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