Ensemble equivalence in the thermodynamic limit
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Transcript Ensemble equivalence in the thermodynamic limit
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
ze
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