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Bose-Einstein Statistics

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

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Bose-Einstein Statistics

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  1. RWL Jones, Lancaster University 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. RWL Jones, Lancaster University Bose-Einstein Statistics Classical limit: Bose-Einstein: Large numbers: gi, ni ni factors

  3. 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 :(ii)‏ dU = 0 i Ei dni = 0 :(iii)‏ Once again, add the (i)+(ii)+(iii) (Lagrange)‏ Thermodymanics gives =-1/kT

  4. 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 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. 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 (from week 1homework) = h F() V 82/c3 d Planck Radiation Formula

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

  7. 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. 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)‏

  9. 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)‏

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