Plotting ln( w(N,s):  The maximum value occurs at

({{({{ln(((N!)/((N/2)!(N/2)!)))≃N ln(N)-N-2*[(N/2) ln(N/2)-N/2]}})}})

Clear[eN,s,lnw,eNmax]

lnw[eN_,s_]:=-(eN/2 + s) * Log[(1 + 2 s/eN)/2]   -(eN/2 -s) Log[(1-2 s/eN)/2]

eNmax[eN_]:= eN*Log[eN]-eN-2*(eN/2 Log[(eN/2.0)]-eN/2.0)

Manipulate[
Plot[lnw[eN,s],{s,-eN/2.0,eN/2.0},AxesLabel->{"s","ln[w(eN,s)]"},PlotRange->All],
{eN,5,100}]

Shape always looks the same! What’s this about narrowing?!







The “Full Width at half max” is a way of measuring the width of a peak.  Plotting a line at the height (Subscript[w, max ])/2, or rather (since we’re plotting ln(w), we should plot a line at height

({{({{ln((Subscript[w, max ])/2)=ln((Subscript[w, max]))-ln 2}})= ln((Subscript[w, max]))-0.69}})

Manipulate[
Plot[
{lnw[eN,s],eNmax[eN]-0.69},
{s,-eN/2.0,eN/2.0},
AxesLabel->{"s","ln[w(eN,s)]"},PlotRange->All],
{eN,2,100}
]

What elementary function does the graph resemble?










Kittel & Croemer quadratic approximation:

kc[eN_,s_]:=eNmax[eN]-2 s^2 /eN

Manipulate[
Plot[
{lnw[eN,s],eNmax[eN]-0.69,kc[eN,s]},
{s,-eN/2.0,eN/2.0},
AxesLabel->{"s","ln[w(eN,s)]"},PlotRange->All],
{eN,2,100}
]

Plotting the gaussian function, (e^(ck(N,s)))...

Manipulate[
Plot[
Exp[kc[eN,s]],
{s,-eN/2.0,eN/2.0},
AxesLabel->{"s","(Subscript[w, ck])(eN,s)]"},PlotRange->All],
{eN,10,200}
]