April, 1962
I1Ear.r CAPACITY AND OPEK A N D CLOSED ENSEMBLE AVUR.~GES
measurements at conceritratioris lower than 0.1 M . This contribution was supported by the Atomic
59 1
Energy Commission under contract AT-(30-1) 1375.
HEAT CAPACITY AND OPEN AND CLOSED ENSEMBLE AVERAGES BYJOSEPH E. MAYER Department of Chemistry, University of California, San Diego, La Jolla, California Received October 7 . 1961
It is well known that the heat capacity is given by Cv = k [ ( ( @ ' ) a ) . 4 ~- ( & ? ~ A V ~ ]with 0 = l / k l ' , where the average is that of an ensemble of closed systems of fixed N, V , 2'. One also may express Cv as the same average in an ensemble of open systems of fixed V , p , T plus calculable correction terms. Any numerical calculation of the mutual potential energy contribution using reduced probability densities always nil1 give only the open ensemble averages, unless explicit account is taken of the long range correlations of order iV-1 in these probability densities for the closed ensemble. A general method of obt.aining these long range correlation ternis to every order of N-I is presented, and it is shown that their ixitroduetion does indeed give the same expression for CV that is obtained by the computation with the ensemble of open systems.
Introduction In statistical mechanics one frequently requires the average value, p5,of the square of some function, F, that is a sum of functions of the coordinates (or of cobrdinates and momenta) of single molecules, or of pairs, triples, etc. Sometimes the average, (F1F2)of the product of two such functions is required. More usually, if not invariably, the quantity of interest is (Fz-I") where E is the average value of F. In general moments such as (n a
(Fa-Fa)) occur. A particular example of such a use is the well known expression for the heat capacity dEI3T = k Y 2 ( ( E - E ) * ) = k T Z ( j F - E2) (1.1) which usually is pro\-ed early in any course on statistical mechanics. The proof usually is made using the ensemble of closed systems of fixed N, V , l', in which case the omitted subscripts in (1) on the partial derivative are V , *V, namely the left-hand side is (bE/b?')v,~and the averages on the right are those of the closed ensemble. One may eqiially well prove (1) for an ensemble of open systems, of fixed V , p, T , with p the chemical potential, in which case one finds on the left of (1) the rather horrible partial derivative a t constant V arid p/lcT, whereas the right-hand side involves the average in the open system. By itself the difference is not alarming, since the averages in the closed ensemble and open ensemble are not expected to be equal. What is, however, disturbing, is that when one attempts to evaluate the averages one finds identical equations in terms of the probability density furictioiis of the molecules. The origin of the discrepancy is riot hard to find. I'or instance, one of the terms of @ is the average of the product of thc potcntinls of two independentl pairs of molecxdes, say 1,2 aiid 3,4, and this term T4, turns out to be, with p4 the four particle probnhility density 7'4
=
4 l l l v l d r d r * d r 3 d r 4P1(rI,r2,r1,r4)U(r12)zI(TU) (1.2)
This iiitegral is prgwtional to N 2 . Subtract from this the square, U 2 , of the average pair potential. is This average, 0,
where we assume the system is fluid so that p2 depends only on the distance rI2 = lrl-r2/. One finds for the diffcrence p-2\k4
y4 -
u2 = 4-'J . . i d ~ ( ~ ) [ p ~ ( r ~ , r -~ , r ~ , r , )
-
Pdrd d r d I u ( T ~ z ) u ( ~ ~ (1.4)
S o w the potentials u(rI2),U ( T ~ are ~ ) short range, so that the integrand of (4) is non-zero only if TIZ aiid r34 are small, but if the distance Tab = '/z Irl 4r2-r3-r4/ is large p4 becomes equal to p z ( r l z ) p 2 ( r 3 4 ) , and the integrand again approaches zero valuc. The integrand is appreciable only when all four molecules are close, and the value of the integral is proportional to the size of the system, N . Sow the statement h,r2,r8,r4) - d r d d r d --+ 0, Tab >> P - ' / J (1.5) is exact for an open system, and the contribution to the integral of ( 3 ) in the open ensemble does come only from the region of integration for which Tab is small. However (5) is correct only to order p 4 / N = p3/V in a closed system, and since the volume for which Tab >> p - ' i 3 is of order V = N/p, thc contribution from the region of large ?"& values is of the same order of magnitude as that from sinall values, even in the limit that N + m , V = N / p -+ a ,for which limit (5) is satisfied. The inference for a hypothetical romputing machine calculation of the term is obvious. Even if p4 were given to many significant decimal digits the diffcrence, ( 5 ) , would be zero to the accuracy of t,he machine, and coniputation would yield the value appropriatc to the open system arid hence to the computation of (dE/bZ')v p / k T and not to Cv. The statement above that (5) is exactly correct for the open system but oiily to order N-1 in the closed system is more intuitively obvious, arid in all detail understandable, if we consider instead p2(r12) - p 2 a t largc r12 values. T h e pair probability density. p2(rl,r2) = pdn2) is the simultaneous prohability density of finding a molerule a t rl and one a t r2. That, of finding the first molecule a t rl may he written as p = .V/V, but then there are only N I molccules left to oc~upythe position r2, and since
-
JOSEPH E. MAYER
592
on the average the first molecule at rl excludes a volume uo, these are distributed in a volume V -v0, so that
T
>> p-’/l
(1.6)
In general then, one may write, for the closed sys-
tem p~(~l)(rl,r2) = pail
+gdfd
- N-Yl
- pudl
+ gz(r1.41
h(O)(rl,r2) = pa[l
(1.7’)
For the closed system the pair probability density, p2(c1) (rlr2), exactly obeys the relation JV d r ~ ~ P ) ( r l , r 2=)
(N
- UP
(1.8)
so that by integration of (7) one has NP
+ p4 S 4&dr
gdd
question of how the “long range” terms of pn(C1) can be evaluated. Thermodynamic Derivatives and Moments.-We use, for convenience, a set of symbols which are tailored t o make the statistical mechanical equations 1ook;most simple. The number density, p , is N / V in the closed ensemble, but N/V in the open one, and we always compare ensembles of equal p . We use
(1.7)
where g2(r) goes strictly to zero a t large r values, and vo is some average volume excluded by a single molecule. The open system is confined to a large but finite volume V , in equilibrium through permeable walls with an infinite reservoir of the same chemical potential, p. Hence its probability density functions pn are those of an infinite closed system, namely that of (7) in the limit N -F so that
- p(1 - pud = ( N - 1)p
Vol, 66
/3 = l / k T
6 = PP (2.1)
v = -pP c = PE/N a = -AIL?
+
6p-l
= BA/N
where p is the chemical potential and A the Helmholtz free energy. The probability densities for the complete set of N molecules in V are then WN(O) = exp
and WN(C’)
-
- [.pV + Nu + BHN]
(2.2)
- [Nu + BHN]
(2.2’)
exp
in the open and closed ensembles, respectively, N Hamiltonian function for the N molewith I ~ the cules. We use { N ) to represent the complete p h a e space, the coordinate-momentum set of all the N molecules, and define an integration operator, $N, as
J’ 4 d d r g2(r) (1.9) B N Z [ * . S ~ ( ~ ” ~ N ! ) - (NI ’~ (2.3) and the “long range” term N - l ( 1- p C ) is determined uniquely by the “short range” function, g2(r), The normalization condition requires that Similarly, the exact relation puo = - p
S dra ~ d r ~ , r=d( N - 2) d
r d
can be used to determine uniquely the terms of order N-1 and of N-* in ps. The author has been unable to succeed in using such a direct method of finding even the terms of order N-I in p 4 . The condition
S
dr4 ~ 4 r : , r 2 , r d= ( N
- 3) aa(r1,rd
(1.11)
is inadequate. One can, however, without too great difficulty, evaluate ( b E ’ / b T ) V , p / k T by thermodynamic means as CVplus a correction, eq. 2.21, and the correction can be evaluated in terms of integrals over probability densities of the open system. Therefore, since CV is given by (1) to which the integral (4) contributes a finite “long range” contribution, and can be evaluated in terms of the “short range” open system integrals, it follows that the “long range” value of the integrand of (4)is uniquely determined by the short range values. In sections 5 and 6 of this paper a method is given whereby these long range correlations in pn, for any n, to order N - k with 1 6 k 6 n- 1, are determined. An entirely different method has been used by Lebowitz and Percusl to determine the terms of order N - l . The purpose of this paper is then threefold: (1) to emphasize that any numerical computation of CV using (1) must be made by directly or indirectly going through the open system ensemble equations; (2) to show exactly what are the open ensemble integrals entering CV; (3) to answer the (1)
(2.4)
(1.10)
J. L. Lebowits and J. K.Peroua, Phur. Rev., 111, 1675 (1961).
$NWN(”) = 1
(2.4’)
for the two cases, and these determine p(v,p) and a(p,P), respectively. The definition of an average of any function, F N ~( N j , of the phase space is that FQ“) = xgNWN‘o’ F N a (2.5) j7a‘Cl’
= gNWN‘c”F N a
(2.5‘)
and, to terms of order N-I if p,/3 are equal in the two cases j?-lFa(C1)
=
N-lh‘O)
+ O(N-’)
(2.6)
We shall hence Gmit the superscripts, and write the averages P = F ( C 1 ) = F. We define moments, I(n a n d J ( n ) , in] = 1 , 2 , . , a,. . , n of functions F N { N~ 1 , in the two ensembles
..
I(%) E
(0)
(2.7)
Since (4) must hold for all v, p, we have that for any change dv, dp 0
8NdWN‘O) =
=
N
=
- C$NWN(’) [Vdp + N d v + I i ~ d p ] N
- [ V d p + dv + EdPl v(ap/aY)B = - R
= -VP
V(a$O/ap)” = -E = - R e p
whereas for the system of constant N , V
(2.8) (2.9)
HEATCAPACITY AND OPEN AND CLOSED ENSEMBLE AVERAGES
April, 1962
0 = gNdWN(") = -LYNWN('I) [ N da
-e
N(da/dO)p
+ H dB]
cy
= -Neb-'
(2.10)
Differentiation of E, computed in the closed ensemble, with respect to p gives
= @'[[3.- 21 = 3/2
i-N
KN =
- ~ N W N ( ' " H N [ da/b8) N( -/- H N ] -J (E21 = - (l/kB*)(aE/aT)v,N
(3.5)
For a system of N molecules the total kinetic energy K N is
(&%/@)v.N = ~ N H N ( ~ W ( ~ ~ ) / ~ B ) = =
593
KI
i-1
so that
from (lo), (7'), and T = l/k& We have, then, for the heat capacity at constant volume, cv, per molecule in units of IC CY = (Nk)-' (d.f.?!/bT)V,N = N-' [J (@E)'] (2.11)
&.I = N@;X= (3/2)N The average of K N is~
On the other hand, using the open ensemble averages, and (8) and (9) for Vd $
which has N terms ~ i when ~ , i = j , and N 2- N terms Q ~j with i # j, whose average kinetic energies are independent. One finds, then F - R2 = N ( 2 - 2) (NT.2) - (Ni)' (3.9) I n the closed system of fixed N the last two terms cancel identically, and
(d~?/bP)v,u =
g"N
(aWN(')/aB)V
N
= =
(dfl/dv)v.~ = =
+
-
-z
~ N W N ( ~[ v) (Z( d~p /Nb @ ) v H N ] N
(E*]
(2.12)
C ~NN(~WN(~'/~V) N
=
c
98"
N
- z [N,E]
(2.14)
- pdv - ped In B B = Z (N 2 ]
= (&/bv)p = - f l - l Z ( N , p E )
(held In S)P = e
I((SE)2)
-fl-l
(2.15) (2.16) (2.17) (2.18)
The quantity cv of (11) is cv =
- (&€/a1np)p + e
(2.19)
The usual partial derivative manipulation gives = (&/a In B). (be/bv)~ (&/a In In S I p = (&/a In sly - ( a e / a ~ )(a~ In p/d In B ) (a ~ In p / a v ) p
+
mcv = Z[(BE)'J
-
(2.20) [Z(N,BEII2 [Z{N21]-1 (2.21)
or with (11) J((BE)2)
=
Z((SE)2) - II(N,B,ElI*[I(N2)1-' (2.22)
Kinetic Energy Moments.-The difference beand J tween the two moments I for the open and closed ensembles, respectively, is particularly easy to understand in detail for the case of a hard sphere fluid in which there is no potential energy. Define ~i = pipi/%
~i
- ( B PiPilPm)
82
- = 3/2 ma
= 15/4
and recognizes the value of cv as
- i2)- Z ( N ' ) i 2 ]
(3.11)
=
Z(N2)fi
(3.12)
.
so that
iiv...fi(.l Pn
( N ! / ( N - .)I)(')
(4.2)
whereas for the closed system the definition is
(3.2)
in the momentum space of a single molecule one finds BK
(3.10)
Had we, however, not counted the terms in K2, but attempted to evaluate the moments by integration over volume elements, we would in both cases have become involved with the integration of ( p s - p Z ) 2 over the volume. The differences in this integral due to the "long range" term, ( p 2 / N ) (1 - pvo) of eq 1.6, would have accounted for the difference between J { ( p K ) 2 ]and I ( p K ] z ] . Since, for the potential energy the average values of the product for two different pairs is not the product of the averages, no simple counting of terms is possible, and we must necessarily recourse to the volume integrals involving such cases as that of (1.4). Reduced Probability Densities and Averages.For a classical system, the distribution in the momentum space of the center of mass of the mole cules is a product of the single molecule functions, Wp(Pi) of (3.2). We will be concerned with reduced probability densities, pn ( n } ,in the coordinate space, in) = rl,. ., rn of the centers of mass of n molecules, defined for the open system by
(3.1)
as the kinetic energy of a single molecule i. With the classical Gaussian distribution wdpi) = (B/2am) *I*exp
I ( ( B K ) 2 )= iVB' [(?
Z(N,BK] (bWN(''/bv)B one verifies (2.22).
(a In p / & ) (a In p/a In p)"
(3.8)
Since, in the open ensemble
so that with p = iT/Vz and eq. 1, 8, and 9, we have dq =
Kj>
-1
+
(2.13)
a
N
-1
Ki
but in the ensemble of open systems they do not, and
N
(am/wv,, = ( ~ E / W ~ =. B-v(awavw = C tf"(dWN'"/b@)'
j=n
(c jx
J((PK)') = N B ' ( K ~- 2)
- C B N W N ' " ) N [ V ( ~ +~ /N~ ]Y )
--I IN')
i=n
=
(3.71
(3.3) (3.4)
and N-1 J ' dr,,tcl) P * I ( (n},r J = [ l -
(n/N)Ipn(C1){nl
(4.4)
is an exact relation. For the open systems, in the
594
JOSEPH
E. ~ I A Y E R
same integral over pa+l, the term of order 8 - 1 does not have the analytical form of pn. It is convenient to introduce cluster functions, G n ( n }and gn in]of the coordinates of n molecules. Use {k(v i ) n } u to indicate a partition of the n numbered molecules of the set ( n }into k unconnected subsets, (pij'n, 1 _< i _< k , Cpi = %, and 2 . ( k ( Y i } n ) u to
Vol. 66 lPK,PUl
CY
I.
indicate the sum over all possible such partitions for given k and n. Dcfine, then k=n
Gn{n) =x
pngn(f1) =
Z
(k
(k{Yl)n}u (-)n-l
-
I)! X
k=l i=k
II
~ u i { ~ i I n (4.5)
1-1
CVU
I: {
~ { Y ~ nI ~ G}v , ~{ v , l n
(4.6)
1=l
and G1(r) = pl(r) =
P , g1 3
1
(4.7)
i n a fluid for which the density is independent of position. In the open system, for which these are defined (except in the neighborhood of a critical point) the functions gn go sufficiently rapidly to zcro if any distance r,j between molecules of the set n goes to infinity, so that thc integral gIl(l)
{nl =
P
$_+:
dr,
gn+ 1 {
{n)rnl, n
2
1
+
p
Jv dr
= i~
(4.9)
J 4nr2 d r g2(r) =
17%+ iTg,(l)
8: - N (4.10)
or (Ay2 -
m2) = I{iYz] = m[l + gl(L)]
(4.11)
The moment 1 { N 2 ) is related to the isothermal compressibility. lirom c2.12) and (2.13) we have -PlZ{.V]
=
( b In
p/&)p
=
= ( 3 1x1 p / d q ) p
=
The Hamiltonian, H N {.VI, is the sum of kinetic plus potcntial energies
+ UN
(4.13)
so that
+ PB Z{.V,PE) = I{.Y,PK) + Z{N,PliJ Z{(PE)2)= Z{(PK)?\+ 21{PK,BII] +Z{(P[l)z] @E= PR
(4.22)
= P-'
*a 9 4
=
l/4
P-'
(4.23) Z{(PC)'I = N(\kz Pa *J 9 2 = p-' f 4nr2dr p2(r)@2U2(r) (4.24) S f drz dr3 P ~ ( ~ ~ , ~ z , ~ ~ ) ( ~ u ( T ~ z )(4.25) Pu(Tz~) S S S dmdrsdr4 [p(rl,r~,r~,rO d r d P Z ( I Pu(r12) ~ P 4 ~ 0(4.26)
-
in whirh all intcgrals can be extended to infinitc limits. The potcntial encrgy, U,consists of + N ( N - 1 ) term, u(r,,),so that $NNz(iv- 1)
U(TI2)WN
N22
(4.27)
+
writing N * ( N- 1) = N ( N - 1 ) ( N - 2 ) 2 Y ( N - 1) we have with (3) for pn, that (LVLi)(o) = S S S drldrz PdrI2)u(tlz) */z SSS drldr2drd pdrl,rz,rd u(r12) from the latter we write, for the subtracting moment
+
z{n-,B,u)= mcxz + X3)
Xz =
p-1
(4.28) J4n~'d~pz(r)&~(r) = 2@O/R (4.29)
With these in (20) we have with (11) that
R-lz{!vZl (4.12')
HN = KN
drzm(r12)Pu(d
that
+ +
( b q / b r ) p= (4.12)
or
- k?'p[l'-'(bI'/bP)T]
f Sv[drl
p2,
-
-( W W a ( b p / b q ) p=
-
( l v ~ ) =" )'/z =
(4.18)
CVU
If (4.21) for U N is squared, one finds for every one of the '/2AV(LV- 1) pairs, i,j , a term u2(qj),for every one of the N(N -- l)(-V - 2)/6 triples, i,j , k , six terms of the type u ( T , ~ ) ~ ( ~ i j (namely ) two each of the three possibilities with the thrce indices i, j , li, occurring twice), and for each of the .V(N - 1) ( N 2) ( N - 3)/24 quadruplets there are six terms of type U(r,j)tL(rk,i). Subtract the square of (22) from the last of these, and notc that the integration ovcr the last coordinate always leads to a factor V . We find
that
Nz + i T p
+
- z2)
N 2 >3>1
(4.8)
exists. In (1.9) we already have used the first of these, namely gl'" = P S 4nr2 dr g d r ) (4.8') Writing p2(r) = p2 11 g&)] from (e), and using (2) we have, since
CVK
-
(K*
we obtain for the average pD, from (1) for
a=k
k=l
=
f12
If the potential, UN, consists of the sum of mutual pair terms only UN(N1 = C U(7iJ) (4.21)
60 ~ a { n l=
(4.17)
= 312 (4.19) R-1{Z{(@U)2) [Z{N,PU}J2[1{iY2)]--1 (4.20) =
CVX
so that k=n
PK I (LV,PUI
=
Use these equations with (3.11) and (3.12) in (2.21) for CV. One finds, as must be the case, that one can write cv as the sum of two independent terms, cvr and mudue to the kinetic and potential energies, respectively
(4.14) (4.15)
(4.18)
The kinetic encrgy avcrages, K , of the singlc molecules arc completely independcnt of their coordinate positions, so that
x?= 1 / ~ - 1 Jdr2dr3 cvU = ~z
+ + + *3
-
[p3(rl,r~,r3) P P Z ( ~ ZBu ) ~ (rld (4.30) *4 (Xz X d Z [ ~ / Z { 1 ~ 2 1 1(4.31)
+
For numerical purposes (31), with thc expressions (24) to (30) for the individnal terms, and with (8) and (11) m-1
z{N21 = I
$-
p-1
f 4arZclr [ p 2 ( r )
- p2]
( 3 32)
are adequate to compute CV, the contribution due to the potential energy to the heat capacity at ($onstaut yoliinie, per molecule in units k . There remains, however, the somewhat inisat isfactory sitiuitiori that cVU = J { (/3fl)2, and this iiivol~esthc "long range" contribution that would appear iu
HEATCAPACITY AND OPENAND CLOSED ENSEMBLE AVERAGES
April, 1962
q.,if closed ensemble probability density functions, pu(Cl), were used. One would l i e to see more directly that these contributions are equal to -(X2 Xd2 [ N / m w . The Difference J { n ) - I { n ) for any n.-I?'e use a mathematical artifice by writing XaPa (5.1) In WN'") = - [qV NU BH]
+
+ + = - [A7a + BH] + C XaFa +
a
In
Wpq(C')
(5.1')
a
so that for X = A,, Xz,. . ., A, 0 the probability densities are those of the equilibrium system. The normalization conditions (2.4) and (2.4') now determine + ( Y , 0,A) and a(p, 0, A), and following the method by which (2.8), (2.9), and (2.10) were obtained we find immediately that V(Bq/dXa)v,p,X=ia Fa (5.2)
R
iY(da/dXa)p,p,x-xa =
(5.2')
With these and the procedure of obtaining (2.11) to (2.14) one finds (5.3) (5.3') ( d F a / d v ) x = (bIi'/dXa)v = Z ( N , a ) (5.3")
(d/'dXJ Zz-'
-
=
J CY,^}
-Iz-Yz(n)
(d/dXa)p
- ( b In
= (d/dX,)" = (d/dXa)v
p / d X a ) ( b In
(5.18)
and since O n l ( n }- 1 = Zln) - Z 2 - 1 Z 1 ( n ) S ( n - 1,l) = Z{n) = Iz-'R(n,k)
we have for n
=
J { 3 } = 113)
(5.20)
3, using (13) and (13l)
- Iz-'S(3,2)
+ SZ2-I
S(3,3) (5.21)
Sirice one can check that 0, and d: commute C".s = c o n (5.22) one has, for any n, that k=n-Z
~ ( n =l I { n l -
(-)k2kz2-zs(n,k k=O
+ 2)
(5.23)
Closed System Ensemble Cluster Functions.The moments defined by (2.7) also may be written k=n
Z ( k ( v i ) n ) u ( w ) ~ -(' k
I ( % )=
-
l)! X
k=l
i=k
11 i=l
p/dv)-I
+ I ( ~ V , C[Z{Ar211-2 Y) (alar)
Cli(n)
+
The rule of differentiation (d/dXa)p
=
6 n - 1, using (11)
OJz-1 S ( n - 1, k ) = 12-I T(n,k) - [.CZi(n)] S(n - 1,k) Z1(n) [b:S(n- l ) , k I = z2--1 (l'n,k) - CIl1(n) S ( n - 1,k)l = 1 2 - 1 Y(n,k) - SR(n,k 1) (5.19)
( d F a / d X a ) v = Z(CY,T)
(dFa/dXy)p
=
we have for any S(n - l,k), k and (12)
595
(
II
Fa)")
(6.1)
aC(vi1n
(5.4)
then gives J { ~ , =B Z~{ ~ , P J- z i x C YIi ~ X , P [I1 ( ~ 2 1 1 - - 1
(5.5)
as a generalization of (2.22). To continue we extend the notation of (2.7) with N held distinct a s a unique function, namely Zr{n} = (')
(5.6)
a,clal
so that (dZ,.c(7b)/dXa)v,~-ia
IF( (n),C Y )
=
- (dZr(nl/dv)x = Iw-lln) whereas, at constant
(5.7) (5.8)
p
J ( { n l ,C Y }
(bJ{n)/dXa)p,x-xa=
(5.9)
Xumber the a's sequentially 1, 2 , . . . , n,. . . and use the notation introduced in (4.6). Define, for given n and giyen IC, with 1 6 k n,the sums
