Transcript Bose-Einstein Statistics - AGH University of Science and

Bose-Einstein Statistics

Applies to a weakly-interacting gas of
indistinguishable Bosons with:
 Fixed N = ini
 Fixed U = iEini


No Pauli Exclusion Principle: ni  0, unlimited
Each group i has:
 gi states, gi-1 possible subgroups, ni to be shared
between them


 g  n  1!
t 
Number of combination to do this is:
n ! g  1!
So number of microstates in distribution {n } states:
i
i
i
i
i
i

g  n  1!
t ({n })  
n ! g  1!
i
j
i
i
i
i
RWL Jones, Lancaster University
Bose-Einstein Statistics

Classical limit:
gi  ni  1!  g  n  1g  n  2g 
i
i
i
i
i
gi  1!
n factors
i
 g i if : ni  g i
ni
ni
gi
t MB ({ n j })  
ni !
i

Bose-Einstein:
 Large numbers: gi, ni
t ({n }) 
j

g  n !

i
i
i
n !g !
i
i
RWL Jones, Lancaster University
Bose-Einstein Distribution


We use the same technique as for Boltzmann,
maximize ln t({ni}) : d ln t ({ni}) = 0
Add to this the constraints:
 dN = 0  idni = 0
 dU = 0  i Ei dni = 0

:(ii)
:(iii)
Once again, add the (i)+(ii)+(iii)
(Lagrange)
n
1
F    
g exp     1
i
i
i

i
Thermodymanics gives =-1/kT
RWL Jones, Lancaster University
Open and Closed Systems


 given by N=igiF(Ei) for a closed system of phoney
bosons (e.g. ground state He4 atom (2p2n2e, each in
up-down spin combinations)
 = -/kT
F   
BE



1
exp      1
kT 

Elementary bosons (not made up of fermions) do not
conserve N – examples are photons and phonons
These correspond to an open system – no fixed n
 no  no 
F   
BE

exp 
1
kT
 1
RWL Jones, Lancaster University
Black Body Radiation

Spectral Energy density is the energy in a photon gas
between E and E+dE = U(E) dE

U

u
()
d


0

Energy in photon gas for photons with frequencies
between  and  + d= u() d= h F(E) g(E) dE

= h F() g() d
(from week 1homework) = h F() V 82/c3 d





 
V
2
h


u
(
)
d

d

c
exp(
h
/
kT
)

1


3
3
2
Planck Radiation Formula
RWL Jones, Lancaster University
Black Body Radiation
u()
u()

h./kT~3

hc./kT~5

In terms of wavelength (= c/)


V
8
hc
1


u
(
)

 
exp(
hc
/
kT
)

1
 
2
5

RWL Jones, Lancaster University
Black Body Radiation

max  hc/5kT



T = Tsun 6000K max  480 nm (yellow light)
T = Troom 300K max  10 m (Infra-red)
T = Tuniverse 3K max  1 mm (microwave background)

Total Energy of Photon Gas:



8
kT
U

u
(
)
d

V

T



15
hc
5

0
4
4
3
RWL Jones, Lancaster University
Radiation Pressure

For massive particles:
P = (2/3) (U/V) (because E ~ k2 and and k ~ V1/3)

For massless particles E ~ K
P = (1/3) (U/V)
RWL Jones, Lancaster University
Classical Limit




In Maxwell-Boltzmann limit, F(E)<<1,
so exp( (E-)/(kBT) ) >> 1
So FMB(E) = exp( -(E-)/(kBT) )
= exp( /(kBT) ) exp( -(E/(kBT) )
= (N/Z) exp( -(E/(kBT) )
So N/Z = exp( /(kBT) )
So chemical potential  = kBT ln(N/Z)
RWL Jones, Lancaster University