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Ensemble equivalence
31
2
31
2
0
( )
NE
NE
e EP E
dE e E
0 20 40 60 80 1000.00
0.02
0.04
0.06
0.08
P(E)
E
N=20kBT=1 ( )P E dE = Probability of finding a system (copy) in
the canonical ensemble with energy in [E,E+dE] example for monatomic ideal
gas
3
2 BU E Nk T
23
2 BN k T
320 1 30
2
example here with N=20, kBT=1
30 5.5 example here with N=20, kBT=1
312
32
B
B
N k T
E NNk T
In the thermodynamic limit N
overwhelming majority of systems in the canonical ensemble has energy U= <E>
Heuristic consideration
The problem of equivalence between canonical and microcanonical ensemble:canonical ensemble contains systems of all energies. How come this leads to the same thermodynamics the microcanonical ensemble generates with fixed E ?
Next we show:22[ ]E Var E E E
1E
E N
and is a general, model independent
result
Brief excursion into the theory of fluctuations
2[ ]Var X X X Measure of: average deviation of the random variable X from its average values <X>
From the definition of <f(x)> as:
( ) ( )f X f X
We obtain:
2 2X X X X
22 2X X X X
22 2X X X X
2X X 1
22X X
Energy fluctuations
Goal:find a general expression for
2 22
2 20
E E E E
UE
We start from:
U E E
E
V EV
e EU
C ET T T e
of the canonical ensemble
2
1E
EB
e E
k T e
TT
2
22
1E E E E
B E
E e e E e E e
k Te
22
2
1E E
E EB
E e E e
k T e e
222
1
B
E Ek T
22VC E E
T
22
2 2V
E E TC
UE
22
2
1E E
NE
U and CV are extensive quantities
E U N VC Nand
and 22
1EE E
E E N
As N almost all systems in the canonical ensemblehave the energy E=<E>=U
Having that said there are exceptions and ensemble equivalencecan be violated as a result
An eye-opening numerical exampleLet’s consider a monatomic ideal gas for simplicity in the classical limit
We ask:What is the uncertainty of the internal energy U, or how much does U fluctuate?
For a system in equilibrium in contact with a heat reservoirU fluctuates around <E> according to
EU E 22
EE E
E E
With the general result 2
B VVT k CTC
U U E B VT k C
For the monatomic ideal gas with 3
2 BE Nk T and3
2V BC Nk
3 3 3 2 / 3 3 0.821 1
2 2 2 2E B B B B BU E Nk T T k Nk Nk T Nk TN N
For a macroscopic system with236 10AN N 1210
Energy fluctuations are completely insignificant
Equivalence of the grand canonical ensemble with fixed particle ensembles
We follow the same logical path by showing:particle number fluctuations in equilibrium become insignificant in the thermodynamic limit
2 22N N N N
We start from: 0
0
0
( )( )
( )
N
N
NN
N
N z Z NN N N
z Z N
0
ln ln ( )NG
N
Z z Z N
With we see 1
0 0
0
0 0
1( ) ( )
1ln ln ( )
( ) ( )
N N
N N NG
N NN
N N
N z Z N N z Z Nz
Z z Z N Nz z zz Z N z Z N
remember fugacity z e
ln Gz Z Nz z z
2 1 1
20 0 0 0 202
0 0
( ) ( ) ( ) ( )( )1 1
( ) ( )
N N N NN
N N N NN
N N
N N
N z Z N z Z N Nz Z N N z Z NN z Z NN N
z z zz Z N z Z N
22GN N z z Ln Z
z z
Remember:
lnB Gk T Z ( , )P T V( , )
ln GB
P T VZ
k T
With z e 1
z z z
22 ( , )G
B
P T VN N z z Ln Z z z
z z z z k T
2
2
1 1 ( , )B
B
P T V Pk TV
k T
With ,,
,TT V
N PV
2 21N P
V v V V
2
2 2
1 1P v
v v
With
P P v
v
and again 1P
v
1 1vPvv
2
2 3
1 1PPvv
Using the definition of the isothermal compressibility 1
TT
v
v P
22N N2
2B
Pk TV
2B T
Vk T
v /B Tk T N v
22
2 0B T
N
N N k T
v NN
Particle fluctuations are
completely insignificant in thethermodynamic limit