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{SECT 0 {EXCHG {PARA 392 "" 0 "" {TEXT 305 69 "Illustration and intuit
ive derivation of the Cluster Variation Method" }}{PARA 392 "" 0 "" 
{TEXT 305 41 "originated by Ryoichi Kikuchi (1919-2003)" }}{PARA 392 "
" 0 "" }{PARA 392 "" 0 "" {TEXT 305 27 "Marcel Sluiter, Dec 09 2004" }
}{PARA 392 "" 0 "" }{PARA 392 "" 0 "" {TEXT 305 9 "Notation:" }}{PARA 
392 "" 0 "" {TEXT 305 36 "spoint, spair, ssquare etc, entropy " }
{TEXT 306 12 "contribution" }{TEXT 305 26 " from a particular cluster"
 }}{PARA 392 "" 0 "" {TEXT 305 23 "kb = Boltzmann constant" }}{PARA 
392 "" 0 "" {TEXT 305 44 "ca = fraction of A atoms, concentration of A
" }}{PARA 392 "" 0 "" {TEXT 305 44 "cb = fraction of B atoms, concentr
ation of B" }}{PARA 392 "" 0 "" {TEXT 305 34 "faa = fraction of pairs \+
of AA type" }}{PARA 392 "" 0 "" {TEXT 305 64 "fab = fraction of pairs \+
of AB type, distinguished from BA pairs!" }}{PARA 392 "" 0 "" {TEXT 
305 23 "z = coordination number" }}{PARA 392 "" 0 "" }{PARA 392 "" 0 "
" {TEXT 305 66 "Point approximation (no correlations between points =>
 randomness)" }}}{EXCHG {PARA 393 "> " 0 "" {MPLTEXT 1 307 36 "spoint:
=-kb*(ca*log(ca)+cb*log(cb));" }}{PARA 394 "" 1 "" {XPPMATH 20 "6#>%'s
pointG,$*&%#kbG\"\"\",&*&%#caGF(-%#lnG6#F+F(F(*&%#cbGF(-F-6#F0F(F(F(!
\"\"" }{TEXT 308 1 " " }}}{EXCHG {PARA 395 "> " 0 "" {MPLTEXT 1 309 
21 "eval(spoint,cb=1-ca);" }}{PARA 394 "" 1 "" {XPPMATH 20 "6#,$*&%#kb
G\"\"\",&*&%#caGF&-%#lnG6#F)F&F&*&,&F&F&F)!\"\"F&-F+6#F.F&F&F&F/" }
{TEXT 308 1 " " }}}{EXCHG {PARA 396 "" 0 "" {TEXT 310 32 "analogously \+
for the pair we have" }}}{EXCHG {PARA 397 "> " 0 "" {MPLTEXT 1 311 71 
"spair:=-kb*0.5*z*(faa*log(faa)+fab*log(fab)+fba*log(fba)+fbb*log(fbb)
);" }}{PARA 394 "" 1 "" {XPPMATH 20 "6#>%&spairG,$**$\"\"&!\"\"\"\"\"%
#kbGF*%\"zGF*,**&%$faaGF*-%#lnG6#F/F*F**&%$fabGF*-F16#F4F*F**&%$fbaGF*
-F16#F8F*F**&%$fbbGF*-F16#F<F*F*F*F)" }{TEXT 308 1 " " }}}{EXCHG 
{PARA 398 "" 0 "" {TEXT 312 20 "In the random limit:" }}}{EXCHG {PARA 
399 "> " 0 "" {MPLTEXT 1 313 54 "eval(spair,[faa=ca*ca,fab=ca*cb,fba=c
b*ca,fbb=cb*cb]);" }}{PARA 394 "" 1 "" {XPPMATH 20 "6#,$**$\"\"&!\"\"
\"\"\"%#kbGF(%\"zGF(,(*&)%#caG\"\"#F(-%#lnG6#*$F-F(F(F(**F/F(F.F(%#cbG
F(-F16#*&F.F(F5F(F(F(*&)F5F/F(-F16#*$F:F(F(F(F(F'" }{TEXT 308 1 " " }}
}{EXCHG {PARA 400 "> " 0 "" {MPLTEXT 1 314 43 "spair_random=-kb*z*(ca*
log(ca)+cb*log(cb));" }}{PARA 394 "" 1 "" {XPPMATH 20 "6#/%-spair_rand
omG,$*(%#kbG\"\"\"%\"zGF(,&*&%#caGF(-%#lnG6#F,F(F(*&%#cbGF(-F.6#F1F(F(
F(!\"\"" }{TEXT 308 1 " " }}}{EXCHG {PARA 401 "" 0 "" {TEXT 315 57 "No
w, spair_random is (z-1) * spoint(_random)  too big!!!!" }}{PARA 402 "
" 0 "" {TEXT 316 95 "Ryoichi Kikuchi's idea in 1951 was to correct for
 this by taking out the point term............" }}}{EXCHG {PARA 403 ">
 " 0 "" {MPLTEXT 1 317 32 "spair_cvm:=spair - (z-1)*spoint;" }}{PARA 
394 "" 1 "" {XPPMATH 20 "6#>%*spair_cvmG,&**$\"\"&!\"\"\"\"\"%#kbGF*%
\"zGF*,**&%$faaGF*-%#lnG6#F/F*F**&%$fabGF*-F16#F4F*F**&%$fbaGF*-F16#F8
F*F**&%$fbbGF*-F16#F<F*F*F*F)*(,&F,F*F*F)F*F+F*,&*&%#caGF*-F16#FCF*F**
&%#cbGF*-F16#FGF*F*F*F*" }{TEXT 308 1 " " }}}{EXCHG {PARA 404 "" 0 "" 
{TEXT 318 77 "we subtract out, rather than divide because in W_point=(
NA+NB)!/(NA! NB!) and" }}{PARA 405 "" 0 "" {TEXT 319 147 "W_pair=(NAA+
NAB+NBA+NBB)!/(NAA! NAB! NBA! NBB!)  we wish to multiple W_pair with a
 correction factor (so in Log(W) we end up adding or subtracting)" }}
{PARA 406 "" 0 "" }{PARA 407 "" 0 "" {TEXT 320 27 "lets apply some alg
ebra...." }}}{EXCHG {PARA 408 "> " 0 "" {MPLTEXT 1 321 50 "eval(spair_
cvm,[fba=fab,fbb=1-2*fab-faa,cb=1-ca]);" }}{PARA 394 "" 1 "" {XPPMATH 
20 "6#,&**$\"\"&!\"\"\"\"\"%#kbGF(%\"zGF(,(*&%$faaGF(-%#lnG6#F-F(F(*(
\"\"#F(%$fabGF(-F/6#F3F(F(*&,(F(F(*&F2F(F3F(F'F-F'F(-F/6#F7F(F(F(F'*(,
&F*F(F(F'F(F)F(,&*&%#caGF(-F/6#F?F(F(*&,&F(F(F?F'F(-F/6#FCF(F(F(F(" }
{TEXT 308 1 " " }}}{EXCHG {PARA 409 "> " 0 "" {MPLTEXT 1 322 26 "s_cvm
:=eval(%,faa=ca-fab);" }}{PARA 394 "" 1 "" {XPPMATH 20 "6#>%&s_cvmG,&*
*$\"\"&!\"\"\"\"\"%#kbGF*%\"zGF*,(*&,&%#caGF*%$fabGF)F*-%#lnG6#F/F*F**
(\"\"#F*F1F*-F36#F1F*F**&,(F*F*F1F)F0F)F*-F36#F:F*F*F*F)*(,&F,F*F*F)F*
F+F*,&*&F0F*-F36#F0F*F**&,&F*F*F0F)F*-F36#FDF*F*F*F*" }{TEXT 308 1 " "
 }}}{EXCHG {PARA 410 "> " 0 "" {MPLTEXT 1 323 9 "e:=fab*J;" }}{PARA 
394 "" 1 "" {XPPMATH 20 "6#>%\"eG*&%$fabG\"\"\"%\"JGF'" }{TEXT 308 1 "
 " }}}{EXCHG {PARA 411 "" 0 "" {TEXT 324 79 "This way the energy of pu
re A and pure B is zero, independent of the value of J" }}{PARA 411 ""
 0 "" {TEXT 324 10 "e = energy" }}{PARA 411 "" 0 "" {TEXT 324 30 "J = \+
effective pair interaction" }}}{EXCHG {PARA 412 "> " 0 "" {MPLTEXT 1 
325 13 "f:=e-t*s_cvm;" }}{PARA 394 "" 1 "" {XPPMATH 20 "6#>%\"fG,&*&%$
fabG\"\"\"%\"JGF(F(*&%\"tGF(,&**$\"\"&!\"\"F(%#kbGF(%\"zGF(,(*&,&%#caG
F(F'F0F(-%#lnG6#F5F(F(*(\"\"#F(F'F(-F86#F'F(F(*&,(F(F(F'F0F6F0F(-F86#F
?F(F(F(F0*(,&F2F(F(F0F(F1F(,&*&F6F(-F86#F6F(F(*&,&F(F(F6F0F(-F86#FIF(F
(F(F(F(F0" }{TEXT 308 1 " " }}}{EXCHG {PARA 413 "" 0 "" {TEXT 326 67 "
Now lets look at the square lattice (Z=4) at equiatomic composition" }
}{PARA 413 "" 0 "" {TEXT 326 15 "f = free energy" }}{PARA 413 "" 0 "" 
{TEXT 326 15 "t = temperature" }}}{EXCHG {PARA 414 "> " 0 "" {MPLTEXT 
1 327 35 "fht:=eval(f,[J=1,kb=1,ca=0.5,z=4]);" }}{PARA 394 "" 1 "" 
{XPPMATH 20 "6#>%$fhtG,&%$fabG\"\"\"*&%\"tGF',(*($\"#S!\"\"F',&$\"\"&F
.F'F&F.F'-%#lnG6#F/F'F.*(F,F'F&F'-F36#F&F'F.$\"+U:Wz?!\"*F.F'F." }
{TEXT 308 1 " " }}}{EXCHG {PARA 415 "> " 0 "" {MPLTEXT 1 328 82 "plot(
[eval(fht,t=0.1),eval(fht,t=0.25),eval(fht,t=0.5),eval(fht,t=1)],fab=0
..0.5);" }}{PARA 416 "" 1 "" {GLPLOT2D 495 495 495 {PLOTDATA 2 "6*-%'C
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hx$!3=Q31'o*p<*)F^q7$F]y$!3?G[31dB(o)F^q7$Fby$!3%)=CkKD#GR)F^q7$Fgy$!3
%\\`3&GrZt!)F^q7$F\\z$!3s7<L'\\hMp(F^q7$Faz$!3i$3ja'>\\tsF^q7$Ffz$!3e#
=p0e?)ynF^q7$F[[l$!3fsEV+%pZC'F^q7$F`[l$!3?b$eIBcVj&F^q7$Fe[l$!3)=?bh
\\(Hd\\F^q7$Fj[l$!3)H]OCi?NE%F^q7$F_\\l$!3/IGM?)\\#pLF^q7$Fd\\l$!3s;sM
Jx))fCF^q7$Fi\\l$!3VG_^1>WS8F^q7$F^]l$!3=mFy(f4yI(F-7$Fc]l$!3IU`>(zC#o
`F67$Fh]l$\"3UFM8V!)[zMF-7$F]^l$\"3e\\'f/P_4(yF-7$Fb^l$\"3H#*)G-A'3E5F
^q7$Fg^l$\"3K=Tl\"p$o$G\"F^q7$F\\_l$\"3wGN\"e*\\8A9F^q7$Fa_l$\"35jj(f?
_)p:F^q7$Ff_l$\"318QBTJ_[;F^q7$F[`l$\"3'\\CQ@v<=t\"F^q7$F``l$\"3OD'>EF
FA#=F^qFd`l-%&COLORG6/%$RGBG$\"#5!\"\"$\"\"!FjfnF[gnF[gnFhfnF[gnFhfnFh
fnF[gnF[gnF[gnFhfn-%+AXESLABELSG6$Q$fab6\"Q!Fagn-%*LINESTYLEG6#F\\gn-%
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{EXCHG {PARA 421 "" 0 "" {TEXT 333 137 "Almost 50% AB pairs at low T, \+
then almost NO AA pairs =>(short/long range) ordering, of course, at h
igh T faa goes towards faa=ca*ca=0.25" }}{PARA 422 "" 0 "" }{PARA 423 
"" 0 "" {TEXT 334 113 "can we get interesting statistical thermodynami
cs?  Lets find the spinodal point for phase separation (d2f/dc2=0)" }}
}{EXCHG {PARA 424 "> " 0 "" {MPLTEXT 1 335 24 "ftc:=eval(f,[kb=1,z=4])
;" }}{PARA 394 "" 1 "" {XPPMATH 20 "6#>%$ftcG,&*&%$fabG\"\"\"%\"JGF(F(
*&%\"tGF(,,*($\"#?!\"\"F(,&%#caGF(F'F0F(-%#lnG6#F1F(F0*($\"#SF0F(F'F(-
F46#F'F(F0*(F.F(,(F(F(F'F0F2F0F(-F46#F<F(F0*(\"\"$F(F2F(-F46#F2F(F(*(F
@F(,&F(F(F2F0F(-F46#FDF(F(F(F0" }{TEXT 308 1 " " }}}{EXCHG {PARA 425 "
> " 0 "" {MPLTEXT 1 336 24 "dftcdfab:=diff(ftc,fab);" }}{PARA 394 "" 
1 "" {XPPMATH 20 "6#>%)dftcdfabG,&%\"JG\"\"\"*&%\"tGF',(*&$\"#?!\"\"F'
-%#lnG6#,&%#caGF'%$fabGF.F'F'*&$\"#SF.F'-F06#F4F'F.*&F,F'-F06#,(F'F'F4
F.F3F.F'F'F'F." }{TEXT 308 1 " " }}}{EXCHG {PARA 426 "> " 0 "" 
{MPLTEXT 1 337 51 "fab0:=solve((ca-fab)*(1-ca-fab)/(fab*fab)=ejt,fab);
" }}{PARA 394 "" 1 "" {XPPMATH 20 "6#>%%fab0G6$,$*(\"\"#!\"\",&\"\"\"F
)%$ejtGF+F),&F+F)*$,,F+F+*&\"\"%F+%#caGF+F)*&F1F+)F2F(F+F+*(F1F+F,F+F2
F+F+*(F1F+F,F+F4F+F)#F+F(F+F+F+,$*(F(F)F*F),&F+F)F.F)F+F+" }{TEXT 308 
1 " " }}}{EXCHG {PARA 427 "> " 0 "" {MPLTEXT 1 338 38 "fab01:=eval(fab
0[1],ejt=exp(0.5*J/t));" }}{PARA 394 "" 1 "" {XPPMATH 20 "6#>%&fab01G,
$*&#\"\"\"\"\"#F(*&,&F(!\"\"-%$expG6#,$*($\"\"&F,F(%\"JGF(%\"tGF,F(F(F
,,&F(F,*$,,F(F(*&\"\"%F(%#caGF(F,*&F:F()F;F)F(F(*(F:F(F-F(F;F(F(*(F:F(
F-F(F=F(F,F'F(F(F(F(" }{TEXT 308 1 " " }}}{EXCHG {PARA 428 "> " 0 "" 
{MPLTEXT 1 339 68 "plot([eval(fab01,[ca=0.5,J=1]),eval(fab01,[ca=0.5,J
=1])],t=0.01..2);" }}{PARA 416 "" 1 "" {GLPLOT2D 495 495 495 
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#Ffn7$$\"3u;a=T@*y/\"Fdv$\"3b\"fgtw)=.AFfn7$$\"3/]7$=#\\3'3\"Fdv$\"3iT
(R3.LN@#Ffn7$$\"3pmTDrTLF6Fdv$\"39Nd(3%y#RA#Ffn7$$\"3-++:#o[*p6Fdv$\"3
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\"3;LLG*42qL\"Fdv$\"3k**p'HjZpE#Ffn7$$\"35+]iud(*z8Fdv$\"3K=v*32kTF#Ff
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3CF%G.7F$3BFfn7$$\"3ummO`SXo;Fdv$\"3SoeU9008BFfn7$$\"3rm\"z!>.m5<Fdv$
\"3\\<AxSkk<BFfn7$$\"3Y$3-c4<Dv\"Fdv$\"3K`G;@o)>K#Ffn7$$\"35++:E\"z4z
\"Fdv$\"3n+;n1szDBFfn7$$\"3KLef`41N=Fdv$\"3>,U/I)o*HBFfn7$$\"3emmcl#)[
u=Fdv$\"3!f=+0^MNL#Ffn7$$\"38](oN+El\">Fdv$\"3%))*)*Rr`<PBFfn7$$\"31]7
t#[hn&>Fdv$\"3;.soIU^SBFfn7$$\"\"#\"\"!$\"3n>78LJ&RM#FfnF&-%&COLORG6)%
$RGBG$\"#5!\"\"$F]^lFf^lFg^lFg^lFd^lFg^l-%+AXESLABELSG6$Q\"t6\"Q!F\\_l
-%*LINESTYLEG6#F]^l-%%VIEWG6$;$Fe^l!\"$$\"#?Ff^l;$!1)ez\"fi!zo%Ffn$\"1
bCz&>K3R#!#;-Fa^l6#%%NONEG-%+PROJECTIONG6#Fd^l-%%FONTG6$%*HELVETICAGFe
^l-%*GRIDSTYLEG6#%,RECTANGULARG" 1 2 2 0 10 1 2 9 1 4 2 1.0 45.0 45.0 
1 0 "Curve 1" }}}}{EXCHG {PARA 429 "" 0 "" {TEXT 340 111 "AB pair prob
ability vs temperature at c=0.5 for homogeneous case, at high temperat
ure fab goes to 0.25 (=ca*cb)" }}}{EXCHG {PARA 430 "> " 0 "" {MPLTEXT 
1 341 24 "fc:=eval(ftc,fab=fab01);" }}{PARA 431 "" 1 "" {XPPMATH 20 "6
#>%#fcG,&*&#\"\"\"\"\"#F(*(,&F(!\"\"-%$expG6#,$*($\"\"&F,F(%\"JGF(%\"t
GF,F(F(F,,&F(F,*$,,F(F(*&\"\"%F(%#caGF(F,*&F:F()F;F)F(F(*(F:F(F-F(F;F(
F(*(F:F(F-F(F=F(F,F'F(F(F4F(F(F(*&F5F(,,*($\"#?F,F(,&F;F(*&F'F(*&F+F,F
6F(F(F,F(-%#lnG6#FEF(F,**$\"+++++?!\"*F(F+F,F6F(-FI6#,$FFF(F(F,*(FCF(,
(F(F(FFF,F;F,F(-FI6#FSF(F,*(\"\"$F(F;F(-FI6#F;F(F(*(FWF(,&F(F(F;F,F(-F
I6#FenF(F(F(F," }{TEXT 342 1 " " }}}{EXCHG {PARA 432 "> " 0 "" 
{MPLTEXT 1 343 36 "plot(eval(fc,[J=1,t=0.35]),ca=0..1);" }}{PARA 416 "
" 1 "" {GLPLOT2D 495 495 495 {PLOTDATA 2 "6*-%'CURVESG6$7hp7$$\"3`****
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7$$\"3U+++S1?k)*Fds$!3sN!3s'Hw'>\"FY7$$\"3;+++50O\"*)*Fds$!3%)[JvQ`oO5
FY7$$\"3/+++X/%\\!**Fds$!3?.8xY]d!\\*F-7$$\"3\"*******z._=**Fds$!3IE6s
1i.`&)F-7$$\"3@+++?.5K**Fds$!3Ta@i!**pGa(F-7$$\"33+++b-oX**Fds$!3%zs7-
L^SW'F-7$$\"3'********=g#f**Fds$!3lkg_(f#GK_F-7$$\"3C+++I,%G(**Fds$!31
_z]q#[i'QF-7$$\"3%)*******4I'z**Fds$!3S%H1D!R+-JF-7$$\"37+++l+U')**Fds
$!3O&*>rv[&)eAF-7$$\"3S+++I+@$***Fds$!33d7oOo@$H\"F-7$%*undefinedGF]hl
-%&COLORG6&%$RGBG$\"#5!\"\"$\"\"!FdhlFehl-%+AXESLABELSG6$Q#ca6\"Q!F[il
-%*LINESTYLEG6#Ffhl-%%VIEWG6$;FehlFbhl;$!13:(p8IDt$!#<$!0@#\\_P0neFds-
F_hl6#%%NONEG-%+PROJECTIONG6#Fbhl-%%FONTG6$%*HELVETICAGFchl-%*GRIDSTYL
EG6#%,RECTANGULARG" 1 2 2 0 10 1 2 9 1 4 2 1.0 45.0 45.0 1 0 "Curve 1"
 }}}}{EXCHG {PARA 433 "" 0 "" {TEXT 344 52 "Now, are we not happy to a
 computer algebra package!" }}{PARA 433 "" 0 "" {TEXT 344 63 "Here we \+
are close to the spinodal!  Lets find it with d2f/dc2=0" }}}{EXCHG 
{PARA 434 "> " 0 "" {MPLTEXT 1 345 18 "dfdc:=diff(fc,ca);" }}{PARA 
431 "" 1 "" {XPPMATH 20 "6#>%%dfdcG,&*&#\"\"\"\"\"%F(**,&F(!\"\"-%$exp
G6#,$*($\"\"&F,F(%\"JGF(%\"tGF,F(F(F,,,F(F(*&F)F(%#caGF(F,*&F)F()F8\"
\"#F(F(*(F)F(F-F(F8F(F(*(F)F(F-F(F:F(F,#F,F;,*F)F,*&\"\")F(F8F(F(*&F)F
(F-F(F(*(FAF(F-F(F8F(F,F(F4F(F(F(*&F5F(,,*($\"#?F,F(,&F(F(*&F'F(*(F+F,
F6F>F?F(F(F,F(-%#lnG6#,&F8F(*&#F(F;F(*&F+F,,&F(F,*$F6FQF(F(F(F,F(F,*,$
\"+++++5!\"*F(F+F,F6F>F?F(-FM6#,$FPF(F(F,*(FGF(,&FJF,F(F,F(-FM6#,(F(F(
FPF,F8F,F(F,*&\"\"$F(-FM6#F8F(F(*&F\\oF(-FM6#,&F(F(F8F,F(F,F(F," }
{TEXT 342 1 " " }}}{EXCHG {PARA 435 "> " 0 "" {MPLTEXT 1 346 22 "d2fdc
2:=diff(dfdc,ca);" }}{PARA 431 "" 1 "" {XPPMATH 20 "6#>%'d2fdc2G,(*&#
\"\"\"\"\")F(**,&F(!\"\"-%$expG6#,$*($\"\"&F,F(%\"JGF(%\"tGF,F(F(F,,,F
(F(*&\"\"%F(%#caGF(F,*&F8F()F9\"\"#F(F(*(F8F(F-F(F9F(F(*(F8F(F-F(F;F(F
,#!\"$F<,*F8F,*&F)F(F9F(F(*&F8F(F-F(F(*(F)F(F-F(F9F(F,F<F4F(F(F,*&#F(F
8F(**F+F,F6#F,F<,&F)F(*&F)F(F-F(F,F(F4F(F(F(*&F5F(,4*($\"#?F,F(,&*&F'F
(*(F+F,F6F?FAF<F(F(*&FFF(*(F+F,F6FHFIF(F(F,F(-%#lnG6#,&F9F(*&#F(F<F(*&
F+F,,&F(F,*$F6FZF(F(F(F,F(F,*(FNF(,&F(F(*&FFF(*(F+F,F6FHFAF(F(F,F<FXF,
F,*,$\"+++++]!#5F(F+F,F6F?FAF<-FV6#,$FYF(F(F(*,$\"+++++5!\"*F(F+F,F6FH
FIF(F`oF(F,*,F]oF(F+F,F6F,FAF<FfnF,F,*(FNF(FPF(-FV6#,(F(F(FYF,F9F,F(F,
*(FNF(,&FjnF,F(F,F<F[pF,F,*&\"\"$F(F9F,F(*&F_pF(,&F(F(F9F,F,F(F(F," }
{TEXT 342 1 " " }}}{EXCHG {PARA 436 "> " 0 "" {MPLTEXT 1 347 20 "eval(
d2fdc2,ca=0.5);" }}{PARA 431 "" 1 "" {XPPMATH 20 "6#,&*,$\"+++++D!#5\"
\"\",&F(!\"\"-%$expG6#,$*($\"\"&F*F(%\"JGF(%\"tGF*F(F(F*F+#F*\"\"#,&\"
\")F(*&F7F(F+F(F*F(F2F(F(*&F3F(,**,$\"+++++5!\"*F(F)F*F+F4F6F(-%#lnG6#
,&F0F(*&#F(F5F(*&F)F*,&F(F**&$\"$+\"!\"#F(F+FDF(F(F(F*F(F(*&$\"#SF*F(F
BF*F*$\"+++++7!\")F(*,F<F(F)F*F+F4F6F(-F@6#,$FCF(F(F*F(F*" }{TEXT 342 
1 " " }}}{EXCHG {PARA 437 "> " 0 "" {MPLTEXT 1 348 13 "solve(%=0,t);" 
}}{PARA 394 "" 1 "" {XPPMATH 20 "6#,$*&$\"+-wt1O!#5\"\"\"%\"JGF(F(" }
{TEXT 308 1 " " }}}{EXCHG {PARA 438 "" 0 "" {TEXT 349 34 "the \"%\" me
ans \"previous equation\"." }}{PARA 438 "" 0 "" {TEXT 349 104 "For squ
are lattice, we get k_B Tc =0.3606737602 J   (where one has to be care
ful about definition of J!)" }}{PARA 439 "" 0 "" {TEXT 350 116 "lets c
ompute the phase diagram, solve for df/dc=0  (because we are completel
y symmetric in composition about ca=0.5)" }}}{EXCHG {PARA 440 "> " 0 "
" {MPLTEXT 1 351 20 "t1:=solve(dfdc=0,t);" }}{PARA 431 "" 1 "" 
{XPPMATH 20 "6#>%#t1G,$*($\"+++++]!#5\"\"\"%\"JGF*,&-%#lnG6#*&,(*&$\"
\"#\"\"!F*%#caGF*F*-%$expG6#-%'RootOfG6#,@**\"\"%F*%#_ZGF*-F86#F@F*F6F
*F**(F4F*-F.6#,(*$)FAF4F*F*FA!\"\"*(F4F*FAF*F6F*F*F*FAF*F**&F4F*FDF*FI
*(\"\"'F*FDF*F6F*F***F?F*FDF*F6F*FAF*FI*(F?F*FDF*)F6F4F*FI*(\"\")F*-F.
6#,(*&F4F*F6F*F*FAF*F*FIF*FPF*F**(\"#7F*FSF*F6F*FI**F?F*FSF*F6F*FAF*F*
*(F?F*FSF*FAF*FI*&F?F*FSF*F**(\"\"$F*-F.6#,$*&,&F*FIF6F*F*F6FIFIF*FAF*
FI*&FgnF*FhnF*F**(\"\"*F*FhnF*F6F*FI*(FMF*FhnF*FPF*F*F*$F*F5FIF*,(F7F*
FaoFIF6F*!\"#F*F:F*FIF*" }{TEXT 342 1 " " }}}{EXCHG {PARA 441 "" 0 "" 
{TEXT 352 102 "specifying that ca and other variables are of type real
 might have simplified the expression above...." }}}{EXCHG {PARA 442 "
> " 0 "" {MPLTEXT 1 353 27 "plot(eval(t1,J=1),ca=0..1);" }}{PARA 416 "
" 1 "" {GLPLOT2D 495 495 495 {PLOTDATA 2 "6*-%'CURVESG6$7[p7$$\"+Q`!eS
$!#7$\"+:#fJp\"!#57$$\"+0k'p3%F*$\"+[!y3u\"F-7$$\"+S&)G\\aF*$\"+8Io?=F
-7$$\"+5G$R<)F*$\"+S->V>F-7$$\"+3x&)*3\"!#6$\"+yOsP?F-7$$\"+N@Ki8F@$\"
+AHj:@F-7$$\"+ilyM;F@$\"+%G`B=#F-7$$\"+*)4D2>F@$\"+G#[4C#F-7$$\"+;arz@
F@$\"+vaL$H#F-7$$\"+)4bQl#F@$\"+wsItBF-7$$\"+!y%*z7$F@$\"+'Q(eUCF-7$$
\"+iW8-OF@$\"+ww$Q]#F-7$$\"+XTFwSF@$\"+.4!)eDF-7$$\"+qMrU^F@$\"+q#)=lE
F-7$$\"+\"z_\"4iF@$\"+/n8aFF-7$$\"+l6m#G(F@$\"+O:+JGF-7$$\"+S&phN)F@$
\"+pg@)*GF-7$$\"+:ddC%*F@$\"+XJbdHF-7$$\"+*=)H\\5F-$\"+%fj2,$F-7$$\"+[
!3uC\"F-$\"+y)Rl4$F-7$$\"+J$RDX\"F-$\"+#)fcrJF-7$$\"+)R'ok;F-$\"+7CxPK
F-7$$\"+1J:w=F-$\"+y-f%H$F-7$$\"+3En$4#F-$\"+V&>_M$F-7$$\"+/RE&G#F-$\"
+-[G%Q$F-7$$\"+D.&4]#F-$\"+)fVHU$F-7$$\"+vB_<FF-$\"+C%3oX$F-7$$\"+v'Hi
#HF-$\"+7PH&[$F-7$$\"+(*ev:JF-$\"+Ma)z]$F-7$$\"+347TLF-$\"+*)zQJNF-7$$
\"+LY.KNF-$\"+\"G1%[NF-7$$\"+\"o7Tv$F-$\"+5[>lNF-7$$\"+$Q*o]RF-$\"+k^^
xNF-7$$\"+\"=lj;%F-$\"+1*H%)e$F-7$$\"+V&R<P%F-$\"+!f&R'f$F-7$$\"+Xh-'e
%F-$\"+#GlAg$F-7$$\"+R\"3Gy%F-$\"+0*3bg$F-7$$\"+.T1&*\\F-$\"+q+t1OF-7$
$\"+(RQb@&F-$\"+'4Gbg$F-7$$\"+=>Y2aF-$\"+=`S-OF-7$$\"+yXu9cF-$\"+H(Qof
$F-7$$\"+\\y))GeF-$\"+C.k)e$F-7$$\"+i_QQgF-$\"+QD8yNF-7$$\"+!y%3TiF-$
\"+EO_lNF-7$$\"+O![hY'F-$\"+5lb[NF-7$$\"+#Qx$omF-$\"+yeZINF-7$$\"+u.I%
)oF-$\"+M\"zz]$F-7$$\"+(pe*zqF-$\"+'[.X[$F-7$$\"+C\\'QH(F-$\"+499bMF-7
$$\"+8S8&\\(F-$\"+SzfBMF-7$$\"+0#=bq(F-$\"+5p/'Q$F-7$$\"+2s?6zF-$\"+r1
;WLF-7$$\"+IXaE\")F-$\"+#e:RH$F-7$$\"+l*RRL)F-$\"+x!p\"QKF-7$$\"+`<.Y&
)F-$\"+av/sJF-7$$\"+8tOc()F-$\"+;#R]4$F-7$$\"+\\Qk\\*)F-$\"+cNE6IF-7$$
\"+5ASg!*F-$\"+m+0cHF-7$$\"+p0;r\"*F-$\"+)=:U*GF-7$$\"+lTAq#*F-$\"+5\"
3?$GF-7$$\"+lxGp$*F-$\"+.GihFF-7$$\"+?-\"\\Z*F-$\"+&[.\\n#F-7$$\"+!oK0
e*F-$\"+oYprDF-7$$\"+l(z5j*F-$\"+Y3O9DF-7$$\"+]oi\"o*F-$\"+\"=k,X#F-7$
$\"+NR<K(*F-$\"+5j5xBF-7$$\"+<5s#y*F-$\"+a#p?H#F-7$$\"+!*3))4)*F-$\"+)
o=(RAF-7$$\"+l2/P)*F-$\"+FY;\"=#F-7$$\"+S1?k)*F-$\"+gI\\9@F-7$$\"+50O
\"*)*F-$\"+n<kO?F-7$$\"+!Q?&=**F-$\"+a5=U>F-7$$\"+b-oX**F-$\"+]zw>=F-7
$$\"+!>g#f**F-$\"+qk-S<F-7$$\"+I,%G(**F-$\"+eB$oj\"F-7$$\"++,jz**F-$\"
+!y7'p:F-7$$\"+l+U')**F-$\"+(\\ED[\"F-7$$\"+I+@$***F-$\"+NZ(=N\"F-7$%*
undefinedGF[dl-%&COLORG6&%$RGBG$\"#5!\"\"$\"\"!FbdlFcdl-%+AXESLABELSG6
$Q#ca6\"Q!Fidl-%*LINESTYLEG6#Fddl-%%VIEWG6$;FcdlF`dl;$\",$Gwx18F@$\",n
<F=l$F@-F]dl6#%%NONEG-%+PROJECTIONG6#F`dl-%%FONTG6$%*HELVETICAGFadl-%*
GRIDSTYLEG6#%,RECTANGULARG" 1 2 2 0 10 1 2 9 1 4 2 1.0 45.0 45.0 1 0 "
Curve 1" }}}}{EXCHG {PARA 443 "" 0 "" {TEXT 354 169 "This is the misci
bility gap in a square lattice for ferromagnetic interactions, if you \+
rotate it 90 degrees you can interpret it as the magnetization versus \+
temperature!" }}{PARA 444 "" 0 "" }{PARA 445 "" 0 "" {TEXT 355 111 "No
w, lets try to do a better job still by using a larger cluster, and re
ally put the CVM to use................" }}{PARA 446 "" 0 "" }{PARA 
447 "" 0 "" {TEXT 356 38 "SQUARE APPROXIMATION on SQUARE LATTICE" }}
{PARA 448 "" 0 "" {TEXT 357 166 "in the pair approximation we had to t
ake care of the fact that each point was part of z pairs, so ultimatel
y we had to subtract out (z-1) times the point entropy term" }}{PARA 
449 "" 0 "" {TEXT 358 57 "in the square approximation we generalize th
is rationale:" }}{PARA 450 "" 0 "" {TEXT 359 147 "each triangle is inc
luded in one and only one square, therefore the triangle entropy contr
ibution is completely covered in the square entropy term." }}{PARA 
451 "" 0 "" {TEXT 360 124 "each second neighbor pair is also part of o
ne and only one square, it also is completely covered in the square en
tropy term." }}{PARA 452 "" 0 "" {TEXT 361 193 "each nearest neighbor \+
pair is part of two squares, its contribution is going to be once over
counted by the square entropy term, so the nearest neighbor pair entro
py term will be subtracted once" }}{PARA 453 "" 0 "" {TEXT 362 179 "th
e point is part of 4 square, and part of 4 nearest neighbor pairs so i
t is counted +4 times and -4 times, or actually, it is not counted at \+
all, so we add the point entropy term" }}{PARA 454 "" 0 "" {TEXT 363 
224 "The cvm entropy in square approximation on the square lattice is \+
thus... +1 times the entropy term of the square, -1 times the entropy \+
term of the nearest neighbor pair, and +1 times the entropy term of th
e point............" }}}{EXCHG {PARA 455 "> " 0 "" {MPLTEXT 1 364 288 
"s_square:=-kb*(faaaa*log(faaaa)+faaab*log(faaab)+faaba*log(faaba)+faa
bb*log(faabb)+fabaa*log(fabaa)+fabab*log(fabab)+fabba*log(fabba)+fabbb
*log(fabbb)+fbaaa*log(fbaaa)+fbaab*log(fbaab)+fbaba*log(fbaba)+fbabb*l
og(fbabb)+fbbaa*log(fbbaa)+fbbab*log(fbbab)+fbbba*log(fbbba)+fbbbb*log
(fbbbb));" }}{PARA 431 "" 1 "" {XPPMATH 20 "6#>%)s_squareG,$*&%#kbG\"
\"\",B*&%&faaaaGF(-%#lnG6#F+F(F(*&%&faaabGF(-F-6#F0F(F(*&%&faabaGF(-F-
6#F4F(F(*&%&faabbGF(-F-6#F8F(F(*&%&fabaaGF(-F-6#F<F(F(*&%&fababGF(-F-6
#F@F(F(*&%&fabbaGF(-F-6#FDF(F(*&%&fabbbGF(-F-6#FHF(F(*&%&fbaaaGF(-F-6#
FLF(F(*&%&fbaabGF(-F-6#FPF(F(*&%&fbabaGF(-F-6#FTF(F(*&%&fbabbGF(-F-6#F
XF(F(*&%&fbbaaGF(-F-6#FfnF(F(*&%&fbbabGF(-F-6#FjnF(F(*&%&fbbbaGF(-F-6#
F^oF(F(*&%&fbbbbGF(-F-6#FboF(F(F(!\"\"" }{TEXT 342 1 " " }}}{EXCHG 
{PARA 456 "" 0 "" {TEXT 365 34 "faaaa = probability of AAAA square" }}
{PARA 457 "" 0 "" {TEXT 366 110 "note that faabb and fabab are not the
 same, we order the sites consecutively along the perimeter of the squ
are" }}}{EXCHG {PARA 458 "> " 0 "" {MPLTEXT 1 367 130 "eval(%,[fbaaa=f
aaab,fabaa=faaab,faaba=faaab,fbaba=fabab,fbbba=fabbb,fbbab=fabbb,fbabb
=fabbb,fbbaa=faabb,fbaab=faabb,fabba=faabb]);" }}{PARA 394 "" 1 "" 
{XPPMATH 20 "6#,$*&%#kbG\"\"\",.*&%&faaaaGF&-%#lnG6#F)F&F&*(\"\"%F&%&f
aaabGF&-F+6#F/F&F&*(F.F&%&faabbGF&-F+6#F3F&F&*(\"\"#F&%&fababGF&-F+6#F
8F&F&*(F.F&%&fabbbGF&-F+6#F<F&F&*&%&fbbbbGF&-F+6#F@F&F&F&!\"\"" }
{TEXT 308 1 " " }}}{EXCHG {PARA 459 "> " 0 "" {MPLTEXT 1 368 54 "eval(
%,fbbbb=1-faaaa-4*faaab-4*faabb-2*fabab-4*fabbb);" }}{PARA 431 "" 1 ""
 {XPPMATH 20 "6#,$*&%#kbG\"\"\",.*&%&faaaaGF&-%#lnG6#F)F&F&*(\"\"%F&%&
faaabGF&-F+6#F/F&F&*(F.F&%&faabbGF&-F+6#F3F&F&*(\"\"#F&%&fababGF&-F+6#
F8F&F&*(F.F&%&fabbbGF&-F+6#F<F&F&*&,.F&F&F)!\"\"*&F.F&F/F&FA*&F.F&F3F&
FA*&F7F&F8F&FA*&F.F&F<F&FAF&-F+6#F@F&F&F&FA" }{TEXT 342 1 " " }}}
{EXCHG {PARA 460 "> " 0 "" {MPLTEXT 1 369 25 "eval(%,fabbb=fbab-fabab)
;" }}{PARA 431 "" 1 "" {XPPMATH 20 "6#,$*&%#kbG\"\"\",.*&%&faaaaGF&-%#
lnG6#F)F&F&*(\"\"%F&%&faaabGF&-F+6#F/F&F&*(F.F&%&faabbGF&-F+6#F3F&F&*(
\"\"#F&%&fababGF&-F+6#F8F&F&*(F.F&,&%%fbabGF&F8!\"\"F&-F+6#F<F&F&*&,.F
&F&F)F>*&F.F&F/F&F>*&F.F&F3F&F>*&F7F&F8F&F&*&F.F&F=F&F>F&-F+6#FBF&F&F&
F>" }{TEXT 342 1 " " }}}{EXCHG {PARA 461 "" 0 "" {TEXT 370 111 "for th
e triangles we follow nearest neighbors so that the 1st and last (3rd)
 site a re second nearest neighbors" }}}{EXCHG {PARA 462 "> " 0 "" 
{MPLTEXT 1 371 25 "eval(%,fabab=faba-faaab);" }}{PARA 431 "" 1 "" 
{XPPMATH 20 "6#,$*&%#kbG\"\"\",.*&%&faaaaGF&-%#lnG6#F)F&F&*(\"\"%F&%&f
aaabGF&-F+6#F/F&F&*(F.F&%&faabbGF&-F+6#F3F&F&*(\"\"#F&,&%%fabaGF&F/!\"
\"F&-F+6#F8F&F&*(F.F&,(%%fbabGF&F9F:F/F&F&-F+6#F>F&F&*&,.F&F&F)F:*&\"
\"'F&F/F&F:*&F.F&F3F&F:*&F7F&F9F&F&*&F.F&F?F&F:F&-F+6#FCF&F&F&F:" }
{TEXT 342 1 " " }}}{EXCHG {PARA 463 "> " 0 "" {MPLTEXT 1 372 25 "eval(
%,faabb=faab-faaab);" }}{PARA 431 "" 1 "" {XPPMATH 20 "6#,$*&%#kbG\"\"
\",.*&%&faaaaGF&-%#lnG6#F)F&F&*(\"\"%F&%&faaabGF&-F+6#F/F&F&*(F.F&,&%%
faabGF&F/!\"\"F&-F+6#F3F&F&*(\"\"#F&,&%%fabaGF&F/F5F&-F+6#F:F&F&*(F.F&
,(%%fbabGF&F;F5F/F&F&-F+6#F?F&F&*&,.F&F&F)F5*&F9F&F/F&F5*&F.F&F4F&F5*&
F9F&F;F&F&*&F.F&F@F&F5F&-F+6#FDF&F&F&F5" }{TEXT 342 1 " " }}}{EXCHG 
{PARA 464 "> " 0 "" {MPLTEXT 1 373 25 "eval(%,faaab=faaa-faaaa);" }}
{PARA 431 "" 1 "" {XPPMATH 20 "6#,$*&%#kbG\"\"\",.*&%&faaaaGF&-%#lnG6#
F)F&F&*(\"\"%F&,&%%faaaGF&F)!\"\"F&-F+6#F/F&F&*(F.F&,(%%faabGF&F0F1F)F
&F&-F+6#F5F&F&*(\"\"#F&,(%%fabaGF&F0F1F)F&F&-F+6#F;F&F&*(F.F&,*%%fbabG
F&F<F1F0F&F)F1F&-F+6#F@F&F&*&,.F&F&F)F&*&F:F&F0F&F1*&F.F&F6F&F1*&F:F&F
<F&F&*&F.F&FAF&F1F&-F+6#FEF&F&F&F1" }{TEXT 342 1 " " }}}{EXCHG {PARA 
465 "> " 0 "" {MPLTEXT 1 374 22 "eval(%,fbab=faa-faab);" }}{PARA 431 "
" 1 "" {XPPMATH 20 "6#,$*&%#kbG\"\"\",.*&%&faaaaGF&-%#lnG6#F)F&F&*(\"
\"%F&,&%%faaaGF&F)!\"\"F&-F+6#F/F&F&*(F.F&,(%%faabGF&F0F1F)F&F&-F+6#F5
F&F&*(\"\"#F&,(%%fabaGF&F0F1F)F&F&-F+6#F;F&F&*(F.F&,,%$faaGF&F6F1F<F1F
0F&F)F1F&-F+6#F@F&F&*&,,F&F&F)F&*&F:F&F0F&F1*&F:F&F<F&F&*&F.F&FAF&F1F&
-F+6#FEF&F&F&F1" }{TEXT 342 1 " " }}}{EXCHG {PARA 466 "> " 0 "" 
{MPLTEXT 1 375 22 "eval(%,faba=gaa-faaa);" }}{PARA 431 "" 1 "" 
{XPPMATH 20 "6#,$*&%#kbG\"\"\",.*&%&faaaaGF&-%#lnG6#F)F&F&*(\"\"%F&,&%
%faaaGF&F)!\"\"F&-F+6#F/F&F&*(F.F&,(%%faabGF&F0F1F)F&F&-F+6#F5F&F&*(\"
\"#F&,(%$gaaGF&*&F:F&F0F&F1F)F&F&-F+6#F;F&F&*(F.F&,,%$faaGF&F6F1F<F1F=
F&F)F1F&-F+6#FAF&F&*&,,F&F&F)F&*&F.F&F0F&F1*&F:F&F<F&F&*&F.F&FBF&F1F&-
F+6#FFF&F&F&F1" }{TEXT 342 1 " " }}}{EXCHG {PARA 467 "" 0 "" {TEXT 
376 96 "gaa = probability for the second nearest neighbor pair (diagon
al of the square) being of AA type" }}}{EXCHG {PARA 468 "> " 0 "" 
{MPLTEXT 1 377 32 "s_square:=eval(%,faab=faa-faaa);" }}{PARA 431 "" 1 
"" {XPPMATH 20 "6#>%)s_squareG,$*&%#kbG\"\"\",.*&%&faaaaGF(-%#lnG6#F+F
(F(*(\"\"%F(,&%%faaaGF(F+!\"\"F(-F-6#F1F(F(*(F0F(,(%$faaGF(*&\"\"#F(F2
F(F3F+F(F(-F-6#F7F(F(*(F:F(,(%$gaaGF(F9F3F+F(F(-F-6#F>F(F(*(F0F(,(*&\"
\"$F(F2F(F(F?F3F+F3F(-F-6#FCF(F(*&,,F(F(F+F(*&F0F(F2F(F3*&F:F(F?F(F(*&
F0F(F8F(F3F(-F-6#FIF(F(F(F3" }{TEXT 342 1 " " }}}{EXCHG {PARA 469 "> "
 0 "" {MPLTEXT 1 378 68 "s_pair:=-kb*2*(faa*log(faa)+fab*log(fab)+fba*
log(fba)+fbb*log(fbb));" }}{PARA 394 "" 1 "" {XPPMATH 20 "6#>%'s_pairG
,$*(\"\"#\"\"\"%#kbGF(,**&%$faaGF(-%#lnG6#F,F(F(*&%$fabGF(-F.6#F1F(F(*
&%$fbaGF(-F.6#F5F(F(*&%$fbbGF(-F.6#F9F(F(F(!\"\"" }{TEXT 308 1 " " }}}
{EXCHG {PARA 470 "> " 0 "" {MPLTEXT 1 379 34 "eval(%,[fbb=1-2*fab-faa,
fba=fab]);" }}{PARA 394 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"%#kbGF&,
(*&%$faaGF&-%#lnG6#F*F&F&*(F%F&%$fabGF&-F,6#F/F&F&*&,(F&F&*&F%F&F/F&!
\"\"F*F5F&-F,6#F3F&F&F&F5" }{TEXT 308 1 " " }}}{EXCHG {PARA 471 "> " 0
 "" {MPLTEXT 1 380 27 "s_pair:=eval(%,fab=ca-faa);" }}{PARA 394 "" 1 "
" {XPPMATH 20 "6#>%'s_pairG,$*(\"\"#\"\"\"%#kbGF(,(*&%$faaGF(-%#lnG6#F
,F(F(*(F'F(,&%#caGF(F,!\"\"F(-F.6#F1F(F(*&,(F(F(*&F'F(F2F(F3F,F(F(-F.6
#F7F(F(F(F3" }{TEXT 308 1 " " }}}{EXCHG {PARA 472 "> " 0 "" {MPLTEXT 
1 381 43 "s_point:=-kb*(ca*log(ca)+(1-ca)*log(1-ca));" }}{PARA 394 "" 
1 "" {XPPMATH 20 "6#>%(s_pointG,$*&%#kbG\"\"\",&*&%#caGF(-%#lnG6#F+F(F
(*&,&F(F(F+!\"\"F(-F-6#F0F(F(F(F1" }{TEXT 308 1 " " }}}{EXCHG {PARA 
473 "> " 0 "" {MPLTEXT 1 382 31 "s_cvm:=s_square-s_pair+s_point;" }}
{PARA 431 "" 1 "" {XPPMATH 20 "6#>%&s_cvmG,(*&%#kbG\"\"\",.*&%&faaaaGF
(-%#lnG6#F+F(F(*(\"\"%F(,&%%faaaGF(F+!\"\"F(-F-6#F1F(F(*(F0F(,(%$faaGF
(*&\"\"#F(F2F(F3F+F(F(-F-6#F7F(F(*(F:F(,(%$gaaGF(F9F3F+F(F(-F-6#F>F(F(
*(F0F(,(*&\"\"$F(F2F(F(F?F3F+F3F(-F-6#FCF(F(*&,,F(F(F+F(*&F0F(F2F(F3*&
F:F(F?F(F(*&F0F(F8F(F3F(-F-6#FIF(F(F(F3*(F:F(F'F(,(*&F8F(-F-6#F8F(F(*(
F:F(,&%#caGF(F8F3F(-F-6#FUF(F(*&,(F(F(*&F:F(FVF(F3F8F(F(-F-6#FZF(F(F(F
(*&F'F(,&*&FVF(-F-6#FVF(F(*&,&F(F(FVF3F(-F-6#F^oF(F(F(F3" }{TEXT 342 
1 " " }}}{EXCHG {PARA 474 "> " 0 "" {MPLTEXT 1 383 22 "f:=J*(ca-faa)-t
*s_cvm;" }}{PARA 431 "" 1 "" {XPPMATH 20 "6#>%\"fG,&*&%\"JG\"\"\",&%#c
aGF(%$faaG!\"\"F(F(*&%\"tGF(,(*&%#kbGF(,.*&%&faaaaGF(-%#lnG6#F4F(F(*(
\"\"%F(,&%%faaaGF(F4F,F(-F66#F:F(F(*(F9F(,(F+F(*&\"\"#F(F;F(F,F4F(F(-F
66#F?F(F(*(FAF(,(%$gaaGF(F@F,F4F(F(-F66#FEF(F(*(F9F(,(*&\"\"$F(F;F(F(F
FF,F4F,F(-F66#FJF(F(*&,,F(F(F4F(*&F9F(F;F(F,*&FAF(FFF(F(*&F9F(F+F(F,F(
-F66#FPF(F(F(F,*(FAF(F1F(,(*&F+F(-F66#F+F(F(*(FAF(F)F(-F66#F)F(F(*&,(F
(F(*&FAF(F*F(F,F+F(F(-F66#FinF(F(F(F(*&F1F(,&*&F*F(-F66#F*F(F(*&,&F(F(
F*F,F(-F66#FcoF(F(F(F,F(F," }{TEXT 342 1 " " }}}{EXCHG {PARA 475 "" 0 
"" {TEXT 384 59 "Now, minimize f with respect to faaaa, faaa, gaa, faa
......" }}}{EXCHG {PARA 476 "> " 0 "" {MPLTEXT 1 385 17 "f1:=eval(f,kb
=1);" }}{PARA 431 "" 1 "" {XPPMATH 20 "6#>%#f1G,&*&%\"JG\"\"\",&%#caGF
(%$faaG!\"\"F(F(*&%\"tGF(,8*&%&faaaaGF(-%#lnG6#F1F(F,*(\"\"%F(,&%%faaa
GF(F1F,F(-F36#F7F(F,*(F6F(,(F+F(*&\"\"#F(F8F(F,F1F(F(-F36#F<F(F,*(F>F(
,(%$gaaGF(F=F,F1F(F(-F36#FBF(F,*(F6F(,(*&\"\"$F(F8F(F(FCF,F1F,F(-F36#F
GF(F,*&,,F(F(F1F(*&F6F(F8F(F,*&F>F(FCF(F(*&F6F(F+F(F,F(-F36#FMF(F,*(F>
F(F+F(-F36#F+F(F(*(F6F(F)F(-F36#F)F(F(*(F>F(,(F(F(*&F>F(F*F(F,F+F(F(-F
36#FZF(F(*&F*F(-F36#F*F(F,*&,&F(F(F*F,F(-F36#F\\oF(F,F(F," }{TEXT 342 
1 " " }}}{EXCHG {PARA 441 "> " 0 "" }}{EXCHG {PARA 477 "" 0 "" {TEXT 
386 100 "analytical solutions get to be tough to derive - even with ma
ple!!!  (can try series expansion etc.)" }}{PARA 478 "" 0 "" }{PARA 
478 "" 0 "" {TEXT 387 11 "POSTSCRIPT:" }}{PARA 478 "" 0 "" {TEXT 387 
89 "This work is not an endorsement of Maple or Mathematica or any oth
er commercial software " }{TEXT 387 96 "package.  There are a number o
f freeware packages also that maybe of interest for your own work:" }}
{PARA 478 "" 0 "" }{PARA 478 "" 0 "" {TEXT 387 22 "http://www.octave.o
rg/" }}{PARA 478 "" 0 "" }{PARA 478 "" 0 "" {TEXT 387 30 "http://maxim
a.sourceforge.net/" }}{PARA 478 "" 0 "" }{PARA 478 "" 0 "" {TEXT 387 
28 "These two appear quite nice." }}}{PARA 479 "" 0 "" }{PARA 480 "" 0
 "" }}{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }