Bose-Einstein Statistics

1
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
• Number of combination to do this is
• So number of microstates in distribution ni
  states

2
Bose-Einstein Statistics

• Classical limit
• Bose-Einstein
• Large numbers gi, ni

ni factors
3
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 (ii)?
• dU 0 ? ?i Ei dni 0 (iii)?
• Once again, add the (i)?(ii)?(iii)
  (Lagrange)?
• Thermodymanics gives ?-1/kT

4
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
• 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 ?

5
Black Body Radiation

• Spectral Energy density is the energy in a photon
  gas between E and EdE U(E) dE
• 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?

Planck Radiation Formula
6
Black Body Radiation

• In terms of wavelength (?? c/?)?

u(?)?
u(?)?
?
?
h?./kT3
hc./?kT5
7
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
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)?

9
Classical Limit

• In Maxwell-Boltzmann limit, F(E)ltlt1,
• so exp( (E-?)/(kBT) ) gtgt 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)?