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:

P r
 

C
0 when E < E r otherwise
Normalization condition:
< E + dE
 r
P
 r
1
1
C =
# of accessible states
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!

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 E
0 is C
0
, a constant
The probability of system A in one specific microscopic state with energy E r
P r

C
0


C
0
 W
'
W

'
 
E
0

E r
 is:
W
0
, E
0
= constant
A d
Q
A’
W r
, E r W ’ , E’

Canonical ensemble: systems interacting with a heat reservoir


Since A’ is much larger than A : ln 
W
'

E
0

E r
   ln
 W
'
 
0

 ln
W
'

E '
 W
'

E
0

E r
  W
'( E
0
  
' E r
)

E r
 ln 
W
'
 
0
A and A’ are in thermal equilibrium:
P r

C
0
 W
'

E
0

E r


'

1 kT
W
0
, E
0
= constant

C
0
 W
'( E
0
  
' E r
)
 exp(
 
E r
)
Normalization:
 r
P
 r
1
A
W r
, E r d
Q
A’
W ’ , E’
 
' E r

Canonical ensemble: systems interacting with a heat reservoir
 The probability of finding A in any of the microscopic states with energy E :
P
E
  W   
E )
Boltzmann factor: exp(
 
E )
Normalization: C

1
Degeneracy factor:
 e
 
E r
W e r '
T

T '
W
W  
E
E
 Ensemble average of extensive variable x : x
  r
  r r

 x e r
 e
 

E r

E r
The sums are performed over all states r



Average energy in a canonical ensemble:
E
  r
P E r
 r


E e r

 e
 

E r

E r
 
 ln
 
Z
Partition function: Z
   e
 
E r
  r E
W  
E
Ensemble average of intensive variable y (conjugate of x ): y



E x

  
E r
 x r
  e

 e
 
E r

E r

 ln
 x
Z e.g. ensemble average of p : p
 





E
V





 ln
V
Z

E
 
 ln Z
  y

 ln
 x
Z
 d

 
  d d
  
 

 ln
 
 d
Z d E
Q
 d
 
  y dx d



E ln
 x
 
Z
  d

 dx E d
  y dx
  
TdS d E
 d
W


 ln Z

E T

 dS

S
 k ln Z

E T
F
  kT ln Z
 
Helmholtz potential

Partition function and Helmholtz potential
 The probability distribution of system energy in a canonical ensemble peaks at: d ln P

0 where
E
P
E

1 Z
 W
E
  
E )
 d ln
 W   
E )
  ln
W
( )
 
E


0
T

T '
'

E
W  
E
E
Z
  
E
W   
E

E
 W   
E  d E
 d ln Z
 d

 ln
W
  dF
 d
  kT ln Z

 

E
0

Properties of canonical partition function


 Classical approximation:
Z
   r e
 
E r
    

1
,..., p f

 dq ...
dp
1 h
0 f f h
0 phase space
Energy values are relative; entropy has absolute values
E
 
 ln Z
 
S
 k
 ln Z
 
E

Weakly interacting systems:
Z tot
 
1,2 exp
  
E
1

E
2
 
E tot

E
1

E
2

1 exp
  
E
1
  
2 exp
  
E
2
 
Z Z
1 2 ln Z tot
 ln Z
1
 ln Z
2






Probability in one microscopic state
Probability in any state with energy r
E :
: P r
P
E
The probability function maximizes when:

1
 exp(

Z

W
Z
 exp(



E r
E )
) dP

E
0 which is equivalent to:
Partition function
Z

Thermodynamic potential dF

0
  r e
 
E r
 
E
F
  kT ln Z
W  
E
 Ensemble average of energy E and intensive variable y :
E
 
 ln Z
  y

 ln
 x
Z

Procedures of calculating macroscopic properties of canonical ensembles


Determine the energy levels of the system
Calculate the partition function
Z
   r e
 
E r
    

1
,..., p f

 dq
1
...
dp h
0 f f
 Evaluate the statistical ensemble average
E
 
 ln Z
  x
  r
  r r

 x e r
 e
 

E r

E r y

 ln
 x
Z

Paramagnetism and the Curie’s Law




Consider one atom with a magnetic dipole:
Two states (+):
Probability (+):
E
 p

  
H , (-):
 exp

E

 
H

  
H
, (-): p


 exp
  
H

Average magnetic dipole:

H

P

  
P

P


P

 
 exp exp




H
H



 exp exp


 
 
H
H


 
H
 tanh
 kT


2
H kT
 
H
 kT

  
H
 kT


M

N

2
H

1 k 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: E

E kinetic p
2
 
2 m p x
2  p y
2  p z
2 m
2
 Probability of the molecule having a coordinate between
( r ; r + dr ) and momentum between ( p ; p + dp ):

3

3
 exp
 
( , )
 3  3  
  exp



3

3 mv
2
2
 exp




 3  3 d r d v
 
( , )
 3  3 
1

C
V

2
 m kT p
2
2 m
 3 2



3

3 d r d p

Maxwell velocity distribution of ideal gas
 Maxwell velocity distribution of N molecules:
( , )
 3  3 
N
V


2
 m kT
 3 2
 exp


 mv
2
2 kT


 3  3 d r d v
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:

( )
  3 
( )
 cos q
 Total molecular flux:

0
  v z

0

( )
   v z

0
  v z

0
( )
 cos q 
( )
 cos q  v
2 sin q q  
1 d d dv nv
4

P
2
 mkT q vdt dA
 Application: impurity incorporation during film deposition


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





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 -
 and exponentiate
Sum over all degrees of freedom (energy levels)
Canonical ensemble: thermal interactions only: T & V constant
Helmholtz potential F exp(
 
F )
 exp(
S k
 
E )

E
 exp(

E
W
S k
 exp(

E
 
)
E )

Z

Grand canonical ensembles: systems with indefinite number of particles d
Q


T ,
 are constant
TD potential: f
= U – TS – 
N dN
 Grand canonical partition function:
System Heat & particle source
 
E exp
  f   
E
W  exp
 
E
 
N
 f   kT ln


Probability to be at one microscopic state with energy E r and particle number N r
: P r
 
1
  exp
 
E r
 
N r

Average energy and particle number:
E
 
 ln
 
 
N N

1


 ln
 

Grand canonical probability distribution
E tot

E r

E ' N tot

N r

N ' r
 r r
  W
'

E tot
 r
, tot

N r
 ln
W
'

E tot
 r
, tot

N r
 ln
W
'

E tot
, N tot
 

E

 ln
W
'
 ln
W
'

E
 
'
 

E r

 ln
W
'

N
(thermal equilibrium)

N r
W r
, E r
 ln
W
'

N
  
'
 

P r
 
1
  exp
(chemical equilibrium)
 
E r
 
N r
 d
Q dN
W ’ , 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:
E


 e


T

T '
'
W  
E
W d 
E

E r

E E
E N
 
E