Higher time derivatives of the generalized entropyAdi R. Bulsara and W. C. Schieve Citation: The Journal of Chemical Physics 65, 2532 (1976); doi: 10.1063/1.433458 View online: http://dx.doi.org/10.1063/1.433458 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/65/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Generalized fractional Schrödinger equation with space-time fractional derivatives J. Math. Phys. 48, 043502 (2007); 10.1063/1.2716203 Higher thermodynamic partial derivatives J. Chem. Phys. 77, 5853 (1982); 10.1063/1.443749 On Lagrangians with Higher Order Derivatives Am. J. Phys. 40, 386 (1972); 10.1119/1.1986557 Time Derivatives of Entropy Production near Equilibrium J. Chem. Phys. 54, 1237 (1971); 10.1063/1.1674961 The Derivatives of the Entropy at Absolute Zero Am. J. Phys. 36, 1018 (1968); 10.1119/1.1974339

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Higher time derivatives of the generalized entropy Adi R. Bulsara and W. C. Schieve

Center for Statistical Mechanics and Thermodynamics. The University of Texas at Austin, Austin, Texas 78712 (Received 8 January 1975)

Using the generalized N-body expression for a Liapunov functional developed by Prigogine, George, and Henin, a condition is obtained whereby the successive time derivatives of this functional alternate in sign for weakly coupled systems. This condition is applied and seen to hold true for generalized Bose and Fermi systems. An N-body entropy is defined in terms of this functional, which contains diagonal as well as off­diagonal elements of the density matrix. This "generalized entropy" is seen to be concave close to equilibrium, similar to the results of Harris. It does not appear that this property holds for the entropy far from equilibrium.

I. INTRODUCTION

It has been shown by Harris1 that the Je function for the weakly coupled Master equation (Pauli equation), is convex near equilibrium. This is a particular case of a conjecture by McKean2 that the property of Boltzmann's form of the Je function which distinguishes it from all other functionals of solutions to the Boltzmann equation is that the successive time derivatives of Je alternate in sign. Harris's choice of Je function is defined solely in terms of the diagonal elements of the denSity matrix, it being tacitly assumed that the off-diagonal elements play no role. Harris has also verified the full alternat­ing property for the Boltzmann JC function of a discrete velocity gas3 and on the first two time derivatives of Je for a hard-sphere gaS.4 Further, Simons5 has investi­gated the alternating property (up to the second time derivative of JC) for thermal conduction and particle dif­fusion problems using linear nonequilibrium thermody­namics. These latter results do not seem to be easily generalized to the full hydrodynamic equations including viscosity. In this connection, Maass6 has shown that for the complete spatially inhomogeneous Boltzmann equation, the second time derivative of Je is in general not semidefinite-in apparent conflict with the conjec­ture of McKean. In addition, for the discrete velocity gas, Maass has by the method of Zubov7 constructed a Liapunov function which is not of the Boltzmann form but has the convexity property. Thus, MCKean's origi­nal conjecture does not seem by itself to determine the Boltzmann form of the entropy. Even though this orig­inal motivation does not exist, the alternating property is of interest in itself and it is our object here to ex­tend Harris's results for general weak coupling quan­tum systems (Bosons or Fermions) and to include the effects of correlations (off-diagonal elements of the density matriX). In addition, we will not use the Boltz­mann form of the entropy.

Recently, Prigogine, George, and Henins have intro­duced a manifestly causal formulation of dynamics wherein the density matrix p satisfies different equa­tions for retarded and advanced components. In this representation, the off-diagonal elements of p vanish at equilibrium and the nonvanishing diagonal elements are all equal. Hence, all accessible quantum states have the same probability and random phases at equilibrium.

The starting point of this work is the Liouville-von

Neumann equation for the density matrix p:

i(ap/at) =Lp ,

where Lp == [H, pi and L is Hermitian. We see that

(1.1)

(1.1) is invariant under the inversion L - - L, t- - t, unlike the Boltzmann equation which is not invariant un­der time reversal. Causality is taken into account by expressing the solutions to (1. 1) as inverse Laplace transforms. In the thermodynamic limit, the retarded solution pT and the advanced solution pG are different (time reversal leads simply from one to the other) and we have a broken symmetry. A nonunitary transforma­tion is introducedB which leads to the causal representa­tion wherein the components of p, namely p(/>lr and p{p)a

obey the separate evolution equations:

ap{t»r fat = - i(il>e + il>0)p(t»r(t) ,

ap(/>la/at =i(il>e _ il>0)p(Pla(t) •

(1.2)

(1.3)

Here, p(Pl{t) is the density matrix in the causal repre­sentation and

(1.4)

is the collision operator which governs the evolution of the retarded density matrix according to (1. 2). It may be seen that for weakly coupled systems [0(;\2) in per­turbation], pIP) reduces to the untransformed density operator, and that (1. 2) is just the Pauli equation for the diagonal elements of p with a suitable generalization for the off-diagonal elements. Further we have

il>e(L) = il>e(_ L)

il>°(L) = - il>0(_ L) , (1. 5)

and iq,e, iq,° are respectively Hermitian and skew Her­mitian. Examples of iil>e in weak coupling will be given in the ensuing discussion. It is seen from (1. 2) and (1. 3) that we have a broken symmetry if il>e(L) * O. Fur­ther it has been shown in weak coupling thatS

iil>e? 0,

and we are led to an N-body Liapunov function:

n=Trlp(p)(t}j2 ,

such that

dn/dt = - 2 Trp(P)·(i<l>e)p(~) ..; 0 .

(1. 6)

(1. 7)

(1- 8)

The functional n leads to a microscopic model for non-

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A. R. Bulsara and W. C:Schieve: Time derivatives of generalized entropy 2533

equilibrium entropy. The system evolves in time until the minimum of Q is reached-this corresponds to the equilibrium distribution. We may now define an entropy

(1. 9)

which is an N-body quantity and does not include the molecular chaos assumption. This entropy is seen to satisfy all the properties of the thermodynamic entropy:

(a) S ~ 0,

dS Trp(j»+(- icJ>e)p(j» (b) dt = - k . Trp(j»+p(j» ~ 0 , (1. 10)

(c) S is additive for independent subsystems.

Further, close to equilibrium, (1. 9) may be shown to reduce to the Boltzmann entropy for the Kac9 and McKean10 models.

In this paper, we will investigate the conditions for the alternation in sign of successive time derivatives of the Liapunov function '1 given by (1. 7). In Sec. II we apply this condition to an N-Boson system and an N­Fermion system for arbitrary interaction and show that it is satisfied in the thermodynamic limit (for any order of derivative). In Sec. ill, the entropy (1. 9) is consid­ered close to equilibrium. This entropy is defined in terms of diagonal as well as off-diagonal elements of the density matrixo We show that near equilibrium S ~ 0 and S ~ 0, which reduces to the results of Harris when the correlations are neglected.

II. THE HIGHER TIME DERIVATIVES OF n In this section, we obtain a general (sufficient) condi­

tion for the higher time derivatives of'1 given by (1. 7), to be semidefinite in weak coupling. Differentiating (1. 8) with respect to time we obtain

n = (- 2)2 Trp(j»+(icJ>~)2p(j» + 2 Trp(j»+[icJ>~, icJ>~Lp(j». (2.1)

Hence, n ~ 0 if in the thermodynamic limit we may ne­glect R == [icJ>8, icJ>°L compared to (icJ>~)2. Let us investi­gate R for the case v'" 0 corresponding to the correlated components of the density matrix. We shall consider weakly coupled systems for which it may be shown8 that

(2.2)

= ~H,(j) , say,

(2.3)

where L~v' are the matrix elements of the interaction part of the Liouville operator in the v-N representa­tionl1:

L~v'(N,N') = (v 1 Je'(N) 1 v')D(N -N') ,

where

(v 1 Je'(N) 1 v') =T(H~_v.(N)1J-v -1J-v·H~_v.(N)1Jv •

Here 1Jv is a displacement operator:

1J±v feN) = feN ± iv) ,

and we define

Ann' =An-n.[i(n +n')] ==Av(N) ,

with v=n-n'; N=i(n+n').

It is seen that the v = 0 components of (2.2) yield the well-known Pauli collision operator. We now express the matrix elements of L' in the v-N representation, setting v' = v - a, v" = v-b. After suitable permutations of the indices a and b which lead to substantial cancella­tions in the contributing terms, we are left with the re­mainder,

R = - 21Ti L D(b· E)<p_l-H~b[N +i(v+b)]H~[N -i(v- b)] a,b a· E

x{IH~[N +i(v+a)] 12+ IH~[N - i(v+a)] 12

-IH~[N+i(v+a+2b)] 12 -IH~[N -i(v+a - 2b)]12}7)2b ,

(2.4) where

D(b • E) == D(Le_b, V-b - Lev) ,

<P(a' E)-1 ==<P(Le-a,v_a - Lev)-1 •

As we shall see in what follOWS, we must investigate the concentration dependence of R given by (2.4), as compared to (icJ>~)2 in the thermodynamic limit. This can be done only if we know the particular matrix ele­ments which contribute to R for a given physical system and, moreover, if we can explicitly calculate these ma­trix elements together with their volume dependences as we will now do.

Let us consider first an arbitrary interacting Bose system, the interaction part of the Hamiltonian being12,13

(2.5)

where a denotes the number of creation and annihilation operators in the interaction, i denotes the number of crea­tion operators in a given term, and we define, 12

C t1"'ta =N L e~1 •• 1 e~.30" L BobC"'(O, x"., xn ••• ) exp[i(ka' x", +ks 0 xn + ••• )]A tl+ta+ ••• a,b,... :Im.s,.. •••

==ND(ku ka, ••• , ku)4tl+ta+oo" say . (2.6)

In the v-N representation we have

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2534 A. R. Bulsara and W. C. Schieve: Time derivatives of generalized entropy

H~(j)(N)=(N+iaIH'WIN-ia) ,

H~~j)(N) =(N - ia IH,<il* IN +ia) ,

whence for instance,

H~(t)(N) =N-a/2 I: Ctl ••• taoatl.IOatz._I··' oata._Il(Ntl +i)(Nt2 +i)'" (Nta +i)]1/2 • Itl···ta

Using the general form for icl>~ given by (2.2) we readily find in the thermodynamic limit (c = :) ,

icl>~= C I-aV1T J .. j dkl '" dka I D(k1, •.• , ka) 12 o(kl + •.• +ka){O (Etl - t Et;) [(Nt1 +ivtl) IT (Nt; +i vt ; + 1)

a a a

+ (Nt1 +iVt1 + 1) II (Nt. +ivt ;) + (Nt1 - i Vtl) II (Nt; - ivt · + 1) + (Nt1 - iVtl + 1) II (Nt; - iVt·) ;=2 • ;=2' ;=2'

+ 0 (E t1 + Et2 - t Etl)[rr (Nt· +ivtj) IT (Nt; +iVt. + 1) + IT (Nt. +ivt · + 1) IT (Nt. +ivtJ \' ;=s i=1 J ;=s • i=1 J J I=S • •

2 a 2 a

+ U (Ntj - i vt ) IT (Nt; - i Vtl + 1) + U (Nti - iVtj + 1) U (Nt; - i vtl )

2 a ] } .!. 1/2 .!. 112 .!. 1/2 .!. 1/2 2 2 -2 -2 -2II(Nt.-2Vti) (NtJ + 2VtJ) II (Nt ;- 2Vt;+1) (Nt .+ 2vt .+1) 1]tl1]t21]ts"'1]ta + ••• , j=1 J 1=3 • •

(2.7a)

(2.7b)

(2.8)

[~t- vI dk in the thermodynamic limit. Also, a tl+t2". - V- I (271Vo(kl + k2 + ••• ).]. Let us now consider the part R' of the remainder R =R' +R", where

Here, ~a includes all (kl , .•• ,ka) modes and ~b includes all (k~,· •• ,k~) modes. It is seen immediately that matrix elements with ata = 0 = bt :., give vanishing contributions and need not be considered in any further calculations. Let us now consider the detailed structure of these matrix elements for one term of the interaction, say, ~Ctl ... ta

Xatlat2'" ata' We have,

H~[N +i(v+a)] =N-alaI: Ct1 • .,t ({N+iv+a}',Ntl +ivtl +atl"'" it} a

(2.10)

The nonvanishing contributions arise for atl = 1, ata = -1 = ••• =at",' The numerical value of (2.10) is of course in­dependent of at'l ••• ato, since we are considering only the (kl , ... ,ka) modes. In fact we readily obtain

a

H~[N + i(v + all =N-ala L C tl ... t oat 1. 1 Oata.-1 •.• Oata.-l II [Nt; + iVt; + at; + 1))1/2 . it} '" 1=1

The contributions to (2.10) from the other terms in (2.5) may be similarly calculated. Let us now consider H~[N + i(v + a + 2b)] for the same term in the interaction

H~[N + i(v+ a+ 2b)] =N-"'/a fu Ctl ... t ", ({N + i(v+ b) + a}', Ntl + i(vtl + btl) + atl' ••. I atlat2 ••• at", I

x{N + i(v<tb)}', Ntl +i(vtl +btl), .•• ,Nt", +i(vt", +bt",) .

We have the following possibilities:

(2.11)

(a) atl = 1, ata = - 1 = ••• = at",; bt;' = 0, '<rIa.'. This contribution identically cancels (2.11) for all terms in the in­teraction (2.5). Thus the sole contributions to R' arise from the cases,

(b) atl = 1, at", = - 1, '<ria. * 1; btl = bti = 1, bt~ = - 1, '<ria. * 1. Here we have taken one wave vector from the set

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A. R. Bulsara and W. C. Schieve: Time derivatives of generalized entropy 2535

(k~,···, k~) equal to one from the set (k1, ••• , k,,). This effectively reduces the number of independent wave vectors by one and gives the dominant contribution to R' in the thermodynamic limit. The lower order terms inR' arise from the cases wherein (k1 = k~, k! = k~), (k1 = k~, k! = k~, k3 = k3) and so on, the number of independent wave vectors being reduced by one in each case. (It should be noted that before the above-mentioned reduction the sets {kJ and {k~}, respectively, contain a -1 and a' -1 independent wave vectors because of the momentum conservation law.) So finally we may write schematically

R' = - 27Ti {L: o(b • E)H~b[N +~(II +b)]H~[N - ~(II- b)]} , ,[-L: a ~ E I H~[N +~(II +a + 2b)] I ~~01)!b] b (t h t2 , •••• t a ) a (t1 ..... ,ta )

- 27Ti [2: o(b • E)H~b[N + ~(II + b)]H~[N - ~(II - b)]] , , b (J'1.ItZ.t3 ••••• t,,)

(2.12)

The first term is of O(V) and the remaining terms are all of lower order in the thermodynamic limit. After simi­larly treating the part R" of the remainder we finally obtain after considering all the terms in (2.5),

R=21Jic2-Z"V J ···f dk1••• dk"dk2••• dk~ID(k1,···,ka)IZID(k~, ••• ,k~,k1)lzO(k1+kz+··· +k,,)

+ <P(Et1 + Etz - t; EltS1 [(Nt1 +~lIt1 + 2)(Ntz +~IIItZ + 1) n (Ntl +~lItl) - (N1t1 +~lIt1 + l)(Ntz +~lIltz)

x f:r (Nit; +~1I1t1 + 1) - (11- - 11)]+' •• [contributions from remaining terms in (2. 5)]} 1=3

x{o (E1t1 - t Elti) r(N1t1 +~lIt1 + 1)1/2(Nt1 - ~lIt1 + l)l/Z f:r (N1t'1 +~lIt'I)1/2(Nt' - ~lIt' )1/2 1=2 L 1=2 I I

" xU (Ntj +~lItj)1/2(Ntj - ~lItj)lIZ + (Nt1 +~lItll2(Nt1 - ~lItl/2(Nt2 +~lIt2)1I2(Nt2 - ~lIt2)1/Z

x f:r (Nt' +~II~ + 1)1/2(N~ - ~II~ + 1)1121)~ 1)~ 1)~ ... 1)-:'] + ••• } + (lower order terms in V) . 1=3 I I I I 1 2 -:I a

(2.13)

Let us now examine the second time derivative of 0 giv­en by (2.1). As may be seen from Eq. (2. 8) (i<)~)2 is of O(V2) , where it should be noted that in considering high­er powers of i<)~ no vector out of the set (k1, ••• ,k",) will be equal to one from the set (kf, ..• ,k~) in general, which is not (as we have seen) the case for R. In the thermodynamic limit, the quantities V-1{i<)~) and V-2(i<)~)2 are well defined. The dominant term in R is of O(V) so that we may neglect the second term on the right-hand side of (2.1) compared to the first one in the V - 00 limit. So, 0 is positive definite.

Let us now consider the higher time derivatives of 0, in particular, the nth derivative. The first term on the right-hand side is always of the form (- 2)n Trp(JI)+(i<)~)n Xp(JI), which by the arguments just given, is of O(Vn). The next lower term is of the form Trp(JI)+(i<)~)n-2Rp(J» with similar terms in i<)e. Thus we have

o(n) = (_ 2)n Trp(J»+(i<)~)np(J» + O(Vn-1) + ••••

In the thermodynamic limit we may neglect all the terms on the right-hand side except the first. So,

sgnO(n) '" sgn(_)n ,

and the alternating property holds for the N-body weak­ly-coupled Bose system a sufficient condition for this being that R is at least O(V) smaller that (i<)~)2.

We now outline a similar proof for an arbitrarily in­teracting Fermi system with the Hamiltonian14

where a typical matrix element in the v-N representa­tion is given byll

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2536 A. R. Bulsara and W. C. Schieve: Time derivatives of generalized entropy

H~(N) = V""! -1 L Vt1 .. ·t",oat1.10at2.1··· Oata/2.10ak(a/2)+1o-1··· Oaka._10kT(~ k; - t kJ IT ONkl .<l/2) t1.·.ta l=l 1=(01/2)+1 1=1

For this case we readily find in the thermodynamic limit [where okT(kl - k j ) - (21T)3 V -10(kl - k J )] ,

i<p~=V1Ta!l(21T)3f···fdk1···dk",lvk1 ••• t 12 o(I: Ekl- t Ekl)O(~kl- t k l\

a 1=1 1=(",/2)+1 1=1 1=(01/2)+1 i)

[

01/2 a ",/2 a

>< IT okT(Nkl +~Vkl -1) IT okT(Ntl +~Vtl) + IT OkT(Ntl -~Vtl) IT okT(Ntl -~Vkl -1) 1=1 1=(a/2)+1 1=1 I=(a /2)+1

0112 a

2IT "kr(N .!. )"kT(N .!. ) IT "kT(N.!. l)"kr(N .!. 1) 2 2 -2 -2] - v tl + 2 Vtl v tl - 2 Vtl v tl + 2 Vt; - v tl - 2 Vtl - 1]t1' •• 1]t(a/2)1]k(a/2)+1 ••• 1]ta 1=1 1=(01/2)+1

(2.15)

a a/2 a } X IT OkT(Nk; +~Vt; -1) + OkT(Nt1 - ~Vt1) IT okT(Ntl - ~Vtl -1) IT /jkr(Ntl - ~Vtl) - (V- - v)

1.(",/2)+1 1=2 1.(01/2)+1

",/2 '"

X /jkT(Nt1 +~Vt1)/jkT(Nt1 - ~Vtl) IT ok1"(Nt j +~Vtj)OkT(Nkj - ~Vtj) IT okr(Ntj +~Vti -l)/jkr(Nt j - ~Vtj - 1) 1.1 Ida/2)+1

X17~ 17~' ... 172t' 17:~ ..• 17:~ + (lower order terms) . ·1 .2 a/2 ·(a/2)+1 .", (2.16)

Once again it is seen that the dominant term in R is O(V) less than (i<p~)2. Hence once again we have

sgnn(") =sgn(-)n , (2.17)

for the N-Fermion system. Thus we conclude that in weak coupling the Liapunov functional n has the re­markable property of alternation in sign of its higher derivatives in the thermodynamic limit for an arbitrary interaction. Details of the above calculation for the special case of the four-Fermion interaction have been given in a forthcoming publication15 wherein the validity of these results for the Friedrichs model is also dis­cussed.

III. DISCUSSION

Let us now consider the generalized entropy (1. 9) close to equilibrium in the light of the results obtained for the Liapunov functional n in the preceding section. We take,

(3.1)

where Pe(N) is the equilibrium density matrix. It should be noted that equilibrium correlations do not playa role. They are generated from the equilibrium distri­bution of the diagonal part of the density matrix by the creation operator C 8;

(3.2)

These may be shown to be of O(X) or higher16 and so, need not be considered. Substituting (3.1) in (1. 9) and retaining terms of 0(E2) only, we obtain readily

One sees immediately that

ddt2~ = - 2kE2 L IPe(N') 1-2{Y+(N)(i<p~)2y(N) NN'

+h+(N)[i<p~(N), i<p~(N)Ly(N)} • (3.4)

We have already shown that the second term on the right hand side may be neglected compared to the first one for an arbitrary N-body interaction in the thermody­namic limit. Hence, using (2.17) we have

(3.5)

In fact, it is apparent that to lowest order in E, the en­tropy has the same form as the Liapunov functional n discussed in Sec. II (apart from time-independent posi­tive constants); so that we may write in general to 0(E2

)

sgn(dnS/dtn) =sgn(_)3n-1 (3.6)

and the full alternating property holds for the entropy as well. It should be mentioned here that for the spe­cial case of a factorized distribution function (molecular chaos), the first term on the right-hand side of Eq. (3.4) vanishes and the second term gives the dominant con­tribution to S. In this case, S is obviously not semidef­inite (the same may be said of all its higher derivatives) and Eqs. (3.5) and (3.6) do not hold. However, this is a very special case corresponding to Bose and Fermi systems with initially uncorrelated modes which is not (in our view) very interesting physically. It does how­ever indicate that the validity of (3.6) depends on the initial conditions imposed on the system.

Now let us consider the special case v = 0, i. e., where n may be redefined only in terms of the diagonal ele­ments of the density operator. Then, Eq. (1. 7) may be

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A. R. Bulsara and W. C. Schieve: Time derivatives of generalized entropy 2537

taken as

(3.7)

with a similar modification of Eq. (1. 9) for the entropy. In this case we may observe that R = 0 which may be verified directly from (2.4). So the alternating proper­ty holds for the restricted Liapunov functional 0 0 to all orders of perturbation and for the entropy So obtained from it, close to equilibrium. It is unnecessary now to use the thermodynamic limit argument of the earlier sections. This result for the entropy So is precisely ·Harris's result1 for weakly coupled systems; to O(E:2

)

our entropy So has the same form as his (the Boltzmann form).

For the case v * 0, 1. e., when 0 and S include corre­lation (off-diagonal) components of the density operator, the alternating property is obtained for 0 in the thermo­dynamic limit for weakly coupled systems even far from equilibrium. The alternating property also holds for the entropy S close to equilibrium in the thermodynamic limit where, as we have seen in (3.4) a functional form similar to 0 is obtained for S. This is an extension of the results of Harris who does not define his JC function to include off-diagonal elements of the density matrix. Our results do not support the conjecture of McKean at least close to global equilibrium for v = 0 or v * 0 since in this case both the Boltzmann form of the entropy con­sidered by Harris and the Prigogine-George-Henin form, Eq. (1. 9), are the same. These results however support the result of Maass given in Sec. I. It does not appear possible to prove the alternating property on the entropy itself to all orders of E:. In fact if we consider only the second time derivative of S given by (1. 9), we have arbitrarily far from equilibrium

s=-- -0--0 .. k(l.. 1 '2) 2 0 0 2

+~[Trp<'I+RP(Pll} , (3.8)

where we have used (1. 8) and (2.1). No cancellation between these terms appears possible except for the special case when the denSity matrix is factorizable in­to a product of one-mode denSity matrices so that Sand consequently S(nl are not semidefinite to higher order than (2. This is also the case for the Boltzmann form of the entropy used by Harris.

ACKNOWLEDGMENTS

The authors would like to thank Professor I. Prigogine for his interest in this work. We also thank Dr. F. Henin of the Universite Libre de Bruxelles for several fruitful comments and would like to acknowledge helpful discussions with Dr. Apolinario Nazarea, Dr. Kenneth Hawker, and John Middleton of the University of Texas at Austin. One of us (W. C. S.) would like to acknowl­edge the partial financial support of NATO Grant No. 644 during the summer of 1974.

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