4.6: Statistical Ensembles from Maximum Entropy
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The basic principle: maximize the entropy,
S=−kB∑nP∗nlnP∗n .
μCE
We maximize S subject to the single constraint
C=∑nP∗n−1=0 .
We implement the constraint C=0 with a Lagrange multiplier, ˉλ≡kBλ, writing
S∗=S−kBλC ,
and freely extremizing over the distribution {P∗n} and the Lagrange multiplier λ. Thus,
δS∗=δS−kBλδC−kBCδλ=−kB∑n[lnP∗n+1+λ]δP∗n−kBCδλ≡0 .
We conclude that C=0 and that
lnP∗n=−(1+λ) ,
and we fix λ by the normalization condition ∑nP∗n=1. This gives
P∗n=1Ω,Ω=∑nΘ(E+ΔE−E∗n)Θ(E∗n−E) .
Note that Ω is the number of states with energies between E and E+ΔE.
OCE
We maximize S subject to the two constraints
C∗1=∑nP∗n−1=0,C∗2=∑nE∗nP∗n−E=0 .
We now have two Lagrange multipliers. We write
S∗=S−kB2∑j=1λ∗jC∗j ,
and we freely extremize over {P∗n} and {C∗j}. We therefore have
δS∗=δS−kB∑n(λ∗1+λ∗2E∗n)δP∗n−kB2∑j=1C∗jδλ∗j=−kB∑n[lnP∗n+1+λ∗1+λ∗2E∗n]δP∗n−kB2∑j=1C∗jδλ∗j≡0 .
Thus, C∗1=C∗2=0 and
lnP∗n=−(1+λ∗1+λ∗2E∗n) .
We define λ∗2≡β and we fix λ∗1 by normalization. This yields
P∗n=1Ze−βE∗n,Z=∑ne−βE∗n .
GCE
We maximize S subject to the three constraints
C∗1=∑nP∗n−1=0,C∗2=∑nE∗nP∗n−E=0,C∗3=∑nN∗nP∗n−N=0 .
We now have three Lagrange multipliers. We write
S∗=S−kB3∑j=1λ∗jC∗j ,
and hence
δS∗=δS−kB∑n(λ∗1+λ∗2E∗n+λ∗3N∗n)δP∗n−kB3∑j=1C∗jδλ∗j=−kB∑n[lnP∗n+1+λ∗1+λ∗2E∗n+λ∗3N∗n]δP∗n−kB3∑j=1C∗jδλ∗j≡0 .
Thus, C∗1=C∗2=C∗3=0 and
lnP∗n=−(1+λ∗1+λ∗2E∗n+λ∗3N∗n) .
We define λ∗2≡β and λ∗3≡−βμ, and we fix λ∗1 by normalization. This yields
P∗n=1Ξe−β(E∗n−μN∗n),Ξ=∑ne−β(E∗n−μN∗n) .