Statistical Physics, Second Revised and Enlarged Edition
48 Gases: the density ofstates 4 .2.2 The dispersion relation The second vital idea is that to make any progress with statistica ...
Summary 49 i.e. ε=(h^2 / 8 Ma^2 )(n^21 +n^22 +n^23 ) (4.8) The energy(h^2 / 8 Ma^2 )=εεj,say, gives a scale for the energy leve ...
50 Gases: the density ofstates States are uniformly distributedink-space, thatisthedensityofstatesink-space is constant, depen ...
5 Gases: the distributions In this chapter the statistical method outlined in section 1. 5 is used to derive the thermalequilibr ...
52 Gases: the distributions Grouped distribution True distribution ith level, energyi containing gistates gi states Fig. 5. 1 D ...
Identical particles –fermions and bosons 53 5 .2 Identical particles – fermions and bosons The next step(stepIII) in the statist ...
54 Gases: the distributions the twoparticles canbewritten as theproduct oftwo one-particle wavefunctions,i.e. ψ(1, 2 )=ψa( 1 )·ψ ...
Counting microstatesfor gases 55 5 .3 Counting microstates for gases Now we are equipped to attack step III of the statistical m ...
56 Gases: the distributions ThesymbolFDisshortfor Fermi–Dirac, since these two physicists were responsible for the first discuss ...
Counting microstatesfor gases 57 Hence our (slightlyapproximate)finalresultfor the number ofmicrostates to an allowed distributi ...
58 Gases: the distributions Againthe numeratoristhe product ofnifactors eachapproximatelyequalto (buta little larger than)gi.Hen ...
The three distributions 59 Substitutinglnt=lntttFDfrom above together with (5.7) and (5.8) forNandUgives afterdifferentiation ∑ ...
60 Gases: the distributions whichrearranges togive n∗i=gi/[exp(−α−βεi)− 1 ] Intermsofthedistributionfunctionthis becomes fffi=ni ...
Summary 61 5. 4 .4 αandβrevisited After the lengthy discussion concerningαandβin Chapter 2, brevity is now in order. The paramet ...
62 Gases: the distributions Use ofStirling’s approximation andLagrangemultipliers, as usedearlier,gives the Fermi–Dirac, Bose–E ...
6 Maxwell–Boltzmann gases As a first application of the groundwork of the two previous chapters, we consider thesimplest situati ...
64 Maxwell–Boltzmanngases where thepartitionfunctionZisdefinedas the sum over allone-particle states of the Boltzmann factors ex ...
The Maxwell–Boltzmann distribution ofspeeds 65 Itis worthputtinginthe numbersfor the worst possible case! Consider^4 Heat1 atmos ...
66 Maxwell–Boltzmanngases inwhichg(v)δvisdefinedas the number ofstatesinthe rangeofinterest,i.e. with speeds betweenvandv+δv. Th ...
The Maxwell–Boltzmann distribution ofspeeds 67 1 2 3 max rms n() ///T Fig. 6. 1 The Maxwell–Boltzmanndistribution ofspe ...
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