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

Generate QR Code in Java When we exponentiate (9.64), we may act as if the two terms in it commute with
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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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
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T):::: N!h 3N
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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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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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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" -
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
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)
= ' + n1 + n2 + n3
n1 = L vij
n3 = -16'7T .. k F F k L
I, j.
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.
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
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;;
1 kT = vz(JvjJz) = v4 [(n 2) _ (n)2] sO
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: