this paper is to start with a. many-body hamittonian in secondquantized form, derive a systematic hierarchy of in~nitely many balance equations for microscopic density, current, energy, and higher-rank tensor operators. Taking slfitabie thermodynamic averages this hierarchy wil! be truncated to yield the wellknown macroscopic balance equations of hydrodyllamics allowing us a term by term comparison of microscopic and macroscop'c quantities. Moreover we will be able to derive a balance equation for the pressure tensor. The significance of higherral~k tensor operators will be discussed.
2o Operator balance equations We start with a hamih_onia~~ i~; second-quantized form 7" + V + U~
(4)
w it t -.(IV/4m) .! ox ~"
[~p'(x~ /17~ (X) +. Aoq7 (x) "~;(x)],
U = ~ dxU (x) "~:'{x.,.~,~.v~,
(5)
v .--- S dx' dx"V(x', x") ~fl(x') ~p(x') ~,~(x") ,<;(x"). Writing V in this i~rm we have absorbed single-particle contributions that result from proper normal ordering> into U~ The field operators ~ (x, t) are subject to
BALANCE EQL AJIONS IN HYDROD~'NAM[CS
587
Heisenberg's e q u a t i o n o f ~ o t i o n
il/a,v' = [,p, ~t].
((9
We assume for si,~plicity that g (x. t) satisfies Bose-Einsteh~ statistics. The non~ local o p e r a t o r S. . .(x', . x") = ,t,,(x ~ ") 7,(x'),
~7)
satisfies the e q u a t i o n ~h c,.,.9 ( x . x ' ) + (h2/,'n) d • '.;¢~2(x o x")
where (~ = ~' 4-. ?"•., ? ..... ~ (~' - g"~. with i?' and d" beirlg the >~.,~.~.~,~ . s with respect, to the variables x' a n d x', respectively. We also have Wz(x',x")
=
•
-
/
5 d v. [ V ( x . y )
t
,
'
e
_V(x . } ) l { . ( y , y ) ( ; ., x . x~ " }' , ~
-
,}
. (~}1
the cut"Iy brackets indicating s y m m e t r i z e d product% Le., {AB} .... .4B + BA. Starti~ag from eq. (8) we can derive a hierarchy o f o p e r a t o r equatior~s by expal?di~lg the nonlocal ope.rator o_~(.~x ,' x ' ) in terms o f the re!ative com~',.Jmate ,,~ := x" - - x ' a r o u n d the cen;.er-of-mass coordi~3ate x = ~-(x' + x"), i.e.. write ,; (x'. x ' ) = ~ (l/n!)
{ ~ • &',;*e) ~'¢9 ( . r ' , ) * . c - . , . ' ,
i:*~ the
"local"
}irnil
x'
= x
= x
we: t:i,~d
the COl~tinuil_y eq~m',ion tbr the o p e r a t o r ~;(x) = [; (at. x} ti v,L~ (x) + ~,h°,,m) 8 .](x) ==: 0 ,
(t0)
where j(x), . . . . i lime_, o. ":e/2 (x', x ' ) is the current operator. C"o!!ecth~g *,.eun>~" " proportionat to # we get lhe e q u a t i o n o f m o t i o n # vddx) . . . . .
- (h~/rm c, 7~a (x) [ G U ( x ) ]. , ; (.x ) . + ..[dv.
[GV(x. .
r)] t.d3' ~' ) ,.,(x)~.
(t 1)
where the kinetic-energy tensor o p e r a t o r is give~ by 7P~,(x)= lime_, o (&'i:G) (a/r?~) ~ (x', x") a n d ~.~ = ~?/&x'~f r o m n o w on. Terms ~ i t h ~': yield ~ ~?,~,
(x)
+
(If/m)
•
,~;/:~, ~ " (x) = [ c" ~ / (x)]j,(x) ~ " + [?, U (x)]j~(x)
- j" dy [G v(x, y)i {&v)y,(x)} -
j~ d y t~.', " J~ (a., "•
39] [,j(y)A(x)},
(l 2)
588
H.J. KREUZER
where ~'~al(x) = --i lira4., o (~/~ff~)(d/O~a) (f~/(?~) 3 (x', x") will in turn couple in its balance equation to yet higher tensor operators namety
- [/~,eke,U(x)] 6(x) + j d3y [0,V(x~ y)l {.dO,)T~,(x)}
+ S daY [OkV(xy)] {~0') T,t(x)} + ~ d3y [~.,V(x,y)l {~(.!') L~(x)}.
(13)
It is obvious that this hierarchy of operator balance equations continues in principle ad infinitum. As we go to higher-rank tensor operators, more microscopic information is exposed through the higher-order derivatives, the complete hierarchy being equivalent to the equation of motion, eq. (8), for the nontocal operator m
'2 (x', x").
3. Macroscopic balance equations In order to define macroscopic quantities like mass density, mass current, energies, etc., we have to be able to define statistical averages oi" the corresponding operators defined in the previous section. To this end we have to know the time evolution of the statistical operator (density matrix) ~ which is only available for simple models (see e.g. reL 11). On the other hand, proceeding phenomenoIogically we can assume the existence of local equilibrium in the system under study and make an Ansatz for the formally defined stat:istical averages. It is this second approach that has been tb!towed in the past and that will also be used in this paper. Followil~g Fr~Shlich~:~) we define ~ {x. ~'" ~:, = ~r (x, ~) e ~;~':°~''b
(/4)
from which, we gel; the macroscopic mass density' ~(x) = m,,x(x, O) and ti~e macroscopic velocity v~:(x) = .h. . . .lira ..
z (ix,
(i5)
It should be remarked that, according to Galasiewiczg), Bogoliubov int:roduced macroscopic velocities at the operator level through a translatio~ e t"
It6)
BALANCE
EQUATIONS
tN HYDRODYNAMICS
589
We can readily generalize Bogo[iubov's m e t h o d by transfi)rming fl'om a fluid at rest to a fluid with a n o n c o n s t a n t macroscopic veh::city field ~:~). Thi.~ i.,~accomplished by defining a unitary transl\ormation U(x)
+/,(x)-+ ,?(x) ~r(x), by ih
- -~?8x
W(x)
=
my (x) W(x}.. .
(i 8)
with the i'\+rma[ solutioa
++.'~x)=
exr
f/ ~d+#/O ~x rCx'), d x i \
'~-'e
~t9)
J
and W(xo) = I. With a similar trap.s(ormatJor~ o~~.e can aiso h~troduce an angular velocily fie!d ~o describe vordces and reIated p h e n o m e n a in turbulence. Usiag eqs. (17)-(I9) we can transform the tensor opera~ors introduced in ti~,e !ast section ',o a m o v i n g fluid. We ,. £et [a ~up~+~.,~pt ~ ' "~,",' (0) refers to a fluid at rest] .otx) . . . =. . V. ?. ( x ) ' ' + '
j(x)
,=
"
'"
' == L'~'" : >I.X~"
j'°'(x) + (re:l,) (kx} vCv),
L,(x) = .}~~?>~x)- (,,+./~),+,~;(x)<,c,-),.:~c,O-- i,,,/,~~,.:,,.i?"+ n.f;%.
F.~,( x'~
~:o)
f.;,~ b~:) + im;iz)
4,~O~(x) +
(m ~,') -,o,~.
and. similar reductions "or highe>ral~k tensors. T o define mac~oscopic quamifie.~ we have to take stafisticaI averages o f the !oca! operators i~ eqs. {.20). This a',eraging procedure ~ecessitates the a s s u m p t i o n of ioca] equilibrium in fi~e fluid, i.e., a r o e n d each point (x~ t) i,~ space and tim.e we assm,a~:: a macroscopic but sti[~ " s m a l l " volume and dine interv~:q over wMci~ averaging is do~'~e wifi~ a n equiiibri um de~ssity matrix, Also to set mac~'oscopic qua.ntilies i'ro:m lhe <4'-.... -~" ~,~o,~. ..... eqs. (20) ,;;e have to multiply an ,xh-rauk te~?sor by h~L because according to dae correspondence 1;-~nciple it is (ili ~ ~:¢:) !:hat correspot~ds Io a macroscopic (classicai)quanti%, (momentum)~ We thus get the macroscopic mas~; deasity ~'x) :.: m -(~(x)),
(2t)
the macroscopic c~rrent j ( x } = h ( j ( x £ > .= !.?(x} v{x)+
(22)
i
)
590
H, J. K R E U Z E K
the macroscopic kinetic energy-stre:~s tensor T,:,(x)
~
-(/?/2m) ( ~ , ( x ) ) = ~,,.Tm)~x).4. ~}~'(x)t:d.'~
(23)
and a macroscopic energy-flux tensor
laity(X)
....
(l}a/2rt't 2) <&~t(X))
= F~(x)
+ v , T ~ ' ( x ) + vkT[iO" (X) + v d]q2~' (x) + -}~4x)r~r~t,~°
(24)
With these expressions we get f r e m eq, (t0) dae macroscopic continuity equation
Go(x) + aev = o.
~25)
Eq, (11) yields the macroscopic equation ol motion or m o m e n t u m balance equation
6 (t)vk) + 26 [T~°'(x) + ~-,,?
(26)
involving the density-density or ~wo-body correlaticm functim~, Usir~g ihe macroscopic equation D { 1 d ~? ~ c~(x) =- • %, ................~,(x) ....... t:!s)° Dt &v~ m &v~
(2'7'~
we can define the stress tensor %~ for ~hict: ~ e find from eq. ~27} by d~e standard expansion of the density-density corre!atio~.,v. ~~) -
F(,") rF] .['~(r)d3~........ c~'~::. F
~..0:;:>}
F
,~r~ ' contains visco~s teems, [ ] ( r ), is t:'~c radiai pair corr:~a~mt~ ' ~' " hmction. We i\~llow here the notation o f ref. 12 wI~ere also a derivation of the Nax-ier-Stokes equations Dora eq. (aTe, "~ ' c a l l be fo.u~d. {n a,~ i~c~,,~r, ie~ ..... fl~lid ~ ,~ = .
i ~.~-
i"r ?V(r) *
O ~ "
¢7Y
This justifies calling T~,t(x) the kinetic part c,f the ene~:g>~::~:ress revisor, To get a balance equation fo': the e~ergy o f the system we {:u~veessemia}~y three differem approaches, We can stall fiom tl~e eq!mtio~ o f m o d o ~ t26} o r f r o m O~,e
BALANCE EQUATIONS iN HYI)RODYNAMI'CS
591
h a m i t t o ~ i a a deasity or *.ko:n eq. (~2) for ~h¢ ki~e~ic
....... ( t':~} ev~ ~:, U .... v~ 5 J3' {7 {:](r} {;{y>~ 71>~-::~~" (x~ y},
(3())
4 i}ee'~e~,> -~ e,,~ e'~7" -~. e', ( 721'" - ~~ (.!::,e- c,,~'~}
:< {*'~{x} [:~I'{:r, ,~ } + e::ix} ,~',,{,{Xo Y}i-
{30a)
use of" eqs. {23)= We ge~
.... v~e 2H,~ i' '.. 4:,
Le~ as {:if:'s~ mves~iga~.e a~ :idea} gas~ ~!-Ee~
where P is ~:~e (scalar! !~ydros~.~.t;c press~.~'~e. T~.t~s ~aki~g ~hc ~race m : % {32} we '~}gd a c,o,!lir~u,ity equat}o~ ~:or &."~a:me!y
C~ t,* + 0~ {ic~ *~) ::,> '~},
{ 34)
592
H.L KREUZER
t h e off-dlagoaa lerms in e% (32) yield a relation (35)
¢oniirming that in an ideal fluid the rate-of-strain tensor Akt vanishes. In a real tl~fid~ h-owever, z~herate-of-strain tensor will cow,tribute to the pressure balance, so will ~ocaliy generated pressure fl,)ws through the term Gt;)~{:'(x). Moreover, the imeractions wilk through correlatior,.s, introduce local sinks and sources for the kir~el.h; part of the pressure. To understand the nature of these sinks and sources let us expat, d the de~sity.-.current correlations for small deviations from equilibrium, !Be expa~sion parameters being the deviations from the equilibrium density i}~,(x. t) :,.-:9 (x, t) ..... ,?~;~and the local velocity field v(x). Neglecting density variat i o ' r ~ we ca~l write .:, . . . . . . . ,.o~...,,, , ('X) } /" ,ig'{Y)Jf {x~;;, ::,: U'~ ,. (x + r ) . , ,.(0)
....
" r~Z', (r) + . . .&9 ...... vtZ'o (r) + cv~ . . . r , ~r'; (r) -4..... ,
(06)
wfwr¢ X~(r) a!~e unspecified functions of the dis*.ance r = tx .... y[. This expansion is ce,rt thfly r~ot complele bu! sufficient for out'purposes. Inserting (36) into (32)we .2 . . . .
dX~ ........ 2G,4~., .... 13, div ,,,3kf -., ( Tpl,O' at,, + T[g' Ovk "~ OX~ OXt / ". . . . .
7
(37)
~
where
l
~. h c ...........7 G
f
;)v ,_,
r 3 dr .......... ~ , ( r ) ,
2 hf
BI. = . . . . . . . . 3 2,,,,
ev_
r 3 d r - 2 - . 1'2(r) • (38) "
Comparir~g eq. (37) with phenomenotogica~ equations for the viscoelastic pressure ~enso:r~4), we carl ider/,:ify G as the shear modulus and B~ as a relaxatior~, bulk modu!us which is equal to .B~ = (~:/.q) G where ~ is the bulk viscosity and ~ the shear viscosity in a certain approximation~4). Thas the sinks for the kinetic wessure are due to the energy dissipation during the viscous deformation (change in vo!:ume)in the fluid. Again we find that ideal or incompressible fluids will satisfy a continuity equation for the pressure without any sinks. As in similar derivations of ~:heNavierStokes equation ~') the expansion (36) allows one to establish the structure of the balance equation (37) but makes no statement on the size or even the sign of the phenomenological coefficients. It should be obvious that our procedm-e opens up a new microscopic approach to visceelastic theory.
593
BALANCE EQUATIONS iN t:,{YDRODYNAM.[CS I
g
Let ,as return to o u r discussion of energy balance eqamtios~es. Defining the hamib. tonian energy density
k(X) .......(h?..m) ''
" "$):(x) + u(x) ?(x) .-~.,~ drY(x, .......~,),.;:x~.: =,..,:( .~
'~.... '.,?
w e ca..~ derive the balance equatiop~ for the Ioml energy (again assumii?g J>::.~ropy of the fluid)
~, <'~.;(x)'>
=
O,T ° (x) + c,c~:v ~ h('
(l
-.
'
., dy r :x, :,) ...... J ~
O?~ov, I Ofx.
Ira)
(~
:;.~7 < i~,(x) ,.(-,~',+. ........ ?. ,
. . . .
<:;;(:r~ ; (.oY ',J[," :.::c,.) :/ }o
:~ (-+0:
/
,b'
Similarly for the iater~a.[ er;< rgy density :.:(:,:)
.t:. ~=v u¢"x) == <.,/ZCt?> " " " ..... ' - ". , . --
(4i)
U(x-),::,. m ' V ~,,
we get 3_,Oz:
::=
..... d i v
(?uv)
l
......
div
,
,(.7(.v),.,] .......
/
k2
\
t'?l
! v
(,
/
:~;~ (vf ~'°' (x))
.... ,....
I
4...... ,/,(x) ~.' VU ,x: q...... div {~; j" d y g ( x , y) <2U(x)'j(y)Y) i17
h
tti
f d~,V,x...j,)f.-,~-q~:{x) ::0,:)> .-~, ,
}
,'(d;Ix)ik(y)),
& i'h
I
h j"
i~ V(x,y)
This compfetes our set of" !'lydrodynamicat balance equations. T o summarize the main points in our derivatioa, let us recall that for ail isotropic :fluid the merarch of tensor balance equatio as can be terminated wit h the s~-',. econd-rank"- kinetic el~ergy:~ r:~o) ..,tess . . . . tensor due to the fact that ~Ji = 0. F r o m the balance equation for the #. Lg: kinetic energy-stress tensor we were able to extract a balance equation f,:~r the kinetic pressure in a viscoela~,;tic m e d i u m . Tracing back the derivation of eqs° (3 l) arid (32) it can be seen that ~ ae anisotropJc m e d i u m the coupling to higher-rank tensors will remain and aploroximate truncation techniques have to be invoked. T h e detailed ~nterpretatior. o f fhe higher-rank tensor quantities, however, remains rather vague at this stage.
594
rL j. K REUZER
ACk~w~dgemen~:
lwouldlikeiolhankK~N
~ " ~ " s in this akamura for checking some of the calcu~atmm
wor'k~
Nefe~em:es ~) R oHaase~ Thermodynamics of Irreversible Processes (Addison~Wesley, New York, 1969). 2~ ,% R de Groot afld: P., M~xzur, Non4?,qt6~ibrh.tm Thermodynamics (North~Holland Publ, Co., 3) t+ Prig~'~gin,e~introduclion to Thermodynamics of h'reversible Processes (Springfield, 1955). 4) J, Mei×ner and H. G. P,eik, Handbuch der Physik II [/2 (Springer, Berlin, 1955), 5} LOr~sager, Phys, Eev, 37 (193:1) 405. f;~) H,O.G~Casifffir, Rev. rood, Phys, 17 (1945) 343, '7) H.SoGreem MoLvcular Theory of Fluids (Norlh-Holland Publ. Co., Amsterdam, 1952). g) H. F;r43Mich: Rivista det Nuovo Cimento 3 (1973) 490. 9) A.M. Galasiewicz, St~perconductiviW and Quantum Fluids (Pergamon, New York, 1970). iO;t H, J. g ret~zer aod K, Nakanmra, Physica 73 (t974.t 600. t 1) ll.. Jo Kreuzer and K. Nakamura, Physica 78 (1974) 131. 12) l-loF~6Mich, Physica 37 (t967) 2t5, :t3) Y oTakahashS, private communicalion~ i4) P,l:{gcB~aff, An In|roduclion tc, lhe L.iquid Stale (Acadet~Mc Press, New York. 1967),