Commun. Math. Phys. 92, 269-294 (1983) Communications in Mathematical

Physk © Springer-Verlag 1983

Debye-Hiickel Theory for Charge Symmetric Coulomb Systems

T o m Kennedy*

Department of Mathematics, University of Virginia, Charlottesville, VA 22903, USA

Abstract. It is proven that the pressure, density and correlation functions of a classical charge symmetric Coulomb system are asymptotic as the plasma parameter e tends to zero to the approximations predicted by the Debye- Htickel theory. These approximations consist of the ideal gas term plus a term of one lower order in e. The sine-Gordon transformation and some new correlation inequalities for the associated functional integrals are used.

1. Introduction

We study a classical charge symmetric system in three dimensions in the limit that e tends to zero. e is the plasma parameter

=fl/E~, (1.1)

where fi is the inverse temperature, and fD is the Debye length

E, = (2flz)- 1/2 (1.2)

z is the chemical activity. Debye and Htickel [4] gave a non-rigorous study of this limit. We will prove that certain predictions of their theory are rigorously correct in this limit.

The Debye-Htickel theory gives an approximation for the pressure P as a function of the density a

1TP~-2a- I-~-~ -3 (1.3t 24u o

(For example, see p. 229 of [11].) We work in the grand canonical ensemble, so the pressure and density are both functions of z and ft. We will show that as e tends to zero the pressure and density are asymptotically given by

l --e-3 a~z+ _ _ f - 3 (1.4) P~2z+ 127r 11 , 167r o •

• Research in partial fulfillment of the requirements of the Ph.D. degree at the University of Virginia

270 T. Kennedy

Combining these approximations yields (1.3). We will also find the first two terms in the asymptotic behavior of the

correlation functions. For example, the correlation function for particles at Yl and Y2 with charges fil and 62 is given by

O(2)(yl,(~l,Y2,(~Z)'-~Z211--fl(~162(--A-k'~D2)-l(yl,y2)+4~D] , (1.5)

and

0(2)(yl ' ~1 ; Y2' ~2) -- ~(1)(y1' ~ 1)~(1)(y2 ' 62) ~ -- g2fl~162( -- A --~ fl/~ 2)- l (y l , Y2)"

The Debye-Hfickel theory predicts that correlation functions of this form should decay exponentially. This is known as Debye screening. Brydges [1] proved that for sufficiently small ~ the correlation functions do indeed decay exponentially. This work was generalized by Brydges and Federbush [2] and by Imbrie [8]. Our result implies that as e tends to zero the correlation functions converge to functions with exponential decay, but this does not imply Debye screening for nonzero e.

To make the Coulomb system stable we must add a short range potential, e.g., hard cores, to the Coulomb potential. No such short range potential appears in the Debye-Hiicket theory, so we will let the short range potential tend to zero as e tends to zero.

One of the main tools we use is the sine-Gordon transformation. It says that the partition function can be expressed as a functional integral

Z= j'd#exp{2z ! :cos[ l/~ck(x)] :d3x}, (1.6)

1 where dp is a Gaussian measure whose covariance is essentially ~ . In units

with fD = 1, e--*0 implies fl--*0 and z ~ ~ with flz fixed. So

: cos [ : 1 - 5 :¢2(x )"

The Debye-Hiickel approximations all follow from this approximation. The use of the sine-Gordon transformation introduces functional integrals that

must be controlled. We do this using some new correlation inequalities. These inequalities give bounds on the moments of the measures that arise from the sine- Gordon transformation.

A natural approach to the problems studied here would be to use the cluster expansion of Brydges and Federbush [2]. Our approach has advantages and disadvantages with respect to the cluster expansion. Our approach is simpler than the cluster expansion. Moreover, we can allow several types of boundary conditions while the cluster expansion has only been carried out for Dirichlet boundary conditions. The disadvantage of our approach is that it requires charge symmetry. The cluster expansion does not.

This paper is organized as follows. We define the Coulomb system and observables in Sect. 2. In Sect. 3 we state our results. The sine-Gordon transfor- mation and Mayer expansion are used in Sect. 4 to express the observables as

Debye-HiJckel Theory 271

functional integrals. Then we give non-rigorous derivations of the results of Sect. 3 using these functional integrals. In Sect. 5 we state and prove the correlation inequalities. Finally, the results stated in Sect. 3 are proved in Sect. 6.

2. Definitions

We consider a system which consists of two species of particles with equal g2

chemical activities z. The species have charges + e. Let fl = ~-~, where k is the

Boltzmann constant and T is the temperature. The particles interact via the two-body potential

76 v ( x , T ; y , 6 ) = - 4-v~(x,7;y,6), (2.1) 4re Ix - Yl

where x,y~lR 3 a r e the positions of the particles and 7 , 6 ~ { - 1 , +1} are their charges. The potential v, is a short range potential depending on e. As e tends to 0, v, tends to zero. [The precise meaning of this statement is given by hypotheses (H1) and (H2) in Sect. 6.]

1 1 The potential ~ is the kernel of _~- , where A has free boundary

conditions. Physically this means that the box containing the particles is an insulator. Our results are true for other boundary conditions, e.g., Dirichlet and periodic. The kernels of operators involving A are simplest with free boundary conditions, so we use free boundary conditions throughout this paper. We leave it to the reader to check that our proofs work for other boundary conditions.

Two examples of v~ are I. Hard cores:

v~(x, 7 ; y, 6)= {O ifotherwise.lX-yI<2Codo, (2.2)

II. Yukawa potential:

- T6 e x p ( - i x - yl/Codo) (2.3) v~(x,7;y,6)= 4rclx-yl '

c o is a constant, fD is included in the definitions so that c o will be dimensionless. For a volume A £1R 3 the grand canonical partition function is

Z(A)= ~ ~ d"xexp[- f lU,(xp. . . ,x , ;71, . . . ,7 , )] . (2.4) n = 0 ~l , . . . , '~n

Each of 71, .--, 7, is summed over -t-1, and

S a"x = f d < . . . I A A A

The potential energy is

U,(x 1,...,x.;71,...,7.) = ~ V(X~'Ti;Xi'h)" l<i<j<n

272 T. Kennedy

The pressure is

k T I ~ log [ Z(A)], P(A)

where IAI is the volume of A. The correlation functions are

(2.5)

Z n

~"O(Yl'""Ym;6p'"'hm)=z"Z(A)-t ~ ~. Z n = O Y l , . . . , Y n

"Sdnxexp[-flU,+,,(xl, "",x,,Yl, "",Ym;Yl, "',~,,61,--', ~m)], (2.6) a

where Yl ..... y,, are distinct points in IR 3 and 61 ... . . 6,.~ { - 1, + 1}. Because of the charge symmetry the two species have the same density (average

number of particles per unit volume). It is given by

1 a(A) = ~ ! dyQ(al)(y; + 1). (2.7)

We will denote the infinite volume limits of the pressure, correlation functions and density by the same letters without a A. For example,

P = lira P(A). (2.8) A~IR3

For simplicity we take the volumes A to be boxes with the ratios of the dimensions of the boxes bounded as A~IR 3.

Lebowitz and Lieb [10] established the existence of the infinite volume limit of the pressure and density. For certain choices of the short range interaction v~ the existence of the infinite volume limits of all the observables was proven by Fr6htich and Park [6]. With Dirichlet boundary conditions and an essentially arbitrary short range interaction v~ these limits were shown to exist by Brydges and Federbush [2]. The existence of some infinite volume limit can always be established by a compactness argument.

3. Results

In the theorems of this section the short range potential v, can be given by either of our two examples, (2.2) and (2.3). These theorems are true for other choices of v~. We state the hypotheses that v~ must satisfy in Sect. 6. In all the theorems of this section the infinite volume limit is taken before the e ~ 0 limit.

The first theorem says that the pressure is asymptotic to its Debye-Hiickel approximation.

1 1 ~-3 in the sense that Theorem 3.1. - ~ P ~ 2 z + 12r~ o

l i m l l p _ 2 z ] f 3 _ 1 ,-*o[kT j o 12n"

Debye-Hiickel Theory 273

The correlation functions are also asymptotic to their Debye-Hiickel approximations.

Theorem 3.2. Let Yl, -.., Ym be distinct points in IR 3 and 61 . . . . ,6,~s { - 1, + 1}. Then

O( ' ) (Yp . . . ,Y , ;6p . . . ,6 , , )~z"[1- - f 3 ~ 616j(--A+d~2)-l(Yi, Yj)+ 8@~o } l <i <j-<m

in the sense that

lim d o r,~('-)(, ~ ;6~, 6,.)- z"] e-~O z m f l T M t ~ t D, " ' " Y r n ~ D " ' "

m 6,6j(- A + 1)- ~(y,, yj) + ~ .

t <i<j<=m

Finally, the density is asymptotic to its Debye-Hiickel approximation.

1 Theorem 3.3. a ~ z + - - d- 3 in the sense that

16re o

1 lim (or- z g 3 = . ~ o 16~

4. The Sine-Gordon Transformation and Mayer Expansion

Following Brydges and Federbush [2] we will apply the sine-Gordon transfor- mation to the long range part of the interaction and use a Mayer expansion for the short range part. The details of the Mayer expansion are in Appendix A.

We split the Coulomb interaction into long and short range parts. They are

Vz(X, 7 ;y, 6) = 76 1 - e x p ( - I x - yl/#dD) 4z Ix -- Yl

(4.1)

vr(x, ? ; Y, 6) = 76 e x p ( - I x - yl/#CD) 4~ rx -- Yl

(4.2)

So v = v L + Vy + v~. We denote the total short range interaction by v s

VS = V y -~" V ~ .

The function

1 - e x p ( - I x - yl/#do) C(x, y) = 4~ Ix - Yl

is the kernel of the positive operator

(4.3)

1 1 C -- (4.4)

- - A - - A ~- ( ]ADD)- 2"

274 T. Kennedy

Hence there exists a Gaussian process with covariance C(x, y), i.e., there exists a probability measure d# and a Gaussian random variable ~b(x) for each xelR 3 such that Id#q~(x)q~(y)= C(x, y). See pp. 16-17 of [12].

The partition function can now be written as Z = ~dl~Z(~), where

Z(dp) = -~. Z d"x exp - fl Vs(i ' j ) _ i ,= o , ...... ~. i ~ i= ?'qS(xi) ' (4.5)

with

Vs(i, J) = Vs(Xi, Yi; x j, ?j), 5 = z exp (/~/8=/*(v). (4.6)

Z(q~) is a partition function with a convergent Mayer expansion

Z(4~) = exp [,~1K,(qi)]. (4.7)

See (A.1) and (A.4) for the definition of K,(qS). The correlation functions are given by

0~)()1 . . . . . )m ;6p . . . , 6,,) = Z(A)- ~"~dl~ f i exp [i V~6jc~(yj)]2(d?), (4.8) j=t

where

with

Z ( ~ b ) = e - ~ r ~ ~ 2 ~ d " x e x p - / ? 2 Vs(i,J) n = 0 ?l , . . - ,7n l<--_i<j<=n

+ A(xi, ~ ) - i 3~40(x~ ) 5=1 ~ i '

(4.9)

E = ~ Vs(Y i, 6i; y j, 61), 1----<i<J----<m (4.10)

A(x, y) = ~ Vs(X , 7 ; Y j, J j). j = l

2(q~) is also a partition function with a convergent Mayer series.

Z(4>)=exp [ - ~ E + ~1/£~(~b)]. (4.11)

/£,(q~) is defined by (A.1) and (A.5). Using the results of the sine-Gordon transformation we can give a non-

rigorous derivation of the Debye-Hiickel approximations for the pressure, density and correlation functions. We will let #--->0 as ~ 0 . Then Vr~0 , and hence vsoO. For n>2, K,(¢) contains at least one factor of v s and so-~0. Thus the important term in the Mayer series is K1(¢).

We have

KI( ) =2z f ax :cos[ :. (4.12) A

Debye-Hi i cke l Theory 275

The normal ordering : : is defined by the requirement that : : be linear and the equation

:exp (~z hg): = exp ( - ~ ~d#~p2) exp (~z ~) ,

where ~E~E and ~ is any Gaussian random variable (see pp. 9-11 of [13]). In particular

:@(x): = 4)2(x)- C(x, x). (4.13)

:cos[ ]//tiff q~(x)] : = exp [~ C(x,x)] cos[ ]/~q~(x)].

In units with Yo = 1,/3 =e and so f i~0. Hence Kl(~b) should be approximately

l Thus

~p(A)~_2z + )-~log{Sd#exp[-½d~2 ! dx:d)2(x):]}. (4.15)

The Gaussian integral in (4.15) can be calculated. We will show in the proof of Theorem 3.1 in Sect. 6 that

/~"*0 A---.~R3 A = 12~z D

1 1 E_ 3 So we have the Debye-Htickel approximation ~ P - 2 z + 12~z o •

For the correlation functions we note that as Vs--,0,/(l(~b)~-Kl(~b). So using (4.14)

O(")(yp..., y,,; 3p..., 3,,) ~- Z(A)- 1~,, S d# fi exp [i ]/])-3fib(yj)] j = l

= z'~Sdfi I~I exp [i V~Sfl~(y)], (4.16) j = l

where dfi is a Gaussian measure whose covariance in the infinite volume limit is (C- 1 +E/~ 2)- l(x ' y). Some computat ion shows that as #--,0 (4.16) is approximately

z~[1 - f l ~ 6iSj(-A+dDz)-I(Y~,Y)+8~o ]" l <=i<j<=m

This is the Debye-HiJckel approximation for the correlation function Q(m) The Debye-Hiickel approximation for the density follows from the case of m = t.

276 T. Kennedy

5. Correlation Inequalities

The sine-Gordon transformation introduces measures of the form ~ d#, where

Z(~b) is given by (4.5). We will define a class of measures which includes the above measures. Then we will prove some correlation inequalities that allow us to bound the moments of these measures. Our techniques are reminiscent of those of Fr6hlich and Park [6]. These bounds will be used in the next section in our proofs of the theorems of Sect. 3.

Our correlation inequalities hold for any Gaussian process. To state them in their full generality we will make use of the idea of a Gaussian process indexed by a Hilbert space (see pp. 15-20 of [13]). The reader who is not familiar with such Gaussian processes should see Remark 1 below. Let ~ be a separable real Hilbert space with inner product (,). Then the Gaussian process indexed by ~ consists of a measure space f2, a probability measure d# and a linear map ~b : ~ L 2 ( d # ) , such that for each Qe ~ , q~(~) is a Gaussian random variable and ~d#4)(Q)~b(Q') = (Q, ( ) for

In our applications of our correlation inequalities the Gaussian process will always be the Gaussian process of Sect. 4. This Gaussian process is equivalent to the Gaussian process indexed by the Hilbert space ~¢~ consisting of all distri- butions / on 1R 3 whose Fourier transform ;g is a function with

H/ll2=SdakL~(k)]2[~_~_ 5 1 ] +&D)_- <

Let 6 x be the delta function centered at x. Then ~b(6x) is equivalent to the random variable that was denoted ~b(x) in Sect. 4.

Definition 5.1. Let ~ be a real Hilbert space. Let (£2, d#,~b) be the Gaussian process indexed by ~ . We will say that a measure ( ) defined on O is a sine- Gordon measure if it can be written in the form

( F( qS ) ) = I d#F( 4)) I dr(o) exp [Rb(Q)], (5.1)

where dv(o) is a finite positive measure on ~ . Furthermore the dv measurable subsets of ~ are such that q~(.) is a jointly measurable function on I2 x J~, and dr(o) is normalized so that

( 1 ) = 1 . (5.2)

d Remarks. 1. In our applications the measure ( ) will be of the form T #' where

Z(q~) is defined by (4.5). This is a sine-Gordon measure. The measure dr(Q) can be thought of as a measure on the configuration space

co @ (ed x { - 1, + 1})". n=O

Given Q=(xl ,7 l ;. . . ;x,,7,),

(1~(~0)= i ]~i~(Xi )" i=1

Debye-Hiickel Theory 277

And if ~ '=(yl ,61 ;... ;y,,, 6,,), then

2 2 i6c(xl, yj). i=l j= l

The reader who is unfamiliar with Gaussian processes indexed by a Hitbert space can follow our proofs by interpreting q~(Q) and (4, ~') in this fashion.

2. In general sine-Gordon measures are complex measures. In the example of the preceding remark the sine-Gordon measure is positive. This follows from Eq. (4.7) for Z(~b) and the fact that the K,(~) are real. They are real because of the charge symmetry of the system. We will make frequent use of the positivity of this sine-Gordon measure. This is why our techniques only work for charge symmetric systems.

3. The assumption that q~(.) is jointly measurable on O x 240 insures that the integrals in (5. t) are defined. Given a Hilbert space ~ f there exists a version of the Gaussian process indexed by ~ which is jointly measurable when the measurable subsets of ~ are taken to be the Borel sets. (For a similar theorem see pp. 60-62 of Doob [5].) So if dv(o) is a Borel measure on J¢~ then the joint measurability assumption of the definition is satisfied simply by choosing the right version of the Gaussian process. This is the case for all the sine-Gordon measures we use in Sect. 6.

Theorem 5.2. Let ( ) be a sine-Gordon measure, Qc~ t~ and ~clR. Then

(cosh [c~qS(~)]) __< ~ d# cosh [c~q~(Q)], (5.3)

and [9]

( - 1)'( : ¢2"(~o) :> > O. (5.4)

I f ( ) is also positive, then

(q~2.(Q)) < dnEj'dM~2(~)] "-- d. II oil z . (5.5)

and

where

( - l f(:q~2,(~) :) ~ d,[~dktq52(~)], = d, tI~Ul 2,, (5.6)

(2n)! e" (5.7) d ,= 2"n"-'

Proof To prove the first inequality (5.3) we begin with the calculation

~d#cosh[~(o(Q)]exp[i(o(~')]=exp[~(O,O)-½(O',Q')]cos[~(Q,Q')]. (5.8)

This implies ~2 l

= {~ d/~ cosh [e4(Q)] } {~ d# exp [i~b(O')] }. (5.9)

Integrating this inequality with respect to dv(o') we obtain (5.3), since <1> = 1.

278 T. Kennedy

To prove the second inequality (5.4) it suffices to show

( - 1)"Sd#: ~b2"(0): exp [iq~(0')] > 0.

This integral equals

(Q, ~,)2, exp [ - 1 , ~(e, e')] => 0,

which proves (5.4). Now we assume that ( ) is positive. Then

0~2n ~,, 2 ~ (2n) ~ . ( 4~ 2,(ff)) =< m=O (4~ Z'(e))

= (cosh [eq~(e)])

_<__ Sd# cosh [czOS(0)]

=exp [~ (e , e)j-

The third inequality (5.5) follows by taking

[ 1 ]- 1/2

To prove the last inequality (5.6), note that by (5.4) each term in

m= 0 ~2m ( (:cos[~4(e)] :) = ~ ~ - 1)"(:42"(0):)

is nonnegative. So ~2n

(2n)~

(5.10)

(5.11)

(5.12)

(5.13)

- - ( - 1)"( : q52"(Q):) < ( :cos [eq~(e)] :)

~2 = exp [~-(~, Q)J (COS E~q~(Q)] ~

--< exp f~-(~, Q) 1 ,

where the last inequality uses the positivity of ( ) . Now choose c~ as before. []

Remarks. 1. Fr6hlich and Park [6] proved inequality (5.4) for n = 1 for a certain class of sine-Gordon measures.

2. Inequality (5.4) can be used to prove a lower bound on the partition function and hence on the pressure for a special choice of v~ (see [9]).

6. Proofs

In this section we prove the theorems stated in Sect. 3. We begin by giving the hypotheses that v, must satisfy for these theorems. Both of the examples of v~ in Sect. 2 satisfy these hypotheses.

Debye-Hi ickel Theory 279

. . . . is repulsive, i.e., We assume that v~- v, + v v where v~

v:(i,j)>=O, (6.1)

and v ° satisfies the stability bound B(e)

[vr(i,j)+v°(i,j)]>= - ~ n (6.2) 1Ni<j~n

for # > e. B(e) is a positive function of e. We assume that v~ is independent of the charges of the particles, and v ° is

unchanged if the charges of both particles are changed, i.e., v°~(x, ~; y, 5) = v°(x, - ? ; y, - 5). Introduce the norms

Ilv°lll = sup Zfdxlv°(x,~;y,~)l. Y,~ ?,

II v~tl, = sup 2 J dx { 1 - exp [ - flv:(x, 7; Y, 5)] }, y,5 7'

II v ° tl 2 = sup [ ~ ~ dx I v°(x, ~;y, 5)121~/2. y,,~ [~,

=2# {/), Note that

(6.3)

(6.4)

= ( # ~ 1 t'2 llvrl[ 2 \ ~ - ] . (6.5)

Hypotheses (H1) and (H2) below say how fast v, must ~ 0 as e-~0. (H3) says that we have a uniform stability bound as e~0. (H4) and (H5) are weak hypotheses of a technical nature. ~o is included in the hypotheses in various places so that the inequalities will be dimensionless.

Hypotheses on v~: (H1) There exist c1,51 >0 such that

t;~ 3 IIv:llr___<Cx ~2÷~1

(H2) There exist Cz,52>0 such that

e ; 2 ilvOll 1 < c2 d +~2.

(H3) There exists B such that

~n(e) <= B.

(H4) There exists c 3 such that

E; i/2 o /)e 2=-~C3 "

(H5) For x+-y

lim v~(x, ?; Y, 5) = O. ~--*0

In Appendix B we verify that the two examples ofv, given in Sect. 2 satisfy (H1) through (H5). Using the bounds of Appendix A we see that (HI) through (H3) imply that the Mayer series of Sect. 4 converge for sufficiently small e and #.

280 T. Kennedy

Dimensional considerations suggest that the theory should be invariant under the scaling fl~fl/d, z ~ z d 3, G~v, , with v~(x, ?; y, fi) = dG(dx, ~ ;dy, 8)

A ~ I _ A d '

yj-.yj/E,

1 P ~ 1 pd a (6.6) kT kT '

G--+ffd 3 ,

Q(m)--. ko(m) d 3m "

Here ~ A = x :x~ A . This invariance is easily checked by a change of variables

in the integrals which define the observables. In our proofs we will "work in units with d o = 1." This simply amounts to

carrying out the above scaling with d =d o since dD~dO/d under the above scaling. Hypotheses (H1) through (H5) were stated in a dimensionless way. Thus they will continue to hold after the scaling (6.6). In the future we will write v~ simply a s b'~.

Notation. Following the notation of field theory we will let ~b denote a point in the measure space on which the Gaussian process is defined. F(~b) will be a function on this measure space, and sup IF(q~)[, the supremum of [FI over the measure space.

4 We will use O(e p) and o(e p) to denote quantities that are O(e p) and o(e p)

uniformly in A. o(A) will denote a quantity that --,0 as A-*IR 3. c, c', and ~ will denote positive constants. The c, c' or c~ in one equation is not necessarily the c, c' or ~ in another equation. However, c o, cl, Cz, and c a do not change from equation to equation.

We will often suppress the argument in integrations with respect to Lebesgue measure o n IR 3. For example

f ex :cos e : = i :cos E : ,

j" d

Proof of Theorem 3.1 (The Pressure). In units with d D = 1 the theorem becomes

l i m [ 1 J _ 1 (6.7) o[kT P - 2z 12~"

Recall that P is the infinite volume pressure. We will work with the finite volume pressure P(A) throughout the proof. At the end we will take the infinite volume limit. Our estimates will be uniform in A and so continue to hold in the infinite volume limit.

We will let #--*0 as e-,0. Then v s will ~0. For n>2, Kn(~b) contains at least one factor of v s and so will - ,0. We would like to let # ~ 0 fast so that Kn(~b)~0 fast.

However, C(x, x) = ~1_. So C(x, x ) ~ o~ as # ~ 0 . Thus our bounds on moments of ~zc#

Debye-Hi~ckel Theory 281

sine-Gordon measures from Theorem 5.2 will be useful only if # does not --+0 too fast. We let

/2~-8 l l 4 + & , (6.8)

with 0 < 6 4 < 1/4. We split the proof into two steps. In the first step we estimate the difference

rP(a) and ~T P(A) with Vs=0. Setting Vs=0 is the same as setting 4

between

K,(¢) = 0 for n > 2. In the second step we estimate the error made by replacing

:cos[l//fl4~(x)] • by i - ~ :Ca(x): in K,(¢).

Step I (The Short Range Interaction). Define an interpolating function on [0, 1] by

Q0

Then I(1) is Z(A) while I(0) is Z(A) with v s set equal to 0. We estimate log[/(1)] -log [I(0)] by bounding the logarithmic derivative of I(s).

Define a measure by

(F(¢))~=I(s)-~d#F(~)exp[KI(¢)+ ,~2 s"K,(q~)] •

Then

I'(s) ~K,(¢))s" (6.10) I(s)- t.:~2 ns"- Since K,(¢) is real, ( )2 is a positive measure. We claim it is a sine-Gordon

measure. Write co

The second factor on the right hand side is Z(¢) with ~ replaced by s~. So by (4.5) it is of the form fdv(~)exp [i¢(Q)]. The first factor on the right hand side is Z(¢) with v s = 0 and ~ replaced by (1 -s)~. So it is also of the form Sdv'(Q')exp [iq~(0')]. Hence their product is fdv(Q)~dv'(Q')exp[i¢(Q + ~')], which proves the claim.

For n > 3 we simply bound K,(¢) as in Appendix A. Using hypotheses (H1), (H2), (H3) and our choice of y Eqs. (A.7) and (A.8) become

with

Hence

IK,($)[ <__ce- lr"- I lAI,

r _ c , r2gi/2+ 2~4 + c g i + a ~ + c g i+az ] -- L ' i 2 "

(6.11)

ns"- 1 [K,(4~)[ = O(ea)lAt. (6.12) n=3

282

When n = 2 there is only one q and

Vs(q)=vr(1,2)+v~(1,2) I u(1,2) fl 1 +u(1,2)"

T. Kennedy

The previous argument shows that the second and third terms contribute O(ea)[AI to [Kz(~b)l. The v r term requires a new argument.

Define

g ( t , 4 , )= - [ds ~, [d2xvr(1,2)exp{-t~s[vr(1,2)+v°~(1,2)]}U(a)cos(tV-fi~), Yl,Y2 A

(6.13)

with ~=ylq~(xl)+72qS(x2). Then the term to be bounded is g(1,~b). By the hypotheses on v~, U(a) is independent of 71 and Y2- But

Z vr(1,2)=0" ?1, )'2

So g(0, ¢ ) = 0. Hence

0

= [dt as 2 IdZxvr(1,2)exp{-tfls[vr(1,2) o 2 ?i,~2 A + V°(1, 2)]} U(a){flsEvr(1, 2) + v°(1, 2)] cos(t V ~ )

+ V-ri • sin(t l//-fi~)}. (6.14)

Our goal is to bound (K/(q~)) s. So we need to bound (g(1, qS))s. Use

[(cos(t ]//fl~))~t =< 1. (6.15)

Since Mn(x)[ = Ix[, Theorem 5.2 and the choice of # (6.8) imply

I( ]//-f14~ sin (t ]/fi4~)hl < fl(~z)~ ---~Ce 3 / 4 - a 4 , (6.16)

where 6 4 < 1/4. Next we bound the integrations over x using hypotheses (H3) and (H4), Eqs.

(6.4) and (6.5), and the Cauchy Schwartz inequality. The final result is

I(K2(4~))~1 = O(e~)tAI. (6.17)

Equations (6.10), (6.12), and (6.17) imply

1 tlog [I(1)] - l o g [I(0)3t = O(eO).

So

~--~P(A)- ~--~ log {~ d# exp [K 1 (q~)] } = O(e~). (6.18)

Debye-Hiickel Theory 283

Step 2 (The Long Range Interaction). Define a second interpolating function on [0,1] by

Z(t)=exp(2zlAl)Sd#exp {2zt-2 ! [:cos(tl/~(o):- l]}. (6.19)

Then

and

Z(1) = j'd# exp [KI(¢)],

}im Z(t):exp(2z[A[)~d# exp ( - 1/2 ~:¢2 :),

by a dominated convergence argument Let

(F(d~))t = N(t)-l~d#F(¢)exp [2zt-2 ! :cos(t ]//fie):],

where N(t) is defined so that ( )t is a probability measure. Then

Z'(t) =/d_2zt_ 2 1 [:c°s(t V i l e ) : - 1]~, Z(t) \dt A

Expanding :cos(t]ffl¢): in a power series this

=2z~dx ~ (2n-2)tz"-3fl"( 1)"( a ,=2 (2n)! - :¢ 2"(x):)t"

Using Theorem 5.2 and the choice of # (6.8) this is

e" / 1 \ " <2z[Al (2n- 2)fl" ~n~-n. i4-~p )

n = 2

=O(eO)lAI . Hence

M • log [Z(1)] - log [lim Z(t)] I = O(e~). L t~° ]I

Combining (6.18), (6.20), (6.21), and (6.24) we have

~--~P(A)-2z-~A[log[~dpexp(-1/2!:¢2:)] = O(e~).

Since the O(e ~) is uniform in A we can let A~IR 3. One can show

)irn m rA-~log[Sd, exp ( -1 /2 ! :q52:)]

_ 1 Sdak 1 1 - log l + k 2 . 2(2x)3 k-2 k2+#-2 k2_~# -

(6.20)

(6.21)

(6.22)

(6.23)

(6.24)

(6.25)

284 T. Kennedy

(See pp. 175-177 of [7] for a similar calculation.) A dominated convergence argument shows that as 12---0 the above

2(2rt) [k

which completes the proof. []

Proof of Theorem 3.2 (The Correlation Functions). The key idea is to do a

complex translation ~b-+~b+iVfito in the functional integral expression for Q("), where i ]/~to is the stationary point of this integral. We will take m = 2. The proof of the general case is essentially identical.

Throughout the proof we will work with finite volume correlation functions. We estimate the difference

z1-2 1-0(a2) - z 2] + fl6162( - A + 1)- 1(yl, Y2) - L 4re"

Our estimate is a sum of two terms. One goes to zero faster than fl uniformly in A. The other term goes to zero as A~IR 3. So taking the infinite volume limit of our estimate proves the theorem.

We let

12=e 1/2+~4, (6.27)

with 0 < 34 < 1/6. As always we work in units with fD = 1. The proof is broken up into seven steps.

Step 1 (Complex Translation). Let C be the covariance operator of d12 (4.4). Let K = ( C - 1 + 1)-1. Let

to = K(61fiy ~ + 626y2), (6.28)

where 6y is the delta function centered at y. With

So K has kernel

and

m~ = 2@ I-1 _ (1 - 4122)1/2],

K=(X_412z)_ 1/2 [ 1 1 ] _ a + m "

K(x, y) - (1 - 4122) - 1/2 exp( - m_ Ix - Yl) - exp( - m+ Ix - y[) 4nix--y]

(6.29)

(6.30)

to(x) = 6lK(x, Yl) + 62K( x, Yz). (6.31)

D e b y e - H i J c k e l T h e o r y 285

We will need two bounds on ~p(x). Since K(x, y) is a positive definite function

IK(x, y)[ < sup K(w, w) w

- - m _ = (1 - 4#2) - 1/2 m+ 4z~

1 < - -

= 2n#

for sufficiently small #. As p ~ 0 , m_ ~ 1, so (6.30) implies

[K(x, y)[ < e x p ( - I x - yl/2) 2rc Ix - Yl

for sufficiently small #. Thus 1

It;(x)l _-< - - , (6.32) ~p

I~p(x)l < e x p ( - Ix - Yl 1/2) + e x p ( - Ix - y2[/2) (6.33) 2rctx- y,l 2rcIx- y21

for sufficiently small #. We perform the complex translation ~(x)~4)(x)+ i]/~p(x) in the functional

integral for Q~I [see (4.8) and (4.11)]. The result is

0(AZ)(Y 1, Y2 ;61, 6 z) = Z(A)-t ~2 S d# exp f - /~E + i ]/-tiff [61 qS(y 1) + 62q~(y2)]

- fl[~l~p(yl)+ 6e~p(yz)] + ~ R,(qS + i]// fl~P)-i]/~ Sq~C- lv? n = l

+/~ 1 ~ p C - ~p}. (6.34)

(See p. 171 of [7] for a discussion of translations in Gaussian measures.) Using the definition of tp (6.28)

o~)(Yl, Yz ;61, 62) = Z(A)- lz2 exp(A + S)Sd# exp [R(qS) + iI(4) + K(q~)], (6.35)

where

A= - ~[6?p~,l)+6ztp(y2)] ~ + ~ [C(Yl, YI) + C(Yz, Y2)],

s = - ~ E + 2 z l c o s h ( ~ ) - 1 - 7 ~ ~ ~,~, A 2 A ~

A ~ = IR 3\A,

n n = l A

I (40=Im /~(~b+iV~tp) + ~/~Iq~p, n

K(~b)= ~ K.(~b). n = l

(6.36)

286 T. Kennedy

Note that Z(A)= Sd#exp[K(qS)]. The definitions are arranged so that z 2 exp(A) converges to the approximation while S, R(q~), and I(4) converge to zero.

Step 2 (Computino A). Using (6.31),

[C(yl, Yl) - K(y~, y~) + C(y 2, Y2) - K(Y2, Yz)] A = - f16162K(y 1, Y2) + ~

A s # ~ 0 , m+~oo , a n d m ~ l . So

exp(-lY~ - Y21) = ( - A + 1)- l(yt, Y2). K(yl'Yz)~ 47rIYl-Y21

A little work shows C(y, y)-K(y, y)~ ~ . Thus

A= -~616z(-A + 1)- l (ypy2)+ 4-~ + o(e). (6.37)

Step 3 (Bounding S). We want to show that S is o(e)+o(A). From (4.10),

E=vs(Ypfl ;yz, fz)=vr(yt,61 ; y2, ~2)q-/)~(yt, 61 ;y2, 62).

The first term --*0 as # 4 0 . The second term ~ 0 as e-*0 by hypothesis (H5). The bounds (6.32) and (6.33) can be used to show

_/?z

<c, z~5/~ ~ ~/2

=O(e~/2).

Thus

Finally,

P {'w2~0~ic as A--fIR 3 . (6.38)

S=o(e) + o(A). (6.39)

Step 4 (Boundin9 I£,(d~ + i l / ~ ) - K,(~)). In this step we show that

[/(,(q~ + i]//-fiV)- K,(~b)[ = o(e). (6.40) rt=2

We apply Lemma A.1 of Appendix A with

i Va(X, 71)= - [/~y~b(x), (6.41)

v'l(x, 7) = yV(x) + A(x, 7), (6.42)

Debye-Hfickel T h e o r y

where

A(x, 7) = Vs(X, 7 ; Y~, ~ 1) + Vs(X, Y ; Y2, 62).

Hypotheses (H1), (H2), (H3), the choice of/~ and Eq. (A.9) imply

r'=o(~),

and

287

(6.43)

(6.44)

jdx rexp [ - flA(x, 7)] - I I = o(e2) • (6.45) Y

Bounds (6.32) and (6.33) imply

~, ~ dx Jexp [ - fl~ip(x)] - 11 = O(e). (6.46)

F rom (6.32), (6.45), and (6.46),

~dx lexp [ - fly' 1 (x, 7)] - lj = O(e). (6.47) y

Lemma A.1 now implies

l£.(q~ + i l / ~ ) - I(.(¢)1_-< c~o(~)[o(~)]"-*

_ _ < c'[o(~)] "-1

which implies (6.40).

Step 5 (Bounding R(~b)). We split R(~b) into two parts, R(~b)=Rl(~b)+ Rz(~b), with

R,((;b) = 2z S [ : cos ( ] /~b) : - 1] [cosh(flW)- 1], A

R2(q~ ) = Re E/~ l(~b + i ]/~tp)] - 2z S :cos(]/~q~): cosh (fl~o) (6.48) A

or)

Using (6.32) and (6.33),

2z I Icosh(fl~p)- iI = O(e;), (6.49) A

sup [Rl(gb)[ = O(e). 4,

Using (6.45),

Re [/£1(~b + i V fllp)] - 2z S :cos(~/flq~) : cosh ~lp.) A

= z ~ S dx : cos (]/fi~b): exp [ - fiT~p(x)] {exp [ - flA(x, 7)] - 1} 7 A

= zo(~ 2)

=o(~).

SO

(6.50)

288 T. K e n n e d y

Along with (6.40) this implies

sup 1R2(4))[ = o(e). (6.51) ¢

Step 6 (Bounding 1(4))). Let 1(4)) = 11(4)) + 12(4)) + 13(4)) + I4(4)), with

11 (4)) = 2z ]" [ ]//-fi4) - : sin (]//f14)) :] sinh (flip), A

12(4) ) = 2z ~ ]//f14) [fl~p - sinh(fl~p)], A

13(4))= ] / f l f 4)q~, (6.52) A e

14(4)) = z ~ ~ :sin (V~?4)): exp ( - flVtp) {exp [ - flA(x, 7)] - 1 } y A

The argument that proved (6.51) shows

sup [I4(4))[ = o(e). (6.53)

Step 7. Define a probabili ty measure by (F(4)))=Z(A)-I~d#F(4))exp[K(4))]. Then (6.35) becomes

Q~)(yl,yz;bl, bz)=zZexp(A+S)(exp[R(4))+iI(4))]). (6.54)

Equat ions (6.37) and (6.39) imply that the proof will be completed by showing

(exp JR(4)) + ii(4))]) - 1 = o(~) + o(A). (6.55)

Since ( ) is even in 4) and I(4)) is odd in 4),

(exp JR(4)) + ii(4))]) = (exp [R(4))] {cos [1(4))] - t } ) + (exp JR(4))] - 1) .

Using (6.50) and (6.51), proving (6.55) reduces to showing

(exp [R 1 (4))]) - 1 = o(e), (6.56)

Let

and

and

So

([cos [1(4))] - 11> = o(e) + o(A).

f(t) = <exp [tR 1(4))] >,

(F(4)))t = f(t)- I (F(4)) exp [tR 1(4))] ) .

(6.57)

1

@xp [Rl(4))] ) - 1 = ~f(t)(Rl(4)))flt. (6.58) 0

Debye-Hiickel Theory

By (6.50)

So (6.56) will follow from

sup f(t)=O(1). O_<t_<l

289

( )t is a positive sine-Gordon measure. So Theorem 5.2 implies

/3" [<:cos[ l / / f l¢(x)] : - l>, l = ~ i2~.t(-1) '<:¢="(x):>,

n=l

/3-:(tt- < Z~ ~n"\4--~) " (6.60)

n=l

By our choice o f# (6.27) this is O(e 1/z-~4) with 64 < 1/6. Combining this with (6.49) proves (6.59).

To prove (6.57) we use

]cos [1(¢)1 - 1[ _-< [cos [I(q~)] - cos [11 (¢)][ + ]cos [11 (~b)] - 11

-< [Iz(¢) + 13(¢) + I,(¢)[ + ½11(¢) 2 , (6.61)

which follows from Icos(a + O) - cos(~)l =< ia[, [cos(O) - 11 ~ ½02. So (6.57) will follow from

([Ii(¢)])=o(e)+o(A) for i=2 ,3 ,4 , (6.62)

( I i (~b) 2 ) = O(g). (6.63)

( ) is a positive sine-Gordon measure so Theorem 5.2 and the Cauchy- Schwartz inequality imply

(~I I/2

= 0(i). (6.64)

Inequalities (6.32) and (6.33) imply

2z S [sinh(/3~p)-/3q~[ = 0(5). (6.65) A

These two bounds imply (6.62) for i = 2. Using (6.64),

(113(~)[> ~ O(1) S [~p[ = o(A), (6.66) A c

which proves (6.62) for i=3. The case of i = 4 follows from (6.53). To prove (6.63)

( { :sin [ ~/flfl ¢(x)] : - V~¢(x)} z >

= < { [exp (/3/8~#) - 1] sin [ ] /~ ¢(x)] + sin [ Vile(x)] - Vile(x)} 2 >

__< [exp(/3/87r#)- 1] I/file(x)[ + ---if- 1¢(x)13 / ) , (6.67)

s u p [ (Rl( (~)) t [ = 0(8). (6.59) o_<t_<l

290 T. K e n n e d y

using Isin (0)1 ~ 10h Isin (0) - 0l N ~-I013. By Theorem 5.2 the above is

0 ( (~ ) ~) = 0(~3/2- 3°~), (6.68)

with (54 < 1/6. (6.32) and (6.33) imply

2z y Isinh(fi~)[ = O(1). A

Combining this with (6.68) proves (6.63). []

Proof of Theorem 3.3 (The Density). The proof is almost immediate from the proof of the previous theorem for m = 1. We actually wrote out the proof for m = 2, but we will refer to it as if it were the proof for m = 1.

Recall

a = a-~3 ~ l i m 1 ! dy~o~)(y; + 1). (6.69)

In the previous proof we estimated a quantity like

1 Q~)(y; __+ 1 ) - z - - 16~-"

Our bounds on this expression were independent of A and y except for two terms.

They were (6.38)~-!c ~p2, and (6.66) (]I3(¢)[).

Using (6.66) it suffices to show

lira 1 ~ dy ~ dxl~(x)lp=0 A"~IR3 H f A A ~

for p = l , 2 . (Remember ~p depends on y.) This follows from (6.33) and our conditions on how A~IR 3. []

Appendix A. The Mayer Expansion

All the Mayer series in this paper have the same two body interaction v s = v r +v°+@ They differ in the one body interaction v 1. Following Brydges and Federbush [3J these Mayer series are given by

~.(vl) = (_/~).-1~.

ZSdaf(q, a) Z fd"xvs(tt) n ~ v l , . . . , yn A

"exp[_--flVs(a)]U(cr)exp[--fli~=l vl(i) 1 , (A.1)

Debye-Hiickel Theory 291

where .1[ Vs(tl) = l--[ vr(i + 1, tl(i)) + v°(i + 1, q(i))

i=1

1 u(i + 1, tl(i)) ] fl l +s,(os,(i)+--~..,siu(~+ l,q(i))]'

u(i, j) = exp [ - flv~(i, j)] - 1,

Vs(~r) = Z sisi + 1"" sj _ 1 [vr(i, J) + v°( i, J)], (A.2) l <=i<j<=n

U ( ~ ) : I-[ [1 "Jr SiSi+l. , , Sj_ lu(i,j)], l <=i<j<=n

n=l 1

I FI I ds,, ,=1 0

n -1

J('l, ~) = H % ) % ) + 1 . . . s,_ l . i=2

Empty products are taken to be 1. The sum over q is a sum over all functions ~/: {1,2, . . . , n - 1}--*{1,2, . . . , n - 1}, such that

t/(i) 5 i. (1.3)

The three Mayer series we use are as follows. K,(~) is oY~,(vi) with

- i vi(i) = V l(Xi, 7/) = ~ ?,d)(xi). (A.4)

/( ,(¢) is Jf,(v 0 with

- i m

/(,(4' + i ] / ~ p ) i s Jff,(Vl) with

- i V 1 (i) : ~ ~;i~)(Xi) ~- ~filP(Xi) "~- j = l ~ Vs(Xi' ~/i ; Yj' ~j)"

As shown in [3]

]oU,(vl) I < ~ exp [fl [Ivl II - + eB(s)] 2 MI r"- 1,

where

(A.5)

(A.6)

(A,7)

[ 1 ] r=~f lexp[ l+f l l j v l l l_+~B(e)] I lvyHl+llv°I t l+~ll~l l , ; (1.8)

II 1tl and II II, are defined by (6.3). IIvllI_ is the sup norm of the negative part of the real part of Vr For each of our three Mayer series

/3 ]l vi 1t - = O(I). (1.9)

292 T. Kennedy

To see this note that the stability bound (6.2) with n = 2 implies

fl ~ Vs(X, 7 ; Y j, 6j) > - 2meB(e) >= - 2roB j = l

by hypothesis (H3). And tp is always such that 11/7~1!~ =o(1). The condition for convergence of the Mayer series is r < 1.

In the proof of Theorem 3.2 we need a bound on the difference of two Mayer series.

Lemma A.1.

with

tS,,(v 1 + ~1) - Jf,(v 1 )1 =< 2~ exp [/7 II vl It - + eB(e)] ~ Sdx

• lexp [ - fivl(x, ?)] - 1 [ (r')"- l , (A.lO)

r' = 2 [exp(fl IIvl [I-) + 1Jr, (A.11)

Proof The difference between the two af,'s is equal to the expression for Jl,(vl) with the factor

{exp[ 2,'1q '} included in the integral. We rewrite this factor as

f i { e x p [ - f l g ~ ( i ) ] - l + l } - l = ~ l ~ { e x p [ - f l g l ( i ) ] - t } , (1.12) i = l S:I=~ i~S

where S is summed over all nonempty subsets of {1, 2,..., n}. Each term in this sum contains at least one factor e x p [ - f l ~ i ( i o ) ] - l . We

bound any other factors of exp [ - flvl(i)] - 1 by exp(fl [I vl It -) + 1. Then we bound the integrations over x in the usual way except that we bound the integration over Xio last. (Think of i o as the base of the tree graph t/.) This last integration gives a factor of

~ d x lexp[--flVl(x, 2)] - 11. y

Since (A.12) has 2 " - 1 terms, the lemma follows. []

Appendix B

In this appendix we verify that the two examples of v~ given in Sect. 2 satisfy hypotheses (H1) through (H5) of Sect. 6.

I. Hard Cores (2.2). Let

v:=v~, v°=0; (B.1)

Debye-Hfickel Theory 293

(H1), (H2), (H4), and (H5) are immediate. Unfor tunate ly (H3) is not true. We must replace our system with an equivalent one in which (H3) holds.

We do this using a generalization to the Yukawa potential of Newton ' s theorem for the Cou lomb potential. Let Ix 1 - x g [ > 2 c o d D. Then the potential energy due to the Yukawa potential

exp ( - Ix - Y[/#~D) (B.2) 4n tx - y]

of charges 71 and 72 at x 1 and x 2 is equal to the potential energy due to this Yukawa potential of two spheres of radius Cod,) with centers at x 1 and x 2 and total charges 91 and 92 distributed uniformly on their surfaces, where

Thus we can redefine vy(xp;q ;x2,~/2) to be the potential energy due to the potential (B.2) of two spheres of radius c o d D at x 1 and x 2 with charges 91 and 92 uniformly distributed on their surfaces. For the long range part of the interaction v L we still treat the charges as point charges. So the s ine-Gordon t ransformat ion is not affected by our redefinition of vy.

is positive definite so the stability bound (6.2) holds with ~ equal to 1/2 of /)y . o

the self energy of these spheres. (H3) follows by comput ing this self energy. Since we have changed the definition of vy we must recompute I] v~ I[ 1 and II v~ II 2.

We leave it to the reader to check that these norms behave essentially as before.

II. Yukawa Potential (2.3). Let v ° =v~, v~ = 0 ; (H1) and (H5) are immediate. (H2) and (H4) take a little calculation.

We can assume that #>Coe, since in the proofs e-+0 faster than #. Hence vy + v ° is positive definite. (H3) follows by comput ing v~(i, i)+ v°(i, i).

Acknowledgement, David Brydges suggested the problems studied here as part of the author's Ph.D. Thesis. The author thanks him for all the help and encouragement he has so freely given.

References

1. Brydges, D.: A rigorous approach to Debye screening in dilute Coulomb systems. Commun. Math. Phys. 58, 313-350 (1978)

2. Brydges, D, Federbush, P. : Debye screening. Commun. Math. Phys. 73, 197-246 (1980) 3. Brydges, D., Federbush, P. : A new form of the Mayer expansion in classical statistical mechanics. J.

Math. Phys. 19, 2064-2067 (1978) 4. Debye, P., Hiickel, E. : Zur Theorie der Elektrolyte. Phys. Z. 24, 185-206 (1923) 5. Doob, J.U : Stochastic processes. New York : Wiley 1953 6. Fr6hlich, J., Park, Y.M. : Correlation inequalities and the thermodynamic limit for classical and

quantum continuous systems. Commun. Math. Phys. 59, 235-266 (1978) 7. Glimm, J., Jaffe, A. : Quantum physics: a functional integral point of view. Berlin, Heidelberg, New

York:Springer 1981 8. Imbrie, J. : Debye screening for jellium and other Coulomb systems. Commun. Math. Phys. 87,

515-565 (1983)

294 T. Kennedy

9. Kennedy, T. : A lower bound on the partition function for a classical charge s)anmetric system. J. Stat. Phys. 28, 633-638 (1982)

10. Lebowitz, J.L., Lieb, E.H. : The constitution of matter : existence of thermodynamics for systems composed of electrons and nuclei. Adv. Math. 9, 316-398 (1972)

11. McConkey, G., Prock, A. : Topics in chemical physics (based on lectures by P. Debye). New York : Elsevier 1962

12. Simon, B. : Functional integration and quantum physics. London, New York: Academic Press 1979

13. Simon, B. : The P(~b)2 Euclidean (quantum) field theory. Princeton: Princeton University Press 1974

Communicated by A. Jaffe

Received August 9, 1983