Transcript Fluctuations - University of Florida

Boltzmann number of 1-particle state >>number of particles
How many 1-particle states?
Remember the sphere used to explain degeneracy?
number of 1-particle states with an energy lower than e = number
of lattice points enclosed by the sphere in the positive octant:
3
2
 e   1  4  R 3   1   8m e  a 3
8 3

6  h
for thermal particles,

3
e  e  kT per particle
2
For the Bolztman condition to hold,

3
2
6
h




  12mk T 
1

 e   1   8m 3kT 
6  h
 e 
N

2 
V
3
2
1  8m 3kT  V

6  h 2  N
which will occur for T  m
and  
3
2
1
to express the energy of N particles,
1
  ln Q  N,V ,T  
  Q  N ,V , T  
E  
 



Q  N ,V , T  


V
V
E 
N
1
 e k  
  e
 
N
!
 
  k
1
 q  N,V ,T  
 N ! 
N

N 1

1
N
 e k 
 e k 
E 

e
    e k  e
N 

k
 q  N,V ,T    N !  k



N
!



 e k 
E 
eke 


q  N,V ,T   k

N
E N e
The average energy of N
particles is N times the
average energy of one particle
e  ek


in terms of the probability , e     k e k  , where  k 
 e j
e
 k


# of available states
N  we have to count more carefully
we have to look at the particle's wavefunction 
i1
i2
i3
Fermions: 2 particles cannot occupy
the same quantum stateoccupation
number nk can only be 0 or 1
P1,2=- antisymmetric
i2
i4
i2
i5
i3
i6
Bosons: 2 particles can occupy the
same quantum stateoccupation
number nk can have any value
P1,2=+ symmetric
There are many ways of counting distributions,
1
fluctuations of physical observables 
X
for large systems, fluctuations are negligible
 all ensembles are equivalent
choose Grand-Canonical Ensemble
 V ,T ,     e
N
  ENJ
e  N
j
ENj  energy states available for a system of N particles
e k  molecular quantum state
nk  E j   molecules in k quantum state, in the E j state
nk 
defines the system at E j  E j   n ke k and N   nk
e
j
 E
NJ
 e
 
ni
  i e n
i i
k
a sum over states becomes
the sum over each distribution
k
 V ,T ,      N  e
N
if we consider
  ENJ
 
N 0
e

n
e
E
NJ
 
   i ei ni
n 

n 
k
  i e n
i i
a sum over states becomes
a sume over each distribution
n 
where
n
k
N
k
  n1e1n 2e 2 ...n k e k ... n1n 2 ...n k ...

m n m  


 e
N 0
     e
N 0
 e
   i ei ni
  e
N 0
-
j
j
N
where we used   e

 e k nk

n 
n1nmax n1nmax

n 0
1

n
2 0
  e
k

 ek nk
where we use the fact that N can have any value (0  )
  V,T,    
k
nk nmax

n 0
k
  e 
 ek nk
starting with
  V,T,    
k
n k nmax

n 0
  e 
 e k n k
k
we look at Fermions, w here n k =0,1
  V, T ,    
k
n k 1
 e


n 0

 e k n k
k

 ek

   e

k 

     e  
 V ,T ,     1    e
k
 ek 1 
0
 e k

Fermi-Dirac
again, we start with   V,T,    
k
we now look at Bosons , where
  V, T ,    
k
nk 
 e


n 0
nk
n k nmax

n 0
  e 
 e k n k
k
 0,1,...N

 ek nk
k
from our outstanding math background …
we recognize the series
 V ,T ,    
k
1
1-   e
 e k

 1 

1

x


m
x
 
m 0
Bose-Einstein
Combining Bose-Einstein with the Fermi-Dirac we obtain a gral. Eq.
 V ,T ,    
k
+
 e k - 1
+
1-  e 
for a Grand-canonical ensemble, N  kt

and using   e kT
 ln  V ,T ,  


1 kT
 d 
e d
kT
 ln  V ,T ,   1
 ln  V ,T ,  
N  kt
 

kT



  +-  ln 1+- e  ek  
e  ek
k


N  

and N = n k

e
+
k 

k
k 1-  e
for each quantum state, the average number of particles
nk

e  ek
1+- e ek 

 which is equivalent to   e  ek
j
 e k

e


k






we can now calculate all other thermo properties:
E  N ek   nk ek  
k
k
e k  e  ek
1+- e ek 
and
PV  kT ln  V ,T ,    kT  ln 1+- e  ek 
k
Keep in mind that even though we started with non-interacting
particles, there wavefunctions are symmetrized they are
NOT INDEPENDENT
to see if these eq. hold for Bolztmann statistics conditions
Boltzmann number of 1-particle state >>number of particles

nk  0, which will occur when   0
for   0
nk
 0
 e  ek
1

 e  ek 
nk = 
 e  ek

 1+- e  ek  


 e  ek

  k (Boltzmann-classical- limit)
N
  e ek
nk
k
e  ek
and for the energy, E   e k
 e k
+
1

e


k
lim E   e k e  ek
 0

k
E
N
 e e e
e 
 e e
k

k
k
k
k
 e e e

 e e
k
k
k
k
k
e   e k k (Boltzmann-classical limit)
k
finally,
pV  kt ln  V ,T ,    kT ln 
+
 e k - 1 +
+
1-  e  - kT
k
for   0 we can expand ln 1  x   x
 ln 1+-  e  e 
k
k
pV  kT   e  ek  kt  q(N,V ,T )
k
pV
ln  V ,T ,   
  V ,T ,    e q (N,V ,T )
kt
Examples: Photon gas, an electromagnetic field in thermal
equilibrium with its container
QM  we know that H =
In this case hi harmonic oscillator
h
i i ei  n
To describe the state of the field, we need to know how many n
are in each oscillator
Photons are bosons  n=0,1,2,3…..
Q N,V,T    e   E 



e    n1e1  n2e2 ... n j e j 
n1 ,n2 ,...,n j
first we want to show that
Q N,V,T  


n1 ,n2 ,...,n j
e    n1e1 e    n2e2 
e
  n j e j 

   e  n j e j 
j
n j 0
Q  N,V,T 
  e   0e1   e   1e1   e    2e1     e   0e2   e   1e 2   e    2e 2  
  e   0e3   e   1e3   e    2e3  
 e   0e1  0e2   e   0e1 1e2   e   0e1  2e2   e   1e1  0e 2   e   1e1 1e 2 
 e   1e1  2e2   e    2e1  0e2   e    2e1 1e 2   e   1e1  2e 2  
  e   0e3   e   1e3   e    2e3  
 e   0e1  0e2  0e3   e   0e1  0e2 1e3   e   0e1  0e2  2e3   e   1e1  0e2 
e   0e1 1e2  0e3   e   0e1 1e2 1e3   e   0e1 1e2  2e3   e   0e1  2e2  0e3 
e   0e1  2e2 1e3   e   0e1  2e2  2e3   e   1e1  0e2  0e3   ...


e    n1e1  n2e2  n3e3 
n1 ,n2 ,...,n j
 e   0e1  0e2  0e3   e   1e1  0e2  0e3   e   0e1 1e2  0e3 
 e   0e1 0e2 1e3   e    2e1  0e2  0e3   e   0e1  2e2 0e3   ...

Q N,V,T     e
  n j e j 
n j 0
j

j
1
where we used
1  e    n j e j  j x j  1 x
1
The average number of photons in a state j…
   n1e1  n2e 2 ...


E

n
e
j

 nje
nj 


 e   E
n
Q

1
nj 
Q
  n j e    n1e1  n2e2 ...
n
   e j n  n j
ln Q  N,V ,T      1 ln 1  e
j

 ln Q  N,V ,T 
 e j
   e j n  n j

 ln Q  N,V ,T 
   e j 
average occupation number is n j = 
1
e  e j
e  e j

1  e  e j
Planck
 1 distribution