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
P(E)
0.06
3
0.04
P(E ) 
2
 
0.02
0.00
= Probability of finding a system (copy) in the
canonical ensemble with energy in [E,E+dE]

0
20
3

E

2
3
2
40
60
3
E
80

N k BT
1
N
3
 E
E
example for monatomic ideal gas
N 1
2
0
example here with N=20, kBT=1
100
U  E 
3
2
N k BT
N 1
E2

 dE e
N k BT
2
30  5.5
e
 E
N k BT 
3
 20  1  30
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
E
2
2
 E
is a general, model independent result
Brief excursion into the theory of fluctuations
V ar [ X ] 
X
 X

Measure of: average deviation of the random
variable X from its average values <X>
2
From the definition of <f(x)> as:
f (X ) 

We obtain:
X  X

2
  f ( X  )


  X   X

2





 X   2 X
2



 X   X

X
2

2
2



X
2
 X   2 X  X  X
1
 X
2

 X
2
Energy fluctuations
Goal:
find a general expression for
E 
0
E
E

2

2
2
 E
2
E
U
2
We start from:
U  E 
  of the canonical ensemble
   E


 U 

C

V



T
T

V
   E



e
e
T

  E
T
k BT

T

2

  E 
e


 

2

1
2

e
  E
E

e
  E





2   E 
  E
  E 
  E

E
e
e


E
e





  E e
 
 
 
 
1
k BT
E



  E

1 

2 
k BT 


1


k BT
2
E e
  E

e
  E

2
E
2
 E






2


E e
  E

e

  E





2





CV 


T
E
2
2
 E

E
2
 E
E
2

2
T CV
U
2
U and CV are extensive quantities
and
E U  N
E
2
 E
E
2
CV  N
2

1
N
and
E
E

E
2
 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

E
For the monatomic ideal gas with
U  E E 
3
2
N k BT  T
kB
3
2
 E
2
2
T CV

U
E
E 
3
2
NkB
N k BT
and

 N k BT  1 

2

3
For a macroscopic system with N  N A  6  10
23
2

T
k B CV
E T
U
CV 
3
2
k B CV
NkB
2/3  3
  N k B T
N  2
 10
12
Energy fluctuations are
completely insignificant
0.82 

1



N


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
 N
2
remember fugacity z  e
2
 N

We start from:

N 

N  (N ) 



N
z Z (N )
N 0

z
ln Z G 

z

ln

N
z Z (N )
N 0

ln Z G  ln
N
N z Z (N )
N 0

N 0
With

z Z (N ) 
N
N 0


we see
N z

1
Z (N )
N 0

N 0
  


z
ln
Z

N
G 

z  z
 z
N 1

N
z Z (N )


z
N
N z Z (N )
N 0


N 0

N
z Z (N )
1
z
N
 
 
 
 
2
N 1
N
N 1
N
  N z Z (N )   z Z (N )    Nz Z (N )   N z Z (N )
 N 0
 N 0
 N 0
 N 0



z

N
N z Z (N )
N 0



 

N
  z Z (N )
 N 0

N
z Z (N )
N 0
N
2
 N
2
 z
2
N
2
ze
 N
2

 z
1
N
2
z

1
N
z
  

z
L
n
Z
G 

z  z

Remember:    k B T ln Z G   P (T ,  ) V
With


z

 
z 

1
ln Z

G
P (T ,  )V
k BT

 z 
  
   P (T ,  )V 

z
L
n
Z

z
z

G 

z  z
z  z
k BT


2
1   1  P (T ,  )V 
 P


  k BT V
2
    
k BT


  
  
,
P








V

T ,

 T ,V
With N   
N
V

1
v
 
2

V 
 
2

V

P

2
 P
 1
2

2


 v
1 v
v 
2
With
P


P v
and again
v 
 P
2

2
P
1

v
3


1
v
v


1 1
v P
v
1
P
v
Using the definition of the isothermal compressibility  T
 P
2
N
2
2
 N
N
2
 k BT V
 N
N
2
2


2
k BT  T
v N
 k BT
V
v
 0
 N  
2
 T  k BT N  T / v
Particle fluctuations are
completely insignificant in the
thermodynamic limit
1  v 
 

v  P T