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4.6: Statistical Ensembles from Maximum Entropy

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The basic principle: maximize the entropy,

S=kBnPnlnPn .

μCE

We maximize S subject to the single constraint

C=nPn1=0 .

We implement the constraint C=0 with a Lagrange multiplier, ˉλkBλ, writing

S=SkBλC ,

and freely extremizing over the distribution {Pn} and the Lagrange multiplier λ. Thus,

δS=δSkBλδCkBCδλ=kBn[lnPn+1+λ]δPnkBCδλ0 .

We conclude that C=0 and that

lnPn=(1+λ) ,

and we fix λ by the normalization condition nPn=1. This gives

Pn=1Ω,Ω=nΘ(E+ΔEEn)Θ(EnE) .

Note that Ω is the number of states with energies between E and E+ΔE.

OCE

We maximize S subject to the two constraints

C1=nPn1=0,C2=nEnPnE=0 .

We now have two Lagrange multipliers. We write

S=SkB2j=1λjCj ,

and we freely extremize over {Pn} and {Cj}. We therefore have

δS=δSkBn(λ1+λ2En)δPnkB2j=1Cjδλj=kBn[lnPn+1+λ1+λ2En]δPnkB2j=1Cjδλj0 .

Thus, C1=C2=0 and

lnPn=(1+λ1+λ2En) .

We define λ2β and we fix λ1 by normalization. This yields

Pn=1ZeβEn,Z=neβEn .

GCE

We maximize S subject to the three constraints

C1=nPn1=0,C2=nEnPnE=0,C3=nNnPnN=0 .

We now have three Lagrange multipliers. We write

S=SkB3j=1λjCj ,

and hence

δS=δSkBn(λ1+λ2En+λ3Nn)δPnkB3j=1Cjδλj=kBn[lnPn+1+λ1+λ2En+λ3Nn]δPnkB3j=1Cjδλj0 .

Thus, C1=C2=C3=0 and

lnPn=(1+λ1+λ2En+λ3Nn) .

We define λ2β and λ3βμ, and we fix λ1 by normalization. This yields

Pn=1Ξeβ(EnμNn),Ξ=neβ(EnμNn) .


This page titled 4.6: Statistical Ensembles from Maximum Entropy is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Daniel Arovas.

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