J. Phys. Chem. 1989, 93, 7019-7022
-
the consequences of the small s divergence in A(s) In (s) using mode coupling theory. On the basis of the same theory one would expect anomalous sound damping in one-dimensional lattice gases, where k2q(s) k2s-’/3with s = ic& (co = sound speed) effectively would be replaced by k5I3 at long wavelength according to the mode coupling analysis of ref 11. Indications of anomalous sound damping in one dimension have been seen in recent computer simulations.’* The equations for the hydrodynamic modes
-
(18) Qian, Y. H.; d’HumiSres, D.; Lallemand, P., preprint, 1988. Qian Y. H., private communication, 1988.
7019
therefore are expected to change drastically in one- and two-dimensional models. From the theoretical point of view, it would therefore be of great interest to have data from computer simulations on the small-k and krge-t behavior of the hydrodynamic correlation functions GaO(k,t). Acknowledgment. The research of J.W.D. was supported by the Pieter Langerhuizen Lambertuszoon Fonds, and the hospitality of the Instituut voor Theoretische Fysica is gratefully acknowledged. Also, M.H.E. is grateful to the organizers for their invitation to participate in the CECAM workshop (Orsay, August 1988), where some preliminary results were obtained.
Hard Sphere Langevin Equations Mordechai Bixon,t J. R. Dorfman,t and James W. Dufty*vl Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742 (Received: April 7, 1989; In Final Form: June 9, 1989)
The Zwanzig projection operator formalism is used to obtain Langevin equations for fluctuations of the microscopic single particle phase space density about its equilibrium value. For hard sphere dynamics the autocorrelation function of the fluctuating force has a singular contribution. The origin of this singular part is identified from an identity relating hard sphere binary collision operators for positive and negative time evolution. Langevin equations with white noise are obtained in two limits. The first is low density, leading to the Boltzmann-Langevin equation; the second is short times, leading to the Enskog-Langevin equation.
I. Introduction The description of fluctuations and transport in simple fluids has been approached from several different points of view. One of the earliest and physically most transparent methods is that of Langevin equations, which provide a unified description of both deterministic dynamics and the fluctuations about the average. An important example is the low-density gas, for which the average dynamics is described by the Boltzmann equation. Almost 20 years ago Zwanzig and Bixon showed how the corresponding Boltzmann-Langevin equation could be constructed to describe fluctuations as well.’,* In that paper they promised a more systematic justification for their result based on the formal Zwanzig-Mori projection operator formalism3s4evaluated at low density. One objective here is to present a brief outline of that derivation. In the analysis of this problem we found an apparent paradox in the fluctuation-dissipation relation for this formalism when hard sphere dynamics is used. A second objective here is to describe the unusual features of hard sphere Langevin equations and to show their consistency with the expected BoltzmannLangevin equation. In addition, our results extend this to obtain a dense-fluid Enskog-Langevin equation. The dynamical variable of interest is the single particle phase space density N
f(x;t) = E:s(x-x,(t)) =fB(X)[l i=l
+ $(x;t)l
(1.1)
where x denotes a point in the phase space of a single particle, Le., x = (7,p’). The second equality in (1.1) defines the relative deviation, $(x;t), from its equilibrium value fB(x) where fB is the Maxwell-Boltzmann distribution normalized to the density n. A formally exact Langevin equation for $(x;t) is obtained in the next section using the Zwanzig-Mori projection operator technique, with the result ‘Permanent address: Tel Aviv University, Tel Aviv, Israel. ‘Also, Department of Physics and Astronomy. 8 Permanent address: Department of Physics, University of Florida, Gainesville. FL. ,
I
,
a
-$(x;t) at
+ S d x ’ Q(x,x’) $(x‘;t) + L ‘ d t ’ Sdx’y(x,x’;t-t’) $(x’;t’) = F(x;t) (1.2)
The precise forms for Q(x,x’), y(x,x’;t), andf(x;t) are given below. Equation 1.2 has the interpretation of a Langevin equation because the “fluctuating source”, F(x;t), on the right side has vanishing average value and vanishing correlation with $(x;O). It represents the additional degrees of freedom in the microscopic dynamics that are not accounted for by the variables $(x;t). To justify this interpretation, it is also required that these degrees of freedom should have a time scale of variation much shorter than that for the average dynamics of $(x;t). Ideally, this would be true if the autocorrelation function for F(x;t) were proportional to a delta function in time (white noise). In general, the exact autocorrelation function can be computed by the Zwanzig-Mori techniques. For continuous potentials it is given by the “fluctuation-dissipation relation” (F(x;t) F(x’)) = - S d x ” y ( x , x ” ; t ) g(x”,x’)
(1.3)
where $(x) = $(x;t=O) is the initial relative deviation from the Maxwellian in (1.1). The brackets denote an equilibrium ensemble average defined in the next section. This close relationship of the fluctuating source, F(x,t), to the matrix characterizing dissipation, y(x,x‘;t), does not extend to the matrix Q(x,x)). The latter is independent of F(x;t) and describes the exact short time reversible (mean field) part of the average dynamics. These quantities can be evaluated at low density directly from their definitions, and the results are quoted in the next section. As expected, agreement (1) Bixon, M.; Zwanzig, R. Phys. Rev. 1969, 187, 267. (2) Fox, R.; Uhlenbeck, G. Phys. Fluids 1970,13, 1893. Logan, J.; Kac, M. Phys. Rev. 1976, 13, 458. (3) Zwanzig, R. Phys. Rev. 1961, 124, 963. (4) Mori, H. Prog. Theor. Phys. 1964, 33, 423. Hansen, J.-P., McDonald, I., Theory of Simple Liquids; Academic Press: New York, 1976.
0 1989 American Chemical Society
7020 The Journal of Physical Chemistry, Vol. 93, No. 19, 1989
with the proposed Boltzmann-Langevin equation of Bixon and Zwanzig is obtained. The derivation of the Boltzmann-Langevin equation is somewhat paradoxical for systems of particles that interact with discontinuous potentials such as hard spheres. Although the formal Langevin equation (1.1) still can be obtained, it is found that the corresponding matrix y(x,x’;t) vanishes to lowest order in the density so that the fluctuation-dissipation relation does not hold in the form (1.3). Instead, there is a dissipative contribution arising from the matrix n(x,x’) for hard spheres. Thus for such systems the fluctuating source is related to both Q and y. Furthermore, since 0 is a local operator with respect to time, the expected fluctuation-dissipation relation requires that the source autocorrelation function have a singular contribution ( F ( x ; t )F(x’)) = F(x,x’) s ( t ) + A(x,x’;f)
(1.5)
Here F(x,x’) denotes the amplitude of the singular part of the autocorrelation function and A(x,x‘;t) is nonsingular at t = 0. In the next section we identify the source of this singular contribution from a new identity relating the hard sphere binary collision operators. The amplitude F(x,x’) is shown to be related to n(x,x’) in precisely the way reuired by a fluctuation-dissipation relation for this component, and the hard sphere BoltzmannLangevin equation is recovered at low density. A second limit, that of short times, is also considered and shown to lead to the dense gas (modified) Enskog-Langevin equation5 In both cases a Markov process is implied by the exact white noise spectrum in these limits. These results are discussed briefly in the last section. 11. Hard Sphere Langevin Equations The generator for the time dependence of 4 ( x ; t ) is a Liouville operator representing the free streaming of particles and the effects of interparticle forces. For hard spheres the form of this operator is somewhat more complex than for continuous potentials because the force is singular. The momentum transfer due to interactions is defined in terms of binary collision operators rather than forces6 The form of the generator is different for positive and negative times since the instantaneous motion of two particles initially in contact is discontinuous for particles moving toward or away from each other. Consequently, the dynamics of 4 ( x ; t ) is given in the form6
p(r) w ; t ) = P ( ~ ) [ ~ x P ( L ~ ~ ) I ~ ( ~ ;(2.1) o) where p(r) is the distribution function characterizing the initial state of the system, taken here to be the equilibrium Gibbs distribution. It has the property of vanishing for overlapping configurations and therefore assures physically meaningful initial conditions for the dynamics. The generators for forward and backward time dynamics, L+ and L-, respectively, are skew-adjoint with respect to the Gibbs distribution J d r ~WWW+B(~)I = J d r
Bixon et al.
N i= 1
( A [ L + B I )= ( [ - L A I B )
(2.2)
It is also convenient to introduce two related operators that are skew adjoints of L, without regard to the Gibbs distribution as weight factor d r pA[LaB] = l d r [-L,pA]B
(2.3)
The significant difference between L+ and Li is that the latter is defined for all configurations, including those corresponding to overlapping spheres. Associated with these four operators are four different binary collision operators T+ and T*: ( 5 ) Dorfman, J. R.; van Beijeren, H.In Statistical Mechanics, Part B; Berne, B., Ed.; Plenum Press: New York, 1977. (6) Ernst, M. H.;Dorfman, J. R.; Hoegy, W.; van Leeuwen, J. M. J. Physica 1969,45, 127. Sengers,J. V.;Ernst, M.H.; Gillespie, D. J . Chew. Phys. 1972, 56, 5583. van Beijeren, H.;Ernst, M. H.J. Srat. Phys. 1979, 21, 125.
(2.4)
i#j
The detailed forms of these binary collision operators and some of their properties are given in the Appendix. With these preliminaries, the Zwanzig-Mori projection operator method can be applied to obtain a formally exact Langevin equation for $ ( x ; t ) . The relevant projection operator is defined by PA = l d x d ( x ) l dx ’ g
(2.5)
(X,X’)(4(X’)A)
where A is an arbitrary phase function and g(x,x’) is defined by (1.4). The following identity is easily verified by direct differentiation with respect to time: erL+ =
+
efL+P
+
L‘
d r e(f-r)L+PL+erL’Q efL’Q, t I0
(2.6)
with Q = 1 - P and L’ = QL+Q. The time derivative of 4 ( x ; t ) can be written as
a
- 4 ( x ; t ) = efL+L++(x;o) at
(2.7)
Then application of (2.6) to (2.7) gives the Langevin equation (1.2) with the definitions Q(xJ? = -Jdx”
y (x,x ’;t) =
-1dx
”
( [L+~“I~(x’?)~-’(x’’,x? ( [L+etL’QL+4(x)]4(x’’) )g-’(x ”,x ’)
F(x;t) = efL’QL+d(x)
(2.8)
Clearly, F(x;t)has the required property for a “fluctuating source”: (F(x;t)4(x3) = 0
(2.9)
The autocorrelation function for this purpose is given by ( F ( x ; t )F ( x 3 ) = ( [e‘L’QL+4(x)l[ Q L + 4 ( ~ 3 1 ) (2.10) The right side of (2.10) is similar to the following quantity related to y(x,x’;t): - 1 d x ” y(x,x”;t)g(x”,x? = ( [etL’QL+4(x)l[QL-d(x?l) (2.1 1)
a. Continuous Potentials. For continuous potentials the operators L+ and L- are same and have the form N
p(r)[-~-4mi~(r)
or, more compactly
N
L* = Ci;,.?,, f y z C T i ( i j )
N
o(i, j ) = [?rfv(rij)I.[epf-
(2.12)
Here = Fi - 7j and V(rij)is the pair potential. All of the above results still hold, with the replacement of Li by L. In this case it is Seen that the right sides of (2.10) and (2.1 1) are equal, leading to the fluctuation-dissipation relation given in (1.3). For practical application, therefore, it is sufficient to know the two quantities n(x,x’) and y(x,x’;t). To illustrate this we quote the results of the low-density expansion for these quantities promised by Bixon and Zwanzig’ f B ( X ) g(X,X’) = 6(X-X’) + f B ( X ) ho(F7’) + ... n(x,x’) = c’?[b(x-x’) - fB(x) ho(sr“)] +
...
y(x,x’;t) = - I d ” ] dxz 6(x’-xl)O( 1,2)dL(1*Z)fB( l)fB(2)e-@”W9(1,2) x [~(x-x,)+ ~(x-xZ)] ... (2.13)
fB(x)
+
where ho(r) = e-@v(r)- 1. The last expression simplifies further in the Boltzmann limits of large space and time scales compared to the force range and the collision time
The Journal of Physical Chemistry, Vol. 93, No. 19, 1989 7021
Hard Sphere Langevin Equations y(x,x’;r)
-
-26(FF’) 6(r) K(C,C’)
(2.14)
where K(3,C’) is the kernel for the usual linearized Boltzmann collision operator, J
441 = Jd” u K(C,C’) 4(C’,%)
collision operators, the left side of (2.22) contains a product of one backward operator T-(ij)and one forward operator T+(iJ). Substitution of (2.21) and (2.22) into (2.20) confirms the assumed singularity and allows identification of F(x,x’);
(2.15)
F(x,x? = -f/zX J d r ~ 4 ( ~ 3 [ T - ( i+ j )T + ( i j ) l d b ) i#j
The detailed calculation of these results is straightforward but lengthy and will not be given here. In summary, for continuous potentials the low-density limit of the formal Langevin equation (1.2) leads to the BoltzmannLangevin equation
[$ + C.9 - J ] 4 ( x ; t )
= F(x;t)
(2.16)
with the fluctuation-dissipation relation (1.3) in the form ( F ( x ; t )F(x’)) = 26(t) 6 ( W ’ ) K(C,C’) (2.17) These are indeed the expressions proposed by Bixon and Zwanzig. b. Hard Spheres. The factor 6 ( t ) in (2.14) and (2.17) is an idealization of a function that decays to zero for times larger than the characteristic collision time and is not the result of any true singularity in the dynamics for continuous potentials. In contrast, for hard spheres the collision time is infinitesimal and such a 6 ( t ) singularity is expected to arise directly from the dynamics without idealization. The primary observations at this point are that the function y(x,x’;t) does not have such a singularity but the source autocorrelation function is indeed singular and has the form assumed in (1.5). Consider first the source correlation function given by (2.10): ( F ( x ; t )F W ) = ( [efL’QL+4(x)l[ Q L + 4 ( x ? l )
=
-1 dr
$ ( x ’)L-p(r)efL’QL+$(x)
(2.18)
where use has been made of (2.3) in the second equality. It is easily verified that the free-streaming contributions to L and QL+4(x) vanish due to the orthogonal projection operator, Q . Therefore from (2.4) we have (F(x;t) F ( x 3 ) = f/4c i#j m # n
J d r 4(x?F-(ij)pefL’QT+(m,n)4(x) (2.19)
The singularity is associated with the dynamics of the same pair of particles occurring in both binary collision operators. Therefore, we isolate these terms as in (1.5) with the singular part identified as the pair contribution to (2.19)
F(x,x’) 6 ( t ) =
y2cI d
= (4(x)[L+ - L-I4(x?) = Jdx”
[ Q ( x , x ” ) g(x”,x’)
+ Q(x’,x’’) g(x”,x)]
(2.23)
As anticipated above, the hard sphere source autocorrelation function is now related to O(x,x’) instead of to y(x,x’;t) as in the continuous potential case. It is shown below that the symmetric form of Q(x,x’) appearing on the right side of (2.23) is related to a dissipative part of this matrix, so that this is correctly identified as a fluctuation-dissipation relation. The close similarity of the right sides of (2.10) and (2.11) suggests that perhaps y(x,x’;t) also has a singularity at t = 0, that in principle could cancel the dissipative part of Q . To see that this is not the case, an analysis similar to that leading to eq 2.19 gives - f d x ” y(x,x”;t) g(x”,x’) =
=
y22 S d r PC#J(X’)F+(ij)pe‘L“(‘j)T+(ij)$(x)+ ...
(2.24)
i#j
where the first term written out on the right side corresponds to the pair contribution which is singular in (2.19). Here, however, this term vanishes as a consequence of the identity ~ + ( i j ) e f ~ ( ’ j ) T += ( i 0j )
(2.25)
This is again a reflection of the fact that isolated pairs can collide only once. Thus y(x,x’;t) does not contain the singularity found in the source autocorrelation function. The pair contribution in (2.24) also contains the leading term in a density expansion. Comparison with (2.12) shows that the essential difference is replacement of the O(ij) operators for the continuous case with hard sphere binary collision operators, T+(iJ) or T + ( i j ) . This difference is significant, however, since in the former case this low-density term leads to the Boltzmann collision operator, whereas for hard spheres the corresponding term vanishes. In place of (2.12), the low-density results for hard spheres are found to be
r 4(x’)F-(ij)pdlc“j)QT+(ij)4(x)
i#j
(2.20) The operator L + ( i j ) is given by (2.4) for the isolated pair of particles i and j . Contributions from the dynamics of three or more particles are included in A(x,x’;t) and are assumed to be nonsingular. To show that the right side of (2.20) is indeed singular, write p = ~fB(i)fB(j)e”“” in (2.20) and consider the resulting operator
y(x,x’;t) = 0
+ ...
(2.26)
The second term in the expression for Q(x,x’) reduces to the hard sphere linearized Boltzmann operator for space scales large compared to the diameter, a. Thus the Boltzmann-Langevin equation (2.16) is regained. The autocorrelation function for the fluctuating source is obtained from (2.23)
F-(ij)fe(i)fB(j)e-BY(‘j)ef4(’j)T+(ij) = fB( i)fB(i) F-( i4) f?b(iJ) T+(id) (2.2 1)
where &(ij) is the free-streaming operator for particles i a n d j . This equality follows from properties of the binary collision operators listed in (A.3) of the Appendix. Essentially, it reflects the fact that two isolated hard spheres cannot collide more than once. An additional new identity is proved in the Appendix: T - ( i j ) e f b ( i J ) T + ( i j=) -6(t)[T-(ij) + T + ( i j ) ] (2.22)
This is a primary result of our analysis, identifying the source of the singularity in the hard sphere Langevin equation. Equation 2.22 is rather surprising yet essential for the proof of the fluctuation dissipation theorem for hard spheres. Unlike the familiar properties listed in (A.3) in which operators on the left side of the equations are either both forward or both backward in time
For space scales large compared to a , the right side of (2.27) reduces to (2.17), so the expected form of the Boltzmann-Langevin equation is established for hard sphere dynamics. In contrast to (2.17), however, the delta function in time is no longer an idealization but rather an exact consequence of hard sphere dynamics. 111. Enskog-Langevin Equation
Equation 2.23 for the amplitude of the singular term of the source autocorrelation function is not restricted to low density or
7022 The Journal of Physical Chemistry, Vol. 93, No. 19, 1989
large space scales. It is therefore of some interest to look for another limit in which this term dominates the additional term, A(x,x’;t). For asymptotically short times this is certainly the case. Furthermore, since y(x,x’;t) is nonsingular near t = 0, the time integral in the formal Langevin equation (1.2) can be neglected compared to Q(x,x? in this limit. The resulting short-time Langevin equation is
a
-4(x;t) at
+ S d x ” Q(x,x? d(x’;t) = F(x;t)
with ( F ( x ; t )F ( x ? ) = 6 ( t ) S d x ’ * [ n ( x , x ” )g(x”,x?
(3.1)
Bixon et al. T & ( i j )= a2Jdb 8(riji,,i.d)liji,~d.l[6(iij-ad)b,(iji,ijj) - 6 ( i i j - a d ) ] (A.2)
where 8 ( x ) = 1 for x > 0 and 0 otherwise, and b,(iji,ij,) replaces the velocities iji,ij, by the values ij*i,ij*j given by = iji - (Cj,-C)C, ij*; = ijj (ijij-C)C (A.3)
+
The operators T , ( i j ) and ( T * ( i j )have the following properties established in ref 6: Ti(ij)e*f~*(iJ)Ti(ij)= T*(jj)e*fb(iJ)T,(jj)= 0 Ti(ij)e*fr*(iJ)Ti(jj)= T*(iJ)e*fb(iJ)T*(iJ)= 0
+ Q(x’,x’? g(x”,x)] ( 3 . 2 )
These equations are completely characterized by g(x,x? and Q ( x , x ? ,both of which can be calculated exactly:’ f d x ) g(XJ? = 6(X-X? + f B ( X ) h(F71 (3.3) Q(X,X? = c-?(d(X-X? - fB(x)[c(+?? - g(a)ho(i-??]) g ( a )S d X ”fB(X ’9F - ( X , X ’9 [6(X-X’? + 6(X’-X’?] (3.4) Here, h(r) = g ( r ) - 1 and g ( r ) is the radial distribution function. Also c(r) is the direct correlation function. Equations 3.1-3.4 provide a remarkably good description of dense fluid transport and fluctuations that are limited only by the short-time approximation. No further assumptions regarding the density or the space scale have been imposed. Because the hard sphere collision time is infinitesimal, this approximation can be understood also as applying for times long compared to a collision time-this is the special theoretical advantage of the hard sphere fluid. In fact, calculations of longtime properties such as transport coefficients using these results are surprisingly good. The deterministic kinetic equation associated with (3.1) is known as the linear modified Enskog equation. The term “modified” is used to indicate that it is an extention of the original Enskog kinetic equation5 to apply on all space scales. Recent comparisons with computer simulation show quite good agreement, even on the scale of the hard sphere diameter.8 Our results extend this modified Enskog kinetic equation to its Langevin form. IV. Discussion We have been able to provide a microscopic derivation for the Boltzman-Langevin equation using projection operator techniques. While the results are very much the same for hard sphere particles and for particles that interact with continuous potentials, the details of the derivations are of necessity quite different. The derivation for hard spheres relies on a rather strange identity, (2.22). However, there are many advantages to using hard spheres over continuous potentials for theoretical models, and we expect the results given here to be quite useful. For example, the discussion given here for the Enskog-Langevin equation extends in a straightforward way to hard sphere models of nonequilibrium stationary state^.^ A useful discussion of the Boltzmann-Langevin equation, outlining the large literature on it, has been given by Ernst and Cohen.lo We hope that this derivation of the Boltzmann-Langevin equation will be of interest and value to other workers in the field, and we dedicate it to our close friend and colleague, Robert Zwanzig, on his 60th birthday. Appendix: Proof of (2.22) The properties of the hard sphere Liouville operators in (2.4) have been discussed in detail elsewhere.6 Here we only define the binary collision operators and outline the proof of (2.22). The binary collision operators for forward and backward in time dynamics are defined by T i ( i j ) = a2Jdd e ( ~ i j i , ~ d ) ~ i j ~ d ~ 6 ( i i j - u d ) [ b u-( i 1j 1i , i j j(A.l) ) (7) Konijnendijk, H. H. U.; van Leeuwen, J. M.J. Physica 1973,64,342. (8) Alley, W.; Alder, B. J.; Yip, S. Phys. Reu. 1983, A27, 3174. (9) Lutsko, J.; Dufty, J.; Das, S. Phys. Rev. 1989, A39, 1311. (10) Emst, M.H.; Cohen, E. G. D.J . Stat. Phys. 1981, 25, 153.
Ti(jJ)e*fWiJ)ho(rij) = ho(rij)e*‘MiJ) T,(ij)
=0
T*(ij) ho(ri,) = ho(rij)T,(ij) = 0 T + ( i j ) e t b ( i J ) T + ( i=j ) O 64.4) where ho(r) = e-avcr) - 1 and &(id) = ijp?,, ij,-Fr,, L,(ij) = L o ( i j ) f T,(ij)
+
L , ( i j ) = L o ( i j ) f Ti(ij) (A.5) The central result of importance for computing the correlation of the fluctuating source for hard spheres is the identity T-(ij)e”‘b(’J)T+(ij)= -6(t)[ T - ( i j ) + T + ( i j ) ] (A.6) To prove this result, we first use the property that the freestreaming operator exp(ftLo) replaces iiand ?, in all functions on which it acts by Tif Zitand 7, f Cjt, respectively. Then the left side of (AS) is given by
T-(i j ) e * f b ( i JT+( ) ij) = a4 d di dC, e(
di) lijif dil{O(-ij’ij-di) X lij:,-Ci16( iij-adi)6(?ijfij{jt-aC,)b,, O(-iji,- ai)Iijij-Ci16( Fij+adi)6( ?i,fiji,t-adj))e*f~ [b, - 1] (A.7) a , &
where ij’i, = bU,Cijin the first term of the brackets. Each of the two terms contains a product of two delta functions, which may be written respectively as 6( ?i,-ui?i) 6( ?ijfij’ijt-aCj) = 6( 7ij-abi) 6 (adifij’ijt-ud,) (A.8) 6( ?ij+adi) 6(?ijfij:jt-adj) = 6(Fij+abi) 6 (-ad,fij ’,,t-udj)
(A.9)
Consider first the delta functions on the right side of (A.8). The vectors adi f ij:jt locate points just inside or outside the interaction sphere about particle j , for small t , while the vectors aSi and aCj locate vectors on the interaction sphere. Thus, if both delta functions on the right side of (A.8) are to be nonzero simultaneously, then t must be equal to zero and Cimust equal d j Then using the fact that ,C: = ijij - 2(ijifdi)di, the delta functions may be analyzed in terms of a set of three mutually orthogonal unit vectors, Cij = iji,/lijul, e(’),and e(*). For example, the argument of the second delta function can be written aCi f Z’,jt - adj = $ , [ a cos (ei) - a cos (ej) =F tliji,ijl cos (28,)] + ;(‘)[a sin (ei)cos ( + i ) - a sin (e,) cos (4,) =F tlui,l sin (2ei) cos (4J] + i?(2)[asin (ei)sin ( 4 J a sin (ej) sin (4) =F flijijijl sin (24) sin (4Jl (A.lO) where we have used 8..= 1J
(eiJ)+ ;(I)sin (eiJ)cos ( $ i j ) + P(2) sin (e,)
sin (4iJ) (A.11) By expressing the second delta function on the right side of (A.8) as a product of three one-dimensional delta functions, one can show that 6(adifij’pzC,) = [Iijifdila2]-’s(t)6(di-d,) (A.12)
Ci, cos
Finally, we use the fact that for di = d,, we get bdibd, = 1. Then it can be shown that the first term in the brackets of (A.6) reduces to - 6 ( t ) T-(ij). It can be shown in a similar way that the second term in the brackets on the right side of (A.6) is equal to -6(t) T+(ij).This confirms the identity (A.6).
