Ensemble equivalence
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 ?
Heuristic consideration
0.08
N=20
kBT=1
P( E ) dE = Probability of finding a system (copy) in the
canonical ensemble with energy in [E,E+dE]
P(E)
0.06
0.04
P( E ) 
2
0

E
20



E
3
N 1
2
3
N 1
2
example for monatomic ideal gas
 E
3
dE
e
E

N kBT
0
2
 30  5.5 example here with N=20, kBT=1
0.02
0.00
e
 E
40
60
E
3
N k BT
1
2

3
N
NkBT
2
80
100
U E 
3
3
Nk BT   20 1  30
2
2
example here with N=20, kBT=1
In the thermodynamic limit N
overwhelming majority of systems in the canonical
ensemble has energy U= <E>
Next we show:  E  Var[ E ] 
E
and
E

1
N
E2  E
2
is a general, model independent result
Brief excursion into the theory of fluctuations
Var[ X ] 
X 
X

Measure of: average deviation of the random
variable X from its average values <X>
2
f ( X )    f ( X  )
From the definition of <f(x)> as:

We obtain:
X  X

2
    X   X


   X   2 X
2

X2

   X   2 X  X  X
2

  X  X

X
2
2
 

1
2

 X2  X
2
Energy fluctuations
Goal:
find a general expression for 0 
E 
E
E

2
2

E2  E
2
U2
We start from:
U  E    E
 of the canonical ensemble

 U 

  CV
 T V
e  E


 


E

   T e 
T 


 E
 E
T
e  E

 
  e 

 E

1
k BT 2



T

 E


2   E 
  E
  E 
 E 2e  E  E e   E

E
e
e


E
e
E e   E













1  
1 












2
  E
  E
2 
k BT 2
k
T
e
e




B

  E


 
e


 

1
2
2

E

E
k BT 2







2





CV 
E
T

2
 E
2

E2  E
E
2
2

TCV
U 2
U and CV are extensive quantities
and
E U  N
E2  E
E
2
CV  N
2

1
N
and
E
E

E2  E
E
2

1
N
As N almost all systems in the canonical ensemble
have the energy E=<E>=U
Having that said there are
exceptions and ensemble equivalence
can be violated as a result
An eye-opening numerical example
Let’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 reservoir
U fluctuates around <E> according to
U  E E
With the general result
E
E
For the monatomic ideal gas with

E2  E
2
TCV T kBCV

U 2
U

E
E 
3
Nk BT
2
and
CV 
 E  T kBCV
3
Nk B
2

3
3
3
2/3  3
 0.82 
U  E   E  Nk BT  T k B Nk B  Nk BT 1 

Nk
T

B 1 

2
2
2
2
N
N




23
For a macroscopic system with N  N A  6 10
 10 12
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
N 
N

2
 N2  N
We start from:
remember fugacity
2

N z

N   N (N ) 

ln Z G  ln  z Z ( N )


ln ZG  ln  z N Z ( N ) 
z
z N 0


z N Z (N )

we see
N z
N 1
Z (N )
N 0


N 0
  
 
z
ln
Z
N
G 

z  z
 z
Z (N )
N 0
N
N 0
N
N 0

N 0
With
z  e 
N
z Z (N )

1 
N z N Z (N )

z N 0


N 0
z N Z (N )

1
N
z
  2 N 1
  N
 
 
N 1
N
N z Z ( N )   N z Z ( N )   z Z ( N )    Nz Z ( N )   N z Z ( N )

 N 0
1 2 1
 N 0
 N 0
 N 0
 N 0



N  N
2

z  N
z
z


N
z Z (N )

z
Z
(
N
)


N 0
 N 0


N
N2  N
Remember:
With
z
  

z
Ln
Z
G 

z  z

  kBT ln ZG   P(T ,  ) V
z  e 
N2  N
2
2
z
ln Z G 
  
1 


z z   z 
P(T ,  )V
k BT
 
   P(T ,  )V 

z
Ln
Z

z
z

G

z  z
z  z k BT


1   1  P(T ,  )V 
2 P


  kBTV 2
     kBT


  
  

 , P  



V

T ,

T ,V
With N   
N 1
 2
 2 P
 


V v
V 
V 
2
2 P  1
1 v

 2
2

 v
v 
With
P P v

 v 
P 1

and again
 v
v
1 1


v P
v
2 P
1 1


 2
v3 P
v
1  v 
Using the definition of the isothermal compressibility T   

v

P
 T
2

P
2
V
N 2  N  kBTV
2  k BT 2  T  k BT N  T / v

v
N2  N
N
2
2

k BT  T
v N
0
 N 
Particle fluctuations are
completely insignificant in the
thermodynamic limit