~~ ra 1-~ ~ c.f-tna fA .rU{ ~l - Eduardo Fradkin...

28
..... 4\ ra 1- "c.f-tna fA .rU{ J -tlu -.f:S2- ...... ::r. ... + s-* ) Ro< --e.qrJR YfU.M..?:> ,'+ l'S. J.So h> .. Oo"? s::= k '-"(£) =- L(E) ikt. fJ +<..c:t- a - 5( - t-L-.- t;) 1 S( f',,.> V)) '\-Lv-

Transcript of ~~ ra 1-~ ~ c.f-tna fA .rU{ ~l - Eduardo Fradkin...

Page 1: ~~ ra 1-~ ~ c.f-tna fA .rU{ ~l - Eduardo Fradkin (Physics)eduardo.physics.illinois.edu/phys504/homepage/gibbs-distribution.pdf · ~Js1-~~

~~ ..... 4\ ra 1-~ ~ "c.f-tna fA .rU{ J ~l.fk

~f~~~ -tlu -.f:S2-...... ~~ n:c.~ ~~C'J .A.u..~~

::r. ... ~(\A!> + s-* ~ ) Ro< V~j --e.qrJR YfU.M..?:> ,'+ l'S. J.So ros~~ h> ~ ~ ~ ~t .. ~ Oo"?

s::= k ~ '-"(£)

~ =- ~ ~ L(E)

ikt. ~4(~ ~ ~s fJ +<..c:t- ~ a lqv~ ~

~ Jc.~t~,V'-? ~~ ~ 1-e\~ +~~1' ~ -

5( -

~~ ~~ t-L-.- ~ (1cJ~'(~ A\,~ ~N)

~r~ (-A-~' )C~ol -h,~ ~n t;) 1 S( f',,.> V))

W~ L~ ~ ~~( 1~C4.>Jv7\.--")

~ '\-Lv- ,,~.~

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\ _(qS) T :::- ~ 2)£ \I

~ T=@!) ~SV

CA·' oJ. -

Fr-ee. ~~; F::: £ - TS

C;"'b(,s F.".u ~: G:: E 1 P V - TS

dv

S(,<c:.(·f.9C l..t4- (~.vr C"'P~"',) Cit ~ ~) = T ~\ toT V Cor)v

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5) -.§cMF'fHhd'-l 1"bf-ec...

c#;t x.,' k -fA '.f~ 1>," 4T iI,'

----

=- S cil"Jr -i [xcCH-E)] 1-1<. ~ d~'

- 5 tI.'Il /\ )p dt\' (1-1 - t) H< 6 ~ ~

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--

<Xi~.,>::--A. R [OtjJ d.~/ldp Qi-c)l O-"J ree,) dE J

H<[;

~t(. r(~)= CU(~)Ll ~

--

~~.~-c~ <£'7~~ .-1)'rj)r. J~~.

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o/-I ~H . ----., ::::. -p, oT,' ( -- ::-

~tV :::) < L Fvi-,,).:- - 3rvJe.T V;::: 1

JoT kO\(~C ~ c::: (3(\1) I k T

CV==~ =- le-~~) \jT')V t

:: \N b

--

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I(e).= J d>' ot~t.Jp

r--

~ln (,r);;. ch r n. -'«> +co

C~~ S S d"x "tV ..... Q)

{o- 1- ~

Sd)':," "" JtA)f~ -..00

-

? J a\ Ii l< y

~ ~harr-)v~ o~ (J. s~r.t ~ f~tl.\v..t ( l.v\

t'\ rJ..A~" cJ'l\1

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--

r(~-+I)

F,y }14 ~ (St;(~ ~V~)

S(£,V)= k~ ICE)

S(S,V)~ k~ c3rJ -~ N~ + #N~r"tr e. ... :/J..] U-2.1T't)J

~ N.':: N ~ N - IV + .. , l N-i (0) ( )1; r ~)

+ ' t •

Page 8: ~~ ra 1-~ ~ c.f-tna fA .rU{ ~l - Eduardo Fradkin (Physics)eduardo.physics.illinois.edu/phys504/homepage/gibbs-distribution.pdf · ~Js1-~~

-

T:: ~)V~ 2. e -=> [ E = o /Vie, r 1 - - -

3 Nie 'l.-

-;:!") [CV:: a N I(. J D~-

, <1 ~ fC1 (" IV.h~) - P-V-I' b

p:>-~l= .. ~ E -2> 1/ s 6 V

61'~~.s P~ .. ,,~ . . ------

c~ .. ~ i W 0 ,'~0.9.. ~ w i f. ~ /oJ, J o • . -cl. tv;.

'f~(~ Wl·+~ v,A?,...- J V, ~ V'2,. "~. w~

Page 9: ~~ ra 1-~ ~ c.f-tna fA .rU{ ~l - Eduardo Fradkin (Physics)eduardo.physics.illinois.edu/phys504/homepage/gibbs-distribution.pdf · ~Js1-~~

~ t:aa;<"f ~ ~lW.c.cl f.o ~~ ~ Cot \/~ V--VJ~V~

v::v,+v ....

~1'-~~'j c~~ l'~

~,~~ c.k~~

~~(~'~~~

VJ~) -==') ~

fis:: N,~ ~C~) oj- N,.1z ~({);>O

-::) -t{..... .o-A-1J ai .. ,-k;t.. 1'" '3<1/.> c.J~~ ~

~+ ,,- ~ ----1~·~'~v:.-. )~~~ u-~ ~lfs ~'l~

r ~ v~ v OJ ....,R it t ~rrfl'~ I\J. TL.."~

~ rc..&.o x- ( G,,~ b..s 'fc:vok>c) W Q/) ~ \J..A:.cJ... ~ 67-1:"t..s

Lv "'-0 t:lJ. u.~ to olt v~·cl.tt 2 (e ) '? if ..l.. ( (.WI~ N~

~ ~h~"c..Jiif"'), 'Tk"'> c~ ~Ju-"~~c:J.. ~d-

, ..... 4u....- ............ <\..~ /?--ee ~ f'f-il:. ~ 1tu.

Page 10: ~~ ra 1-~ ~ c.f-tna fA .rU{ ~l - Eduardo Fradkin (Physics)eduardo.physics.illinois.edu/phys504/homepage/gibbs-distribution.pdf · ~Js1-~~

~"'- ,'(" -". .... ~4 ........ k.. f{,... Jl.-)<"c.I..CU(? -1 L'~~ rarty~. ~

T~IA.~"c" ~ ~ C>~\cJ. Ch~tA

~ ~ C~U ~~ ~ -a..t.t..CO() c'J

(kat {I'IC-ec£ t:xd ~ T~~~ r'> . ItJ-

~ het ~', -h, rf-cl ~ ~tJlA..., ~ c. ~

Cd {+(.. ~ ~ (oS

If' ( e-).:= ...!... e - S IkT ~

60 -

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(;/

-fk-~CUAA.~ F;1A.£. t: ......... !J- C H....f ...... t..."..UJ-)

(!) F,.. ~ J.?ctA..t.A.i1'.-( ~ ~ ~ 1w 0 ~ c~d ~

c') ,.., ~ H J. + M ~

C0 L~t (..( '> c.~ .... E:::- (Hi»

~f _ k~t; - itT 'dt--» - - • ';)T "l aT

, ~e I t!ri- ) e - En /", - - - L ~ - -2 1'\

'OT

dF - == aT

) < t-I) ::. - T(~ ) + F "" T 1/

Page 12: ~~ ra 1-~ ~ c.f-tna fA .rU{ ~l - Eduardo Fradkin (Physics)eduardo.physics.illinois.edu/phys504/homepage/gibbs-distribution.pdf · ~Js1-~~

F=. <I-f) T T(~\ lb,)v

(H);::- E (1~t(~kl1l~ ~(tj)

~ r -CEJ)V:::- .s I '-.s it... -e--ry'/iJ

~ ~ f~'t;~ ~c1t0'--\ 2 ---7 ~

~4 ~ ~!J r-:: - kT~"t c..V; V)

'l?:::- - (~) dV T

£~ -~)v G:: F + T>V (6)~b{,s tv~~ )

~ E. = (ttl=:" t= 1" T S C l~f"~~ ~c.ru )

Page 13: ~~ ra 1-~ ~ c.f-tna fA .rU{ ~l - Eduardo Fradkin (Physics)eduardo.physics.illinois.edu/phys504/homepage/gibbs-distribution.pdf · ~Js1-~~

6s -Tk ~1""!A ~ ~ ~\'1'~ MQt.,.(~

~ ~ t.~cc:JL ~ ~ ~~~ Mclh'~"r A

Z ::- ~ e-{iti =: L: e- {l£h

S~

w"- cJt.,..o ~."'~ M-.o..:f ~ ~("~..s c..~ ~

c~~w<.cR. {4--~ ~ T(~ ~ld ~

s ~ - (£1:1 bT )\/

=> $= ~ ~~ -\- ~ ~ = k~3 ~ + v 2. 2)T -r

.., ..-"\ ( "" ,Cl.I-J ) ""'" "'" ) v:::- (E';>::: (H'> = ...L "'t:r H e-r = tr~ I-{

~

0", ~ ~~ kc.e.v--cL. ~ ~~Sh'-'-.

::!) ~ == - ~ t{f ~f ) 1 "" - k. ~ [!" ~ 'P,,] ( v ff"\ rveU\'-'<II\~) T

r(O~ ~,lt'S 1 ~ ~~c.b .

Ttw ... \'~ "'" G<>~ ---.-R 1-..............t ~ r S.

Page 14: ~~ ra 1-~ ~ c.f-tna fA .rU{ ~l - Eduardo Fradkin (Physics)eduardo.physics.illinois.edu/phys504/homepage/gibbs-distribution.pdf · ~Js1-~~

e~~ F\'-t~~~~: Co~d1d'-\ ~t~\ ~ M.tf(.r~~~ ~tLt ~"~~~

~ ~w",,~.j, ~

~~i U$ c.~ ~ ~ ~'-c..qV~,p.lu.c cr~c:S/~

C va '(\\~q ) ~ ~ ~~ ~ ~ c.~l\c-.Q.

~J.R;..t . t...~tt 1).::::. (H,) ~ +l..t ~ +~...Q.

~

)

Page 15: ~~ ra 1-~ ~ c.f-tna fA .rU{ ~l - Eduardo Fradkin (Physics)eduardo.physics.illinois.edu/phys504/homepage/gibbs-distribution.pdf · ~Js1-~~

F= -- k."T ~6

d~~_ J

-of?> - 2 2E: = J.. ~ tr e-r;f1 dP> ~ df'

::: -:.1 t ... (H e-foH ) = - (H'/ 2

d2~ _t _ 2... "",1.-'2 ~ _ .l- (g '\ + ..L v ~ - ofO'l,- - - 2:" "d~) 6 d~"'-

;~ :: ~ [_ Vr 01 e-jH1 )j

== + t ... t "1. e-f.>tI )

-

?<.~ ~::. .( 142"> - <..~ .... /:::: < Ctt - U- )t.) '7 (l ~~1-

::: - --L E1 (- d'2-) ~ --L -tr Q4 2. e-(31-f ) Z 7- d~ 0[0 ~

:= .( H )~ - ~ ",,1.) =- _ <. Cti -U\'"")

Page 16: ~~ ra 1-~ ~ c.f-tna fA .rU{ ~l - Eduardo Fradkin (Physics)eduardo.physics.illinois.edu/phys504/homepage/gibbs-distribution.pdf · ~Js1-~~

-

~') ~T1.(~\::: < (tt--VJ'-):: <H~) - <tl,/~ \...oT/V

') "f\ .... ~~ ~~ ~ cp+()2.:=. <t-I z >- <1-1')2-

~.~ 'f~-c~ -to ~ ¥~'f"c.. \'U..c:t (k.e4 c.~ c.'~ )

1-c-, ~"H) 0( N

-.::-) Ali 0( ~

L... .!t..... ~l.--l ~,+- v- l$ C>. .k"'~t-?­

~ i"'~b ~ C AH -I 0), /JoN<JL ~

KcZt ~~ ~~~ '('"-!..7v.A~ ~ Cv <.00 .

-rlMn. ~ tltv~ C $ V'ltrij-(,cJ... ttd-c.... cV'-l1~ r~C4J-( TrC1.C..( ~P'v,.

Page 17: ~~ ra 1-~ ~ c.f-tna fA .rU{ ~l - Eduardo Fradkin (Physics)eduardo.physics.illinois.edu/phys504/homepage/gibbs-distribution.pdf · ~Js1-~~

~ (VJ>..~;fV ...... ~ T $(£) -t:, ~c.~ g~Orl"rJ;r

~ . b '-'-> ~< " r ~ s« J.J.J.. r'i-

N :f ()(") e ~

- "'0 -~

w~ '>(0/ , f (xo ) ::: 0

I -t ~ \\ C'G» < 0

Page 18: ~~ ra 1-~ ~ c.f-tna fA .rU{ ~l - Eduardo Fradkin (Physics)eduardo.physics.illinois.edu/phys504/homepage/gibbs-distribution.pdf · ~Js1-~~

2£ 1 =: 0 :::'! (T (~) - 1 ) of. N ~oE

1J \I

(~) = 1... =-) \Z)c; V T

(d2s ) _ d ('dS) _ d fl) '" _...L (~\ aB"" - ~ ~u- - ~ IT - T1, oEI

E:: V-\ - --- ,1.

-

"\ 'loS I ) ~ s-(t.) - C; = [T S( t.r) - (r J + ~ e -ri) T ~() .) +,.-

2.. O£'- E-=rr

=: (!S('CJ) - U- ) ~ ) _ ' a--tr) 'L-t • ( . "2. \ <:"

Page 19: ~~ ra 1-~ ~ c.f-tna fA .rU{ ~l - Eduardo Fradkin (Physics)eduardo.physics.illinois.edu/phys504/homepage/gibbs-distribution.pdf · ~Js1-~~

~)

f.> C 7 S (U) - U) J -~ C c,- (,)) 2-

e rJ. E. e .27'Cv [ It· ::J

F tr --- ': --... N tJ

v - T S(u) - .!=2 ~ ~ffIc.1'cv) t··

.. T stO ) _ -tv

'2..

k T_ ~ 8Tik. rt.cV ) ,. ...

d..N

=") +\..... tcJ ~ cf-,. ~ ~i\M·V.....Q-." 1- ho tL"

~~~ IJ ~ ~r~~ 1- ~ ~t("1 WI",- ~ 1tu.. ~~ . T~~ fo\lnvs

fr~ ~ ~~~ o,f-~ ~~ 1r ~ (.vet:) rV e E

Page 20: ~~ ra 1-~ ~ c.f-tna fA .rU{ ~l - Eduardo Fradkin (Physics)eduardo.physics.illinois.edu/phys504/homepage/gibbs-distribution.pdf · ~Js1-~~

~Js1-~~ <--"",.+~, Vot~l~~ ~l..v-1 ~r1j~ ..

T~ ~ ~~ i5",~~

W...( \A~v< dc~CM..s.k.J ~ fo.r -tw() ~ 1 S,h::d-L'lf/r..f)

~sc.U. ~'cJ-> (~ ~~~) :~)~ ~~cr<~~..J)

~\4. ) ~ wlM.c.~ tL.... JU A Q.r~ t'J <p~)e~of ~ C\..

'\'lu +~rJ2.-t'~ .:> d2.."l'rtc;;( ~~ I 0J4 (6) ~

c~cJ ~ ~ c..0~C~ ~ t-t2.A.M~

C,> ~~~ \'..4" ~ ~ f'v..d"~~ . \I..Ll ~

~ <:4..vJ +~ ~ ~ -+~~c. 0;. .... -+ bu~

¥ +tu. ~ lx.r 1 pa r h'cJa.,.. tv ~1C-< cJ .

W JL l,..t9d \ t-N<w-7\ ,~ <\)...() t.J ~ c~ ~ c..v ~"t ~ c.u ~

F,h'c..tz., ') Nl. ,;» N1 . TLu. kr~ ~~ ~~

v~ Vl. >':> \/, ,~ v~ .... ~ 1~ ~

~L.\.., • fk.... +O~ H~o.&I-~I~~ ,'J

Page 21: ~~ ra 1-~ ~ c.f-tna fA .rU{ ~l - Eduardo Fradkin (Physics)eduardo.physics.illinois.edu/phys504/homepage/gibbs-distribution.pdf · ~Js1-~~

L<Ut" N~ to \ i- tJz.. ,) v=' V. +V2..

N

2:N

( v) T) ::: r ~1.J( Vi)')~ N::-o •

1

'T\.u. 'r'0 ~l,.,'k~ ~ ~ ~

~ l~i ~ ~'-~ ~J P,

~Nl (Vt;r) -'2M (V,,)

r.~ lVl.) T) 'l.

~ (N - )J I) V-v I ) 7) - F (N > V ) C ') = - N" fA. +- V~ 'P of.'.

(~~ tJ» IV, ) V').) V,

c.ukMt r- t~ t..~ ~ ck...,~"~ed.:

fA- "'" 2£CIJ"V,iJj) ,', ~ , l-t.... 1~ u-.:t ttf o.~ d NL ""'~~ 0- ~r h~ _

~ P.::. -. .Q.f I rt \.1 ~ \.1 _ 1.1

Page 22: ~~ ra 1-~ ~ c.f-tna fA .rU{ ~l - Eduardo Fradkin (Physics)eduardo.physics.illinois.edu/phys504/homepage/gibbs-distribution.pdf · ~Js1-~~

F~ ~o. ""'ts : ~= e~~ =, ~ t; ~ PfA

.=, PC7J, PJ N).::: ~N e-p:f v -t3 HC 1,pJ -(VJ

~ ?~Y~ffcM T~c1)'cM

c:>o tv % fil;) V) ,) ==- LN::::" () Z =6N CV)7)

Lt id.w,,'1;( t<u w~ kJ2.. .QJI ~ ~ ~ ~ ,,'.J

~ .... :: _ ..:i ~ ~ (~}v) T}

Cf-> CI-E:'1r~ ~ ~

ctf:~ = T &. S - P c:i \I t- fA ctJJ ~ ~ PDt~d ~ .f2. = F - fJ. IV

cl.Jl ::: - oS' d. T - tV of}J- - P ct V

12.

Page 23: ~~ ra 1-~ ~ c.f-tna fA .rU{ ~l - Eduardo Fradkin (Physics)eduardo.physics.illinois.edu/phys504/homepage/gibbs-distribution.pdf · ~Js1-~~

Tk,.. a-,·Io~S ~t~ Ite>J (f~~) r N f;'~~

o~ 4 G:= _ 5 ctT +- V dp

~ G- CF> J V ~ tJ) =- fA I\J

~ Sl-:=. F- fA tJ "" i="- &~l __ l'Vd2. J ~ n.. &a.t~'f~~ ~ -eiu..~~ 1- st-c.k

xv ~ ~ 1: (~)V) T) kT- J

~ <~> = N =- k T ?~ z; 'Of>

Page 24: ~~ ra 1-~ ~ c.f-tna fA .rU{ ~l - Eduardo Fradkin (Physics)eduardo.physics.illinois.edu/phys504/homepage/gibbs-distribution.pdf · ~Js1-~~

KN.:: j <@rJ'/)::: /<N-) y k T.!!:I 1

- -- ...." ---tv

(<A1l» = ?

J~ (/ (.,(.a. 'b~cJ'-

~ < (PN)\..) =

0-

J -

( J( T -f> 1>0) C

v -.. e>O J Tc ~:> < (p e ) 'l..) --, ~

Page 25: ~~ ra 1-~ ~ c.f-tna fA .rU{ ~l - Eduardo Fradkin (Physics)eduardo.physics.illinois.edu/phys504/homepage/gibbs-distribution.pdf · ~Js1-~~

------

IV .... "l.

~rJ( V) T)= VIII J c:t)"'p -z Pc..'

e C:=, -- ~ l?1k.T N{ (3iTt)3'" ,

aN iCV

VN .... - J ~\' e. -"'! ~

.... Q>

J~J ~N

- ~N[ ~Ti - .. N' .)T71i" •

V N -Vt.J IN ~N J

tk~)~ ( QU'o; ~ /.. ; ") ~""h -c::..,.... - - -N~ N~ 2. 7Ti;'"

::::-) Ff\} (V)T)= - k7 ~ ~N

rrJCII)T):= - kr Ne-~~V-.f-l ~ ~~J

-l-k.T N~kJ-11

Page 26: ~~ ra 1-~ ~ c.f-tna fA .rU{ ~l - Eduardo Fradkin (Physics)eduardo.physics.illinois.edu/phys504/homepage/gibbs-distribution.pdf · ~Js1-~~

r--- (00 ~i)'~ F,.e....cA" .... ottc""l

(N'>- <N)~=-«~N) >=:

~ fA-= j(v)T) - 1> ~.f('l)JT) » ocr

j.J. ==- ~ (0) T)

~= :£ ('D"l -r)

g? ~? - ~ J... --- - »

"d/, ~ 1)-

0'-'"

""" -

v-

1<- -I .. (),f~~ c.cI\oN'\~S~'~~~.

Page 27: ~~ ra 1-~ ~ c.f-tna fA .rU{ ~l - Eduardo Fradkin (Physics)eduardo.physics.illinois.edu/phys504/homepage/gibbs-distribution.pdf · ~Js1-~~

v-= V -tv

Ftv

(V) T):::= N S C 0-) T)

f (~) T)== - kT[~ ~-f'lhC~;:J -t tJ cs ~ ~~~\~.

C~~LJ. ~~V~ f=- (2 F.) = (2) cr) 'OJV V~7 ~ p} T

)J-::::. it-r +- 1- (V~ T)

jA--= - ~ T [~ v -t~ ~;;JJ

Page 28: ~~ ra 1-~ ~ c.f-tna fA .rU{ ~l - Eduardo Fradkin (Physics)eduardo.physics.illinois.edu/phys504/homepage/gibbs-distribution.pdf · ~Js1-~~

c(c."9i~ -f:Jro.-J.. - p&\ r ttty@1 F ~J).1'1 : 1ch-~ ~.s. ------------------

~ =:- e t-A,t: '7

~ 10" ~[.g.(~~~~f/~]~~G>!)