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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:
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RWL Jones, Lancaster University Bose-Einstein Statistics Classical limit: Bose-Einstein: Large numbers: g i, n i n i factors
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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
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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
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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
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RWL Jones, Lancaster University Black Body Radiation In terms of wavelength ( = c/ ) h./kT~3 hc./ kT~5 u( )
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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:
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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)
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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)
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