Bose-Einstein distribution function

awfulhihatUrban and Civil

Nov 15, 2013 (3 years and 11 months ago)

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Bose
-
Einstein distribution function





i i
i
n
G
n
i
Z e
 
 



Armed with

We calculate



1 2
(,,...,,...)
i i
i
n
i
G
e
n n n
Z
  

 


Probability to find the state

1 2
(,,...,,...)
i
n n n
i
n the
grandcanonical

ensemble

With


we can calculate thermal averages:

1 2
1 2
(,,...,,...)
(,,...,,...)
i
i i i
n n n
n n n n n



(,,)
i i
i i
N n n N T V

  
 
1 2
1 2 1 2
(,,...,,...)
(,,...,,...) (,,...,,...)
i
i i i i
n n n i
U E n n n n n n n
 
 
 
Again complete thermodynamics via potential

ln
B G
k T Z

 
i
n
alternative ways to derive using either


or



i i
i
U n



1

calculate

f
or
B
ose gas identifying average occupation #

i
n
w
ith Bose
-
Einstein distribution

U TS N
 
  
dF SdT PdV dN

   
f
rom Legendre transformation of Helmholtz free energy

( )
SdT PdV d N Nd
 
    
( ) ( )
d F N d U TS N SdT PdV Nd
  
       

using

U TS N
 
  
and

,
V
S
T



 
 
 

 
,
T V
N


 

 
 

 
i
i
N n


and



0
i i
i
n
G
n
i
Z e
 

 




For non
-
interacting

b
osons
with

0,1,2,3,...
i
n



1
1
i
i
e
 
 



ln
B G
k T Z

 


1
ln
1
i
B
i
k T
e
 
 
 




1
ln
1
i
B
i
k T
e
 
 
 


,
T V
N


 

 
 

 




1
1
i
i
i
e
e
 
 


 
 

 






ln 1
i
B
i
k T e
 
 
 



1
1
i
i
i i
n
e
 

 

 


1
1
i
i
n
e
 



Bose
-
Einstein distribution function









1
ln 1 ln 1
i i
B
i i
U k T e N e
   
 
 
   
 

    
 

 
 
T
T


 
 
 


,
1
1
i
i
T V
N
e
 



 

  
 


 

Simplifying and using

,
V
U N T
T


 

 
  
 

 














2
1 1
ln 1 ln 1
1
i
i i
i
i
B
i i i
e
U k T e N e
e
 
   
 
 
 
 
 
   
 
 

      
 
 

 
  






1
i
i
i
i
e
U N
e
 
 
 

 
 

 




1
i
i
i
e
 





i i
i
n



As a crosscheck we show from

t
hat in fact

i i
i
U n







ln 1
i
B
i
k T e
 

 
 

With













1 2
(,,...,,...)
1
ln
1 1 1
ln 1
1 1
i i
i
i
j
i
j j
n
j j G
n n n
G j
j
i
j
e
n n Z
Z
e
e n
e e
  
 
 
   
 

  
 
 
 
  


  


    

 


2

Calculate thermal average using



1 2
1 2
(,,...,,...)
(,,...,,...)
i
j j i
n n n
n n n n n





1 2
(,,...,,...)
i i
i
i
n
j
n n n
G
e
n
Z
  
 



1 2
,,...,,...
ln ln
i i i
i i
i
n n
G
n n n
Z e
  
 
 
 
 
 
 


From

1 2
1 2 1 2
,,...,,...
,,...,,...,,...,,...
ln ln
1
i i i
i i
i
i i i i i i
i i i i
i i
n n
G
n n n
j j
n n n n
j
n n n n n n
G j G
Z e
e n e
Z Z
  
     
 


 
 
 
 
 
   
   
   
   
   
 
 
 
 
   

 


 
Click here
for on
-
line animation and downloadable
live version& source code in
Mathematica

E


Visualizing & discussing the Bose
-
Einstein distribution function

Inspection of the summations in the partition functions show that
convergence requires

0
i
 
 
If lowest single particle energy

0
0


0


c
hemical potential

We also see for N=const

0
( 0)
T
 
 
Careful analysis (
see text page 426
) shows that for N=const the fraction
N
0
/N of bosons condensed into the lowest single
-
particle energy state has
the T
-
dependence:

3/2
0
1
c
N T
N T
 
 
 
 
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.0
0.2
0.4
0.6
0.8
1.0


N
0
/N
T/T
c
Bose
-
Einstein condensation

Phase transition setting at a

Critical temperature
T
c


Examples (
although with interaction
):


Superconductivity


Superfluidity


The Bose
-
Einstein distribution function in the limit

,0
,
T
N fixed


 

s
uch that



0
1
 
 


1
1
i
i
n
e
 





i
i
e e
 

 

 
Boltzmann distribution

Let’s find the normalizing factor:



i
i
i i
N n e
 
 
 
 
i
i
N e e





i
i
i
i
e
n N
e






normalized Boltzmann factor