When we exponentiate (9.64), we may act as if the two terms in it commute with

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*The exact expansion is known as the Baker-Campbell-Hausdorf theorem, For an elementary derivation see R, M. Wilcox, J. Math. Phys. 8, 962 (1967).

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GENERAL PROPERTIES OF THE PARTITION FUNCTION

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each other because the correction belongs to a higher order in 13 than we are considering. With this in mind, we substitute (9.64) into (9.63), and again use free-particle states to calculate the trace. The operator "Vi may then be replaced by ip;/h. Thus we have

QN(V,

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T):::: N!h 3N

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l<I>p(l, ... , N) 1

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h2 p2 if3h f3 N xexp -f3L ( _J_ + - G p ) -f3L ( v - - w )] [ " "2m 2m J J "" IJ 2m IJ J-I I<J

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(9.66) where

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(9.67)

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The momentum integrations can be done, again in term of the function f(r) of (9.51). We then obtain

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x == L8p [J(r 1 p

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+ GO ... f(r N

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PrN + G~)]

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(9.68)

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where

f3h -"w 2m

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..,

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(9.69)

Making an expansion similar to (9.53), and an approximation similar to (9.56), we write

X ::::

f(r; - r + G:)f(r r + Gj) n f (G' ) n [1 + -'---'-------,--------,--'---,--------''-- ] f(G;)f(Gj)

J" -

(9.70)

The first product can be rewritten as exp L Fji Fik 16'7T I, ]"k " . which gives rise to an effective three-body force among the particles. The second product is generally complicated, involving many-body forces. We shall assume that the range and depth of the intermolecular potential is such that we can

STATISTICAL MECHANICS

neglect the terms

G: in this product. Then we can state

1 Q ( V T) :::: - - f d 3Npd 3Nre-f3 ;'ff N' N!h3N

(9.71) (9.72)

(9.73)

= ' + n1 + n2 + n3

n1 = L vij

(9.74)

n3 = -16'7T .. k F F k L

I, j.

'}.,2

(9.75)

9.3 SINGULARITIES AND PHASE TRANSITIONS

Phase transitions are manifested in experiments by the occurrence of singularities in thermodynamic functions, such as the pressure in a liquid-gas system, or the magnetization in a ferromagnet. How is it possible that such singularities arise from the partition function, which seems to be an analytic function of its arguments The answer lies in the fact that a macroscopic body is close to the idealized thermodynamic limit-the limit of infinite volume with particle density held fixed. As we approach this limit, the partition function can develop singularities, because the limit function of a sequence of analytic functions need not be analytic. Yang and Lee propose a definite scenario for the occurrence of singularities in the thermodynamic limit, which we shall now describe. It is formal in character, and belongs to a field sometimes known as "rigorous statistical mechanics."* As a concrete model consider a classical system consisting of N molecules in volume V, interacting with one another through a pairwise potential as depicted in Fig. 9.4. Each molecule is taken to be a hard sphere surrounded by an attractive potential of finite range. Thus, a finite volume V can accommodate at most a finite number of molecules M(V). For N > M(V) the partition function vanishes because at least two molecules must "touch," rendering the energy infinite: for N > M(V) (9.76) where we have suppressed the temperature to simplify the notation. The grand

*c. N. Yang and T. D. Lee, Phys. Rev. 87, 404 (1952); T. D. Lee and C. N. Yang, Phys. Rev. 87, 410 (1952). For rigorous stuff see also D. Ruelle, Statistical Mechanics (Benjamin, New York, 1969), s 3 and 5; and 1. Glimm and A Jaffe, Quantum Physics (Springer-Verlag, New York, 1981), 2.

GENERAL PROPERTIES OF THE PARTITION FUNCTION

v(r)

I------\C=------i'----.+ r

Fig. 9.4 Idealized interparticle potential.

partition function is a polynomial of degree M(V) in the fugacity z:

!2(z, V) = 1

+ ZQI(V) + z2Q2(V) + '" +ZMQM

(9.77)

Since all the coefficients QN(V) are positive, the polynomial has no real positive roots. The parametric form of the equation of state is p - = V-I 10g!2 ( z, V) kT (9.78) 1 J Iz-Iog!2(z, V) - = Vv Jz For any fipite value of V, however large, both P and v are analytic functions of z in a region of the complex z plane that includes the entire real axis. Therefore P is an analytic function of v in a region of the complex v plane that include all physical values of v, i.e., the real axis. Hence all thermodynamic functions must be free of singularities. From (9.78) and (9.77) we see that P > 0, and

a; a;;

JP Jz

1 kT = vz(JvjJz) = v4 [(n 2) _ (n)2] sO

(9.79)

where n is the density. To have the possibility of singularities, we must go to the limit V --+ 00 at fixed v- the thermodynamic limit: