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
Add quick response code for java
using barcode creator for java control to generate, create qr codes image in java applications.
*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).
Bar Code encoder on java
using java toadd barcode for asp.net web,windows application
GENERAL PROPERTIES OF THE PARTITION FUNCTION
Java barcode reader for java
Using Barcode recognizer for Java Control to read, scan read, scan image in Java applications.
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
Control qr code 2d barcode size with c#.net
qr code jis x 0510 size on c#
QN(V,
Aspx.cs Page qrcode development in .net
using asp.net web toaccess denso qr bar code in asp.net web,windows application
T):::: N!h 3N
Embed qr code iso/iec18004 in .net
use .net crystal qr bidimensional barcode integrating topaint qr in .net
l<I>p(l, ... , N) 1
Control qrcode image with visual basic.net
use visual studio .net qr code maker todisplay qrcode with vb
h2 p2 if3h f3 N xexp -f3L ( _J_ + - G p ) -f3L ( v - - w )] [ " "2m 2m J J "" IJ 2m IJ J-I I<J
Code 128 Code Set B implement on java
generate, create code 128 code set c none in java projects
(9.66) where
Control barcode code 128 size on java
to access code 128c and code128b data, size, image with java barcode sdk
G; =
Control qr code 2d barcode data in java
qr bidimensional barcode data on java
(9.67)
Control ean-13 supplement 5 image in java
use java ean / ucc - 13 encoder todevelop ean13 with java
(j*i)
Java matrix barcode creation on java
using java toencode matrix barcode on asp.net web,windows application
The momentum integrations can be done, again in term of the function f(r) of (9.51). We then obtain
Leitcode barcode library with java
using barcode maker for java control to generate, create leitcode image in java applications.
x == L8p [J(r 1 p
VS .NET Crystal pdf417 integration with .net
using barcode generator for visual .net crystal control to generate, create barcode pdf417 image in visual .net crystal applications.
+ GO ... f(r N
Asp.net Web Crystal qr-codes creator with c#.net
using barcode development for web.net crystal control to generate, create qr code image in web.net crystal applications.
PrN + G~)]
Incoporate ucc.ean - 128 for vb.net
generate, create ean/ucc 128 none for visual basic projects
(9.68)
Code 128C encoder in microsoft word
using barcode writer for office word control to generate, create code 128b image in office word applications.
where
Control barcode pdf417 image for excel spreadsheets
generate, create pdf417 none with excel projects
f3h -"w 2m
Control qr code jis x 0510 image for .net
using barcode development for .net winforms control to generate, create denso qr bar code image in .net winforms applications.
..,
An Asp.net Form Crystal barcode 3/9 writer on visual basic.net
using asp.net website crystal todraw barcode 3/9 on asp.net web,windows application
(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: