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BeginPackage["IsingScalingFunction`"]
g::usage = "g[θ0, gs][θ] gives the Schofield coordinate transformation defined in (14)."
ut::usage = "ut[θ] gives the scaling field u_t as a function of Schofield coordinates."
uh::usage = "uh[θ0, gs][θ] gives the scaling field u_h as a function of Schofield coordinates."
η::usage = "η[θ0, gs][θ] gives the invariant scaling combination η."
ξ::usage = "ξ[θ0, gs][θ] gives the invariant scaling combination ξ."
ReScriptF::usage = "ReScriptF[θ0, θYL, B, C0, CYL, Gs][θ] gives the free energy scaling function defined in (19)."
ScriptF::usage = "ScriptF[θ0, θYL, B, C0, CYL, Gs][θ] gives the free energy scaling function defined in (35)."
DScriptFPlusMinusDξθ0List::usage =
"DScriptFPlusMinusDξθ0List computes the first m derivatives of the scaling function F_- evaluated at θ_0."
DScriptFPlusMinusDξList::usage =
"DScriptFPlusMinusDξList computes the first m derivatives of the scaling function F_+/-."
DScriptF0DηList::usage =
"DScriptF0DηList computes the first m derivatives of the scaling function F_0."
uf::usage = "uf computes the singular free energy u_f."
DufDut::usage =
"DufDut computes derivatives of the singular free energy u_f with respect to the temperature-like scaling field u_t."
DufDuh::usage =
"DufDuh computes derivatives of the singular free energy u_f with respect to the temperature-like scaling field u_h."
ruleB::usage = ""
ruleAL::usage = ""
Begin["Private`"]
β := 1/8
δ := 15
Δ := β δ
OverlineS := 2^(1/12) Exp[-1/8] Glaisher^(3/2)
Φs := {
-Gamma[1/3]Gamma[1/5]Gamma[7/15]/(2 π Gamma[2/3]Gamma[4/5]Gamma[8/15])(4 π^2 Gamma[13/16]^2 Gamma[3/4]/(Gamma[3/16]^2 Gamma[1/4]))^(8/15),
-0.31881012489061,
Around[0.110886196683, 2.0 10^-12],
Around[0.01642689465, 2.0 10^-11],
Around[-2.639978 10^-4, 1.0 10^-10],
Around[-5.140526 10^-4, 1.0 10^-10],
Around[2.08865 10^-4, 1.0 10^-9],
Around[-4.4819 10^-5, 1.0 10^-9],
Around[3.16 10^-7, 1.0 10^-9],
Around[4.31 10^-6, 0.01 10^-6],
Around[-1.99 10^-6, 0.01 10^-6]
}
Gls := {
0,
-OverlineS,
−1.000960328725262189480934955172097320572505951770117 Sqrt[2]/((2 )^(-7/8) (2^(3/16)/OverlineS)^2)/2/(12 \[Pi]),
Around[ 0.038863932, 3.0 10^(-9)],
Around[−0.068362119, 2.0 10^(-9)],
Around[ 0.18388370, 1.0 10^(-8)],
Around[-0.6591714, 1.0 10^(-7)],
Around[ 2.937665, 3.0 10^(-6)],
Around[-15.61, 1.0 10^(-2)],
Around[ 96.76, 1.0 10^(-2)],
Around[-6.79 10^2, 1.0],
Around[ 5.34 10^3, 10.],
Around[-4.66 10^4, 0.01 10^4],
Around[ 4.46 10^5, 0.01 10^5],
Around[-4.66 10^6, 0.01 10^6]
}
Ghs := {
0,
0,
-1.000815260440212647119476363047210236937534925597789 Sqrt[2]/((2 )^(-7/8) (2^(3/16)/OverlineS)^2)/2,
0,
Around[ 8.333711750, 5.0 10^(-9)],
0,
Around[-95.16896, 1.0 10^(-5)],
0,
Around[1457.62, 3.0 10^(-2)],
0,
Around[-2.5891 10^4, 2.0],
0,
Around[5.02 10^5, 0.01 10^5],
0,
Around[-1.04 10^7, 0.01 10^7]
}
t[θ_] := θ^2 - 1
g[θ0_, gs_][θ_] := (1 - (θ/θ0)^2) Total[MapIndexed[Function[{gi, i}, gi θ^(2*i[[1]]-1)], gs]]
ut[R_, θ_] := R t[θ]
uh[θ0_, gs_][R_, θ_] := R^Δ g[θ0, gs][θ]
η[θ0_, gs_][θ_] := t[θ] / RealAbs[g[θ0, gs][θ]]^(1 / Δ)
ξ[θ0_, gs_][θ_] := g[θ0, gs][θ] / RealAbs[t[θ]]^Δ
ScriptR[θc_, B_][θ_] := (θc Exp[1/(B θc)] ExpIntegralEi[-1/(B θc)] + (θ - θc) Exp[-1/(B (θ - θc))] ExpIntegralEi[1/(B (θ - θc))]) / π
ReScriptF0[C0_, θc_, B_][θ_] := C0 (ScriptR[θc, B][θ] + ScriptR[θc, B][-θ])
ScriptFYL[θYL_, CYL_][θ_] := CYL ((-I θ + θYL)^(5/6) + (I θ + θYL)^(5/6) - 2 θYL^(5/6))
ReScriptFRegular[θ0_, θYL_, B_, C0_, CYL_, Gs_][θ_] := C0 ScriptR[θ0, B][-θ] + ScriptFYL[θYL, CYL][θ] + Total[MapIndexed[Function[{G, i}, G θ^(2*i[[1]])], Gs]]
ReScriptF[θ0_, θYL_, B_, C0_, CYL_, Gs_][θ_] := ReScriptFRegular[θ0, θYL, B, C0, CYL, Gs][θ] + C0 ScriptR[θ0, B][θ]
DReScriptFIrregular[θ0_, B_, C0_][m_] := Piecewise[{{C0 m! Gamma[m - 1] B^(m - 1) / π, m > 1}, {C0 θ0 Exp[1/(B θ0)] ExpIntegralEi[-1/(B θ0)] / π, m == 0}}, 0]
ScriptF[θ0_, θYL_, B_, C0_, CYL_, Gs_][θ_] := ReScriptF[θ0, θYL, B, C0, CYL, Gs][θ] + C0 I Sign[Im[θ]] ((θ-θ0)Exp[-1/(B(θ-θ0))]-(-θ-θ0)Exp[-1/(B(-θ-θ0))])
ScriptFPlusMinus[ScriptF_][θ_] := ScriptF[θ] / t[θ]^2 - 1/(8 \[Pi]) Log[t[θ]^2]
ScriptF0[θ0_, gs_][ScriptF_][θ_] := RealAbs[g[θ0, gs][θ]]^(-2 / Δ) ScriptF[θ] - η[θ0, gs][θ]^2 Log[g[θ0, gs][θ]^2] / (8 π Δ)
uf[params__][R_, θ_] := R^2 ReScriptF[params][θ] + t[θ]^2 R^2 / (8 π) Log[R^2]
EfficientDerivativeList[n_][f_][x_] := Module[
{xp}, NestList[Function[g, D[g, xp]], f[xp], n] /. xp -> x
]
InverseDerivativeList[n_][f_][x_] := Module[
{xp, dfs, fp, Pns},
dfs = Rest[EfficientDerivativeList[n][f][x]];
Pns = FoldList[Function[{Pm, m},
fp'[xp] D[Pm, xp] - (2 m - 1) fp''[xp] Pm], 1, Range[n - 1]] /.
Derivative[m_][fp][xp] :> dfs[[m]];
MapIndexed[{Pn, i} \[Function] Pn/dfs[[1]]^(2 i[[1]] - 1), Pns]
]
CompositeFunctionDerivativeList[G_, F_, X_, FSupp_:(0&)][m_, θ_] := Module[
{ ds, dF, df, fp },
ds = InverseDerivativeList[m+1][X][θ];
dF = EfficientDerivativeList[m][F][θ] + FSupp /@ Range[0, m];
df = EfficientDerivativeList[m][G[fp]][θ] /.
Map[Derivative[#][fp][θ] -> dF[[# + 1]] &, Range[0, m]];
Table[Sum[df[[k+1]] BellY[j, k, ds[[;; j - k + 1]]], {k, 0, j}]/(j!), {j, 0, m}]
]
DScriptFPlusMinusDξθ0List[θ0_, θYL_, B_, C0_, CYL_, Gs_, gs_][m_] := CompositeFunctionDerivativeList[
ScriptFPlusMinus, ReScriptFRegular[θ0, θYL, B, C0, CYL, Gs],
ξ[θ0, gs], DReScriptFIrregular[θ0, B, C0]
][m, θ0]
DScriptFPlusMinusDξList[θ0_, θYL_, B_, C0_, CYL_, Gs_, gs_][m_, θ_] := CompositeFunctionDerivativeList[
ScriptFPlusMinus, ReScriptF[θ0, θYL, B, C0, CYL, Gs], ξ[θ0, gs]
][m, θ]
DScriptF0DηList[θ0_, θYL_, B_, C0_, CYL_, Gs_, gs_][m_, θ_] := CompositeFunctionDerivativeList[
ScriptF0[θ0, gs], ReScriptF[θ0, θYL, B, C0, CYL, Gs], η[θ0, gs]
][m, θ]
DScriptFPlusMinusDξθ0[params__][m_] := Last[DScriptFPlusMinusDξθ0List[params][m]]
DScriptFPlusMinusDξ[params__][m_, θ_] := Last[DScriptFPlusMinusDξList[params][m, θ]]
DScriptF0Dη[params__][m_, θ_] := Last[DScriptF0DηList[params][m, θ]]
DufDut[θ0_, θYL_, B_, C0_, CYL_, Gs_, gs_][m_][R_, θ_] := RealAbs[uh[θ0, gs][R, θ]]^(2 / Δ - m / Δ) DScriptF0Dη[θ0, θYL, B, C0, CYL, Gs, gs][m, θ] + Log[uh[θ0, gs][R, θ]^2] / (8 π Δ) Derivative[m][Function[utp, utp^2]][ut[R, θ]]
DufDuh[θ0_, θYL_, B_, C0_, CYL_, Gs_, gs_][m_][R_, θ_] := RealAbs[ut[R, θ]]^(2-m Δ) DScriptFPlusMinusDξ[θ0, θYL, B, C0, CYL, Gs, gs][m, θ] + ut[R, θ]^2 / (8 π) Log[ut[R, θ]^2]
ruleB[θ0_, gs_] := (2 * OverlineS / π) * (- g[θ0, gs]'[θ0] / t[θ0]^Δ)
ruleAL[θ0_, gs_] := Exp[Δ t[θ0]^(Δ - 1) t'[θ0] / (2 OverlineS / π g[θ0, gs]'[θ0]) - t[θ0]^Δ g[θ0, gs]''[θ0] / (4 OverlineS / π g[θ0, gs]'[θ0]^2)] t[θ0]^(1/8) OverlineS / (2 π) * g[θ0, gs]'[θ0]
End[]
EndPackage[]
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