Upload
madeline-franklin
View
223
Download
0
Embed Size (px)
Citation preview
MSEG 803Equilibria in Material Systems
8: Statistical Ensembles
Prof. Juejun (JJ) Hu
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