Bose-Einstein distribution function
ZG    e
Armed with
i
    i    ni
ni

We calculate
 (n1 , n2 ,..., ni ,...) 
e
i  ni
i
ZG
Probability to find the state (n1 , n2 ,..., ni ,...) in the grandcanonical ensemble
With  we can calculate thermal averages:

ni 
ni  (n1 , n2 ,..., ni ,...)
( n1 , n2 ,..., ni ,...)
N
n
i
  ni  N (T ,V ,  )
i
U
i

( n1 ,n2 ,...,ni ,...)
E(n1, n2 ,..., ni ,...)  (n1, n2 ,..., ni ,...)   i ni
i
Again complete thermodynamics via potential
  U  TS   N
  kBT ln ZG
from Legendre transformation of Helmholtz free energy
dF  SdT  PdV  dN  SdT  PdV  d ( N )  Nd 
d ( F   N )  d (U  TS   N )  SdT  PdV  Nd 

alternative ways to derive
1
calculate U
and
   i ni
i
N   ni
ni
using either  or 
for Bose gas identifying average occupation #
with Bose-Einstein distribution
i
  

   T ,V
using N   
  U  TS   N
  

 T   ,V
and S   
ni
For non-interacting bosons with ni  0,1, 2, 3, ...

ZG    e
i
   i    ni
ni 0

i
1
1 e
   i   
  kBT ln ZG  k BT ln 
i
 kBT

 ln 1  e
i
  
N  

   T ,V
   i   
1
1 e
   i   
 k BT
 ln
i
1
1 e
   i   

e  i 
  
 i 1  e   i   
   
1
ni 

i
1
e
  i   
1
e
  i   
1
Bose-Einstein distribution function
1
  ni
i
  
U    N T 


T

  ,V
As a crosscheck we show from
U    i ni
that in fact
i
With

  kBT  ln 1  e    
i
U  kBT
i

 ln 1  e
i
   i   


 N  
T
U  kBT
 ln 1  e
   i   
i
Simplifying and using
U  N  
i

 1
 N     2
 


 ln 1  e
   i   
i
i
i
   i   

i
e
  i   




T

  
1
N  

i e i    1

   T ,V
  i    e     
1 e
  1
   i    
ln
1

e
 

   i

1
  i ni
i
  
1
i
  i    e      
i
1 e
   i   



2
Calculate thermal average using 


nj 
n j  (n1 , n2 ,..., ni ,...)
( n1 , n2 ,..., ni ,...)
From

ln ZG  ln
e
n1 , n2 ,..., ni ,...


ln ZG 
ln  e
 j
 j n1 ,n2 ,...,ni ,...
1 
ZG  j

e

 



ni  i  

 



i
i

 



ni 




i
nj 

nj
e
1 

  j
i

 ln 1  e
i
i

ZG
( n1 , n2 ,..., ni ,...)

i
 i   ni
   i   

i
ni  i  
n1 , n2 ,..., ni ,...

ni i  

ZG



ni 


e
nj
i  ni
i
ZG
( n1 , n2 ,..., ni ,...)

ni 




n je

 



ni i  
i

i

ni 


n1 , n2 ,..., ni ,...
1 

ln ZG
  j


1 e

   j 
 1 e


   j 

1

e
  j   
1
 nj
Visualizing & discussing the Bose-Einstein distribution function
Click here for on-line animation and downloadable
live version& source code in Mathematica
E
Inspection of the summations in the partition functions show that
convergence requires  i    0
If lowest single particle energy  0  0
We also see for N=const
chemical potential
 (T  0)   0
 0
Careful analysis (see text page 426) shows that for N=const the fraction
N0/N of bosons condensed into the lowest single-particle energy state has
the T-dependence:
Bose-Einstein condensation
1.0
3/2
T 
N0
Phase transition setting at a
0.8
 1  
Critical temperature Tc
N
T
c

N0/N

0.6
Examples (although with interaction):
•Superconductivity
•Superfluidity
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
T/Tc
The Bose-Einstein distribution function in the limit
T  , 0
N fixed ,   
  0     1
1
   
 e  i   e i
ni      
e i 1
such that
Boltzmann distribution


ni  e
Let’s find the normalizing factor: N 
i
i
e i
ni  N
normalized Boltzmann factor
  i
e

i
   i   
N  e  e i
i