MSEG 803Equilibria in Material Systems

8: Statistical Ensembles

Prof. Juejun (JJ) Hu

hujuejun@udel.edu

Micro-canonical ensemble: isolated systems

Fundamental postulate: given an isolated system in equilibrium, it is found with equal probability in each of its accessible microstates

Probability of finding the state in a microstate r is:

When considering two system A and A’ that only interact with each other, we can always treat the composition system A + A’ as an isolated system!

rP

C

0

when E < Er < E + dE

otherwise

1rr

P Normalization condition:# of accessible states

1C =

Canonical ensemble: systems interacting with a heat reservoir

A small system A kept in thermal equilibrium with a large heat reservoir A’ (DOF of A << DOF of A’)

The probability of the isolated system A + A’ in a microscopic state with total energy E0 is C0 , a constant

The probability of system A in one specific microscopic state with energy Er is:

A A’dQ

W’ , E’Wr , Er

W0 , E0 = constant

0

0 0

' '

'

r

r

P C E

C E E

Canonical ensemble: systems interacting with a heat reservoir

Since A’ is much larger than A:

A and A’ are in thermal equilibrium:

0 0 0

ln 'ln ' ln ' ln ' '

'r r rE E E E E EE

0 0

0 0

'

'( ) exp( ' )

exp( )

r r

r

r

P C E E

C E E

E

A A’dQ

W’ , E’Wr , Er

W0 , E0 = constant

0 0' '( ) exp( ' )r rE E E E

' 1 kT

1rr

P Normalization:

Canonical ensemble: systems interacting with a heat reservoir

The probability of finding A in any of the microscopic states with energy E :

( ) exp( )EP C E E

Boltzmann factor: exp( )E

Normalization:

Degeneracy factor: ( )E

( )ErEe

( ) EE e

E

' 'T T

Ensemble average of extensive variable x:

r

r

Er

r r Er

x ex P x

e

The sums are performed over all states r

1 rEC e

Average energy and intensive variables

Average energy in a canonical ensemble:

Ensemble average of intensive variable y (conjugate of x):

e.g. ensemble average of p:

lnr

r

Er

r r Er

E e ZE P E

e

( , ) ( )rE E

r E

Z x e E e

1 lnr

r

Er r

E

E x eE Zy

x e x

1 lnE Zp

V V

Partition function:

Partition function and Helmholtz potential

1 ln Zy

x

ln lnln ( , )

Z Zd Z x d dx E d y dx

x

d E dE y dx d E dE W

d E Q d E TdS

ln ZE

lnd k Z E T dS

lnF kT Z E TS Helmholtz potential

lnS k Z E T

The probability distribution of system energy in a canonical ensemble peaks at:

Partition function and Helmholtz potential

( ) EE e

E

' 'T T

1 ( ) exp( )EP Z E E ln 0Ed P where

ln ( ) exp( ) ln ( ) 0d E E d E E

ln lnd Z d E E

( , ) ( )

( )

E

E

E

Z x E e

E ed

E

E

E ln 0dF d kT Z

Properties of canonical partition function

Classical approximation:

where is the (arbitrary) volume of one state in the phase space

Energy values are relative; entropy has absolute values

Weakly interacting systems:

11

0

...( , ) ... exp ,...,r fE

f fr

dq dpZ x e E q p

h

0fh

ln ZE

lnS k Z E

1 2totE E E

1 2 1 2 1 21,2 1 2

exp exp exptotZ E E E E Z Z

1 2ln ln lntotZ Z Z

Summary of canonical ensembles

Probability in one microscopic state r :

Probability in any state with energy E :

The probability function maximizes when:

Partition function

Thermodynamic potential

Ensemble average of energy E and intensive variable y:

1exp( )r rP E

Z

exp( )EP EZ

( , ) ( )rE E

r E

Z x e E e 0EdP 0dF which is equivalent to:

lnF kT Z

ln ZE

1 ln Zy

x

Procedures of calculating macroscopic properties of canonical ensembles

Determine the energy levels of the system Calculate the partition function

Evaluate the statistical ensemble average

11

0

...( , ) ... exp ,...,r fE

f fr

dq dpZ x e E q p

h

ln ZE

1 ln Zy

x

r

r

Er

r r Er

x ex P x

e

Paramagnetism and the Curie’s Law

Consider one atom with a magnetic dipole: Two states (+): , (-): Probability (+): , (-): Average magnetic dipole:

E H E H

expp H expp H

exp exptanh

exp expH

P P H H H

P P H H kT

2H

kT

H kT

H kT

2 1N H

Mk T

Curie’s law

Maxwell velocity distribution of ideal gas

One classical ideal gas molecule enclosed in a rigid container at constant temperature T

Energy of gas:

Probability of the molecule having a coordinate between (r ; r + dr) and momentum between (p ; p + dp):

2 2 22

2 2x y z

kinetic

p p ppE E

m m

2

3 3 3 3 3 3exp exp2

pP d r d p E d r d p d r d p

m

2

3 3 3 3( , ) exp2

mvP r v d r d v C d r d v

3 3

( ) ( )( , ) 1

r vP r v d r d v

3 21

2

mC

V kT

Maxwell velocity distribution of ideal gas

Maxwell velocity distribution of N molecules:3 2 2

3 3 3 3( , ) exp2 2

N m mvf r v d r d v d r d v

V kT kT

T = 298 K (25 °C)

Number of molecules striking a surface

The # of molecules with velocity between

v and v + dv which strike a unit area of

the wall per unit time:

Total molecular flux:

Application: impurity incorporation during film deposition

dA

vdt

q

3 3( ) ( ) cosv d v d v f v v

3 30 0 0

2

0

( ) ( ) cos

1( ) cos sin

4 2

z z

z

v v

v

v d v f v v d v

Pf v v v d d dv nv

mkT

Maxwell’s Demon

A demon opens the door only to allow the “hot” molecules to pass to the right side and the “cold” molecules to pass to the left side → S decrease!

Maxwell’s Demon in action: he is devilishly COOL

Partition functions for general ensembles

Evaluate the boundary conditions for the system Determine the variables that are kept constant

Determine the thermodynamic potential for the system Multiply the TD potential by - b and exponentiate Sum over all degrees of freedom (energy levels)

Boundary condition TD potential Multiply &

exponentiateSum over

energy levels

Canonical ensemble:thermal interactions only: T & V constant

Helmholtz potential F

exp( )

exp( )

F

SE

k

exp( )

exp( )E

E

SE

k

E

Z

Grand canonical ensembles: systems with indefinite number of particles

T, m are constant TD potential: f = U – TS – mN Grand canonical partition function:

Probability to be at one microscopic state with energy Er and particle number Nr :

Average energy and particle number:

System Heat & particle source

dQ

dN

exp expE E

E N

1 expr r rP E N

lnE N

1 lnN

lnkT

Grand canonical probability distribution

dQ

dN

W’ , E’Wr , Er

'tot rE E E 'tot rN N N

, ' ,r r r tot r tot rP E N E E N N

ln ' ,

ln ' ln 'ln ' ,

tot r tot r

tot tot r r

E E N N

E N E NE N

ln '

'E

ln '' '

N

(thermal equilibrium)

(chemical equilibrium)

1 expr r rP E N

Equivalence of ensembles

( ) EE e

E

' 'T T

E N E

E

rE E

Microcanonical and canonical ensembles are equivalent in the thermodynamic weak coupling limit

The constant T (canonical ensemble)

and the constant E (microcanonical

ensemble) are connected by:

r

r

E kTr

E kT

E eE

e