3.the canonical ensemble 1.equilibrium between a system & a heat reservoir 2.a system in the...
TRANSCRIPT
![Page 1: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/1.jpg)
3. The Canonical Ensemble
1. Equilibrium between a System & a Heat Reservoir
2. A System in the Canonical Ensemble
3. Physical Significance of Various Statistical Quantities in the Canonical Ensemble
4. Alternative Expressions for the Partition Function
5. The Classical Systems
6. Energy Fluctuations in the Canonical Ensemble:
Correspondence with the Microcanonical Ensemble
7. Two Theorems: the “Equipartition” & the “Virial”
8. A System of Harmonic Oscillators
9. The Statistics of Paramagnetism
10. Thermodynamics of Magnetic Systems: Negative Temperatures
![Page 2: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/2.jpg)
Reasons for dropping the microcanonical ensemble:
1. Mathematical: Counting states of given E is difficult.
2. Physical: Experiments are seldom done at fixed E.
Canonical ensemble : System at constant T through contact with a heat reservoir.
Let r be the label of the microstates of the system.
Probablity Pr ( Er ) can be calculated in 2 ways:
1. Pr # of compatible states in reservoir.
2. Pr ~ distribution of states in energy sharing ensemble.
![Page 3: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/3.jpg)
3.1. Equilibrium between a System & a Heat
ReservoirIsolated composite system A(0) = ( System of interest A ) + ( Heat reservoir A )
Heat reservoir : , T = const.
Let r be the label of the microstates of A.
0r rE E E r with 0
r rE E E
r rP E Probability of A in state r is 0
rE E
0
0 0lnln lnr r
E E
E E E EE
rconst E
,
ln
N VE
1rE
rP eZ
rE
r
Z e
Classical mech (Gibbs –corrected ):
3 3 ( , )3
1
!N N H q p
NZ d p d q e
N h
![Page 4: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/4.jpg)
3.2. A System in the Canonical Ensemble
Consider an ensemble of N identical systems sharing a total energy E.
Let nr = number of systems having energy Er ( r = 0,1,2,... ).
rr
n Nr r
r
n E U E N
U EN
= average energy per system
Number of distinct configurations for a given E is
!
!r rr
r
W n W nn
N
{ nr* } = most probable distribution
Equal a priori probabilities 1r rP n W n
* maxr rW n W n
,
r
rn
W n N E
,1
r
s s rn
n n W n N E (X) means sum includes only
terms that satisfy constraint on X.
0 1 2, , ,rn n n n
![Page 5: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/5.jpg)
Method of Most Probable Values
!
!rr
r
W nn
N ln ln ! !r
r
W n N
ln ! lnn n n n ln ln lnr r
r
W n n N N
rr
n N
To maximize lnW subjected to constraints
ln 1 0r rn E
rr
n N r rr
n E E
, are Lagrange multipliers
1* rErn e rEC e
is equivalent to minimize, without constraint ln r r rr r
W n n E N E
![Page 6: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/6.jpg)
* rErn C e
*r
r
n N *r r
r
n E E
rE
r
C e NrE
rr
C e E E
r
r
Er
rE
r
e EU
e
EN
* r
r
Er E
r
n ee
N
Same as sec 3.1* 1
rErr
ne P
Z
N
rE
r
Z e Let
E.g.
r rr
U P E
1
kT
and set
with
![Page 7: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/7.jpg)
Method of Mean Values
Let 0
0
!
!
rnr
rr
rr
W nn
N
Thus 1r
X X
,,
r
rn
U W n N EN r
r
r rr
n
n E U
N
E NConstraints:
,1
r
s s rn
n n W n N E
1r
s
s
,
r
rn
W n N E
ln 1s s
s s
1r Note
:
,1
r
s rn
n W n N E
r in { nr } is a dummy variable that runs from 0 to , including s.
0
!!
rnr
r rn
N
,
0
!!
r
r
nr
n r rn
N E
N
1
lnr
ss
~ means “depend on {r } ”.
![Page 8: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/8.jpg)
Method of Steepest Descent ( Saddle Point )
,
0
, !!
r
r
nr
n r r
Un
N E
N N is difficult to evaluate due to the energy constraint.
Its asymptotic value ( N ) can be evaluate by the MSD.
Define the generating function 0
, , U
U
G z U z
NN N
,
0 0
, !!
rr
r
nEr
nU r r
zG z
n
N E
N N
r rr
n E U E N
Binomial theorem
r rn EU
r
z zN
U removes the energy constraint.
, rEr
r
G z z
N
N
0
!!
rr
r
nEr
n r r
z
n
N= N
f z N
rErr
f z zwhere
![Page 9: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/9.jpg)
0
, , rEUr
U r
G z U z z f z
NNNN N
N U = integers = coefficient of zN U in power expansion of .
This is the case if all Er , except the ground state E0 = 0, are integer multiples
of a basic unit.
1
1,
2 UC
f zU d z
i z
N
NN
C : |z| < R
analytic for |z| < R
Let 1
g z
U
f ze
z
N
NN
( For { r ~ 1 }, sharp min at z = x0 )
1ln lng z f z U z
N
,U N G
f z
Mathematica
![Page 10: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/10.jpg)
1ln lng z f z U z
N
1 1fg U
zf
N
0
0
00
1 10
f xU
xf x
N
2
22
1 1f fg U
zf f
N
N >>1
00
0
f xU x U
f x
2
00 2
0 00
f x U Ug x
x xf x
Fo z real, has sharp min at x0
20 0 0
1
2g x i y g x g x y
0 0g x 0 0g x
For z complex : max along ( i y )-axis
x0 is a saddle point of .
g
g
![Page 11: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/11.jpg)
20 0 0
1
2g x i y g x g x y
1
g z
U
f ze
z
N
NN
0 20 01
0
1exp
2U
f x i yg x g x y
x i y
N
N N N
0 2
010
1exp
2U
f xg x y
x
N
N N
1
1,
2 UC
f zU d z
i z
N
NN
MSD: On C, integrand has sharp max near x0 .
0
0
1
0
1
2 UNear x
f x i yi d y
i x i y
N
N
0 2
010
1 1exp
2 2U
f xg x y d y
x
N
N N
Gaussian dies quickly
0
10 0
1 2
2 U
f x
x g x
N
N N
![Page 12: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/12.jpg)
0
10 0
1 2,
2 U
f xU
x g x
N
NNN
20
0 00
1ln ln ln 2
2U
f xg x x
x N N
0
0
lnU
f x
x
N
N
rErr
f z z
0 0ln ln lnf x U x N
0
0
s
r
E
E
r
x
x
N
0
0
s
r
Es
Er
r
x
x
N
1
lnr
s ss
n
10 00 0s rE E
r rrs s
f x xx E x
00
0
sE
s
U xx f
x
0
0
r
r
Er r
rE
rr
E xU U
x
0 0 0
0 0
lnsE
s ss s s
x U x U x
x xf
N
0sE
s
x
fN
![Page 13: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/13.jpg)
0x e
0
0
s
r
E
s E
r
xn
x
N
0
0
r
r
Er r
rE
rr
E xU
x
0 0ln ln lnrEr
r
x U x
N
C.f.* 1
rErr
ne P
Z
N
With { r = 1 } :
r
r
Er
rE
r
E eU
e
ln ln rE
r
e U N
rr
r
nE N
1s
s
r
EEs
E
r
n ee
e Z
Nso that
ln Z U N
rE
r
Z e
0 1r
Z f x
rE
(r) r is a dummy variable
sEs rs
sr r sr
nn eP P
Z
N N
r rr
E P
![Page 14: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/14.jpg)
Fluctuations
0
!!
r
r r
nr
rn n r r
W nn
N
2
1
1
r
s s ss s
n
1r
ss
s
n
22
s s sn n n 22 2s s s sn n n n
22s sn n
ln ss s s
s s s s
22
2
1 ss s
s s s
22
1
ln
r
s s s ss s
n n
2
sn
where
![Page 15: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/15.jpg)
0lnsE
ss
s
x
f
N
0rE
r
Z x
0 0s sE E
s
x xn
Z f N N
00
0
sE
s s
f U xx f
x
0lnsE
ss s s
s s s
x
f
N
10 0 0 0 0
020
s s s
s
E E EEs s s
ss s
x E x x x U xx f
xf f f
N
2
0
0
s ss s s s s
s
n xn n E U
x
N
0
0
r
r
Er r
rE
rr
E xU U
x
0rE
r rrs s
fE x U
12 0 00 0 0
0
s srE EEs r r
r s s
x U xE x E x U x f
x
1r
![Page 16: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/16.jpg)
12 0 00 0 0
0
s srE EEs r r
r s s
x U xE x E x U x f
x
20
0
0
1rE
r rsr
ss
E xnx
UU U Ex f
N
2 2 2
2
s s ss
r
n n E Un
E U
N N
2
0
01r
s ss s s s s
s
n xn n E U
x
N
0
2 20 1
1
r
ss
s r
nU Ex
x E U
N
2
1
ln
r
s s ss s
n
2
ss
r
nU E
E U
N
2 2
2
1 1 1 ss
s sr
E Un
n n E U
N NRelative fluctuation
0sand hence n
N
* if non-zeror r rn n n
rU E
![Page 17: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/17.jpg)
3.3. Physical Significance of Various Statistical Quantities in the Canonical Ensemble
Canonical distribution :rE
rr
n eP
Z
N
1
rErr
r
U E E eZ
rE
r
Z e
ln Z
Helmholtz free energy A ( T, V, N ) :
dA dU Td S SdT
A U T S
SdT PdV d N
,V N
AS
T
,T N
AP
V
,V T
A
N
U A T S ,V N
AA T
T
2
,V N
AT
T T
,V N
A
![Page 18: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/18.jpg)
lnU Z
,V N
A
lnA kT Z
= Partition function ( Zustandssumme / sum over states ) ,NZ Q V T
; , ,rEr
r
Z e Z T E V N A , & hence lnZ , must be extensive.
,
V
N V
UC
T
,N V
AA T
T T
2
2
,N V
AT
T
G A PV
Gibbs free energy G ( T, P, N ) :
,T N
AA V
V
N,T V
AN
N
Prob 3.5
![Page 19: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/19.jpg)
P
lnA kT Z
,T N
AP
V
rE
r
Z e
1rEr
r N
Ee
Z V
1rE
r Nr
PdV e dEZ
r r Nr
P dE 1rE
rP eZ
r rrr
U E P E , rr rN P N
r
dU P dE PdV
,N S
UP
V
c.f.
F
Er is indep of T
( Fixed { Pr } = Fixed S )
![Page 20: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/20.jpg)
S1
rErP e
Z ln lnr rP Z E
ln lnr rP Z E A U
lnA kT Z
T SS
k
ln rS k P
lnr rr
S k P P
T = 0, non-degenerate ground state 0r rP 0S ( 3rd law )
1rP
1
1ln
r
S k
lnk ( microcanonical
)
Disorder Unpredictability S Information theory (Shannon)
![Page 21: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/21.jpg)
3.4. Alternative Expressions for the Partition
Function ,, , ,rE N V
Nr
Z N V T e Q V T Non-degenerate systems:
Degenerate systems: rEr
r
Z g e gr = degeneracy of Er
r rr
X X P 1rE
r rr
g X eZ
rE
rr
g eP
Z
Thermodynamic limit ( N , V ) continuum approx. :
EZ d E g E e 1 EP E g E eZ
X dE X E P E 1 Ed E X E g E eZ
![Page 22: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/22.jpg)
0
EZ d E g E e Z( > 0 ) = Laplace transform of g(E)
Inverse transform:
1
2
i E
ig E e Z d
i
If g diverges, then > 0 is realsuch that all poles of Z are to the left of
1
2i Ee Z i d
![Page 23: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/23.jpg)
3.5. The Classical Systems
Quantum Classical states = d
1rE
r r rr r
X X P X eZ
3 3
3 3
, ,
,
N N
N N
d q d p X q p q pX
d q d p q p
H
H
d X e
d e
where 3 3N Nd d q d p
Gibbs’ prescription: 3
1
!H
NZ d e
N h
,H H q p
,NQ V T3! N
dd
N h
![Page 24: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/24.jpg)
Ideal Gas2
1 2
Ni
i
Hm
p
2
3 33
1 1
1, , exp
! 2
N Ni
i iNi i
Z T V N d q d pN h m
p
( In Cartesian coordinates, sum has 3N terms )
23
3exp
! 2
NN
N
Vd p
N h m
p
2 23 2
0
exp 4 exp2 2
pd p d p p
m m
p 3/22 m k T
2
10
1 1 1
2 2n x
n
nd x x e
where
3 1 1
2 2 2 2
3
1,
!
N
N
VZ Q T V
N
1
1,
!N
Q T VN
3h
![Page 25: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/25.jpg)
3
1,
!
N
N
VZ Q T V
N
lnA kT Z 3ln ln
VkT N N N NkT
3
ln 1N
A NkTV
3
,
ln 1T V
A N NkTkT
N V N
3
lnN
kTV
,T N
A NkTP
V V
3
,
3ln 1
2N V
A N NkTS Nk
T V T
3
5ln
2
VNk
N
![Page 26: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/26.jpg)
lnU Z
lnA kT Z ,N V
A
,N V
AA
,N V
AU A T
T
A T S
3
2U NkT
3
ln 1N
A NkTV
3
5ln
2
VS Nk
N
![Page 27: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/27.jpg)
Non-interacting (free) particles : 1
1, , , ,
!N
NZ N T V Q T V Q T VN
g EE
3 /2
3
2
3 / 2 !
NN mEV
h N
( from sec 1.4 )
3 /2 3 /2 1
3
2
3
1
! / 2 1 !
NN Nm EVg E
h NN
( Gibbs factor added by hand )
EZ d E g E e
3 /2
3 /2 1
30
2
3 / 2 1 !
1
!
NNN EmV
dE E eh NN
3 /2 1 3 /2
0
3 / 2 1 !N E Nd E E e N
3 /2
3
1
!
2NN
V mZ
hN
3
1
!
N
N
V
![Page 28: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/28.jpg)
1-particle DOS : 3/2 1/2
3
2
/ 2
mVa
h
1
0
Q d a e
3/2
3
2V m
h
1
1
!, N
NZ Q T VN
Q 3 /2
3
1
!
2NN
V
N
m
h
( same as before )
1
2
i E
ig E e Z d
i
3 /2 3 /2
3
12
2
1
!
NiN E N
i
Vm d
hNe
i
11 0
Res 01!
20 0
s x ns xs i nn ss i
e xxe
d s s ni s
x
3 /2 3 /2 1
2
20
3 / 2 1 !
0
!
0
N NNV m EE
g E hN N
E
contour closes on the left
contour closes on the right
( same as before )
Prob 3.15
3
V
![Page 29: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/29.jpg)
3.6. Energy Fluctuations in the Canonical Ensemble: Correspondence with the Microcanonical Ensemble
rU E E 1rE
rr
E eZ
rE
r
Z e
2
22
,
1 1r rE E
r rr rN V
UE e E e
Z Z
22E E 2E
2
,N V
UE
2
,N V
UkT
T
,r rE E N V
2VkT C
Relative root-mean-square fluctuation in E : 2
2V
E kT C
E U
1
N
Almost all systems in a canonical ensemble have energy U .
( Just like the microcanonical ensemble )
![Page 30: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/30.jpg)
P(E)
EP E g E e
max P at E* satisfies : 0E EP ge g e
E E
*
0E E
gg
E
*
ln
E E
g
E
or
lnS k g *E E
Sk
E
1
T
c.f.,
1
N V
S
U T
*E U E
( Every system in ensemble has same N & V )
i.e., Most probable E = mean E
![Page 31: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/31.jpg)
*g E E U
2
2
2
1ln ln ln
2E U E
E U
g E e g U e g E e E UE
lnS k g U
2
2
2
1ln
2E
E U
SU g E e E U
k E
1 S
k U
2 2
2 2
1ln E
E U
Sg E e
E k U
ln lnE
E U
g E e g U UE U
, ,S S U N V ,
1
N V
S S
U U T
2
2 2
, ,
1
N V N V
S T
U T U
2
1
VT C
2
1
Vk T C
2
2
1ln
2E
V
g E e U T S E Uk T C
Everything, except E, are kept const.
![Page 32: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/32.jpg)
2
2
1ln
2E
V
g E e U T S E Uk T C
EP E g E e 2
2exp
2U T S
V
E Ue
k T C
P(E) is a Gaussian with mean U and dispersion (rms) 2 2VE E k T C
P(E/U ) is a Gaussian with mean 1 and dispersion (rms)
2 2
1 Vk T CE
U U
1~O
N
P(E) (E U ) as N
![Page 33: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/33.jpg)
Ideal Gas
3 /2 3 /2 1
2
2
! 3 / 2 1 !
N NNV m Eg E
N h N
EP E g E e 2
2exp
2U T S
V
E Ue
k T C
*
ln
E E
g
E
13 / 2 1
*N
E
3 / 2 1*
NE
0
E dE P E E
0
0
E
E
d E g E e E
d E g E e
3 /2
0
3 /2 1
0
E N
NE
d E e E
d E e E
3 / 2N
*E E U for N >> 1
![Page 34: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/34.jpg)
N = 10, = 1
3 /2 1
3 /2 1
N
N
g E E
g U U
3 / 2 1*
NE
3 / 2N
U
2VkT C 2
1 3
2Nk
k
3 / 2
U
N
Mathematica
![Page 35: 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical](https://reader030.vdocuments.us/reader030/viewer/2022013004/5697bf911a28abf838c8ea3f/html5/thumbnails/35.jpg)
Z
2
2exp
2U T SE
V
E Ug E e e
k T C
0
, ENZ Q V T dE g E e 2
20exp
2U T S
V
E Ue d E
k T C
22U T SVe k T C
lnA kT Z 21ln 2
2 VU T S kT k T C lnU T S O N
O(N)
U T S
2
0
22U T S xV x
e k T C d x e 0 22 V
Ux O N
k T C
2xd x e