Microscopic derivation of hydrodynamic balance equations in the presence of an electromagnetic field

Physica 83A (1976) 573-583 © North-Holland Publishing Co.

MICROSCOPIC DERIVATION OF HYDRODYNAMIC BALANCE EQUATIONS IN THE PRESENCE OF AN ELECTROMAGNETIC FIELD* H.J. KREUZER and C.S. ZASADA Theoretical Physics Institute and Department of Physics, University of Alberta, Edmonton, Alberta, Canada Received 27 May 1975 Revised 2 December 1975

Using field-theoretic methods we derive balance equations for a charged fluid in an external electromagnetic field the effects of which are included by minimal coupling. An infinite hierarchy of balance equations for tensor operators is derived. A macroscopic velocity field is introduced by a unitary transformation on the field operators. Suitable statistical averages in the local equilibrium approximation yield macroscop!c balance equations. The significance of new terms is discussed.

1. Introduction I n a recent p a p e r 1) one of the a u t h o r s (H.J.K.) has p r o p o s e d a new m e t h o d for the derivation of h y d r o d y n a m i c balance equations from a microscopic m a n y - b o d y theory. We start with a h a m i l t o n i a n in second quantized form H=

T+

V+

U

(1)

with h2 4m

f dx [~f(x) d~v (x) + / 1 ~ t (x) ~p(x)],

U = # d x U ( x ) ~0t(x) ~o(x),

(2)

v = S dx' dx" Z (x', x") W*(x') W(x') ¢(x") W(x"). * Work supported in part by the National Research Council of Canada. 573

574

H.J. K R E U Z E R A N D C.S. ZASADA

Writing V in this form we have absorbed single-particle contributions that result from proper normal ordering, into U. The field operators ~0(x, t) being subject to Heisenberg's equation of motion ih Ot~o = [~p, H ] .

(3)

We assume for simplicity that ~p(x, t) satisfies Bose-Einstein statistics. Next we construct a nonlocal density operator

(4)

(x', x") = ,e*(x") v,(x') which satisfies the equation ih O,~" (x ' , x") + -hZ - O . O~Q (x', x") m

= [te(x') - U(x")]

~ (x', x") + w~ (x', x"),

(5)

where 0 = O' + 0", 0¢ = ½ (0" - 0") with O' and 0" being the gradients with respect to the variables x' and x", respectively. We also have W2 (x', x") = - S dy [ V ( x ' , y) - V ( x " , y)] {~ (y, y) ~ (x', x")}

(6)

the curly brackets j indicating symmetrized products, i.e. {AB} = A B + BA. Starting from eq. (5) we can derive a hierarchy of operator equations by expanding the nonlocal operator ~ (x', x") in terms of the relative coordinate ¢ -- x' - x " around the center-of-mass coordinate x = ½ (x' + x"), i.e. write

~(x',x")=

n!

¢"

~(x',x")l~,:x,,.

(7)

The local limit x' = x" = x yields the continuity equation for the local density operator ~(x) = lim¢-,o ~ (x', x"), terms proportional to ¢ give the equation of motion, i.e. the balance equation for the current operator j ( x ) = - i lim¢oo O~d x (x', x"). The next order yields a balance equation for the kinetic energy tensor f~,(x) = lira !

~ ~ (x', x").

To proceed from these operator balance equations to macroscopic equations a unitary operator transformation is performed on the field operators to introduce a velocity field in the fluid. After that suitable statistical averages are taken, we then arrive at an (infinite) hierarchy of hydrodynamical balance equations. For an isotropic fluid the hierarchy terminates with the energy balance, whereas in

DERIVATION OF HYDRODYNAMIC BALANCE EQUATIONS

575

the anisotropic case the energy balance iS coupled to higher tensor quantities like energy flows etc. As an example for the new physics to be learnt in this approach, we derived in ref. I a balance equation for the kinetic pressure tensor in a viscoelastic medium. It is the intention of this work to include electromagnetic effects in the above theoryZ). The physical situation we have in mind is a mixture of fluids in an external electromagnetic field. The fluid molecules might carry electric charges, and electric and magnetic dipoles, but are assumed to behave nonrelativistically. This leads to a number of modifications in the hamiltonian, eqs. (1) and (2). The generalization to a mixture of k components is done by introducing field operators for each one; this amounts to treating the field operators in eqs. (1) and (2) as k-dimensional vectors. In such a notation all products of field operators have to be understood as tensor products. In the following, however, we will use the same notation as for a one-component system. The effect of the external electromagnetic field on the charges will be included by minimal coupling, i.e. by replacing canonical momenta r - , t, -

q

A(x)

¢

(8)

in the kinetic energy and adding a term

}/el -----q S qgo(X) ~t(x) ~(X) d3x

(9)

to the hamiltonian, where A(x) and ~bo(x) are the vector and scalar potential of the external electromagnetic field and q is the molecular charge. The interaction between the dipoles and the external field adds terms Hdip = --~ dax [,u. B(x) + ~. E(x)] ~t(x) (v(x),

(io)

where/~ and 0~are the molecular magnetic and electric dipole moments. Following de G r o o t and Suttorp 2) a multipole expansion of the classical hamiltonian of a composite particle in a uniform electromagnetic field up to order I/c yields the additional term Hdip

--

¢

~X ×

"B

(11)

arising from the motion of the electric dipole in a magnetic field. It should be remarked that a Foldy-Wouthuysen transformation on a relativistically invariant interaction term a~v Fv, will lead straightforwardly to the same expressions. Such a procedure, of course, has to assume the existence of intrinsic dipole moments and does not bring out their physical origin as nicely as the G r o o t - S u t t o r p expan-

576

H . J . K R E U Z E R A N D C.S. Z A S A D A

sion. In toto, our hamiltonian is then given by (12)

H = T' + U' + V' + Hair, + Hajo,

where T' is the kinetic energy modified by minimal coupling, Iv" is the sum total of all two-body interactions, and U' = U + q 5 qSo(X)~°t(x) ~o(x) d3x.

(13)

We would like to point out that the effect of Hjjp can be incorporated into all subsequent calculations by an extension of the minimal coupling scheme, i.e. by doing the replacement p-+p

- - -q- A + e

- 1- o~ × (V x A ) c

(14)

in the original hamiltonian. This trick simplifies our algebra considerably. In the next section we will derive the hierarchy of operator balance equations in the presence of an electromagnetic field. In section 3, we will introduce a velocity field by a unitary transformation on the field operators and take statistical averages to obtain macroscopic equations, which we will discuss in some detail.

2. O p e r a t o r b a l a n c e e q u a t i o n s

We proceed now with the derivation of the tensor hierarchy of operator balance equations for a one-component fluid of charged particles. The necessary modifications due to other components in a fluid mixture and due to electric and magnetic dipoles on the particles will be indicated at the end of the calculations. Including the effects of the external electromagnetic field on the charges in the fluid by minimal coupling, i.e. via eq. (8), we derive the equation of motion for the nonlocal density operator ~ (x', x"), eq. (4), from eq. (3) to be

~t

= (~'(x') - Cl'(x")) ~ (x', x")

q2

+ ----

( A ~ ( x ') -

m

~

~ ~ (x', x")

- -0~ ~ (x', x") + ~ (x', x")

W~ (x', x")

~x~ ~

iqh I 2 ( A , ( x ' )

A 2 ( x " ) ) ~ (x', x") + - -

2me z

c~x~

h2

- Ai(x"))

2me

(n,(x') - A,(x")) + (A,(x') + n,(x"))

~ (x', x") + ½6 (x', x") ~3 (A,(x') + A,(x"))]. l 6xi

(15)

D E R I V A T I O N OF H Y D R O D Y N A M I C BALANCE EQUATIONS

577

U'(x) has been defined in eq. (13) and W~ (x', x") is given by an expression like eq. (6) but including electromagnetic two-body interactions. To generate the tensor hierarchy of balance equations use the expansion, eq. (7). The local limit, i.e. terms independent of ~, yield " ~(x) + O. (hemj (x)) = O,

(16)

Ot where the local density operator is defined by ~(x) = iim ~ (x', x"),

(17)

4--,0

and the current operator in the presence of an electromagnetic field is given by emj(x) = j ( x ) -

q A ( x ) ~(x), he

08)

with ] ( x ) = - i lim mQ ~ (x', x").

~ o ~:

(19)

Obviously eq. (18) can be gotten by using minimal coupling in eq. (19), i.e. by replacing say -ih

- -

- -

(3

q

c~x'

(3 --+ - i h

t~x'

c

A(x').

(20)

In the absence of a field we have previously 1) defined a kinetic energy tensor Tk,(x) = lim c~

0 ~ (x', x").

(21)

With the prescription, eq. (20), we can define a similar tensor in the presence of a field, namely, q

°mi"k,(x) = ~,(x) + ~ c [A,(x)°mA(x) + a~(x)j,(x)]

(22) which is still a symmetric tensor.

578

H.J. KREUZER AND C.S. ZASADA

Collecting terms linear in ~ in the equation of motion for ~ (x', we get a balance equation for the momentum operator

~(hemjk(X))

8t

x"),

eq. (15),

G ( ~ - emTik(X))

~x~

= - ( ~-~xkU(x)) ~(x) + f d3y (-~xk V (x, Y)) {~(Y) ~(x)} + qE (x) ~(x) + q(~j(x)×c

B(x)) .k

(23)

Thus minimal coupling leads naturally to the appearance of the Lorentz force in the equation of motions besides altering the definitions of the dynamical tensors. To get a balance equation for the kinetic energy tensor we collect terms proportional to ~2 in eq. (15), and find

~ (~--2 emFikl(X)) ~,(? - (---~ emTki(X)) -{- ~Xi

= (--q Ek(X) hemfl (X) \ m 1 m

"~ l m

~U(x)

(hem..]/(x))

Oxk

d3y - -8 V (x, y) {~(y) h*mj, (x)) cqxk

+ qh2 Fti(x) em~ik(X))

?~12C

,

/k~*l

(24) where the right-hand side has to be symmetrized with respect to the indices k and l and where

r,i(x) = ~Ai 8xl

~At 8x~

We defined

q [Atem~'ig + emFikt(X) = FikI(X) -- -~C

AkemTit + hiTk,

q AkAtemj, + 1 Q2A~ ~(x)] he 4 8xk 8xt _J,

(25)

and the energy flow tensor in the absence of a field is as usual given by ~) 8 8 ffikl(X) = --i lim - ~-~o a~L 8 ~

8 ~ (x', x'). 8~i

(26)

DERIVATION OF HYDRODYNAMIC BALANCE EQUATIONS

579

It should be noted that introducing minimal coupling into eq. (26) obviously makes the order of differentiation important in the definition of en'Pikt(X). Our definition, eq. (25), arises naturally in the derivation of the energy balance, eq. (24). The fact that this energy flow tensor is no longer symmetric in the presence of an external vector field can be understood as a consequence of the anisotropy o f space in such a situation. A more detailed discussion of eq. (24) will be given after we have taken macroscopic averages. Obviously the energy flow tensor will in its balance equation couple to a fourth rank tensor etc., the hierarchy of tensor equations continuing 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. (15), for the nonlocal density operator ~ (x', x").

3. Macroscopic balance equations We will next introduce space- and time-dependent macroscopic quantities like mass density, mass current, energies, etc. by defining formally statistical averages of the corresponding operators defined in the previous section. Assuming local equilibrium in the system under study we can introduce a macroscopic velocity field v (x, t) by a unitary transformation U(x) on the operators, ~ (x, t), which transforms them from a fluid at rest to a fluid in motion in an external electromagnetic field, i.e.

(o(x) ~ ~(x) O(x),

(27)

where U(x) satisfies the equation

-ih

O c~x

q A(x)'] (](x) = my (x) O(x), / c

(28)

with the initial conditions O(x0) = 1 at some arbitrary point Xo. In the absence of a field this transformation reduces to the one used previously~). Our task now is to transform all tensor operators, defined previously, to a moving fluid in the presence of an electromagnetic field. The local density operator ~(x), of course, does not change; we will therefore outline the calculation with the current operator. Starting from equations (18) and (19), we find for the current operator in a moving frame using eq. (28)

jk(x) = - i lim

c~

Ot(~") ¢(x") ~(x') V(x')

= i(*°)(x) + ( ~ ) 6(x) vk(x) + q Ak(x)

(29)

580

H.J. KREUZER AND C.S. ZASADA

where the superscript zero on j~°)(x) refers to the rest frame with zero velocity and zero field. Thus the full current operator is given by

¢mjk(x) = j~°) (X) + ( h ) ~(X) Vk(X) ,

(30)

where v(x) is the macroscopic velocity in the presence of a field, in a similar fashion, we find for the kinetic energy operator of a moving fluid in an external electromagnetic field

em~'kl(X) = Z.~°)(x)-- ( ~ )

[l)k(X)ff(lO)(x) +Vl(X) j(kO'(x)] -- ( ~ )

Vg(X) t)l(X)~(X). (31)

Again the structure of this tensor is unchanged from the field-free Situation, if the effects of the electromagnetic field are absorbed in the definition of the velocity field v(x). However, for the third rank tensor we find

*mlW/k/(X) = ff}:)l(X) ~- (~)

,~(0) 4,(0) 1 [/)k~'[ O) -4- Vill~ ! JV I)IIU¢ 1

,(0) ] [V,Vkj~°) + vivtj(k°) + VkVlJ~

--

+ --4

C Vl "g~O

~

+

--

9ViVkVt

.(0) [At,A*J~°) + AiAkJ~ °) + A*AtJk ]. (32)

Next we define formally macroscopic tensor quantities by suitable statistical averages in the local equilibrium approximation. We get for the macroscopic mass density o(x) = in (~(x)>,

(33)

for the macroscopic mass current em/(x) = h (em/(x)) = V(X)e(X),

(34)

for the macroscopic kinetic energy tensor

¢"~r,,,(x)

= - -h2 - (*mT'k,(x)) = T~°)(x) + ½~ (x) vdx ) v,(x)

2m

(35)

and finally

emFikl(X ) : - - -

h3

2m 2


= F(t°l'(x) + [viT~°) + VkT[°) + v~T[°'] + ½Ov,vkv,.

(36)

DERIVATION

OF HYDRODYNAMIC

BALANCE

EQUATIONS

581

It should be noted that at the macroscopic level most terms in the operators, eqs. (30) to (32), vanish due to the zero ensemble average of the current operator A"tO)ix) t in zero field in the rest frame. Moreover, in the expression for emffm(X), eq. (32), we have a term involving the second derivative of the velocity field v(x) which depends obviously on the order in which the derivatives are taken in the original definition of ff~k~, eq. (26) as discussed at that stage. The point is that on the macroscopic level this term is small of order h z due to the fact that in all our tensor operators (h 8/8x) is a macroscopic operator. Thus ~ikl(X) involving three derivatives has to be multiplied by h 3. This remarkable "coincidence" makes the nonsymmetric part of ff~kt a second order quantum correction. Using eqs. (33) and (34) in eq. (16) we recover the macroscopic continuity equation for mass conservation i.e. -

-

8t

o(x) + div (O(x) v(x)) = 0.

(37)

Eq. (23). yields after averaging the macroscopic equation of motion

~t

(ovO + 2 a (r[O)(x) + ½o (x) v,vO ax~ = _l¢(x) m

8 U(x) + 8xk

A

d3y 8xk

A

)

v ' (x, y ) {Co(y),o(x))~

l(q~ ) + (x) E(x) + q-- e(x) v(x) × a(x) m c k"

(38)

The divergence part on the left involves the convective momentum flow ~v~vk and the kinetic part of the pressure 2Tt(°) (x). The dynamic or correlation part of the pressure can be extracted phenomenologically from the integral on the right involving the two-body forces (including electromagnetic interactions as indicated by the prime) and the density correlations1,3). Changes in momentum can also be caused by external forces of non-electromagnetic origin, due to U(x) and by the Lorentz force (last term on the right). If the molecules carry intrinsic electric and magnetic dipole moments, additional contributions to the right side of eq. (15) arise from the commutators of Haip and H,~p with Q (x', x"). Hence, additional external forces arise and eq. (38) must be supplemented on the right by 8Ei

s, =--

CXk

+/~,

8B~ OXk

1 8

+ -- --

C ~t

1

8

( s × B)~ + - - - -

c Ox~

[v: [~ × B i d ,

where c~, = S dax7't (x) &Ao(x).

(39)

582

H.J. KREUZER AND C.S. ZASADA

Proceeding in our hierarchy of balance equations we get from eq. (24)

Ot

(T 2°) (x) + ½ovkv,) + ~ (F~°](x) + lv,vkv,9 + VkT[°) + v,T(i°) + v J 2°)) " Ox~ ~U

1

c~xk

~

3

(~

+ -- j V(x, y) 2m d y ~xk

+ ½qgv~ (Ek(x) + ½ (v x B)k) +

h), (x)>}

q F.T~ °) (x)) mc k~l

(40)

where the right-hand side has to be symmetrized with respect to the tensor indices k and l. On the other hand, we can multiply the balance equation for jk, eq. (38), by vt and get after symmetrization a balance equation for the kinetic energy T(O) + vk ~? T¢°) + 0 (-}ViVkVlg) Q (-IzOVRVt)+ V, g ~,k C~t OXi CXi OXi 1 v~q ~ ~ --~m

=

U(x) + ½vt f day ~ 0

+ iqqvt (Ek(x) + __1 (v × B)k~ C

V ( x , y ) <{~(y)~(x)}>

.

(41)

!/ k ~ l

Here the last two terms represent the rate at which the electromagnetic field does mechanical work on the charged particles. Subtracting eq. (41) from eq. (40) we are left with a balance equation for the kinetic part of the pressure tensor (o)

at

(x) +

1

6~

(o) + v,r , (x))

d3y ~?xk

+ q F,T(~°'~ me

~xi

(42)

Jk~! ~

where again the right-hand side has to be symmetrized with respect to the tensor indices k and l. It has been shown previously 1) that the first term on the right-hand side of eq. (42) represents local sinks and sources in the kinetic pressure arising essentially from viscous deformations in the fluid. To interpret the last term, observe that in an isotropic fluid T~°) is a diagonal tensor, and recall that F , is antisymmetric. Thus the last term will be zero in such a situation. In an anisotropic

DERIVATION OF HYDRODYNAMIC BALANCE EQUATIONS

583

fluid, however, it will be the electromagnetic contribution to the local rate of strain tensor. It should thus be a corrective term in a visco-elastic theory of charged fluids. In addition to eqs. (40)-(42) we can derive a balance equation for the total energy starting from the hamiltonian energy density, and by proper subtraction we get an internal energy balance. These equations do, however, not exhibit any new features over previous results2). To summarize we have derived an infinite tensor hierarchy of operator balance equations for a charged fluid in an external electromagnetic field. Using minimal coupling we recover all standard results in a straightforward manner. In addition, having several independent approaches to balance equations for various energies in the fluid we derive a balance equation for the kinetic part of the pressure tensor for which the local sinks and sources are modified by electromagnetic contributions to the rate of strain tensor for visco-elastic deformations.

References

1) H.J.Kreuzer, Physica 80A (1975) 585. 2) S.R.de Groot and L.G.Suttorp, Foundations of Electrodynamics (North-Holland Publ. Comp., Amsterdam, 1972). This monograph gives a rather complete survey of the original literature on the subject. 3) H.Fr6hlich, Riv. del Nuovo Cim. 3 (1973) 490.