RWL Jones, Lancaster University Bose-Einstein Statistics  Applies to a weakly-interacting gas of indistinguishable Bosons with:  Fixed N =  i n i  Fixed U =  i E i n i  No Pauli Exclusion Principle: n i  0, unlimited  Each group i has:  g i states, g i -1 possible subgroups, n i to be shared between them  Number of combination to do this is:  So number of microstates in distribution {n i } states:

RWL Jones, Lancaster University Bose-Einstein Statistics  Classical limit:  Bose-Einstein:  Large numbers: g i, n i n i factors

RWL Jones, Lancaster University Bose-Einstein Distribution  We use the same technique as for Boltzmann, maximize ln t({n i }) : d ln t ({n i }) = 0  Add to this the constraints:  dN = 0   i dn i = 0 :(ii)‏  dU = 0   i E i dn i = 0 :(iii)‏  Once again, add the (i)+  (ii)+  (iii) (Lagrange)‏  Thermodymanics gives  =-1/kT

RWL Jones, Lancaster University Open and Closed Systems   given by N=  i g i F(E i ) for a closed system of phoney bosons (e.g. ground state He 4 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 

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  Energy in photon gas for photons with frequencies between and + d  = u( ) d  = h F(E) g(E) dE  = h F( ) g( ) d  = h F( ) g( ) d (from week 1homework) = h F( ) V 8  2 /c 3 d (from week 1homework) = h F( ) V 8  2 /c 3 d Planck Radiation Formula

RWL Jones, Lancaster University Black Body Radiation  In terms of wavelength (  = c/ )‏ h./kT~3 hc./ kT~5 u( )‏

RWL Jones, Lancaster University Black Body Radiation  max  hc/5kT  T = T sun 6000K max  480 nm (yellow light)‏  T = T room 300K max  10  m (Infra-red)‏  T = T universe 3K max  1 mm (microwave background)  Total Energy of Photon Gas:

RWL Jones, Lancaster University Radiation Pressure  For massive particles: P = (2/3) (U/V) (because E ~ k 2 and and k ~ V 1/3 )‏ P = (2/3) (U/V) (because E ~ k 2 and and k ~ V 1/3 )‏  For massless particles E ~ K P = (1/3) (U/V)‏ P = (1/3) (U/V)‏

RWL Jones, Lancaster University Classical Limit  In Maxwell-Boltzmann limit, F(E)<<1, so exp( (E-  )/(k B T) ) >> 1 so exp( (E-  )/(k B T) ) >> 1  So F MB (E) = exp( -(E-  )/(k B T) ) = exp(  /(k B T) ) exp( -(E/(k B T) )‏ = exp(  /(k B T) ) exp( -(E/(k B T) )‏ = (N/Z) exp( -(E/(k B T) )‏ = (N/Z) exp( -(E/(k B T) )‏  So N/Z = exp(  /(k B T) )  So chemical potential  = k B T ln(N/Z)‏