Log - Gamma and Spherical Nhd of Uniformity

q[N_,ν_,τ_]:=(((ν^τ) ((Abs[Log[(1/N)]])^(τ-1))(N^(ν-1)))/(Gamma[τ]))

q[N,1,1]//FullSimplify

Plot3D[q[N,1/((2^(1/τ))-1),τ]//Evaluate,{N,0.01,0.99},{τ,0.3,20},PlotRange->All,Mesh->False,PlotPoints->50];

q[0.5,.3,3]

(∫_0^1)(((ν^τ) (((Log[(1/N)]))^(τ-1))(N^(ν-1)))/(Gamma[τ]))\[DifferentialD]N//Re[ν]>0&&Re[τ]>0

Mean

Integrate[(((ν^τ) (((Log[(1/N)]))^(τ-1))(N^(ν-1)))/(Gamma[τ]))N ,{N,0,1},Assumptions->τ>0&&ν>0]

Plot3D[ ((((ν/(1+ν))))^τ),{ν,0,3},{τ,0.0,3},PlotRange->All]

Variance

V[ν_,τ_]:=Integrate[(((ν^τ) (((Log[(1/N)]))^(τ-1))(N^(ν-1)))/(Gamma[τ]))(N^2) ,{N,0,1},Assumptions->τ>0&&ν>0]- ((((ν/(1+ν))))^(2τ))

V[1,1.]

Plot3D[-((((ν/(1+ν))))^(2 τ))+((((ν/(2+ν))))^τ),{ν,0,9},{τ,0,3},PlotRange->All,PlotPoints->50]

Plot3D[(Sqrt[-((((ν/(1+ν))))^(2 τ))+((((ν/(2+ν))))^τ)]),{ν,0,9},{τ,0,3},PlotRange->All,PlotPoints->50]

cv

((-((((ν/(1+ν))))^(2 τ))+((((ν/(2+ν))))^τ))/((((ν/(1+ν))))^(2τ)))//FullSimplify

(Sqrt[-1+(ν^(-τ)) (((1+ν))^(2 τ)) (((2+ν))^(-τ))])

Plot3D[(Sqrt[-1+(ν^(-τ)) (((1+ν))^(2 τ)) (((2+ν))^(-τ))]),{ν,0.1,3},{τ,0,3},PlotRange->All]

Entropy via gamma pdf (e^(-x)) = N

We have log-gamma entropy =  -(∫_0^1)g(N)   Log[g(N) ]   dN = -(∫_0^∞)f(x)  (e^(-x))  (  Log[f(x) ] + x   )dx

f[x_,ν_,τ_]:= (((E^(-x ν)) (x^(-1+τ)) (ν^τ))/(Gamma[τ]))

FullSimplify[(((E^(-x ν)) (x^(-1+τ)) (ν^τ))/(Gamma[τ])) (E^(-x ))Log[(((E^(-x ν)) (x^(-1+τ)) (ν^τ))/(Gamma[τ]))],Assumptions->τ>0&&ν>0]

Integrate[(((E^(-x (1+ν))) (x^(-1+τ)) (ν^τ)( Log[(( (ν^τ))/(Gamma[τ]))]-x ν +(τ-1)Log[x]))/(Gamma[τ])),{x,0,∞},Assumptions->τ>0&&ν>0]

Integrate[x(((E^(-x (1+ν))) (x^(-1+τ)) (ν^τ) )/(Gamma[τ])),{x,0,∞},Assumptions->τ>0&&ν>0]

FullSimplify[(1/(Gamma[τ]))(ν^τ) (((1+ν))^(-1-τ)) (ν Gamma[1+τ]+(1+ν) Gamma[τ] (τ Log[1+ν]+Log[(((ν^(-τ)) Gamma[τ])/(1+ν))]-(-1+τ) PolyGamma[0,τ]))-(ν^τ) (((1+ν))^(-1-τ)) τ,Assumptions->τ>0&&ν>0]

(Subscript[S, LG]) [ν_,τ_]:= (ν^τ) (((1+ν))^(-1-τ)) (τ (-1+ν+(1+ν) Log[1+ν])+(1+ν) Log[(((ν^(-τ)) Gamma[τ])/(1+ν))]-(1+ν) (-1+τ) PolyGamma[0,τ])

TeXForm[(Subscript[S, LG])[ν,τ]]

(ν^τ) (((1+ν))^(-1-τ)) (τ (-1+ν+(1+ν) Log[1+ν])+(1+ν) Log[(((ν^(-τ)) Gamma[τ])/(1+ν))]-(1+ν) (-1+τ) PolyGamma[0,τ])

τ (-1+ν+(1+ν) Log[1+ν])-(1+ν) (-1+τ) PolyGamma[0,τ]//TeXForm

(ν^τ) (((1+ν))^(-1-τ))//TeXForm

+(1+ν) Log[(((ν^(-τ)) Gamma[τ])/(1+ν))]//TeXForm

Limit[(Subscript[S, LG])[ν,τ],τ->∞]

Limit[(ν^τ) (((1+ν))^(-1-τ)) (τ (-1+ν+(1+ν) Log[1+ν])+(1+ν) Log[(((ν^(-τ)) Gamma[τ])/(1+ν))]-(1+ν) (-1+τ) PolyGamma[0,τ]),τ->∞]

(Subscript[S, LG])[ν,1]//PowerExpand//FullSimplify

Plot3D[(Subscript[S, LG])[ν,τ],{ν,0.,3},{τ,0.1,.53},PlotRange->All]

Plot[{(Subscript[S, LG])[ν,.289],(Subscript[S, LG])[ν,2.24]},{ν,0.,3},PlotRange->All]

LogGamEntCont=ContourPlot[(Subscript[S, LG])[ν,τ], {ν,0.4,3},{τ,0.1,3},Contours->50,ColorFunction->(Hue[#]&), ClippingStyle->Automatic]

LogGamGradFlow=GradientFieldPlot[(Subscript[S, LG])[ν,τ], {ν,0.4,3},{τ,0.1,3},GridLines->Automatic]

Show[LogGamEntCont,LogGamGradFlow]

Approximating Entropy-- - (Direct Integral intractable)

Integrate[(((ν^τ) ((Abs[Log[(1/N)]])^(τ-1))(N^(ν-1)))/(Gamma[τ])) Log[(((ν^τ) ((Abs[Log[(1/N)]])^(τ-1))(N^(ν-1)))/(Gamma[τ]))],{N,0,1},Assumptions->τ>0&&ν>0]

Integrate[(((ν^τ) ((Abs[Log[(1/N)]])^(τ-1))(N^(ν-1)))/(Gamma[τ])) Log[(((ν^τ) (N^(ν-1)))/(Gamma[τ]))],{N,0,1},Assumptions->τ>0&&ν>0]

Integrate[(((ν^τ) ((Abs[Log[(1/N)]])^(τ-1))(N^(ν-1)))/(Gamma[τ])) (τ-1)Log[Log[(1/N)]],{N,0,1},Assumptions->τ>0&&ν>0]

Plot[{Log[Abs[Log[(1/N)]]],3.4(0.36-N)},{N,0,1}]

Integrate[(((ν^τ) ((Abs[Log[(1/N)]])^(τ-1))(N^(ν-1)))/(Gamma[τ])) (τ-1)3.4(0.36-N),{N,0,1},Assumptions->τ>0&&ν>0]

S[ν_,τ_]:=((τ-ν τ+ν Log[((ν^τ)/(Gamma[τ]))])/ν)-0.136 (-9+25 ((((ν/(1+ν))))^τ)) (-1+τ)

Plot3D[S[ν,τ],{ν,0.4,3},{τ,0.1,3},PlotRange->All]

Plot[{S[1,τ],S[5,τ],S[1.5,τ],S[2,τ] },{τ,0.4,.95},PlotRange->{{0,3},{-.5,2.5}}]

Needs["VectorAnalysis`"]
Needs["VectorFieldPlots`"]

LogGamEntContNuTau=ContourPlot[S[ν,τ], {ν,0.4,3},{τ,0.1,3},Contours->50,ColorFunction->(Hue[#]&), ClippingStyle->Automatic]

GradFlowNuTau=GradientFieldPlot[S[ν,τ], {ν,0.4,3},{τ,0.1,3},GridLines->Automatic]

Show[LogGamEntContNuTau,GradFlowNuTau]

Solving grad flow curves in μ, κ

{(∂_ν)(S[ν,τ]),(∂_τ)(S[ν,τ])}//FullSimplify

Limit[(((-1.`+(ν^2)+ν ((((ν/(1+ν))))^τ) (3.4000000000000004`-3.4000000000000004` τ)) τ)/((ν^2) (1.`+ν))),ν->∞]

NDSolve[{x'[t]==(1/((1+x[t])((x[t])^2)))(((x[t])^2)-1+x[t](((((x[t])/(1+x[t]))))^(y[t]))3.4  y[t](1-y[t])),
y'[t]==(1/(x[t]))(1+0.224 x[t]+x[t] Log[x[t]]+x[t] (((((x[t])/(1+x[t]))))^(y[t])) (-3.4+(3.4-3.4y[t]) Log[((x[t])/(1+x[t]))])- x[t] PolyGamma[0,y[t]]),x[0]==.1,y[0]==1},{x[t],y[t]},{t,0,.50}]

NDSolve[{x'[t]==((-1+((x[t])^2)+3.4` x[t] (((((x[t])/(1+x[t]))))^(y[t])) (1-y[t]) y[t])/(((x[t])^2) (1+x[t]))),y'[t]==(1/(x[t]))(1+0.224` x[t]+Log[x[t]] x[t]-PolyGamma[0,y[t]] x[t]+x[t] (((((x[t])/(1+x[t]))))^(y[t])) 3.4(-1+Log[((x[t])/(1+x[t]))] (1- y[t]))),x[0]==0.4,y[0]==1.02},{x[t],y[t]},{t,0,40}]
L0410=ParametricPlot[Evaluate[{x[t],y[t]}/.%],{t,0,.030},Frame->True,GridLines->Automatic,PlotStyle->Thickness[0.015],PlotRange->All]

Tubular Nhd of τ = 1

AffGamma[ν_,τ_]:={ν,τ,Log[Gamma[τ]]-τ Log[ν]}

G=ParametricPlot3D[AffGamma[ν,τ]//Evaluate,
{ν,0.05,3},{τ,0.2,4}, ViewPoint->{3,2,1}, BoxRatios->{1,1,1},Mesh->None,PlotRange->{-3,4}]

c[t_]:={t,1,-Log[t]}

tubecurve[c_][r_][t_,θ_]:={t-((r Cos[θ])/((Sqrt[1+(1/(t^2))]) (Sqrt[(1/(t^4))]) (t^3))),1-((r Sin[θ])/((Sqrt[(1/(t^4))]) (t^2))),-((r Cos[θ])/((Sqrt[1+(1/(t^2))]) (Sqrt[(1/(t^4))]) (t^2)))-Log[t]}

R=ParametricPlot3D[tubecurve[
c][1.][t,θ]
//Evaluate,
{t,0.08,4},{θ,-1.2Pi/2,.98Pi},
PlotPoints->{20,20},ViewPoint->{3,2,1}, 
BoxRatios->{1,1,1},Mesh->None,PlotRange->{{0,3},{0,3},{-4,2}}]

U=ParametricPlot3D[{{t,1,-Log[t]},{1,t,Log[Gamma[t]]}},
{t,0.08,4},ViewPoint->{3,2,1}, 
BoxRatios->{1,1,1},PlotStyle->Thickness[0.01],PlotRange->{-2,3}]

<<Graphics`Graphics3D`
<<Graphics`Graphics`

scatt=ScatterPlot3D[{{0.1,0.289,Log[Gamma[0.289]]-0.289 Log[0.1]+.2},{2.75,2.24,Log[Gamma[2.24]]-2.24 Log[2.75]+.2}},BoxRatios->{1,1,1},PlotRange->All,
ViewPoint->{15,3,15},BoxRatios->{1,1,1},PlotStyle->{RGBColor[0,0,0],PointSize[0.04]} ];

B=ParametricPlot3D[{1-1.6 2 Cos[u] Cos[v],1.6 1-2 Cos[v] Sin[u],1.6 2 Sin[v]},{u,.3Pi,3.99Pi/2},{v,Pi/2,3Pi/2},Mesh->None,ViewPoint->{3,2,1}, 
BoxRatios->{1,1,1},PlotRange->{-2,3}];
Show[scatt,G,U,B,PlotRange->{-3,4}]