(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 1429311, 64026]*) (*NotebookOutlinePosition[ 1429964, 64049]*) (* CellTagsIndexPosition[ 1429920, 64045]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Second revision of analytical calculation for a single velocity step\ \>", "Subtitle"], Cell["\<\ This notebook corrects the omission of the second ratio from the ratio factor \ R as done is stochasflow4.nb. This makes negligible difference to the \ numerical results, as illustrated below in some key plots, so all the numeric \ work is not repeated. Also, the various variable transformations from \ stochasflow4.nb are consolidated and used from the start. This allows direct \ simplification to be applied and the use of limits to obtain compact formulas \ is not needed any more. \ \>", "Subsubtitle"], Cell[TextData[{ "This notebook uses initially the ratio factor ", Cell[BoxData[ \(\(\(z + \((1 - f)\) w2\)\/\(z + \((1 - f)\) wg\)\) \(z\ f\ + \((1 - f)\) wg\ \[Phi]g\)\/\(z\ f\ + \((1 \ - f)\) w2\ \[Phi]2\)\)]], " where wg and \[Phi]g contain the as yet undetermined velocity Vg , which \ is the velocity argument of the g-function, and is expected to be somewhere \ between the extremes V1 and V2. Its value is to be determined below from the \ conservation requirtement.\n" }], "Text"], Cell[CellGroupData[{ Cell["Formulas needed from stochasflow3.nb", "Section"], Cell[CellGroupData[{ Cell["\<\ Consider a single step where the flow velocity increases at x=x1 by 50%:\ \>", "Subsubsection"], Cell[BoxData[ \(\(Upstep = {S -> 0.1, \[Xi] -> 0.2, \[Gamma] -> \@0.05, t0 -> \(-\(S\^2\/\[Gamma]\^2\)\), \[IndentingNewLine]V1 -> 0.1, x1 -> 1.0, V2 -> 0.15, \[Theta]1 -> \(x1 - \[Xi]\)\/V1};\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ A downwards step where the flow velocity decreases at x=x1 by 50%:\ \>", "Subsubsection"], Cell[BoxData[ \(\(Downstep = {S -> 0.1, \[Xi] -> 0.2, \[Gamma] -> \@0.05, t0 -> \(-\(S\^2\/\[Gamma]\^2\)\), \[IndentingNewLine]V1 -> 0.1, x1 -> 1.0, V2 -> 0.05, \[Theta]1 -> \(x1 - \[Xi]\)\/V1};\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["For checking, a case of a virtual step (no velocity change):", \ "Subsubsection"], Cell[BoxData[ \(\(Nostep = {S -> 0.1, \[Xi] -> 0.2, \[Gamma] -> \@0.05, t0 -> \(-\(S\^2\/\[Gamma]\^2\)\), \[IndentingNewLine]V1 -> 0.1, x1 -> 1.0, V2 -> 0.1, \[Theta]1 -> \(x1 - \[Xi]\)\/V1};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify[\[PartialD]\_f\ \(\(z + \((1 - f)\) w2\)\/\(z + \((1 - f)\) w1\)\)]\)], "Input"], Cell[BoxData[ \(\(\((w1 - w2)\)\ z\)\/\((w1 - f\ w1 + z)\)\^2\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["The first dimensionless variable transformation:", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(\(zdef = {z -> \((x - x1)\)\/\(\[Gamma] \@\( 2 \((t - t0)\)\)\), \ \[Phi]1 -> \(\[Theta]1 - t0\)\/\(t - t0\), \[Phi]2 -> \(\((V1/V2)\) \[Theta]1 \ - t0\)\/\(t - t0\), w2 -> \(V2 \@\( t - \ t0\)\)\/\(\[Gamma] \@ 2\), w1 -> \(V1 \@\( t - \ t0\)\)\/\(\[Gamma] \@ 2\), Z2 \[Rule] \(V2\ \((t - \[Theta]1)\)\)\/\(\@2\ \@\(t - t0\)\ \ \[Gamma]\)};\)\)], "Input"], Cell[BoxData[ \(General::"spell1" \(\(:\)\(\ \)\) "Possible spelling error: new symbol name \"\!\(\[Phi]1\)\" is similar \ to existing symbol \"\!\(\[Theta]1\)\"."\)], "Message"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Various auxiliary functions:", "Subsubsection"], Cell["\<\ The old definition of the f function, incorporating sign information:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f[y_] = \(y\^2 + y \@\( y\^2 + 4 z\ w2\ \[Phi]2\) + \(\(2\) \(w2\)\(\ \ \)\(\[Phi]2\) \((z + w2\ \[Phi]2)\)\(\ \)\)\)\/\(2 \((y\^2 + \((z + w2\ \ \[Phi]2)\)\^2)\)\)\)], "Input"], Cell[BoxData[ \(\(y\^2 + 2\ w2\ \[Phi]2\ \((z + w2\ \[Phi]2)\) + y\ \@\(y\^2 + 4\ w2\ z\ \ \[Phi]2\)\)\/\(2\ \((y\^2 + \((z + w2\ \[Phi]2)\)\^2)\)\)\)], "Output"] }, Open ]], Cell[BoxData[ \(F[y_] := y\^2 + Collect[\((\(w1\^2\) \((\[Phi]1 - \ f[y])\)\^2 - \(w2\^2\) \((\ \[Phi]2 - \ f[y])\)\^2)\)/f[y], f[y], Simplify]\)], "Input"], Cell["The normalised gaussian:", "Text"], Cell[BoxData[ \(g[x_, \[Sigma]_] := 1\/\(\[Sigma] \@\( 2 \[Pi]\)\)\ Exp[\(-\(x\^2\/\(2 \[Sigma]\^2\)\)\)]\ \)], "Input"], Cell["The peak position in region 2:", "Text"], Cell[BoxData[ \(Xd[t_, Step_] := x1 + V2 \((t - \[Theta]1)\) //. Step\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Approximation formulas :", "Subsubsection"], Cell["For ytilde, the y value at the peak:", "Text"], Cell[BoxData[ \(yap[ t_] := \(\(\((V2 - V1)\) \[Theta]1\)\/\(\(\[Gamma]\) \(\@\( 2 \((\[Theta]1 - t0)\)\)\)\(\ \)\)\) \@\(\(t - \[Theta]1\)\/\(t - t0\)\ \)\)], "Input"], Cell["Expressed back into the f-variable, this becomes", "Text"], Cell[BoxData[ \(ftil[t_] := f[yap[t]]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Formulas from stochasflow2.nb", "Subsubsection"], Cell[BoxData[ \(C1[x_, t_, Step_] := \(1\/\@\[Pi]\) \(1\/\@\(2 \((S\^2 + \(\[Gamma]\^2\) \ t)\)\)\) Exp[\(-\(\((x - \[Xi] - V1\ t)\)\^2\/\(2 \((S\^2 + \(\[Gamma]\^2\) t)\)\)\)\)] /. Step\)], "Input"], Cell[BoxData[ \(Px1[tt_, x_, t_, Step_] := \(\[Gamma]\^2\/\@\[Pi]\) \(\(x - x1 + V2 \((t - tt)\)\)\/\((2 \( \[Gamma]\^2\) \((t - tt)\))\)\^\(3/2\)\ \) Exp[\(-\(\((x - x1 - V2 \((t - tt)\))\)\^2\/\(2 \( \[Gamma]\^2\) \((t - tt)\)\)\)\)] /. Step\)], "Input"], Cell[BoxData[ \(\(g[xx_, tt_, x_, t_, V_] = \(\((t - tt)\) \((xx - \[Xi] - V\ t0)\) + \((tt - t0)\) \((x \ - xx)\)\)\/\(\((tt - t0)\) \((x - xx + V \((t - tt)\))\)\);\)\)], "Input"], Cell[BoxData[ \(C2[x_, t_, Step_] := NIntegrate[ Evaluate[\((V1/V2)\) \((Px1[tt, x, t, Step] C1[x1, tt, Step] g[x1, tt, x, t, V1])\) //. Step], {tt, t0 + 0.0001 //. Step, t - 0.0001}]\)], "Input"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Check the effect of using the full ratio factor in a numerical \ calculation:\ \>", "Section"], Cell["Use Vg = V1 here as explained below", "Text"], Cell[BoxData[ \(C2ap[xx_, tt_, Step_] := Module[{ymin}, ymin = FindRoot[\((\(F'\)[y] //. 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Cell[BoxData[ \(\(y\^2 + 2\ w2\ \[Phi]2\ \((z + w2\ \[Phi]2)\) + \@\(y\^4 + 4\ w2\ y\^2\ \ z\ \[Phi]2\)\)\/\(2\ \((y\^2 + \((z + w2\ \[Phi]2)\)\^2)\)\)\)], "Output"] }, Open ]], Cell["\<\ The change will be used later on to simplify the taking of limits. However, \ the downside is that by taking the y inside the square root factor, \ information about the sign of y is lost. Looking back at the early part of \ stochasflow3.nb, shows that selecting the correct sign of y is crucial to \ make everything work. So a sign in front of the square root will be \ reinserted manually later on, when the \"expansion\" transformation is \ defined..\ \>", "Text"], Cell[BoxData[ \(\(Fseries = Series[F[y] + \((z + w2\ \[Phi]2 - w2)\)\^2, {z, w2 - w2\ \[Phi]1, 2}];\)\)], "Input"], Cell["\<\ The coefficients a0,a1 and a2 are simplified by expressing them in terms of \ Z2 as well:\ \>", "Text"], Cell[BoxData[ \(\(a0 = SeriesCoefficient[Fseries, 0] /. {w2\ \[Phi]1 \[Rule] w2 - Z2};\)\)], "Input"], Cell[BoxData[ \(\(a1 = SeriesCoefficient[Fseries, 1] /. {w2\ \[Phi]1 \[Rule] w2 - Z2};\)\)], "Input"], Cell[BoxData[ \(\(a2 = SeriesCoefficient[Fseries, 2] /. {w2\ \[Phi]1 \[Rule] w2 - Z2};\)\)], "Input"], Cell["\<\ A new set of variables and constants are defined based on the outcome of \ stochasflow4.nb::\ \>", "Text"], Cell[BoxData[ \(\(TGrep = {T \[Rule] \@\(\(t - t0\)\/\(\[Theta]1 - t0\)\), \[Alpha] \ \[Rule] \(\[Theta]1 - t0\)\/\(-t0\), \[Beta] \[Rule] \[Gamma]\/\(V1\ \@\(\ \[Theta]1 - t0\)\), \[CapitalDelta] \[Rule] \(V2 - V1\)\/\(V2 + V1\)};\)\)], \ "Input"], Cell["\<\ To write the equations in terms of them we need the transformation\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(GTrep = {t \[Rule] t0 + \((\[Theta]1 - t0)\)\ T\^2, \[Gamma] \[Rule] \[Beta]\ V1\ \@\(\[Theta]1 \ - t0\), \[Theta]1 \[Rule] t0 - \[Alpha]\ t0, V2 \[Rule] V1 \( 1 + \[CapitalDelta]\)\/\(1 - \[CapitalDelta]\)};\)\)], \ "Input"], Cell[BoxData[ \(General::"spell1" \(\(:\)\(\ \)\) "Possible spelling error: new symbol name \"\!\(GTrep\)\" is similar to \ existing symbol \"\!\(TGrep\)\"."\)], "Message"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Tzrep = Simplify[{z \[Rule] \((x - x1)\)\/\(\[Gamma] \(\@ 2\) \(\@\(\[Theta]1 - \ t0\)\) T\), \[Phi]1 \[Rule] 1\/T\^2, \[Phi]2 \[Rule] \(\((V1/V2)\)\ \[Theta]1 - \ \ t0\)\/\(\((\[Theta]1 - t0)\) T\^2\), w1 \[Rule] \(\(\(\ \)\(V1 \@\( \[Theta]1 - t0\)\)\)\/\(\[Gamma] \ \@ 2\)\) T, w2 \[Rule] \(\(\(\ \)\(V2 \@\( \[Theta]1 - t0\)\)\)\/\(\[Gamma] \ \@ 2\)\) T, Z2 \[Rule] \(\(\(\ \)\(V2 \@\( \[Theta]1 - t0\)\)\)\/\(\[Gamma] \ \@ 2\)\) \((T - 1\/T)\)} //. GTrep]\)], "Input"], Cell[BoxData[ \({z \[Rule] \(\(-x\) + x1\)\/\(\@2\ T\ t0\ V1\ \[Alpha]\ \[Beta]\), \ \[Phi]1 \[Rule] 1\/T\^2, \[Phi]2 \[Rule] \(\[Alpha] + 2\ \[CapitalDelta] - \[Alpha]\ \ \[CapitalDelta]\)\/\(T\^2\ \[Alpha]\ \((1 + \[CapitalDelta])\)\), w1 \[Rule] T\/\(\@2\ \[Beta]\), w2 \[Rule] \(-\(\(T\ \((1 + \[CapitalDelta])\)\)\/\(\@2\ \[Beta]\ \ \((\(-1\) + \[CapitalDelta])\)\)\)\), Z2 \[Rule] \(\((\(-\(1\/T\)\) + T)\)\ \((1 + \ \[CapitalDelta])\)\)\/\(\@2\ \((\[Beta] - \[Beta]\ \[CapitalDelta])\)\)}\)], \ "Output"] }, Open ]], Cell[BoxData[ \(\(contraction = {\@\(yy\^2 + 4\ w2\ yy\ Z2\ \[Phi]2\) \[Rule] R, 1\/\@\(yy\^2 + 4\ w2\ yy\ Z2\ \[Phi]2\) -> 1\/R, 1/\((yy\^2 + 4\ w2\ yy\ Z2\ \[Phi]2)\)\^\(3/2\) \[Rule] 1/R\^3, 1/\((yy\^2 + 4\ w2\ yy\ Z2\ \[Phi]2)\)\^\(5/2\) \[Rule] 1/R\^5, 1/\((yy\^2 + 4\ w2\ yy\ Z2\ \[Phi]2)\)\^\(7/2\) \[Rule] 1/R\^7, z \[Rule] p\/T, \[Phi]1 \[Rule] 1\/T\^2, \[Phi]2 \[Rule] r\/T\^2, w1 \[Rule] s\ T, w2 \[Rule] q\ T, Z2 \[Rule] q\ \((T - 1\/T)\), yy \[Rule] u \((1 - 1\/T\^2)\), R \[Rule] v \((1 - 1\/T\^2)\)};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[{\((\(\((\(-V1\) + V2)\)\ \@\(\(t - \[Theta]1\)\/\(t - t0\)\)\ \ \[Theta]1\)\/\(\@2\ \[Gamma]\ \@\(\(-t0\) + \[Theta]1\)\))\)\^2, \(\((\(-1\) \ + T\^2)\)\ \((V2 - V1)\)\ \[Theta]1\ \((\(-2\)\ t0\ V2 + \((V1 + V2)\)\ \ \[Theta]1)\)\)\/\(2 T\^2\ \[Gamma]\^2\ \((\[Theta]1 - t0)\)\)} //. GTrep]\)], "Input"], Cell[BoxData[ \({\(2\ \((\(-1\) + T\^2)\)\ \((\(-1\) + \[Alpha])\)\^2\ \ \[CapitalDelta]\^2\)\/\(T\^2\ \[Alpha]\^2\ \[Beta]\^2\ \((\(-1\) + \ \[CapitalDelta])\)\^2\), \(2\ \((\(-1\) + T\^2)\)\ \((\(-1\) + \[Alpha])\)\ \ \[CapitalDelta]\ \((\[Alpha] + \[CapitalDelta])\)\)\/\(T\^2\ \[Alpha]\^2\ \ \[Beta]\^2\ \((\(-1\) + \[CapitalDelta])\)\^2\)}\)], "Output"] }, Open ]], Cell[BoxData[ \(\(expansion = {p \[Rule] \(-\((x - x1)\)\)\/\(\@2\ \ t0\ V1\ \[Alpha]\ \ \[Beta]\), q \[Rule] \(1 + \[CapitalDelta]\)\/\(\@2\ \[Beta]\ \((1 - \ \[CapitalDelta])\)\), r \[Rule] \(\[Alpha] + 2\ \[CapitalDelta] - \[Alpha]\ \ \[CapitalDelta]\)\/\(\(\ \)\(\[Alpha]\ \((1 + \[CapitalDelta])\)\)\), s \[Rule] 1\/\(\@2\ \[Beta]\), u \[Rule] \(2\ \ \((1 - \[Alpha])\)\^2\ \[CapitalDelta]\^2\)\/\(\(\ \ \)\(\[Alpha]\^2\ \[Beta]\^2\ \((1 - \[CapitalDelta])\)\^2\)\), v \[Rule] \(\(-2\)\ \((1 - \ \[Alpha])\)\ \((\[Alpha] + \ \[CapitalDelta])\)\ \[CapitalDelta]\)\/\(\(\ \)\(\[Alpha]\^2\ \[Beta]\^2\ \ \((1 - \[CapitalDelta])\)\^2\)\)};\)\)], "Input"], Cell["Check that it is correct:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[expansion /. TGrep]\)], "Input"], Cell[BoxData[ \({p \[Rule] \(x - x1\)\/\(\@2\ \[Gamma]\ \@\(\(-t0\) + \[Theta]1\)\), q \[Rule] \(V2\ \@\(\(-t0\) + \[Theta]1\)\)\/\(\@2\ \[Gamma]\), r \[Rule] \(t0\ V2 - V1\ \[Theta]1\)\/\(t0\ V2 - V2\ \[Theta]1\), s \[Rule] \(V1\ \@\(\(-t0\) + \[Theta]1\)\)\/\(\@2\ \[Gamma]\), u \[Rule] \(-\(\(\((V1 - V2)\)\^2\ \[Theta]1\^2\)\/\(2\ \[Gamma]\^2\ \ \((t0 - \[Theta]1)\)\)\)\), v \[Rule] \(\((V1 - V2)\)\ \[Theta]1\ \((\(-2\)\ t0\ V2 + \((V1 + V2)\)\ \ \[Theta]1)\)\)\/\(2\ \[Gamma]\^2\ \((t0 - \[Theta]1)\)\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Formulas for the a-coefficients", "Subsection"], Cell[BoxData[ \(D2y[f_] := 2 D[f /. y -> \@yy, {yy, 1}] + 4\ yy\ D[f /. y -> \@yy, {yy, 2}]\)], "Input"], Cell[BoxData[ \(\(a0con = Simplify[\((a0 /. y -> \@yy)\) //. contraction];\)\)], "Input"], Cell[BoxData[ \(\(a1con = Simplify[\((a1 /. y -> \@yy)\) //. contraction];\)\)], "Input"], Cell[BoxData[ \(\(a2con = \((a2 /. y -> \@yy)\) //. contraction;\)\)], "Input"], Cell[BoxData[ \(\(a011con = D2y[a0] //. contraction;\)\)], "Input"], Cell[BoxData[ \(\(a111con = D2y[a1] //. contraction;\)\)], "Input"], Cell[BoxData[ \(\(a211con = D2y[a2] //. contraction;\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[a0con //. expansion]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[a1con //. expansion]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell["\<\ These demonstrate that the solution for ytilde constructed previously, is \ after all, exact.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(a2 = Simplify[a2con //. expansion]\)], "Input"], Cell[BoxData[ \(\(4\ \[Alpha]\^2\ \[CapitalDelta] + T\^4\ \((\[Alpha] + \ \[CapitalDelta])\)\^2 + T\^2\ \[Alpha]\ \((\(-2\)\ \[CapitalDelta]\ \((1 + \ \[CapitalDelta])\) + \[Alpha]\ \((\(-1\) - 4\ \[CapitalDelta] + \ \[CapitalDelta]\^2)\))\)\)\/\(T\^2\ \((\(-1\) + T\^2)\)\ \((\[Alpha] + \ \[CapitalDelta])\)\^2\)\)], "Output"] }, Open ]], Cell[BoxData[ \(\(a2 = \((4\ \[Alpha]\^2\ \[CapitalDelta] + T\^4\ \((\[Alpha] + \[CapitalDelta])\)\^2 + T\^2\ \[Alpha]\ \((\(-2\)\ \[CapitalDelta]\ \((1 + \ \[CapitalDelta])\) + \[Alpha]\ \((\(-1\) - 4\ \[CapitalDelta] + \ \[CapitalDelta]\^2)\))\))\)/\((T\^2\ \((\(-1\) + T\^2)\)\ \((\[Alpha] + \[CapitalDelta])\)\^2)\);\)\)], \ "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[1 - a2]\)], "Input"], Cell[BoxData[ \(\(-\(\(\[CapitalDelta]\ \((\[Alpha]\^2\ \((4 + T\^2\ \((\(-4\) + \[CapitalDelta])\))\) + T\^2\ \[CapitalDelta] - 2\ T\^2\ \[Alpha]\ \[CapitalDelta])\)\)\/\(T\^2\ \((\(-1\) + T\^2)\)\ \((\[Alpha] + \[CapitalDelta])\)\^2\)\)\)\)], \ "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Collect[Numerator[%], \[CapitalDelta], Simplify]\)], "Input"], Cell[BoxData[ \(4\ \((\(-1\) + T\^2)\)\ \[Alpha]\^2\ \[CapitalDelta] - T\^2\ \((\(-1\) + \[Alpha])\)\^2\ \[CapitalDelta]\^2\)], "Output"] }, Open ]], Cell[BoxData[ \(\(aa2 = 1 - \(4\ \[Alpha]\^2\ \[CapitalDelta]\)\/\(T\^2\ \ \((\[Alpha] + \ \[CapitalDelta])\)\^2\) + \(\((1 - \[Alpha])\)\^2\ \[CapitalDelta]\^2\)\/\(\(\ \ \)\(\((T\^2 - 1)\)\ \((\[Alpha] + \[CapitalDelta])\)\^2\)\);\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[aa2 \[Equal] a2]\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(a011 = Simplify[a011con //. expansion]\)], "Input"], Cell[BoxData[ \(\(2\ \[Alpha]\^2\ \((T\^2\ \((\(-1\) + \[CapitalDelta])\)\^2 + 4\ \ \[CapitalDelta])\)\)\/\(T\^2\ \((\[Alpha] + \[CapitalDelta])\)\^2\)\)], \ "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(a111 = Simplify[a111con //. expansion]\)], "Input"], Cell[BoxData[ \(\(-\(\((\@2\ \[Alpha]\^2\ \[Beta]\ \((\(-1\) + \[CapitalDelta])\)\ \ \[CapitalDelta]\ \((2\ \[Alpha]\ \((12 - 22\ T\^2 + T\^4\ \((7 - 3\ \[CapitalDelta])\))\)\ \[CapitalDelta] + 3\ T\^4\ \[CapitalDelta]\ \((1 + \[CapitalDelta])\) + \ \[Alpha]\^2\ \((24 + 4\ T\^2\ \((\(-8\) + 3\ \[CapitalDelta])\) + T\^4\ \((8 - 9\ \[CapitalDelta] + 3\ \[CapitalDelta]\^2)\))\))\))\)/\((T\^3\ \ \((\(-1\) + T\^2)\)\ \((\[Alpha] + \[CapitalDelta])\)\^4)\)\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(a211 = Simplify[a211con //. expansion]\)], "Input"], Cell[BoxData[ \(Simplify::"time" \(\(:\)\(\ \)\) "Time spent on a transformation exceeded \!\(300\) seconds, and the \ transformation was aborted. Increasing the value of TimeConstraint option may \ improve the result of simplification."\)], "Message"], Cell[BoxData[ \(Simplify::"time" \(\(:\)\(\ \)\) "Time spent on a transformation exceeded \!\(300\) seconds, and the \ transformation was aborted. Increasing the value of TimeConstraint option may \ improve the result of simplification."\)], "Message"], Cell[BoxData[ \(Simplify::"time" \(\(:\)\(\ \)\) "Time spent on a transformation exceeded \!\(300\) seconds, and the \ transformation was aborted. Increasing the value of TimeConstraint option may \ improve the result of simplification."\)], "Message"], Cell[BoxData[ \(General::"stop" \(\(:\)\(\ \)\) "Further output of \!\(Simplify :: \"time\"\) will be suppressed during \ this calculation."\)], "Message"], Cell[BoxData[ \(\((3\ \[Alpha]\^2\ \[Beta]\^2\ \((\(-1\) + \[CapitalDelta])\)\^2\ \ \[CapitalDelta]\ \((2\ T\^6\ \[CapitalDelta]\^3 - 2\ T\^6\ \[Alpha]\ \[CapitalDelta]\^2\ \((3 + 7\ \[CapitalDelta])\) - 2\ \[Alpha]\^3\ \[CapitalDelta]\ \((\(-64\) - 24\ T\^2\ \((\(-7\) + \[CapitalDelta])\) + 4\ T\^4\ \((\(-34\) + 15\ \[CapitalDelta])\) + T\^6\ \((29 - 33\ \[CapitalDelta] + 10\ \[CapitalDelta]\^2)\))\) + \[Alpha]\^2\ \ \[CapitalDelta]\ \((64\ \[CapitalDelta] - 192\ T\^2\ \[CapitalDelta] + 196\ T\^4\ \[CapitalDelta] + T\^6\ \((\(-3\) - 56\ \[CapitalDelta] + 27\ \[CapitalDelta]\^2)\))\) + \[Alpha]\^4\ \((64 + 48\ T\^2\ \((\(-3\) + \[CapitalDelta])\) + 4\ T\^4\ \((24 - 20\ \[CapitalDelta] + 5\ \[CapitalDelta]\^2)\) + T\^6\ \((\(-16\) + 29\ \[CapitalDelta] - 20\ \[CapitalDelta]\^2 + 5\ \[CapitalDelta]\^3)\))\))\))\)/\((2\ T\^4\ \ \((\(-1\) + T\^2)\)\^2\ \((\[Alpha] + \[CapitalDelta])\)\^6)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(LeafCount[a211]\)], "Input"], Cell[BoxData[ \(177\)], "Output"] }, Open ]], Cell["\<\ This is a vast simplification of the original expression - originally a211 \ had a leafcount of 21000 !\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Expanding ftilde around Z2", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(f[y]\)], "Input"], Cell[BoxData[ \(\(y\^2 + 2\ w2\ \[Phi]2\ \((z + w2\ \[Phi]2)\) + \@\(y\^4 + 4\ w2\ y\^2\ \ z\ \[Phi]2\)\)\/\(2\ \((y\^2 + \((z + w2\ \[Phi]2)\)\^2)\)\)\)], "Output"] }, Open ]], Cell[BoxData[{ \(\(fseri = Series[f[y], {z, Z2, 2}];\)\), "\[IndentingNewLine]", \(\(f0 = SeriesCoefficient[fseri, 0] /. y -> \@yy;\)\), "\[IndentingNewLine]", \(\(f1 = SeriesCoefficient[fseri, 1] /. y -> \@yy;\)\), "\[IndentingNewLine]", \(\(f2 = SeriesCoefficient[fseri, 2] /. y -> \@yy;\)\)}], "Input"], Cell[CellGroupData[{ Cell[BoxData[{ \(f0 = Simplify[\((f0 //. contraction)\) /. expansion]\), "\[IndentingNewLine]", \(f1 = Simplify[\((f1 //. contraction)\) /. expansion]\), "\[IndentingNewLine]", \(f2 = Simplify[\((f2 //. contraction)\) /. expansion]\)}], "Input"], Cell[BoxData[ \(1\/T\^2\)], "Output"], Cell[BoxData[ \(\(\@2\ \[Alpha]\ \[Beta]\ \((\(-1\) + \[CapitalDelta])\)\)\/\(T\^3\ \((\ \[Alpha] + \[CapitalDelta])\)\)\)], "Output"], Cell[BoxData[ \(\(-\(\(\[Alpha]\^2\ \[Beta]\^2\ \((\(-1\) + \[CapitalDelta])\)\^2\ \((\ \[Alpha]\ \((4 + T\^2\ \((\(-4\) + \[CapitalDelta])\))\) + \((4 - 5\ T\^2)\)\ \[CapitalDelta])\)\)\/\(2\ T\^4\ \((\(-1\) \ + T\^2)\)\ \((\[Alpha] + \[CapitalDelta])\)\^3\)\)\)\)], "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Expanding the modulation factor", "Section"], Cell[CellGroupData[{ Cell["Determination of Vg", "Subsection"], Cell["Use the ratio factor as in equation 18.17:", "Text"], Cell[BoxData[ \(modulag[ z_] := \(\(z + \((1 - ftilde[z])\) w2\)\/\(z + \((1 - ftilde[z])\) wg\)\) \(\(z\ ftilde[z] + \((1 - ftilde[z])\) wg\ \[Phi]g\)\/\(z\ ftilde[z] + \((1 - ftilde[z])\) w2\ \[Phi]2\)\) \((w1\/w2)\) \(1\/\@a2\) \@\(2/F11[z]\)\)], \ "Input"], Cell[BoxData[ \(ftilde[z_] := f0 + f1 \((z - Z2)\) + f2 \((z - Z2)\)\^2\)], "Input"], Cell[BoxData[ \(F11[z_] := a011 + a111 \((z - Z2)\) + a211 \((z - Z2)\)\^2\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(mseri = Series[modulag[z], {z, Z2, 2}];\)\)], "Input"], Cell[BoxData[ \(General::"spell1" \(\(:\)\(\ \)\) "Possible spelling error: new symbol name \"\!\(mseri\)\" is similar to \ existing symbol \"\!\(fseri\)\"."\)], "Message"] }, Open ]], Cell["\<\ As seen in stochasflow4.nb, and confirmed below, at T \[Rule] \[Infinity] \ higher terms in the series vanish and to maintain solute conservation we must \ have b0 = 1. So we calculate only b0 here in order to enforce this condition.\ \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(Timing[ b0[T_] = Simplify[ SeriesCoefficient[mseri, 0] /. Tzrep, {T > 0, t0 < 0, \[Theta]1 > 0, \[CapitalDelta] > \(-1\), \[CapitalDelta] < 1}]]\)\(\[IndentingNewLine]\)\)\)], "Input"], Cell[BoxData[ \({5.647999999999911`\ Second, \((T\^3\ \[Alpha]\ \((\(-1\) + \ \[CapitalDelta])\)\ \@\(\((\[Alpha] + \[CapitalDelta])\)\^2\/\(\[Alpha]\^2\ \ \((T\^2\ \((\(-1\) + \[CapitalDelta])\)\^2 + 4\ \[CapitalDelta])\)\)\)\ \ \((\(-\@2\) - \@2\ \[CapitalDelta] - 2\ T\ wg\ \[Beta]\ \[Phi]g + 2\ T\ wg\ \[Beta]\ \[CapitalDelta]\ \[Phi]g)\))\)/\((\((\ \[Alpha] + \[CapitalDelta])\)\ \((\(-2\)\ wg\ \[Beta]\ \((\(-1\) + \ \[CapitalDelta])\) + \@2\ T\ \((1 + \[CapitalDelta])\))\)\ \[Sqrt]\((\((4\ \ \[Alpha]\^2\ \[CapitalDelta] + T\^4\ \((\[Alpha] + \[CapitalDelta])\)\^2 + T\^2\ \[Alpha]\ \((\(-2\)\ \[CapitalDelta]\ \((1 + \ \[CapitalDelta])\) + \[Alpha]\ \((\(-1\) - 4\ \[CapitalDelta] + \[CapitalDelta]\^2)\))\ \))\)/\((\((\(-1\) + T\^2)\)\ \((\[Alpha] + \[CapitalDelta])\)\^2)\))\))\ \)}\)], "Output"] }, Open ]], Cell[TextData[{ "Replace the unknown velocity related factors with their explicit T \ dependence and take the infinite time limit. \[Beta]g here is defined as ", Cell[BoxData[ \(TraditionalForm\`\(\[Gamma] \@ \[Theta]1\)\/\(Vg\ \[Theta]1\)\)]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[ Limit[b0[T] /. {wg -> T\/\(\@2\ \[Beta]g\), \[Phi]g -> \(\(\[Beta]g\/\[Beta]\) \ \[Theta]1 - t0\)\/\(\((\[Theta]1 - t0)\) T\^2\)}, T \[Rule] \[Infinity]], {T > 1, \[CapitalDelta] > \(-1\), \[CapitalDelta] < 1, \[Alpha] > 1, t0 < 0}]\)], "Input"], Cell[BoxData[ \(General::"spell" \(\(:\)\(\ \)\) "Possible spelling error: new symbol name \"\!\(\[Beta]g\)\" is similar \ to existing symbols \!\({\[Beta], \[Phi]g}\)."\)], "Message"], Cell[BoxData[ \(\(t0\ \((\[Beta]\ \((\(-1\) + \[CapitalDelta])\) - \[Beta]g\ \((1 + \ \[CapitalDelta])\))\) + 2\ \[Beta]g\ \[Theta]1\)\/\(\((\[Beta]\ \((\(-1\) + \ \[CapitalDelta])\) - \[Beta]g\ \((1 + \[CapitalDelta])\))\)\ \((t0 - \ \[Theta]1)\)\)\)], "Output"] }, Open ]], Cell["\<\ Now we can finally choose \[Beta]g and by implication Vg to ensure \ conservation:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Beta]grep = Simplify[Solve[% \[Equal] 1, \[Beta]g]]\)], "Input"], Cell[BoxData[ \({{\[Beta]g \[Rule] \[Beta]}}\)], "Output"] }, Open ]], Cell["\<\ Well, it turns out then that we have to choose Vg = V1 (which is used to \ define \[Beta], eq 18.25). This was foreshadowed in the text on logical \ grounds, but now it is confirmed as the only choice that will guarantee \ solute conservation.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Expansion using Vg = V1 and full ratio factor", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(modula[ z_] := \(\(z + \((1 - ftilde[z])\) w2\)\/\(z + \((1 - ftilde[z])\) w1\)\) \(\(z\ ftilde[z] + \((1 - ftilde[z])\) w1\ \[Phi]1\)\/\(z\ ftilde[z] + \((1 - ftilde[z])\) w2\ \[Phi]2\)\) \((w1\/w2)\) \(1\/\@a2\) \@\(2/F11[z]\)\)], \ "Input"], Cell[BoxData[ \(General::"spell1" \(\(:\)\(\ \)\) "Possible spelling error: new symbol name \"\!\(modula\)\" is similar \ to existing symbol \"\!\(modulag\)\"."\)], "Message"] }, Open ]], Cell[BoxData[ \(ftilde[z_] := f0 + f1 \((z - Z2)\) + f2 \((z - Z2)\)\^2\)], "Input"], Cell[BoxData[ \(F11[z_] := a011 + a111 \((z - Z2)\) + a211 \((z - Z2)\)\^2\)], "Input"], Cell[BoxData[ \(\(mseri = Series[modula[z], {z, Z2, 2}];\)\)], "Input"], Cell["\<\ This is a series expansion in powers of (z-Z2). However, we really want an \ expansion in terms of the u-functions of Gaussionsuperposition.nb:\ \>", "Text"], Cell[BoxData[ \(u[x_, n_, \[Sigma]_] := \(\(\ \)\(2\^\(\(-n\)/2\)\ \@\[Pi]\ \((x\/\[Sigma])\ \)\^n\ gg[x, \[Sigma]]\)\)\/Gamma[\(1 + n\)\/2]\)], "Input"], Cell[TextData[{ "The gaussian factor in the z-coordinate is, from equations 17.36 and 17.50 \ (as also used in 17.54) ", Cell[BoxData[ \(TraditionalForm\`g(z - Z2, 1/\@\(2\ a2\))\)]], " and this allows us to identify the \[Sigma] needed in the expression for \ u above. . So find the coefficients needed for the first 3 terms:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(u[z - Z2, 0, 1\/\@\(2\ A2\)]\), "\[IndentingNewLine]", \(u[z - Z2, 1, 1\/\@\(2\ A2\)]\), "\[IndentingNewLine]", \(u[z - Z2, 2, 1\/\@\(2\ A2\)]\)}], "Input"], Cell[BoxData[ \(gg[z - Z2, 1\/\(\@2\ \@A2\)]\)], "Output"], Cell[BoxData[ \(\@A2\ \@\[Pi]\ \((z - Z2)\)\ gg[z - Z2, 1\/\(\@2\ \@A2\)]\)], "Output"], Cell[BoxData[ \(2\ A2\ \((z - Z2)\)\^2\ gg[z - Z2, 1\/\(\@2\ \@A2\)]\)], "Output"] }, Open ]], Cell["\<\ So incorporating these factors, the first 3 coefficients in the un expansion \ are:\ \>", "Text"], Cell["\<\ So here now are the final formulas for the expansion coefficients in the \ un-function basis:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(Timing[ b0[T_] = Simplify[ SeriesCoefficient[mseri, 0] /. Tzrep, {T > 0, \[Alpha] > 1, \[CapitalDelta] > \(-1\), \[CapitalDelta] < 1}]]\)\(\[IndentingNewLine]\)\)\)], "Input"], Cell[BoxData[ \({5.807999999999993`\ Second, \(-\(\(T\^2\ \((\(-1\) + \ \[CapitalDelta])\)\ \((\[Alpha] + \[CapitalDelta])\)\ \@\(1\/\(T\^2\ \ \((\(-1\) + \[CapitalDelta])\)\^2 + 4\ \[CapitalDelta]\)\)\)\/\@\(\(4\ \ \[Alpha]\^2\ \[CapitalDelta] + T\^4\ \((\[Alpha] + \[CapitalDelta])\)\^2 + \ T\^2\ \[Alpha]\ \((\(-2\)\ \[CapitalDelta]\ \((1 + \[CapitalDelta])\) + \ \[Alpha]\ \((\(-1\) - 4\ \[CapitalDelta] + \[CapitalDelta]\^2)\))\)\)\/\(\(-1\ \) + T\^2\)\)\)\)}\)], "Output"] }, Open ]], Cell[BoxData[ \(\(b0[ T_] = \(T\^2\ \((1 - \[CapitalDelta])\)\ \((\[Alpha] + \ \[CapitalDelta])\)\ \@\(1\/\(T\^2\ \((\(-1\) + \[CapitalDelta])\)\^2 + 4\ \ \[CapitalDelta]\)\)\)\/\@\(\(4\ \[Alpha]\^2\ \[CapitalDelta] + T\^4\ \((\ \[Alpha] + \[CapitalDelta])\)\^2 + T\^2\ \[Alpha]\ \((\(-2\)\ \[CapitalDelta]\ \ \((1 + \[CapitalDelta])\) + \[Alpha]\ \((\(-1\) - 4\ \[CapitalDelta] + \ \[CapitalDelta]\^2)\))\)\)\/\(\(-1\) + T\^2\)\);\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(Timing[ b1[T_] = Simplify[\(1\/\@\(\[Pi]\ a2\)\) SeriesCoefficient[mseri, 1] /. Tzrep, {T > 0, \[Alpha] > 1, \[CapitalDelta] > \(-1\), \[CapitalDelta] < 1}]]\)\(\[IndentingNewLine]\)\)\)], "Input"], Cell[BoxData[ \({7.7309999999999945`\ Second, \(-\(\((T\^2\ \[Beta]\ \((\(-1\) + \ \[CapitalDelta])\)\^2\ \[CapitalDelta]\ \((1\/\(T\^2\ \((\(-1\) + \ \[CapitalDelta])\)\^2 + 4\ \[CapitalDelta]\))\)\^\(3/2\)\ \((T\^2\ \ \[CapitalDelta]\ \((\(-8\)\ \[CapitalDelta] + T\^2\ \((1 + 10\ \[CapitalDelta] + \[CapitalDelta]\^2)\))\) \ + 2\ \[Alpha]\ \[CapitalDelta]\ \((\(-4\)\ \((\(-3\) - 2\ \[CapitalDelta] + \[CapitalDelta]\^2)\) - T\^2\ \((23 + 9\ \[CapitalDelta] - 5\ \[CapitalDelta]\^2 + \[CapitalDelta]\^3)\) + T\^4\ \((10 - \[CapitalDelta] - 2\ \[CapitalDelta]\^2 + \ \[CapitalDelta]\^3)\))\) + \[Alpha]\^2\ \((\(-8\)\ \((\(-3\) - 2\ \[CapitalDelta] + \[CapitalDelta]\^2)\) - 2\ T\^2\ \((17 + 5\ \[CapitalDelta] - 7\ \[CapitalDelta]\^2 + \[CapitalDelta]\^3)\) + T\^4\ \((10 - 5\ \[CapitalDelta] - 4\ \[CapitalDelta]\^2 + 3\ \[CapitalDelta]\^3)\))\))\))\)/\((2\ \@\(2\ \ \[Pi]\)\ \((1 + \[CapitalDelta])\)\ \((4\ \[Alpha]\^2\ \[CapitalDelta] + T\^4\ \((\[Alpha] + \[CapitalDelta])\)\^2 + T\^2\ \[Alpha]\ \((\(-2\)\ \[CapitalDelta]\ \((1 + \ \[CapitalDelta])\) + \[Alpha]\ \((\(-1\) - 4\ \[CapitalDelta] + \ \[CapitalDelta]\^2)\))\))\))\)\)\)}\)], "Output"] }, Open ]], Cell[BoxData[ \(\(b1[ T_] = \(-\((T\^2\ \[Beta]\ \((\(-1\) + \[CapitalDelta])\)\^2\ \ \[CapitalDelta]\ \((1\/\(T\^2\ \((\(-1\) + \[CapitalDelta])\)\^2 + 4\ \ \[CapitalDelta]\))\)\^\(3/2\)\ \((T\^2\ \[CapitalDelta]\ \((\(-8\)\ \ \[CapitalDelta] + T\^2\ \((1 + 10\ \[CapitalDelta] + \ \[CapitalDelta]\^2)\))\) + 2\ \[Alpha]\ \[CapitalDelta]\ \((\(-4\)\ \((\(-3\) - 2\ \[CapitalDelta] + \[CapitalDelta]\^2)\) - T\^2\ \((23 + 9\ \[CapitalDelta] - 5\ \[CapitalDelta]\^2 + \[CapitalDelta]\^3)\) \ + T\^4\ \((10 - \[CapitalDelta] - 2\ \[CapitalDelta]\^2 + \ \[CapitalDelta]\^3)\))\) + \[Alpha]\^2\ \((\(-8\)\ \((\(-3\) - 2\ \[CapitalDelta] + \[CapitalDelta]\^2)\) - 2\ T\^2\ \((17 + 5\ \[CapitalDelta] - 7\ \[CapitalDelta]\^2 + \[CapitalDelta]\^3)\) \ + T\^4\ \((10 - 5\ \[CapitalDelta] - 4\ \[CapitalDelta]\^2 + 3\ \[CapitalDelta]\^3)\))\))\))\)\)/\((2\ \ \@\(2\ \[Pi]\)\ \((1 + \[CapitalDelta])\)\ \((4\ \[Alpha]\^2\ \[CapitalDelta] \ + T\^4\ \((\[Alpha] + \[CapitalDelta])\)\^2 + T\^2\ \[Alpha]\ \((\(-2\)\ \[CapitalDelta]\ \((1 + \ \[CapitalDelta])\) + \[Alpha]\ \((\(-1\) - 4\ \[CapitalDelta] + \ \[CapitalDelta]\^2)\))\))\))\);\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(Timing[ b2[T_] = Simplify[\(1\/\(2 a2\)\) SeriesCoefficient[mseri, 2] /. Tzrep, {T > 0, \[Alpha] > 1, \[CapitalDelta] > \(-1\), \[CapitalDelta] < 1}]]\)\(\[IndentingNewLine]\)\)\)], "Input"], Cell[BoxData[ \({47.97900000000004`\ Second, \(-\(\((T\^2\ \[Beta]\^2\ \((\(-1\) + \ \[CapitalDelta])\)\^3\ \[CapitalDelta]\ \((1\/\(T\^2\ \((\(-1\) + \ \[CapitalDelta])\)\^2 + 4\ \[CapitalDelta]\))\)\^\(5/2\)\ \ \[Sqrt]\((\(1\/\(\(-1\) + T\^2\)\)\((4\ \[Alpha]\^2\ \[CapitalDelta] + T\^4\ \((\[Alpha] + \[CapitalDelta])\)\^2 + T\^2\ \[Alpha]\ \((\(-2\)\ \[CapitalDelta]\ \((1 + \ \[CapitalDelta])\) + \[Alpha]\ \((\(-1\) - 4\ \[CapitalDelta] + \[CapitalDelta]\^2)\))\ \))\))\)\ \((64\ \[Alpha]\^2\ \[CapitalDelta]\ \((\[Alpha] + \[CapitalDelta])\ \)\^2\ \((3 - \[CapitalDelta] - 2\ \[CapitalDelta]\^2 + 2\ \[CapitalDelta]\^3)\) + 32\ T\^2\ \[Alpha]\ \((\(-4\)\ \[CapitalDelta]\^4\ \((3 - \ \[CapitalDelta] + \[CapitalDelta]\^2)\) + \[Alpha]\ \[CapitalDelta]\^2\ \ \((\(-12\) - 55\ \[CapitalDelta] + 13\ \[CapitalDelta]\^2 - 13\ \[CapitalDelta]\^3 - 7\ \[CapitalDelta]\^4 + 2\ \[CapitalDelta]\^5)\) + 2\ \[Alpha]\^2\ \[CapitalDelta]\ \((\(-12\) - 34\ \[CapitalDelta] + 13\ \[CapitalDelta]\^2 - 4\ \[CapitalDelta]\^3 - 7\ \[CapitalDelta]\^4 + 2\ \[CapitalDelta]\^5)\) + \[Alpha]\^3\ \ \((\(-12\) - 25\ \[CapitalDelta] + 17\ \[CapitalDelta]\^2 + \[CapitalDelta]\^3 - 7\ \[CapitalDelta]\^4 + 2\ \[CapitalDelta]\^5)\))\) + 8\ T\^4\ \[Alpha]\ \((2\ \[CapitalDelta]\^3\ \((21 + 87\ \[CapitalDelta] - 5\ \[CapitalDelta]\^2 + 17\ \[CapitalDelta]\^3 - 4\ \[CapitalDelta]\^4)\) + \[Alpha]\ \ \[CapitalDelta]\^2\ \((187 + 456\ \[CapitalDelta] - 147\ \[CapitalDelta]\^2 + 140\ \[CapitalDelta]\^3 - 7\ \[CapitalDelta]\^4 - 10\ \[CapitalDelta]\^5 + \[CapitalDelta]\^6)\) \ + 2\ \[Alpha]\^2\ \[CapitalDelta]\ \((127 + 197\ \[CapitalDelta] - 134\ \[CapitalDelta]\^2 + 91\ \[CapitalDelta]\^3 + 8\ \[CapitalDelta]\^4 - 10\ \[CapitalDelta]\^5 + \[CapitalDelta]\^6)\) \ + \[Alpha]\^3\ \((109 + 106\ \[CapitalDelta] - 149\ \[CapitalDelta]\^2 + 58\ \[CapitalDelta]\^3 + 9\ \[CapitalDelta]\^4 - 10\ \[CapitalDelta]\^5 + \[CapitalDelta]\^6)\))\ \) - 4\ T\^6\ \((24\ \[CapitalDelta]\^4\ \((1 + \[CapitalDelta])\) + \[Alpha]\ \ \[CapitalDelta]\^3\ \((167 + 314\ \[CapitalDelta] - 28\ \[CapitalDelta]\^2 + 56\ \[CapitalDelta]\^3 - 23\ \[CapitalDelta]\^4 + 2\ \[CapitalDelta]\^5)\) + \[Alpha]\^2\ \ \[CapitalDelta]\^2\ \((418 + 581\ \[CapitalDelta] - 294\ \[CapitalDelta]\^2 + 244\ \[CapitalDelta]\^3 - 66\ \[CapitalDelta]\^4 + 3\ \[CapitalDelta]\^5 + 2\ \[CapitalDelta]\^6)\) + \[Alpha]\^4\ \((148 \ + 39\ \[CapitalDelta] - 206\ \[CapitalDelta]\^2 + 128\ \[CapitalDelta]\^3 - 28\ \[CapitalDelta]\^4 + 5\ \[CapitalDelta]\^5 + 2\ \[CapitalDelta]\^6)\) + \[Alpha]\^3\ \ \[CapitalDelta]\ \((417 + 324\ \[CapitalDelta] - 460\ \[CapitalDelta]\^2 + 328\ \[CapitalDelta]\^3 - 77\ \[CapitalDelta]\^4 + 4\ \[CapitalDelta]\^6)\))\) + T\^8\ \((3\ \[CapitalDelta]\^3\ \((1 + 35\ \[CapitalDelta] + 35\ \[CapitalDelta]\^2 + \[CapitalDelta]\^3)\) \ + 4\ \[Alpha]\^3\ \[CapitalDelta]\ \((98 - 22\ \[CapitalDelta] - 97\ \[CapitalDelta]\^2 + 107\ \[CapitalDelta]\^3 - 49\ \[CapitalDelta]\^4 + 11\ \[CapitalDelta]\^5)\) + \[Alpha]\^4\ \((104 \ - 78\ \[CapitalDelta] - 94\ \[CapitalDelta]\^2 + 159\ \[CapitalDelta]\^3 - 91\ \[CapitalDelta]\^4 + 27\ \[CapitalDelta]\^5 - 3\ \[CapitalDelta]\^6)\) + 4\ \[Alpha]\ \[CapitalDelta]\^2\ \((3 + 92\ \[CapitalDelta] + 71\ \[CapitalDelta]\^2 - 37\ \[CapitalDelta]\^3 + 20\ \[CapitalDelta]\^4 - 7\ \[CapitalDelta]\^5 + 2\ \[CapitalDelta]\^6)\) + 2\ \[Alpha]\^2\ \[CapitalDelta]\ \((3 + 277\ \[CapitalDelta] + 89\ \[CapitalDelta]\^2 - 227\ \[CapitalDelta]\^3 + 170\ \[CapitalDelta]\^4 - 62\ \[CapitalDelta]\^5 + 14\ \[CapitalDelta]\^6)\))\))\))\)/\((32\ \((1 \ + \[CapitalDelta])\)\ \((\[Alpha] + \[CapitalDelta])\)\ \((4\ \[Alpha]\^2\ \ \[CapitalDelta] + T\^4\ \((\[Alpha] + \[CapitalDelta])\)\^2 + T\^2\ \[Alpha]\ \((\(-2\)\ \[CapitalDelta]\ \((1 + \ \[CapitalDelta])\) + \[Alpha]\ \((\(-1\) - 4\ \[CapitalDelta] + \ \[CapitalDelta]\^2)\))\))\)^2)\)\)\)}\)], "Output"] }, Open ]], Cell[BoxData[ \(\(b2[ T_] = \(-\((T\^2\ \[Beta]\^2\ \((\(-1\) + \[CapitalDelta])\)\^3\ \ \[CapitalDelta]\ \((1\/\(T\^2\ \((\(-1\) + \[CapitalDelta])\)\^2 + 4\ \ \[CapitalDelta]\))\)\^\(5/2\)\ \[Sqrt]\((\(1\/\(\(-1\) + T\^2\)\) \((4\ \[Alpha]\^2\ \[CapitalDelta] + T\^4\ \((\[Alpha] + \[CapitalDelta])\)\^2 + T\^2\ \[Alpha]\ \((\(-2\)\ \[CapitalDelta]\ \((1 + \ \[CapitalDelta])\) + \[Alpha]\ \((\(-1\) - 4\ \[CapitalDelta] + \ \[CapitalDelta]\^2)\))\))\))\)\ \((64\ \[Alpha]\^2\ \[CapitalDelta]\ \((\ \[Alpha] + \[CapitalDelta])\)\^2\ \((3 - \[CapitalDelta] - 2\ \[CapitalDelta]\^2 + 2\ \[CapitalDelta]\^3)\) + 32\ T\^2\ \[Alpha]\ \((\(-4\)\ \[CapitalDelta]\^4\ \((3 - \ \[CapitalDelta] + \[CapitalDelta]\^2)\) + \[Alpha]\ \[CapitalDelta]\^2\ \ \((\(-12\) - 55\ \[CapitalDelta] + 13\ \[CapitalDelta]\^2 - 13\ \[CapitalDelta]\^3 - 7\ \[CapitalDelta]\^4 + 2\ \[CapitalDelta]\^5)\) + 2\ \[Alpha]\^2\ \[CapitalDelta]\ \((\(-12\) - 34\ \[CapitalDelta] + 13\ \[CapitalDelta]\^2 - 4\ \[CapitalDelta]\^3 - 7\ \[CapitalDelta]\^4 + 2\ \[CapitalDelta]\^5)\) + \[Alpha]\^3\ \ \((\(-12\) - 25\ \[CapitalDelta] + 17\ \[CapitalDelta]\^2 + \[CapitalDelta]\^3 - 7\ \[CapitalDelta]\^4 + 2\ \[CapitalDelta]\^5)\))\) + 8\ T\^4\ \[Alpha]\ \((2\ \[CapitalDelta]\^3\ \((21 + 87\ \[CapitalDelta] - 5\ \[CapitalDelta]\^2 + 17\ \[CapitalDelta]\^3 - 4\ \[CapitalDelta]\^4)\) + \[Alpha]\ \ \[CapitalDelta]\^2\ \((187 + 456\ \[CapitalDelta] - 147\ \[CapitalDelta]\^2 + 140\ \[CapitalDelta]\^3 - 7\ \[CapitalDelta]\^4 - 10\ \[CapitalDelta]\^5 + \ \[CapitalDelta]\^6)\) + 2\ \[Alpha]\^2\ \[CapitalDelta]\ \((127 + 197\ \[CapitalDelta] - 134\ \[CapitalDelta]\^2 + 91\ \[CapitalDelta]\^3 + 8\ \[CapitalDelta]\^4 - 10\ \[CapitalDelta]\^5 + \ \[CapitalDelta]\^6)\) + \[Alpha]\^3\ \((109 + 106\ \[CapitalDelta] - 149\ \[CapitalDelta]\^2 + 58\ \[CapitalDelta]\^3 + 9\ \[CapitalDelta]\^4 - 10\ \[CapitalDelta]\^5 + \ \[CapitalDelta]\^6)\))\) - 4\ T\^6\ \((24\ \[CapitalDelta]\^4\ \((1 + \ \[CapitalDelta])\) + \[Alpha]\ \[CapitalDelta]\^3\ \((167 + 314\ \[CapitalDelta] - 28\ \[CapitalDelta]\^2 + 56\ \[CapitalDelta]\^3 - 23\ \[CapitalDelta]\^4 + 2\ \[CapitalDelta]\^5)\) + \[Alpha]\^2\ \ \[CapitalDelta]\^2\ \((418 + 581\ \[CapitalDelta] - 294\ \[CapitalDelta]\^2 + 244\ \[CapitalDelta]\^3 - 66\ \[CapitalDelta]\^4 + 3\ \[CapitalDelta]\^5 + 2\ \[CapitalDelta]\^6)\) + \[Alpha]\^4\ \ \((148 + 39\ \[CapitalDelta] - 206\ \[CapitalDelta]\^2 + 128\ \[CapitalDelta]\^3 - 28\ \[CapitalDelta]\^4 + 5\ \[CapitalDelta]\^5 + 2\ \[CapitalDelta]\^6)\) + \[Alpha]\^3\ \ \[CapitalDelta]\ \((417 + 324\ \[CapitalDelta] - 460\ \[CapitalDelta]\^2 + 328\ \[CapitalDelta]\^3 - 77\ \[CapitalDelta]\^4 + 4\ \[CapitalDelta]\^6)\))\) + T\^8\ \((3\ \[CapitalDelta]\^3\ \((1 + 35\ \[CapitalDelta] + 35\ \[CapitalDelta]\^2 + \ \[CapitalDelta]\^3)\) + 4\ \[Alpha]\^3\ \[CapitalDelta]\ \((98 - 22\ \[CapitalDelta] - 97\ \[CapitalDelta]\^2 + 107\ \[CapitalDelta]\^3 - 49\ \[CapitalDelta]\^4 + 11\ \[CapitalDelta]\^5)\) + \[Alpha]\^4\ \ \((104 - 78\ \[CapitalDelta] - 94\ \[CapitalDelta]\^2 + 159\ \[CapitalDelta]\^3 - 91\ \[CapitalDelta]\^4 + 27\ \[CapitalDelta]\^5 - 3\ \[CapitalDelta]\^6)\) + 4\ \[Alpha]\ \[CapitalDelta]\^2\ \((3 + 92\ \[CapitalDelta] + 71\ \[CapitalDelta]\^2 - 37\ \[CapitalDelta]\^3 + 20\ \[CapitalDelta]\^4 - 7\ \[CapitalDelta]\^5 + 2\ \[CapitalDelta]\^6)\) + 2\ \[Alpha]\^2\ \[CapitalDelta]\ \((3 + 277\ \[CapitalDelta] + 89\ \[CapitalDelta]\^2 - 227\ \[CapitalDelta]\^3 + 170\ \[CapitalDelta]\^4 - 62\ \[CapitalDelta]\^5 + 14\ \[CapitalDelta]\^6)\))\))\))\)\)/\((32\ \ \((1 + \[CapitalDelta])\)\ \((\[Alpha] + \[CapitalDelta])\)\ \((4\ \ \[Alpha]\^2\ \[CapitalDelta] + T\^4\ \((\[Alpha] + \[CapitalDelta])\)\^2 + T\^2\ \[Alpha]\ \((\(-2\)\ \[CapitalDelta]\ \((1 + \ \[CapitalDelta])\) + \[Alpha]\ \((\(-1\) - 4\ \[CapitalDelta] + \ \[CapitalDelta]\^2)\))\))\)^2)\);\)\)], "Input"], Cell["\<\ See what they look like for the example steps. Also include the low time \ cutoff taken from stochasflow6.nb:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Tc[\[Alpha]_, \[CapitalDelta]_] = \(\[Alpha] \((1 + \ Abs[\[CapitalDelta]])\)\)\/\(\[Alpha] + Abs[\[CapitalDelta]]\)\)], "Input"], Cell[BoxData[ \(\(\[Alpha]\ \((1 + Abs[\[CapitalDelta]])\)\)\/\(\[Alpha] + Abs[\ \[CapitalDelta]]\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Plot3D[ Tc[\[Alpha], \[CapitalDelta]], {\[Alpha], 1, 50}, {\[CapitalDelta], 0, 0.8}]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .81114 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% SurfaceGraphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 5.55112e-017 1.04977 -0.0679587 1.04977 [ [.1361 .20347 -11.9648 -9 ] [.1361 .20347 .03518 0 ] [.26236 .15379 -11.4542 -9 ] [.26236 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[(1.6)] -0.00309 .43563 1 -0.35844 Mshowa .125 Mabswid .03419 .27388 m .03786 .27246 L s .03202 .28756 m .0357 .28614 L s .02984 .30135 m .03351 .29994 L s .02541 .3293 m .02909 .32791 L s .02316 .34347 m .02685 .34208 L s .0209 .35776 m .02459 .35639 L s .01631 .38674 m .02 .38538 L s .01398 .40142 m .01768 .40008 L s .01163 .41625 m .01533 .41491 L s .00687 .4463 m .01057 .44499 L s .00445 .46154 m .00816 .46023 L s .00201 .47692 m .00573 .47563 L s .25 Mabswid .03716 .25514 m 0 .48963 L s 0 .48963 m .39787 .81114 L s .39787 .81114 m .40529 .59895 L s .40529 .59895 m .03716 .25514 L s .68874 0 m .96935 .42924 L s .96935 .42924 m 1 .6535 L s 1 .6535 m .70298 .24544 L s .70298 .24544 m .68874 0 L s .03716 .25514 m 0 .48963 L s 0 .48963 m .70298 .24544 L s .70298 .24544 m .68874 0 L s .68874 0 m .03716 .25514 L s .40529 .59895 m .96935 .42924 L s .96935 .42924 m 1 .6535 L s 1 .6535 m .39787 .81114 L s .39787 .81114 m .40529 .59895 L s 0 0 m 1 0 L 1 .81114 L 0 .81114 L closepath clip newpath .5 Mabswid .934 .569 .295 r .38313 .58323 .40512 .60371 .43857 .72902 .41624 .7005 Metetra .271 .505 .897 r .41624 .7005 .43857 .72902 .47693 .74773 .45472 .71643 Metetra .544 .624 .882 r .45472 .71643 .47693 .74773 .51653 .7499 .49449 .71711 Metetra .599 .64 .863 r .49449 .71711 .51653 .7499 .55695 .74635 .53511 .71253 Metetra .619 .645 .854 r .53511 .71253 .55695 .74635 .59808 .74008 .57646 .70546 Metetra .629 .646 .849 r .57646 .70546 .59808 .74008 .6399 .73226 .6185 .69696 Metetra .634 .647 .845 r .6185 .69696 .6399 .73226 .68238 .72344 .66122 .68752 Metetra .638 .647 .843 r .66122 .68752 .68238 .72344 .72554 .71391 .70462 .67742 Metetra .64 .647 .841 r .70462 .67742 .72554 .71391 .76937 .70383 .74871 .66681 Metetra .642 .647 .84 r .74871 .66681 .76937 .70383 .8139 .69331 .79351 .65576 Metetra .643 .647 .839 r .79351 .65576 .8139 .69331 .85914 .68241 .83903 .64435 Metetra .644 .647 .839 r .83903 .64435 .85914 .68241 .90509 .67118 .88527 .63261 Metetra .644 .647 .838 r .88527 .63261 .90509 .67118 .95178 .65963 .93227 .62056 Metetra .645 .647 .837 r .93227 .62056 .95178 .65963 .99922 .6478 .98003 .60822 Metetra .93 .556 .271 r .36056 .56222 .38313 .58323 .41624 .7005 .39343 .67115 Metetra .328 .534 .903 r .39343 .67115 .41624 .7005 .45472 .71643 .43205 .68434 Metetra .557 .629 .878 r .43205 .68434 .45472 .71643 .49449 .71711 .47201 .68353 Metetra .604 .642 .861 r .47201 .68353 .49449 .71711 .53511 .71253 .51284 .67794 Metetra .622 .645 .852 r .51284 .67794 .53511 .71253 .57646 .70546 .5544 .67006 Metetra .631 .647 .847 r .5544 .67006 .57646 .70546 .6185 .69696 .59667 .66088 Metetra .636 .647 .844 r .59667 .66088 .6185 .69696 .66122 .68752 .63963 .65083 Metetra .639 .647 .842 r .63963 .65083 .66122 .68752 .70462 .67742 .68328 .64015 Metetra .641 .647 .841 r .68328 .64015 .70462 .67742 .74871 .66681 .72764 .62898 Metetra .642 .647 .84 r .72764 .62898 .74871 .66681 .79351 .65576 .77271 .6174 Metetra .643 .647 .839 r .77271 .6174 .79351 .65576 .83903 .64435 .81851 .60547 Metetra .644 .647 .838 r .81851 .60547 .83903 .64435 .88527 .63261 .86505 .59321 Metetra .645 .647 .838 r .86505 .59321 .88527 .63261 .93227 .62056 .91235 .58064 Metetra .645 .647 .837 r .91235 .58064 .93227 .62056 .98003 .60822 .96043 .56779 Metetra .924 .539 .242 r .33741 .54067 .36056 .56222 .39343 .67115 .37013 .64097 Metetra .378 .559 .905 r .37013 .64097 .39343 .67115 .43205 .68434 .40892 .65142 Metetra .569 .633 .875 r .40892 .65142 .43205 .68434 .47201 .68353 .44908 .64915 Metetra .61 .643 .859 r .44908 .64915 .47201 .68353 .51284 .67794 .49011 .64255 Metetra .625 .646 .851 r .49011 .64255 .51284 .67794 .5544 .67006 .5319 .63387 Metetra .633 .647 .846 r .5319 .63387 .5544 .67006 .59667 .66088 .5744 .624 Metetra .637 .647 .844 r .5744 .624 .59667 .66088 .63963 .65083 .6176 .61333 Metetra .64 .647 .842 r .6176 .61333 .63963 .65083 .68328 .64015 .66151 .60207 Metetra .642 .647 .84 r .66151 .60207 .68328 .64015 .72764 .62898 .70613 .59034 Metetra .643 .647 .839 r .70613 .59034 .72764 .62898 .77271 .6174 .75147 .57822 Metetra .644 .647 .838 r .75147 .57822 .77271 .6174 .81851 .60547 .79756 .56575 Metetra .644 .647 .838 r .79756 .56575 .81851 .60547 .86505 .59321 .8444 .55295 Metetra .645 .647 .837 r .8444 .55295 .86505 .59321 .91235 .58064 .89202 .53986 Metetra .645 .647 .837 r .89202 .53986 .91235 .58064 .96043 .56779 .94042 .52647 Metetra .915 .517 .204 r .31365 .51855 .33741 .54067 .37013 .64097 .34633 .60991 Metetra .422 .58 .904 r .34633 .60991 .37013 .64097 .40892 .65142 .38531 .61766 Metetra .58 .637 .872 r .38531 .61766 .40892 .65142 .44908 .64915 .42568 .61395 Metetra .615 .645 .857 r .42568 .61395 .44908 .64915 .49011 .64255 .46693 .60634 Metetra .628 .647 .849 r .46693 .60634 .49011 .64255 .5319 .63387 .50895 .59686 Metetra .635 .648 .845 r .50895 .59686 .5319 .63387 .5744 .624 .55168 .5863 Metetra .639 .648 .843 r .55168 .5863 .5744 .624 .6176 .61333 .59513 .575 Metetra .641 .648 .841 r .59513 .575 .6176 .61333 .66151 .60207 .63929 .56315 Metetra .642 .647 .84 r .63929 .56315 .66151 .60207 .70613 .59034 .68418 .55085 Metetra .644 .647 .839 r .68418 .55085 .70613 .59034 .75147 .57822 .7298 .53818 Metetra .644 .647 .838 r .7298 .53818 .75147 .57822 .79756 .56575 .77618 .52516 Metetra .645 .647 .838 r .77618 .52516 .79756 .56575 .8444 .55295 .82332 .51182 Metetra .645 .647 .837 r .82332 .51182 .8444 .55295 .89202 .53986 .87125 .49818 Metetra .646 .647 .837 r .87125 .49818 .89202 .53986 .94042 .52647 .91999 .48424 Metetra .902 .488 .156 r .28926 .49583 .31365 .51855 .34633 .60991 .32201 .57795 Metetra .461 .597 .901 r .32201 .57795 .34633 .60991 .38531 .61766 .36121 .58303 Metetra .59 .64 .868 r .36121 .58303 .38531 .61766 .42568 .61395 .4018 .57789 Metetra .62 .646 .855 r .4018 .57789 .42568 .61395 .46693 .60634 .44328 .56929 Metetra .631 .647 .848 r .44328 .56929 .46693 .60634 .50895 .59686 .48552 .55902 Metetra .637 .648 .844 r .48552 .55902 .50895 .59686 .55168 .5863 .5285 .54776 Metetra .64 .648 .842 r .5285 .54776 .55168 .5863 .59513 .575 .57219 .53582 Metetra .642 .648 .84 r .57219 .53582 .59513 .575 .63929 .56315 .61661 .52337 Metetra .643 .647 .839 r .61661 .52337 .63929 .56315 .68418 .55085 .66177 .5105 Metetra .644 .647 .838 r .66177 .5105 .68418 .55085 .7298 .53818 .70768 .49726 Metetra .645 .647 .838 r .70768 .49726 .7298 .53818 .77618 .52516 .75435 .48368 Metetra .645 .647 .837 r .75435 .48368 .77618 .52516 .82332 .51182 .8018 .46978 Metetra .646 .647 .837 r .8018 .46978 .82332 .51182 .87125 .49818 .85005 .45558 Metetra .646 .647 .836 r .85005 .45558 .87125 .49818 .91999 .48424 .89911 .44108 Metetra .881 .447 .091 r .2642 .4725 .28926 .49583 .32201 .57795 .29716 .54505 Metetra .495 .612 .897 r .29716 .54505 .32201 .57795 .36121 .58303 .33661 .5475 Metetra .6 .643 .865 r .33661 .5475 .36121 .58303 .4018 .57789 .37743 .54095 Metetra .624 .647 .853 r .37743 .54095 .4018 .57789 .44328 .56929 .41914 .53137 Metetra .634 .648 .847 r .41914 .53137 .44328 .56929 .48552 .55902 .46162 .5203 Metetra .638 .648 .843 r .46162 .5203 .48552 .55902 .5285 .54776 .50483 .50835 Metetra .641 .648 .841 r .50483 .50835 .5285 .54776 .57219 .53582 .54878 .49577 Metetra .643 .648 .84 r .54878 .49577 .57219 .53582 .61661 .52337 .59347 .48272 Metetra .644 .647 .839 r .59347 .48272 .61661 .52337 .66177 .5105 .6389 .46925 Metetra .645 .647 .838 r .6389 .46925 .66177 .5105 .70768 .49726 .68509 .45543 Metetra .645 .647 .837 r .68509 .45543 .70768 .49726 .75435 .48368 .73205 .44128 Metetra .646 .647 .837 r .73205 .44128 .75435 .48368 .8018 .46978 .77981 .42681 Metetra .646 .647 .836 r .77981 .42681 .8018 .46978 .85005 .45558 .82838 .41204 Metetra .646 .647 .836 r .82838 .41204 .85005 .45558 .89911 .44108 .87778 .39696 Metetra .845 .387 0 r .23846 .44853 .2642 .4725 .29716 .54505 .27177 .51119 Metetra .525 .624 .892 r .27177 .51119 .29716 .54505 .33661 .5475 .31149 .51104 Metetra .608 .645 .862 r .31149 .51104 .33661 .5475 .37743 .54095 .35256 .50312 Metetra .628 .648 .851 r .35256 .50312 .37743 .54095 .41914 .53137 .3945 .49256 Metetra .636 .648 .845 r .3945 .49256 .41914 .53137 .46162 .5203 .43722 .4807 Metetra .64 .648 .842 r .43722 .4807 .46162 .5203 .50483 .50835 .48068 .46805 Metetra .642 .648 .84 r .48068 .46805 .50483 .50835 .54878 .49577 .52489 .45482 Metetra .644 .648 .839 r .52489 .45482 .54878 .49577 .59347 .48272 .56984 .44115 Metetra .645 .647 .838 r .56984 .44115 .59347 .48272 .6389 .46925 .61554 .42709 Metetra .645 .647 .838 r .61554 .42709 .6389 .46925 .68509 .45543 .66202 .41267 Metetra .646 .647 .837 r .66202 .41267 .68509 .45543 .73205 .44128 .70928 .39793 Metetra .646 .647 .837 r .70928 .39793 .73205 .44128 .77981 .42681 .75735 .38288 Metetra .646 .647 .836 r .75735 .38288 .77981 .42681 .82838 .41204 .80625 .36751 Metetra .646 .646 .836 r .80625 .36751 .82838 .41204 .87778 .39696 .85599 .35184 Metetra .78 .295 0 r .212 .42389 .23846 .44853 .27177 .51119 .24583 .47632 Metetra .551 .633 .886 r .24583 .47632 .27177 .51119 .31149 .51104 .28585 .47364 Metetra .616 .647 .858 r .28585 .47364 .31149 .51104 .35256 .50312 .32717 .46436 Metetra .632 .649 .849 r .32717 .46436 .35256 .50312 .3945 .49256 .36935 .45283 Metetra .638 .649 .844 r .36935 .45283 .3945 .49256 .43722 .4807 .41232 .44018 Metetra .642 .648 .841 r .41232 .44018 .43722 .4807 .48068 .46805 .45603 .42682 Metetra .643 .648 .84 r .45603 .42682 .48068 .46805 .52489 .45482 .50049 .41295 Metetra .644 .648 .839 r .50049 .41295 .52489 .45482 .56984 .44115 .5457 .39865 Metetra .645 .647 .838 r .5457 .39865 .56984 .44115 .61554 .42709 .59169 .38397 Metetra .646 .647 .837 r .59169 .38397 .61554 .42709 .66202 .41267 .63845 .36896 Metetra .646 .647 .837 r .63845 .36896 .66202 .41267 .70928 .39793 .68602 .35361 Metetra .646 .647 .836 r .68602 .35361 .70928 .39793 .75735 .38288 .7344 .33796 Metetra .647 .647 .836 r .7344 .33796 .75735 .38288 .80625 .36751 .78363 .32198 Metetra .647 .646 .836 r .78363 .32198 .80625 .36751 .85599 .35184 .83371 .3057 Metetra .658 .149 0 r .18479 .39856 .212 .42389 .24583 .47632 .21933 .44042 Metetra .574 .641 .88 r .21933 .44042 .24583 .47632 .28585 .47364 .25967 .43525 Metetra .623 .649 .855 r .25967 .43525 .28585 .47364 .32717 .46436 .30125 .42464 Metetra .635 .65 .847 r .30125 .42464 .32717 .46436 .36935 .45283 .34368 .41216 Metetra .64 .649 .843 r .34368 .41216 .36935 .45283 .41232 .44018 .38689 .39872 Metetra .643 .649 .841 r .38689 .39872 .41232 .44018 .45603 .42682 .43086 .38466 Metetra .644 .648 .839 r .43086 .38466 .45603 .42682 .50049 .41295 .47558 .37012 Metetra .645 .648 .838 r .47558 .37012 .50049 .41295 .5457 .39865 .52106 .35518 Metetra .646 .647 .837 r .52106 .35518 .5457 .39865 .59169 .38397 .56732 .33988 Metetra .646 .647 .837 r .56732 .33988 .59169 .38397 .63845 .36896 .61438 .32425 Metetra .646 .647 .836 r .61438 .32425 .63845 .36896 .68602 .35361 .66225 .30829 Metetra .647 .647 .836 r .66225 .30829 .68602 .35361 .7344 .33796 .71095 .29201 Metetra .647 .646 .836 r .71095 .29201 .7344 .33796 .78363 .32198 .76051 .27541 Metetra .647 .646 .835 r .76051 .27541 .78363 .32198 .83371 .3057 .81094 .2585 Metetra 0 .073 .563 r .1568 .3725 .18479 .39856 .21933 .44042 .19225 .40345 Metetra .594 .648 .873 r .19225 .40345 .21933 .44042 .25967 .43525 .23294 .39586 Metetra .63 .651 .852 r .23294 .39586 .25967 .43525 .30125 .42464 .27478 .38395 Metetra .639 .65 .845 r .27478 .38395 .30125 .42464 .34368 .41216 .31747 .37052 Metetra .642 .649 .842 r .31747 .37052 .34368 .41216 .38689 .39872 .36093 .35629 Metetra .644 .649 .84 r .36093 .35629 .38689 .39872 .43086 .38466 .40515 .34152 Metetra .645 .648 .838 r .40515 .34152 .43086 .38466 .47558 .37012 .45013 .32631 Metetra .646 .648 .838 r .45013 .32631 .47558 .37012 .52106 .35518 .49589 .31072 Metetra .646 .647 .837 r .49589 .31072 .52106 .35518 .56732 .33988 .54243 .29479 Metetra .647 .647 .836 r .54243 .29479 .56732 .33988 .61438 .32425 .58979 .27852 Metetra .647 .647 .836 r .58979 .27852 .61438 .32425 .66225 .30829 .63796 .26193 Metetra .647 .647 .836 r .63796 .26193 .66225 .30829 .71095 .29201 .68699 .24501 Metetra .647 .646 .835 r .68699 .24501 .71095 .29201 .76051 .27541 .73688 .22777 Metetra .647 .646 .835 r .73688 .22777 .76051 .27541 .81094 .2585 .78766 .21021 Metetra 0 .348 .809 r .128 .34568 .1568 .3725 .19225 .40345 .16458 .36537 Metetra .612 .653 .867 r .16458 .36537 .19225 .40345 .23294 .39586 .20564 .35542 Metetra .635 .652 .85 r .20564 .35542 .23294 .39586 .27478 .38395 .24777 .34224 Metetra .642 .651 .844 r .24777 .34224 .27478 .38395 .31747 .37052 .29071 .32789 Metetra .644 .65 .841 r .29071 .32789 .31747 .37052 .36093 .35629 .33442 .31287 Metetra .645 .649 .839 r .33442 .31287 .36093 .35629 .40515 .34152 .3789 .29738 Metetra .646 .648 .838 r .3789 .29738 .40515 .34152 .45013 .32631 .42415 .28149 Metetra .646 .648 .837 r .42415 .28149 .45013 .32631 .49589 .31072 .47017 .26524 Metetra .647 .647 .836 r .47017 .26524 .49589 .31072 .54243 .29479 .517 .24866 Metetra .647 .647 .836 r .517 .24866 .54243 .29479 .58979 .27852 .56465 .23174 Metetra .647 .647 .836 r .56465 .23174 .58979 .27852 .63796 .26193 .61314 .2145 Metetra .647 .646 .835 r .61314 .2145 .63796 .26193 .68699 .24501 .66248 .19693 Metetra .647 .646 .835 r .66248 .19693 .68699 .24501 .73688 .22777 .71271 .17903 Metetra .647 .646 .835 r .71271 .17903 .73688 .22777 .78766 .21021 .76384 .16079 Metetra .264 .574 .94 r .09834 .31807 .128 .34568 .16458 .36537 .13631 .32614 Metetra .627 .658 .861 r .13631 .32614 .16458 .36537 .20564 .35542 .17776 .31392 Metetra .641 .653 .847 r .17776 .31392 .20564 .35542 .24777 .34224 .22017 .2995 Metetra .644 .651 .842 r .22017 .2995 .24777 .34224 .29071 .32789 .26338 .28422 Metetra .646 .65 .84 r .26338 .28422 .29071 .32789 .33442 .31287 .30735 .26842 Metetra .646 .649 .838 r .30735 .26842 .33442 .31287 .3789 .29738 .35208 .2522 Metetra .647 .648 .837 r .35208 .2522 .3789 .29738 .42415 .28149 .39759 .23563 Metetra .647 .648 .837 r .39759 .23563 .42415 .28149 .47017 .26524 .4439 .21871 Metetra .647 .647 .836 r .4439 .21871 .47017 .26524 .517 .24866 .49101 .20146 Metetra .647 .647 .836 r .49101 .20146 .517 .24866 .56465 .23174 .53896 .18388 Metetra .647 .647 .835 r .53896 .18388 .56465 .23174 .61314 .2145 .58775 .16597 Metetra .647 .646 .835 r .58775 .16597 .61314 .2145 .66248 .19693 .63743 .14772 Metetra .647 .646 .835 r .63743 .14772 .66248 .19693 .71271 .17903 .68799 .12914 Metetra .647 .646 .835 r .68799 .12914 .71271 .17903 .76384 .16079 .73948 .1102 Metetra .516 .693 .949 r .0678 .28963 .09834 .31807 .13631 .32614 .10743 .28573 Metetra .64 .661 .855 r .10743 .28573 .13631 .32614 .17776 .31392 .14929 .27131 Metetra .645 .654 .844 r .14929 .27131 .17776 .31392 .22017 .2995 .192 .25569 Metetra .647 .651 .84 r .192 .25569 .22017 .2995 .26338 .28422 .23547 .2395 Metetra .647 .65 .839 r .23547 .2395 .26338 .28422 .30735 .26842 .27969 .22291 Metetra .647 .649 .837 r .27969 .22291 .30735 .26842 .35208 .2522 .32469 .20597 Metetra .648 .648 .837 r .32469 .20597 .35208 .2522 .39759 .23563 .37046 .18869 Metetra .648 .647 .836 r .37046 .18869 .39759 .23563 .4439 .21871 .41704 .17109 Metetra .648 .647 .836 r .41704 .17109 .4439 .21871 .49101 .20146 .46445 .15316 Metetra .648 .647 .835 r .46445 .15316 .49101 .20146 .53896 .18388 .51269 .1349 Metetra .648 .646 .835 r .51269 .1349 .53896 .18388 .58775 .16597 .5618 .1163 Metetra .648 .646 .835 r .5618 .1163 .58775 .16597 .63743 .14772 .6118 .09736 Metetra .648 .646 .835 r .6118 .09736 .63743 .14772 .68799 .12914 .66271 .07807 Metetra .648 .646 .835 r .66271 .07807 .68799 .12914 .73948 .1102 .71456 .05842 Metetra .664 .741 .908 r .03634 .26033 .0678 .28963 .10743 .28573 .07793 .24409 Metetra .652 .664 .849 r .07793 .24409 .10743 .28573 .14929 .27131 .12022 .22757 Metetra .65 .655 .842 r .12022 .22757 .14929 .27131 .192 .25569 .16322 .21077 Metetra .649 .652 .839 r .16322 .21077 .192 .25569 .23547 .2395 .20695 .19369 Metetra .649 .65 .838 r .20695 .19369 .23547 .2395 .27969 .22291 .25144 .17632 Metetra .648 .649 .837 r .25144 .17632 .27969 .22291 .32469 .20597 .29669 .15864 Metetra .648 .648 .836 r .29669 .15864 .32469 .20597 .37046 .18869 .34274 .14065 Metetra .648 .647 .836 r .34274 .14065 .37046 .18869 .41704 .17109 .3896 .12235 Metetra .648 .647 .835 r .3896 .12235 .41704 .17109 .46445 .15316 .43729 .10372 Metetra .648 .647 .835 r .43729 .10372 .46445 .15316 .51269 .1349 .48583 .08476 Metetra .648 .646 .835 r .48583 .08476 .51269 .1349 .5618 .1163 .53526 .06546 Metetra .648 .646 .835 r .53526 .06546 .5618 .1163 .6118 .09736 .58558 .0458 Metetra .648 .646 .834 r .58558 .0458 .6118 .09736 .66271 .07807 .63684 .02578 Metetra .648 .646 .834 r .63684 .02578 .66271 .07807 .71456 .05842 .68905 .00539 Metetra 0 g .25 Mabswid .68874 0 m .96935 .42924 L s .96935 .42924 m 1 .6535 L s 1 .6535 m .70298 .24544 L s .70298 .24544 m .68874 0 L s .03716 .25514 m 0 .48963 L s 0 .48963 m .70298 .24544 L s .70298 .24544 m .68874 0 L s .68874 0 m .03716 .25514 L s .03716 .25514 m .68874 0 L s .14536 .21277 m .14998 .21742 L s [(10)] .1361 .20347 .99414 1 Mshowa .27119 .1635 m .2756 .16835 L s [(20)] .26236 .15379 .90904 1 Mshowa .40336 .11175 m .40753 .11681 L s [(30)] .39501 .10162 .82394 1 Mshowa .54236 .05732 m .54626 .06259 L s [(40)] .53456 .04676 .73884 1 Mshowa .68874 0 m .69233 .00549 L s [(50)] .68156 -0.01098 .65374 1 Mshowa .125 Mabswid .17004 .20311 m .17279 .20592 L s .19496 .19335 m .19768 .19619 L s .22012 .1835 m .22282 .18636 L s .24553 .17355 m .2482 .17644 L s .2971 .15335 m .29972 .15629 L s .32327 .14311 m .32586 .14607 L s .3497 .13276 m .35226 .13575 L s .37639 .1223 m .37893 .12532 L s .43059 .10108 m .43307 .10414 L s .45811 .09031 m .46055 .0934 L s .4859 .07942 m .48831 .08254 L s .51399 .06843 m .51636 .07157 L s .57103 .04609 m .57333 .04928 L s .59999 .03475 m .60226 .03797 L s .62926 .02329 m .63149 .02653 L s .65884 .01171 m .66104 .01497 L s .12091 .22234 m .12371 .22511 L s .0967 .23182 m .09952 .23457 L s .07271 .24122 m .07556 .24394 L s .04896 .25052 m .05182 .25322 L s % End of Graphics 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The question of validity \ near T = 1 will be further discussed below\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["The infinite time limit", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[ Series[b0[T], {T, \[Infinity], 4}], {T > 0, \[Alpha] > 1, \[CapitalDelta] > \(-1\), \[CapitalDelta] < 1}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{ "1", "-", \(\(\((\[CapitalDelta]\^2\ \((1 - \[Alpha]\ \((\(-3\) + \ \[CapitalDelta])\) + \[CapitalDelta])\)\^2)\)\ \((1\/T)\)\^2\)\/\(2\ \ \((\((\(-1\) + \[CapitalDelta])\)\^2\ \((\[Alpha] + \ \[CapitalDelta])\)\^2)\)\)\), "+", InterpretationBox[\(O[1\/T]\^4\), SeriesData[ T, DirectedInfinity[ 1], {}, 0, 4, 1]]}], SeriesData[ T, DirectedInfinity[ 1], {1, 0, Times[ Rational[ -1, 2], Power[ Plus[ -1, \[CapitalDelta]], -2], Power[ \[CapitalDelta], 2], Power[ Plus[ \[Alpha], \[CapitalDelta]], -2], Power[ Plus[ 1, Times[ -1, \[Alpha], Plus[ -3, \[CapitalDelta]]], \[CapitalDelta]], 2]]}, 0, 4, 1]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[ Series[b1[T], {T, \[Infinity], 4}], {T > 0, \[Alpha] > 1, \[CapitalDelta] > \(-1\), \[CapitalDelta] < 1}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{\(\((\[Beta]\ \[CapitalDelta]\ \((\[CapitalDelta]\ \((1 + 10\ \[CapitalDelta] + \[CapitalDelta]\^2)\) + 2\ \[Alpha]\ \[CapitalDelta]\ \((10 - \[CapitalDelta] - 2\ \[CapitalDelta]\^2 + \[CapitalDelta]\^3)\) + \ \[Alpha]\^2\ \((10 - 5\ \[CapitalDelta] - 4\ \[CapitalDelta]\^2 + 3\ \[CapitalDelta]\^3)\))\))\)/\((2\ \@\(2\ \[Pi]\)\ \ \((\[Alpha] + \[CapitalDelta])\)\^2\ \((\(-1\) + \[CapitalDelta]\^2)\)\ \ T)\)\), "-", \(\((\((\[Beta]\ \[CapitalDelta]\ \((2\ \[CapitalDelta]\^4\ \((7 \ + 22\ \[CapitalDelta] + 7\ \[CapitalDelta]\^2)\) + 2\ \[Alpha]\ \[CapitalDelta]\^2\ \((\(-1\) + 28\ \[CapitalDelta] + 77\ \[CapitalDelta]\^2 + 18\ \[CapitalDelta]\^3 - \[CapitalDelta]\^4 - 2\ \[CapitalDelta]\^5 + \[CapitalDelta]\^6)\) + \ \[Alpha]\^2\ \[CapitalDelta]\ \((\(-1\) + 88\ \[CapitalDelta] + 169\ \[CapitalDelta]\^2 + 76\ \[CapitalDelta]\^3 - 59\ \[CapitalDelta]\^4 + 19\ \[CapitalDelta]\^6 - 4\ \[CapitalDelta]\^7)\) + 2\ \[Alpha]\^3\ \[CapitalDelta]\ \((37 + 21\ \[CapitalDelta] + 57\ \[CapitalDelta]\^2 - 63\ \[CapitalDelta]\^3 + \[CapitalDelta]\^4 + 29\ \[CapitalDelta]\^5 - 11\ \[CapitalDelta]\^6 + \[CapitalDelta]\^7)\) + \ \[Alpha]\^4\ \((24 - 13\ \[CapitalDelta] + 64\ \[CapitalDelta]\^2 - 79\ \[CapitalDelta]\^3 + 2\ \[CapitalDelta]\^4 + 45\ \[CapitalDelta]\^5 - 22\ \[CapitalDelta]\^6 + 3\ \[CapitalDelta]\^7)\))\))\)\ \((1\/T)\)\^3)\)/\ \((2\ \((\@\(2\ \[Pi]\)\ \((\(-1\) + \[CapitalDelta])\)\^3\ \((1 + \ \[CapitalDelta])\)\ \((\[Alpha] + \[CapitalDelta])\)\^4)\))\)\), "+", InterpretationBox[\(O[1\/T]\^5\), SeriesData[ T, DirectedInfinity[ 1], {}, 1, 5, 1]]}], SeriesData[ T, DirectedInfinity[ 1], { Times[ Rational[ 1, 2], Power[ Times[ 2, Pi], Rational[ -1, 2]], \[Beta], \[CapitalDelta], Power[ Plus[ \[Alpha], \[CapitalDelta]], -2], Power[ Plus[ -1, Power[ \[CapitalDelta], 2]], -1], Plus[ Times[ \[CapitalDelta], Plus[ 1, Times[ 10, \[CapitalDelta]], Power[ \[CapitalDelta], 2]]], Times[ 2, \[Alpha], \[CapitalDelta], Plus[ 10, Times[ -1, \[CapitalDelta]], Times[ -2, Power[ \[CapitalDelta], 2]], Power[ \[CapitalDelta], 3]]], Times[ Power[ \[Alpha], 2], Plus[ 10, Times[ -5, \[CapitalDelta]], Times[ -4, Power[ \[CapitalDelta], 2]], Times[ 3, Power[ \[CapitalDelta], 3]]]]]], 0, Times[ Rational[ -1, 2], Power[ Times[ 2, Pi], Rational[ -1, 2]], \[Beta], Power[ Plus[ -1, \[CapitalDelta]], -3], \[CapitalDelta], Power[ Plus[ 1, \[CapitalDelta]], -1], Power[ Plus[ \[Alpha], \[CapitalDelta]], -4], Plus[ Times[ 2, Power[ \[CapitalDelta], 4], Plus[ 7, Times[ 22, \[CapitalDelta]], Times[ 7, Power[ \[CapitalDelta], 2]]]], Times[ 2, \[Alpha], Power[ \[CapitalDelta], 2], Plus[ -1, Times[ 28, \[CapitalDelta]], Times[ 77, Power[ \[CapitalDelta], 2]], Times[ 18, Power[ \[CapitalDelta], 3]], Times[ -1, Power[ \[CapitalDelta], 4]], Times[ -2, Power[ \[CapitalDelta], 5]], Power[ \[CapitalDelta], 6]]], Times[ Power[ \[Alpha], 2], \[CapitalDelta], Plus[ -1, Times[ 88, \[CapitalDelta]], Times[ 169, Power[ \[CapitalDelta], 2]], Times[ 76, Power[ \[CapitalDelta], 3]], Times[ -59, Power[ \[CapitalDelta], 4]], Times[ 19, Power[ \[CapitalDelta], 6]], Times[ -4, Power[ \[CapitalDelta], 7]]]], Times[ 2, Power[ \[Alpha], 3], \[CapitalDelta], Plus[ 37, Times[ 21, \[CapitalDelta]], Times[ 57, Power[ \[CapitalDelta], 2]], Times[ -63, Power[ \[CapitalDelta], 3]], Power[ \[CapitalDelta], 4], Times[ 29, Power[ \[CapitalDelta], 5]], Times[ -11, Power[ \[CapitalDelta], 6]], Power[ \[CapitalDelta], 7]]], Times[ Power[ \[Alpha], 4], Plus[ 24, Times[ -13, \[CapitalDelta]], Times[ 64, Power[ \[CapitalDelta], 2]], Times[ -79, Power[ \[CapitalDelta], 3]], Times[ 2, Power[ \[CapitalDelta], 4]], Times[ 45, Power[ \[CapitalDelta], 5]], Times[ -22, Power[ \[CapitalDelta], 6]], Times[ 3, Power[ \[CapitalDelta], 7]]]]]]}, 1, 5, 1]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[ Series[b2[T], {T, \[Infinity], 4}], {T > 0, \[Alpha] > 1, \[CapitalDelta] > \(-1\), \[CapitalDelta] < 1}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{\(\((\[Beta]\^2\ \[CapitalDelta]\ \((3\ \[CapitalDelta]\^3\ \ \((1 + 35\ \[CapitalDelta] + 35\ \[CapitalDelta]\^2 + \[CapitalDelta]\^3)\) + 4\ \[Alpha]\^3\ \[CapitalDelta]\ \((98 - 22\ \[CapitalDelta] - 97\ \[CapitalDelta]\^2 + 107\ \[CapitalDelta]\^3 - 49\ \[CapitalDelta]\^4 + 11\ \[CapitalDelta]\^5)\) + \[Alpha]\^4\ \((104 - 78\ \[CapitalDelta] - 94\ \[CapitalDelta]\^2 + 159\ \[CapitalDelta]\^3 - 91\ \[CapitalDelta]\^4 + 27\ \[CapitalDelta]\^5 - 3\ \[CapitalDelta]\^6)\) + 4\ \[Alpha]\ \[CapitalDelta]\^2\ \((3 + 92\ \[CapitalDelta] + 71\ \[CapitalDelta]\^2 - 37\ \[CapitalDelta]\^3 + 20\ \[CapitalDelta]\^4 - 7\ \[CapitalDelta]\^5 + 2\ \[CapitalDelta]\^6)\) + 2\ \[Alpha]\^2\ \[CapitalDelta]\ \((3 + 277\ \[CapitalDelta] + 89\ \[CapitalDelta]\^2 - 227\ \[CapitalDelta]\^3 + 170\ \[CapitalDelta]\^4 - 62\ \[CapitalDelta]\^5 + 14\ \[CapitalDelta]\^6)\))\)\ \((1\/T)\)\^2)\)/\((32\ \ \((\(-1\) + \[CapitalDelta])\)\^2\ \((1 + \[CapitalDelta])\)\ \((\[Alpha] + \ \[CapitalDelta])\)\^4)\)\), "-", \(\((\((\[Beta]\^2\ \[CapitalDelta]\ \((\(-3\)\ \ \[CapitalDelta]\^5\ \((1 - 51\ \[CapitalDelta] - 670\ \[CapitalDelta]\^2 - 670\ \[CapitalDelta]\^3 - 51\ \[CapitalDelta]\^4 + \[CapitalDelta]\^5)\) + 2\ \[Alpha]\ \[CapitalDelta]\^4\ \((\(-18\) + 463\ \[CapitalDelta] + 5839\ \[CapitalDelta]\^2 + 3680\ \[CapitalDelta]\^3 + 56\ \[CapitalDelta]\^4 + 7\ \[CapitalDelta]\^5 + 55\ \[CapitalDelta]\^6 - 6\ \[CapitalDelta]\^7 + 4\ \[CapitalDelta]\^8)\) + \[Alpha]\^2\ \ \[CapitalDelta]\^3\ \((\(-114\) + 3070\ \[CapitalDelta] + 26957\ \[CapitalDelta]\^2 + 8309\ \[CapitalDelta]\^3 - 724\ \[CapitalDelta]\^4 - 912\ \[CapitalDelta]\^5 + 1605\ \[CapitalDelta]\^6 - 435\ \[CapitalDelta]\^7 + 196\ \[CapitalDelta]\^8 - 32\ \[CapitalDelta]\^9)\) + 2\ \[Alpha]\^5\ \[CapitalDelta]\ \((1652 + 2128\ \[CapitalDelta] - 638\ \[CapitalDelta]\^2 - 1034\ \[CapitalDelta]\^3 - 3573\ \[CapitalDelta]\^4 + 8283\ \[CapitalDelta]\^5 - 6888\ \[CapitalDelta]\^6 + 3140\ \[CapitalDelta]\^7 - 745\ \[CapitalDelta]\^8 + 75\ \[CapitalDelta]\^9)\) + \[Alpha]\^6\ \((768 - 80\ \[CapitalDelta] + 432\ \[CapitalDelta]\^2 - 1174\ \[CapitalDelta]\^3 - 1458\ \[CapitalDelta]\^4 + 5063\ \[CapitalDelta]\^5 - 5081\ \[CapitalDelta]\^6 + 2696\ \[CapitalDelta]\^7 - 812\ \[CapitalDelta]\^8 + 135\ \[CapitalDelta]\^9 - 9\ \[CapitalDelta]\^10)\) + 4\ \[Alpha]\^3\ \[CapitalDelta]\^2\ \((\(-24\) + 1389\ \[CapitalDelta] + 7930\ \[CapitalDelta]\^2 + 27\ \[CapitalDelta]\^3 - 265\ \[CapitalDelta]\^4 - 1332\ \[CapitalDelta]\^5 + 2126\ \[CapitalDelta]\^6 - 1073\ \[CapitalDelta]\^7 + 419\ \[CapitalDelta]\^8 - 83\ \[CapitalDelta]\^9 + 6\ \[CapitalDelta]\^10)\) + \[Alpha]\^4\ \ \[CapitalDelta]\ \((\(-24\) + 5840\ \[CapitalDelta] + 18704\ \[CapitalDelta]\^2 - 4604\ \[CapitalDelta]\^3 - 1073\ \[CapitalDelta]\^4 - 10213\ \[CapitalDelta]\^5 + 18386\ \[CapitalDelta]\^6 - 12366\ \[CapitalDelta]\^7 + 5027\ \[CapitalDelta]\^8 - 1041\ \[CapitalDelta]\^9 + 84\ \[CapitalDelta]\^10)\))\))\)\ \ \((1\/T)\)\^4)\)/\((64\ \((\((\(-1\) + \[CapitalDelta])\)\^4\ \((1 + \ \[CapitalDelta])\)\ \((\[Alpha] + \[CapitalDelta])\)\^6)\))\)\), "+", InterpretationBox[\(O[1\/T]\^5\), SeriesData[ T, DirectedInfinity[ 1], {}, 2, 5, 1]]}], SeriesData[ T, DirectedInfinity[ 1], { Times[ Rational[ 1, 32], Power[ \[Beta], 2], Power[ Plus[ -1, \[CapitalDelta]], -2], \[CapitalDelta], Power[ Plus[ 1, \[CapitalDelta]], -1], Power[ Plus[ \[Alpha], \[CapitalDelta]], -4], Plus[ Times[ 3, Power[ \[CapitalDelta], 3], Plus[ 1, Times[ 35, \[CapitalDelta]], Times[ 35, Power[ \[CapitalDelta], 2]], Power[ \[CapitalDelta], 3]]], Times[ 4, Power[ \[Alpha], 3], \[CapitalDelta], Plus[ 98, Times[ -22, \[CapitalDelta]], Times[ -97, Power[ \[CapitalDelta], 2]], Times[ 107, Power[ \[CapitalDelta], 3]], Times[ -49, Power[ \[CapitalDelta], 4]], Times[ 11, Power[ \[CapitalDelta], 5]]]], Times[ Power[ \[Alpha], 4], Plus[ 104, Times[ -78, \[CapitalDelta]], Times[ -94, Power[ \[CapitalDelta], 2]], Times[ 159, Power[ \[CapitalDelta], 3]], Times[ -91, Power[ \[CapitalDelta], 4]], Times[ 27, Power[ \[CapitalDelta], 5]], Times[ -3, Power[ \[CapitalDelta], 6]]]], Times[ 4, \[Alpha], Power[ \[CapitalDelta], 2], Plus[ 3, Times[ 92, \[CapitalDelta]], Times[ 71, Power[ \[CapitalDelta], 2]], Times[ -37, Power[ \[CapitalDelta], 3]], Times[ 20, Power[ \[CapitalDelta], 4]], Times[ -7, Power[ \[CapitalDelta], 5]], Times[ 2, Power[ \[CapitalDelta], 6]]]], Times[ 2, Power[ \[Alpha], 2], \[CapitalDelta], Plus[ 3, Times[ 277, \[CapitalDelta]], Times[ 89, Power[ \[CapitalDelta], 2]], Times[ -227, Power[ \[CapitalDelta], 3]], Times[ 170, Power[ \[CapitalDelta], 4]], Times[ -62, Power[ \[CapitalDelta], 5]], Times[ 14, Power[ \[CapitalDelta], 6]]]]]], 0, Times[ Rational[ -1, 64], Power[ \[Beta], 2], Power[ Plus[ -1, \[CapitalDelta]], -4], \[CapitalDelta], Power[ Plus[ 1, \[CapitalDelta]], -1], Power[ Plus[ \[Alpha], \[CapitalDelta]], -6], Plus[ Times[ -3, Power[ \[CapitalDelta], 5], Plus[ 1, Times[ -51, \[CapitalDelta]], Times[ -670, Power[ \[CapitalDelta], 2]], Times[ -670, Power[ \[CapitalDelta], 3]], Times[ -51, Power[ \[CapitalDelta], 4]], Power[ \[CapitalDelta], 5]]], Times[ 2, \[Alpha], Power[ \[CapitalDelta], 4], Plus[ -18, Times[ 463, \[CapitalDelta]], Times[ 5839, Power[ \[CapitalDelta], 2]], Times[ 3680, Power[ \[CapitalDelta], 3]], Times[ 56, Power[ \[CapitalDelta], 4]], Times[ 7, Power[ \[CapitalDelta], 5]], Times[ 55, Power[ \[CapitalDelta], 6]], Times[ -6, Power[ \[CapitalDelta], 7]], Times[ 4, Power[ \[CapitalDelta], 8]]]], Times[ Power[ \[Alpha], 2], Power[ \[CapitalDelta], 3], Plus[ -114, Times[ 3070, \[CapitalDelta]], Times[ 26957, Power[ \[CapitalDelta], 2]], Times[ 8309, Power[ \[CapitalDelta], 3]], Times[ -724, Power[ \[CapitalDelta], 4]], Times[ -912, Power[ \[CapitalDelta], 5]], Times[ 1605, Power[ \[CapitalDelta], 6]], Times[ -435, Power[ \[CapitalDelta], 7]], Times[ 196, Power[ \[CapitalDelta], 8]], Times[ -32, Power[ \[CapitalDelta], 9]]]], Times[ 2, Power[ \[Alpha], 5], \[CapitalDelta], Plus[ 1652, Times[ 2128, \[CapitalDelta]], Times[ -638, Power[ \[CapitalDelta], 2]], Times[ -1034, Power[ \[CapitalDelta], 3]], Times[ -3573, Power[ \[CapitalDelta], 4]], Times[ 8283, Power[ \[CapitalDelta], 5]], Times[ -6888, Power[ \[CapitalDelta], 6]], Times[ 3140, Power[ \[CapitalDelta], 7]], Times[ -745, Power[ \[CapitalDelta], 8]], Times[ 75, Power[ \[CapitalDelta], 9]]]], Times[ Power[ \[Alpha], 6], Plus[ 768, Times[ -80, \[CapitalDelta]], Times[ 432, Power[ \[CapitalDelta], 2]], Times[ -1174, Power[ \[CapitalDelta], 3]], Times[ -1458, Power[ \[CapitalDelta], 4]], Times[ 5063, Power[ \[CapitalDelta], 5]], Times[ -5081, Power[ \[CapitalDelta], 6]], Times[ 2696, Power[ \[CapitalDelta], 7]], Times[ -812, Power[ \[CapitalDelta], 8]], Times[ 135, Power[ \[CapitalDelta], 9]], Times[ -9, Power[ \[CapitalDelta], 10]]]], Times[ 4, Power[ \[Alpha], 3], Power[ \[CapitalDelta], 2], Plus[ -24, Times[ 1389, \[CapitalDelta]], Times[ 7930, Power[ \[CapitalDelta], 2]], Times[ 27, Power[ \[CapitalDelta], 3]], Times[ -265, Power[ \[CapitalDelta], 4]], Times[ -1332, Power[ \[CapitalDelta], 5]], Times[ 2126, Power[ \[CapitalDelta], 6]], Times[ -1073, Power[ \[CapitalDelta], 7]], Times[ 419, Power[ \[CapitalDelta], 8]], Times[ -83, Power[ \[CapitalDelta], 9]], Times[ 6, Power[ \[CapitalDelta], 10]]]], Times[ Power[ \[Alpha], 4], \[CapitalDelta], Plus[ -24, Times[ 5840, \[CapitalDelta]], Times[ 18704, Power[ \[CapitalDelta], 2]], Times[ -4604, Power[ \[CapitalDelta], 3]], Times[ -1073, Power[ \[CapitalDelta], 4]], Times[ -10213, Power[ \[CapitalDelta], 5]], Times[ 18386, Power[ \[CapitalDelta], 6]], Times[ -12366, Power[ \[CapitalDelta], 7]], Times[ 5027, Power[ \[CapitalDelta], 8]], Times[ -1041, Power[ \[CapitalDelta], 9]], Times[ 84, Power[ \[CapitalDelta], 10]]]]]]}, 2, 5, 1]]], "Output"] }, Open ]], Cell[TextData[{ "It is seen that bn approaches the infinite limit in even/odd powers of \ (1/T), starting at ", Cell[BoxData[ \(TraditionalForm\`T\^\(-n\)\)]], " . This means that all distortions of the Gaussian die away at large \ times, only a modulation \"amplitude\" of 1 remains. Furthermore, the \ variance of Gaussian is determined by the behaviour of a2, and " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[ Limit[a2, T \[Rule] \[Infinity]], {T > 0, \[Alpha] > 1, \[CapitalDelta] > \(-1\), \[CapitalDelta] < 1}]\)], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell["\<\ This means that the Gaussian also loses the initial kinetic \ compression/expansion and returns to the dispersion it would have had in the \ absence of a velocity step. The time scale over which this happens is to be \ discussed further below.\ \>", "Text"], Cell[TextData[{ "It appears that the coefficients in the expansions are not correct. Assume \ that for a given step size \[CapitalDelta] , a T can be chosen so that all \ 4th order terms can be neglected compared to 2nd order terms. Then at such a \ time the conservation requirement ", Cell[BoxData[ \(TraditionalForm\`\[Sum]b\_\(2 n\) = 1\)]], " reduces to b0+b2 = 1, and that cannot be satisfied because the 2nd order \ terms in b0 and b2 above are not equal. In that respect the approximations \ made are not adequate; it seems possible that the choice of Vg that was made \ to guarantee conservation at infinite times may not have been the value \ needed to give conservation at intermediate times, or that a more complex \ functional form for g is needed at a step, or one of the other assumptions \ like a constant R and quadratic expansion of the exponent breaks down. The \ deviations from exact conservation is anyway only in the 2nd order in 1/T and \ study of the dependence on step parameter below may also shed further light \ on whether this is important." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Comment added later", "Subsubtitle"], Cell["\<\ This argument about normalisation is not correct, because it relies on using \ the Un expansion over the entire spatial axis when the normalisation integral \ is evaluated at a fixed time. Actually, there is a lower cutoff -Z2(T) - in \ effect the position of the step - which only reduces to -\[Infinity] when T\ \[Rule]\[Infinity]. So the odd n terms do not cancel, and the even ones do \ not give a bn sum to 1. Disregard remarks above and elsewhere about \ discrepancies between the limits of the various orders. \ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ ------------------------------------------------------------------------------\ --\ \>", "Subsubtitle"], Cell["\<\ The expressions above are rather unwieldy. That can be improved by making use \ of the result below that the bn are very insensitive toi the value of \ \[Alpha], and that practical values of \[Alpha] seem to be >> 1. So we might \ just as well take the limit \[Alpha] \[Rule] \[Infinity] before looking at \ the infinite time limit:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(Simplify[ Limit[b0[T], \[Alpha] \[Rule] \[Infinity]], {T > 1, \[CapitalDelta] > \(-1\), \[CapitalDelta] < 1}]\), "\[IndentingNewLine]", \(Simplify[ Series[%, {T, \[Infinity], 4}], {T > 0, \[CapitalDelta] > \(-1\), \[CapitalDelta] < 1}]\)}], "Input"], Cell[BoxData[ \(\(-\(\(T\^2\ \((\(-1\) + \[CapitalDelta])\)\)\/\@\(\(\((T\^2\ \((\(-1\) \ + \[CapitalDelta])\)\^2 + 4\ \[CapitalDelta])\)\ \((T\^4 + 4\ \[CapitalDelta] \ + T\^2\ \((\(-1\) - 4\ \[CapitalDelta] + \[CapitalDelta]\^2)\))\)\)\/\(\(-1\) \ + T\^2\)\)\)\)\)], "Output"], Cell[BoxData[ InterpretationBox[ RowBox[{ "1", "-", \(\(\((\((\(-3\) + \[CapitalDelta])\)\^2\ \[CapitalDelta]\^2)\ \)\ \((1\/T)\)\^2\)\/\(2\ \((\(-1\) + \[CapitalDelta])\)\^2\)\), "+", InterpretationBox[\(O[1\/T]\^3\), SeriesData[ T, DirectedInfinity[ 1], {}, 0, 3, 1]]}], SeriesData[ T, DirectedInfinity[ 1], {1, 0, Times[ Rational[ -1, 2], Power[ Plus[ -3, \[CapitalDelta]], 2], Power[ Plus[ -1, \[CapitalDelta]], -2], Power[ \[CapitalDelta], 2]]}, 0, 3, 1]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(Simplify[ Limit[b1[T], \[Alpha] \[Rule] \[Infinity]], {T > 1, \[CapitalDelta] > \(-1\), \[CapitalDelta] < 1}]\), "\[IndentingNewLine]", \(Simplify[ Series[%, {T, \[Infinity], 4}], {T > 0, \[CapitalDelta] > \(-1\), \[CapitalDelta] < 1}]\)}], "Input"], Cell[BoxData[ \(\(-\(\((T\^2\ \[Beta]\ \((\(-1\) + \[CapitalDelta])\)\^2\ \ \[CapitalDelta]\ \((\(-8\)\ \((\(-3\) - 2\ \[CapitalDelta] + \[CapitalDelta]\^2)\) - 2\ T\^2\ \((17 + 5\ \[CapitalDelta] - 7\ \[CapitalDelta]\^2 + \[CapitalDelta]\^3)\) + T\^4\ \((10 - 5\ \[CapitalDelta] - 4\ \[CapitalDelta]\^2 + 3\ \[CapitalDelta]\^3)\))\))\)/\((2\ \@\(2\ \[Pi]\)\ \ \((1 + \[CapitalDelta])\)\ \((T\^2\ \((\(-1\) + \[CapitalDelta])\)\^2 + 4\ \ \[CapitalDelta])\)\^\(3/2\)\ \((T\^4 + 4\ \[CapitalDelta] + T\^2\ \((\(-1\) - 4\ \[CapitalDelta] + \ \[CapitalDelta]\^2)\))\))\)\)\)\)], "Output"], Cell[BoxData[ InterpretationBox[ RowBox[{\(\(\[Beta]\ \[CapitalDelta]\ \((10 - 5\ \[CapitalDelta] - 4\ \[CapitalDelta]\^2 + 3\ \[CapitalDelta]\^3)\)\)\/\(2\ \@\(2\ \[Pi]\)\ \((\(-1\) \ + \[CapitalDelta]\^2)\)\ T\)\), "-", \(\((\((\[Beta]\ \[CapitalDelta]\ \((24 - 13\ \[CapitalDelta] + 64\ \[CapitalDelta]\^2 - 79\ \[CapitalDelta]\^3 + 2\ \[CapitalDelta]\^4 + 45\ \[CapitalDelta]\^5 - 22\ \[CapitalDelta]\^6 + 3\ \[CapitalDelta]\^7)\))\)\ \((1\/T)\)\^3)\)/\((2\ \((\ \@\(2\ \[Pi]\)\ \((\(-1\) + \[CapitalDelta])\)\^3\ \((1 + \[CapitalDelta])\))\ \))\)\), "+", InterpretationBox[\(O[1\/T]\^5\), SeriesData[ T, DirectedInfinity[ 1], {}, 1, 5, 1]]}], SeriesData[ T, DirectedInfinity[ 1], { Times[ Rational[ 1, 2], Power[ Times[ 2, Pi], Rational[ -1, 2]], \[Beta], \[CapitalDelta], Power[ Plus[ -1, Power[ \[CapitalDelta], 2]], -1], Plus[ 10, Times[ -5, \[CapitalDelta]], Times[ -4, Power[ \[CapitalDelta], 2]], Times[ 3, Power[ \[CapitalDelta], 3]]]], 0, Times[ Rational[ -1, 2], Power[ Times[ 2, Pi], Rational[ -1, 2]], \[Beta], Power[ Plus[ -1, \[CapitalDelta]], -3], \[CapitalDelta], Power[ Plus[ 1, \[CapitalDelta]], -1], Plus[ 24, Times[ -13, \[CapitalDelta]], Times[ 64, Power[ \[CapitalDelta], 2]], Times[ -79, Power[ \[CapitalDelta], 3]], Times[ 2, Power[ \[CapitalDelta], 4]], Times[ 45, Power[ \[CapitalDelta], 5]], Times[ -22, Power[ \[CapitalDelta], 6]], Times[ 3, Power[ \[CapitalDelta], 7]]]]}, 1, 5, 1]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(Simplify[ Limit[b2[T], \[Alpha] \[Rule] \[Infinity]], {T > 1, \[CapitalDelta] > \(-1\), \[CapitalDelta] < 1}]\), "\[IndentingNewLine]", \(Simplify[ Series[%, {T, \[Infinity], 4}], {T > 0, \[CapitalDelta] > \(-1\), \[CapitalDelta] < 1}]\)}], "Input"], Cell[BoxData[ \(\((T\^2\ \[Beta]\^2\ \((\(-1\) + \[CapitalDelta])\)\^3\ \[CapitalDelta]\ \ \@\(\(T\^4 + 4\ \[CapitalDelta] + T\^2\ \((\(-1\) - 4\ \[CapitalDelta] + \ \[CapitalDelta]\^2)\)\)\/\(\(-1\) + T\^2\)\)\ \((\(-64\)\ \[CapitalDelta]\ \ \((3 - \[CapitalDelta] - 2\ \[CapitalDelta]\^2 + 2\ \[CapitalDelta]\^3)\) - 32\ T\^2\ \((\(-12\) - 25\ \[CapitalDelta] + 17\ \[CapitalDelta]\^2 + \[CapitalDelta]\^3 - 7\ \[CapitalDelta]\^4 + 2\ \[CapitalDelta]\^5)\) - 8\ T\^4\ \((109 + 106\ \[CapitalDelta] - 149\ \[CapitalDelta]\^2 + 58\ \[CapitalDelta]\^3 + 9\ \[CapitalDelta]\^4 - 10\ \[CapitalDelta]\^5 + \[CapitalDelta]\^6)\) + 4\ T\^6\ \((148 + 39\ \[CapitalDelta] - 206\ \[CapitalDelta]\^2 + 128\ \[CapitalDelta]\^3 - 28\ \[CapitalDelta]\^4 + 5\ \[CapitalDelta]\^5 + 2\ \[CapitalDelta]\^6)\) + T\^8\ \((\(-104\) + 78\ \[CapitalDelta] + 94\ \[CapitalDelta]\^2 - 159\ \[CapitalDelta]\^3 + 91\ \[CapitalDelta]\^4 - 27\ \[CapitalDelta]\^5 + 3\ \[CapitalDelta]\^6)\))\))\)/\((32\ \((1 + \ \[CapitalDelta])\)\ \((T\^2\ \((\(-1\) + \[CapitalDelta])\)\^2 + 4\ \ \[CapitalDelta])\)\^\(5/2\)\ \((T\^4 + 4\ \[CapitalDelta] + T\^2\ \((\(-1\) - \ 4\ \[CapitalDelta] + \[CapitalDelta]\^2)\))\)\^2)\)\)], "Output"], Cell[BoxData[ InterpretationBox[ RowBox[{\(-\(\((\((\[Beta]\^2\ \[CapitalDelta]\ \((\(-104\) + 78\ \[CapitalDelta] + 94\ \[CapitalDelta]\^2 - 159\ \[CapitalDelta]\^3 + 91\ \[CapitalDelta]\^4 - 27\ \[CapitalDelta]\^5 + 3\ \[CapitalDelta]\^6)\))\)\ \((1\/T)\)\^2)\)/\((32\ \ \((\((\(-1\) + \[CapitalDelta])\)\^2\ \((1 + \[CapitalDelta])\))\))\)\)\), "+", \(\((\[Beta]\^2\ \[CapitalDelta]\ \((\(-768\) + 80\ \[CapitalDelta] - 432\ \[CapitalDelta]\^2 + 1174\ \[CapitalDelta]\^3 + 1458\ \[CapitalDelta]\^4 - 5063\ \[CapitalDelta]\^5 + 5081\ \[CapitalDelta]\^6 - 2696\ \[CapitalDelta]\^7 + 812\ \[CapitalDelta]\^8 - 135\ \[CapitalDelta]\^9 + 9\ \[CapitalDelta]\^10)\)\ \((1\/T)\)\^4)\)/\((64\ \ \((\(-1\) + \[CapitalDelta])\)\^4\ \((1 + \[CapitalDelta])\))\)\), "+", InterpretationBox[\(O[1\/T]\^5\), SeriesData[ T, DirectedInfinity[ 1], {}, 2, 5, 1]]}], SeriesData[ T, DirectedInfinity[ 1], { Times[ Rational[ -1, 32], Power[ \[Beta], 2], Power[ Plus[ -1, \[CapitalDelta]], -2], \[CapitalDelta], Power[ Plus[ 1, \[CapitalDelta]], -1], Plus[ -104, Times[ 78, \[CapitalDelta]], Times[ 94, Power[ \[CapitalDelta], 2]], Times[ -159, Power[ \[CapitalDelta], 3]], Times[ 91, Power[ \[CapitalDelta], 4]], Times[ -27, Power[ \[CapitalDelta], 5]], Times[ 3, Power[ \[CapitalDelta], 6]]]], 0, Times[ Rational[ 1, 64], Power[ \[Beta], 2], Power[ Plus[ -1, \[CapitalDelta]], -4], \[CapitalDelta], Power[ Plus[ 1, \[CapitalDelta]], -1], Plus[ -768, Times[ 80, \[CapitalDelta]], Times[ -432, Power[ \[CapitalDelta], 2]], Times[ 1174, Power[ \[CapitalDelta], 3]], Times[ 1458, Power[ \[CapitalDelta], 4]], Times[ -5063, Power[ \[CapitalDelta], 5]], Times[ 5081, Power[ \[CapitalDelta], 6]], Times[ -2696, Power[ \[CapitalDelta], 7]], Times[ 812, Power[ \[CapitalDelta], 8]], Times[ -135, Power[ \[CapitalDelta], 9]], Times[ 9, Power[ \[CapitalDelta], 10]]]]}, 2, 5, 1]]], "Output"] }, Open ]], Cell["\<\ One important result that can be read off from these equations is that both \ b1 and b2 are proportional to \[CapitalDelta], and so change sign when \ \[CapitalDelta] changes sign, as is also apparent in the plots below.\ \>", "Text"], Cell[CellGroupData[{ Cell["The deterministic limit", "Subsection"], Cell["\<\ Note that as \[Alpha], \[Beta] and T all depend on t0, these have to be \ substituted before taking the limit t0 \[Rule] -\[Infinity] :\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[ Limit[b0[T] /. TGrep, t0 \[Rule] \(-\[Infinity]\)], {V1 > 0, V2 > 0}]\)], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[ Limit[b1[T] /. TGrep, t0 \[Rule] \(-\[Infinity]\)], {V1 > 0, V2 > 0}]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[ Limit[b2[T] /. TGrep, t0 \[Rule] \(-\[Infinity]\)], {V1 > 0, V2 > 0}]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[TextData[{ "This behaviour is easy to understand, because in the deterministic limit \ both T and \[Alpha] reduce to 1, while \[Beta] becomes 0, and it is seen that \ bn is proportional to ", Cell[BoxData[ \(TraditionalForm\`\[Beta]\^n\)]], "." }], "Text"], Cell["\<\ So in the deterministic limit, the modulation factor reduces to 1 at all \ times and all positions - i.e., there is no modulation, which is the correct \ result obtained also by directly calculating the integral in the \ deterministic case.\ \>", "Text"], Cell["\<\ Notice the instability of the limits at T = 1, depending on the order in \ which the two limits \[Gamma] \[Rule] 0 and T \[Rule] 1 are taken. If \ \[Gamma] \[Rule] 0 is taken first, as here, b0 is 1 and all other b's 0; but \ when T \[Rule] 1 is taken for \[Gamma] > 0, b0 is 1 and the others diverge. \ This is another manifestation (and partly due to) the instability in a2 \ because of the discontinuity at the step, as discussed in a previous \ notebook.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Behaviour as function of step size and stochasticity", "Subsection"], Cell[TextData[{ "The step size parameter \[CapitalDelta]can in pronciple vary between -1 \ and 1. However, from its definition, ", Cell[BoxData[ \(TraditionalForm\`\[CapitalDelta] = \(V2 - V1\)\/\(V1 + V2\)\)]], " it is clear that these limits are only reached if one (but not both) of \ the velocities become either 0 or \[Infinity], which are not physically \ meaningful (e.g., if either velocity is 0, there cannot in fact be any step \ penetration at all. It therefore seems plausible to restrict the range of \ \[CapitalDelta] to say, (-0.8,0.8) which represents a 9-fold increases or \ reduction of the velocity at the step." }], "Text"], Cell["\<\ First choose a fixed time T = 2 (because in both examples above that would \ avoid the region where the behaviour was dominated by the divergences at T = \ 1). The restriction on \[Alpha] is \[Alpha] > 1. \[Beta] only occurrs as an \ overall coefficient in b1 and b2 so we can just take it as 1. See how each \ coefficient behaves as function \[Alpha] and \[CapitalDelta] :\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Plot3D[ b0[2], {\[CapitalDelta], \(-0.8\), 0.8}, {\[Alpha], 1.1, 5}\ ]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .81114 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% SurfaceGraphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0 1.04977 -0.0679587 1.04977 [ [.13842 .20256 -23.9105 -9 ] [.13842 .20256 .0895 0 ] [.3345 .12542 -5.5867 -9 ] [.3345 .12542 .4133 0 ] [.54622 .04218 -15.5873 -9 ] [.54622 .04218 2.41266 0 ] [.77714 .11187 0 -6.14746 ] [.77714 .11187 6 2.85254 ] [.85293 .22792 0 -6.03643 ] [.85293 .22792 6 2.96357 ] [.92079 .33185 0 -5.93942 ] [.92079 .33185 6 3.06058 ] [.98191 .42546 0 -5.85393 ] [.98191 .42546 6 3.14607 ] [.01757 .3061 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.60396 .26669 .57813 Metetra .425 .649 .952 r .26669 .57813 .29318 .60396 .33329 .60693 .30698 .5815 Metetra .432 .658 .955 r .30698 .5815 .33329 .60693 .37443 .6094 .34835 .58431 Metetra .452 .673 .958 r .34835 .58431 .37443 .6094 .41664 .61065 .39083 .58579 Metetra .489 .695 .96 r .39083 .58579 .41664 .61065 .45993 .60964 .43446 .58488 Metetra .544 .722 .956 r .43446 .58488 .45993 .60964 .50425 .60498 .47918 .58015 Metetra .611 .75 .944 r .47918 .58015 .50425 .60498 .5495 .59493 .52489 .5698 Metetra .682 .774 .92 r .52489 .5698 .5495 .59493 .59543 .5775 .57136 .55183 Metetra .748 .79 .887 r .57136 .55183 .59543 .5775 .64172 .55082 .61825 .52439 Metetra .801 .798 .849 r .61825 .52439 .64172 .55082 .68793 .51369 .66514 .48634 Metetra .839 .8 .814 r .66514 .48634 .68793 .51369 .73363 .46619 .71157 .43787 Metetra .864 .8 .786 r .71157 .43787 .73363 .46619 .77849 .40999 .7572 .38075 Metetra .878 .799 .768 r .7572 .38075 .77849 .40999 .82238 .34805 .80189 .31805 Metetra .885 .799 .759 r .80189 .31805 .82238 .34805 .86538 .2838 .84571 .25325 Metetra .417 .631 .943 r .20014 .54692 .22741 .57471 .26669 .57813 .23952 .55093 Metetra .414 .638 .948 r .23952 .55093 .26669 .57813 .30698 .5815 .27996 .55486 Metetra .421 .649 .953 r .27996 .55486 .30698 .5815 .34835 .58431 .32152 .55816 Metetra .443 .666 .957 r .32152 .55816 .34835 .58431 .39083 .58579 .36426 .56001 Metetra .482 .69 .959 r .36426 .56001 .39083 .58579 .43446 .58488 .40821 .55929 Metetra .54 .719 .956 r .40821 .55929 .43446 .58488 .47918 .58015 .45332 .55452 Metetra .61 .749 .944 r .45332 .55452 .47918 .58015 .52489 .5698 .4995 .54387 Metetra .683 .774 .92 r .4995 .54387 .52489 .5698 .57136 .55183 .54652 .52533 Metetra .749 .79 .886 r .54652 .52533 .57136 .55183 .61825 .52439 .59404 .49706 Metetra .802 .798 .848 r .59404 .49706 .61825 .52439 .66514 .48634 .64162 .45802 Metetra .839 .799 .813 r .64162 .45802 .66514 .48634 .71157 .43787 .6888 .40851 Metetra .864 .799 .785 r .6888 .40851 .71157 .43787 .7572 .38075 .73523 .35044 Metetra .878 .798 .767 r .73523 .35044 .7572 .38075 .80189 .31805 .78075 .28698 Metetra .884 .798 .76 r .78075 .28698 .80189 .31805 .84571 .25325 .82542 .22165 Metetra .403 .614 .936 r .17226 .51719 .20014 .54692 .23952 .55093 .21169 .52205 Metetra .399 .622 .943 r .21169 .52205 .23952 .55093 .27996 .55486 .25222 .52678 Metetra .407 .636 .949 r .25222 .52678 .27996 .55486 .32152 .55816 .29394 .53078 Metetra .43 .656 .955 r .29394 .53078 .32152 .55816 .36426 .56001 .3369 .53318 Metetra .473 .684 .958 r .3369 .53318 .36426 .56001 .40821 .55929 .38115 .53279 Metetra .534 .716 .956 r .38115 .53279 .40821 .55929 .45332 .55452 .42665 .52806 Metetra .608 .748 .944 r .42665 .52806 .45332 .55452 .4995 .54387 .47331 .51712 Metetra .684 .774 .919 r .47331 .51712 .4995 .54387 .54652 .52533 .52089 .49795 Metetra .751 .79 .884 r .52089 .49795 .54652 .52533 .59404 .49706 .56905 .46877 Metetra .804 .797 .846 r .56905 .46877 .59404 .49706 .64162 .45802 .61735 .42865 Metetra .84 .799 .811 r .61735 .42865 .64162 .45802 .6888 .40851 .66531 .37802 Metetra .864 .798 .784 r .66531 .37802 .6888 .40851 .73523 .35044 .71257 .31897 Metetra .877 .797 .767 r .71257 .31897 .73523 .35044 .78075 .28698 .75894 .25475 Metetra .883 .797 .76 r .75894 .25475 .78075 .28698 .82542 .22165 .80448 .18895 Metetra .386 .591 .925 r .14385 .48487 .17226 .51719 .21169 .52205 .18323 .49097 Metetra .38 .601 .934 r .18323 .49097 .21169 .52205 .25222 .52678 .22377 .49688 Metetra .388 .618 .943 r .22377 .49688 .25222 .52678 .29394 .53078 .26558 .50192 Metetra .414 .643 .951 r .26558 .50192 .29394 .53078 .3369 .53318 .30873 .50515 Metetra .461 .675 .957 r .30873 .50515 .3369 .53318 .38115 .53279 .35325 .50529 Metetra .527 .712 .956 r .35325 .50529 .38115 .53279 .42665 .52806 .39913 .50071 Metetra .606 .747 .944 r .39913 .50071 .42665 .52806 .47331 .51712 .44626 .4895 Metetra .685 .775 .919 r .44626 .4895 .47331 .51712 .52089 .49795 .49442 .46965 Metetra .754 .79 .883 r .49442 .46965 .52089 .49795 .56905 .46877 .54325 .43946 Metetra .805 .797 .844 r .54325 .43946 .56905 .46877 .61735 .42865 .59229 .39813 Metetra .841 .797 .809 r .59229 .39813 .61735 .42865 .66531 .37802 .64106 .34631 Metetra .864 .796 .783 r .64106 .34631 .66531 .37802 .71257 .31897 .68916 .28623 Metetra .876 .795 .767 r .68916 .28623 .71257 .31897 .75894 .25475 .73642 .22128 Metetra .881 .796 .761 r .73642 .22128 .75894 .25475 .80448 .18895 .78288 .15505 Metetra .362 .556 .908 r .11507 .44885 .14385 .48487 .18323 .49097 .15425 .4568 Metetra .355 .57 .92 r .15425 .4568 .18323 .49097 .22377 .49688 .19468 .4645 Metetra .362 .591 .933 r .19468 .4645 .22377 .49688 .26558 .50192 .23648 .47114 Metetra .391 .622 .945 r .23648 .47114 .26558 .50192 .30873 .50515 .27973 .47566 Metetra .443 .662 .954 r .27973 .47566 .30873 .50515 .35325 .50529 .32447 .47665 Metetra .518 .706 .956 r .32447 .47665 .35325 .50529 .39913 .50071 .37071 .47241 Metetra .604 .746 .945 r .37071 .47241 .39913 .50071 .44626 .4895 .41832 .46097 Metetra .687 .775 .918 r .41832 .46097 .44626 .4895 .49442 .46965 .46707 .44036 Metetra .756 .791 .881 r .46707 .44036 .49442 .46965 .54325 .43946 .51659 .40902 Metetra .807 .796 .841 r .51659 .40902 .54325 .43946 .59229 .39813 .5664 .36635 Metetra .842 .796 .807 r .5664 .36635 .59229 .39813 .64106 .34631 .616 .31323 Metetra .863 .794 .781 r .616 .31323 .64106 .34631 .68916 .28623 .665 .2521 Metetra .875 .794 .766 r .665 .2521 .68916 .28623 .73642 .22128 .71317 .18644 Metetra .88 .794 .762 r .71317 .18644 .73642 .22128 .78288 .15505 .76056 .11987 Metetra .331 .504 .878 r .08626 .40702 .11507 .44885 .15425 .4568 .12499 .41789 Metetra .319 .522 .896 r .12499 .41789 .15425 .4568 .19468 .4645 .16509 .42843 Metetra .325 .55 .914 r .16509 .42843 .19468 .4645 .23648 .47114 .20669 .43764 Metetra .357 .59 .933 r .20669 .43764 .23648 .47114 .27973 .47566 .24991 .44424 Metetra .418 .641 .949 r .24991 .44424 .27973 .47566 .32447 .47665 .2948 .44666 Metetra .504 .696 .955 r .2948 .44666 .32447 .47665 .37071 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.5507 .46184 .52838 .43713 Metetra .732 .762 .877 r .52838 .43713 .5507 .46184 .59233 .43957 .57048 .41445 Metetra .706 .756 .891 r .57048 .41445 .59233 .43957 .63448 .4208 .6131 .39523 Metetra .679 .747 .902 r .6131 .39523 .63448 .4208 .67731 .40533 .65638 .3793 Metetra .655 .738 .911 r .65638 .3793 .67731 .40533 .72091 .39234 .70044 .36583 Metetra .643 .733 .914 r .70044 .36583 .72091 .39234 .76532 .38045 .74531 .35347 Metetra .644 .732 .913 r .74531 .35347 .76532 .38045 .81047 .36825 .79095 .34085 Metetra .651 .735 .911 r .79095 .34085 .81047 .36825 .85631 .35492 .83731 .32715 Metetra .659 .739 .909 r .83731 .32715 .85631 .35492 .90281 .34047 .88435 .31235 Metetra .866 .74 .722 r .24902 .66441 .2754 .69096 .32001 .61149 .29436 .58646 Metetra .828 .757 .785 r .29436 .58646 .32001 .61149 .36218 .56273 .33713 .53813 Metetra .802 .764 .818 r .33713 .53813 .36218 .56273 .40372 .52479 .37921 .50021 Metetra .784 .767 .837 r .37921 .50021 .40372 .52479 .44513 .49216 .42115 .4674 Metetra 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.46423 m .00774 .46292 L s .03309 .2808 m .03677 .27938 L s .03492 .2693 m .03859 .26787 L s .03673 .25788 m .04039 .25644 L s .25 Mabswid .03716 .25514 m 0 .48963 L s 0 .48963 m .39787 .81114 L s .39787 .81114 m .40529 .59895 L s .40529 .59895 m .03716 .25514 L s .68874 0 m .96935 .42924 L s .96935 .42924 m 1 .6535 L s 1 .6535 m .70298 .24544 L s .70298 .24544 m .68874 0 L s .03716 .25514 m 0 .48963 L s 0 .48963 m .70298 .24544 L s .70298 .24544 m .68874 0 L s .68874 0 m .03716 .25514 L s .40529 .59895 m .96935 .42924 L s .96935 .42924 m 1 .6535 L s 1 .6535 m .39787 .81114 L s .39787 .81114 m .40529 .59895 L s 0 0 m 1 0 L 1 .81114 L 0 .81114 L closepath clip newpath .5 Mabswid .405 .659 .962 r .38308 .58438 .40512 .60371 .44151 .60826 .41969 .5888 Metetra 0 .318 .78 r .41969 .5888 .44151 .60826 .4785 .63077 .4568 .61113 Metetra 0 .189 .67 r .4568 .61113 .4785 .63077 .51674 .65935 .49516 .63949 Metetra 0 .217 .697 r .49516 .63949 .51674 .65935 .55643 .68805 .53498 .66792 Metetra 0 .36 .815 r .53498 .66792 .55643 .68805 .5975 .71161 .57625 .69116 Metetra .221 .56 .936 r .57625 .69116 .5975 .71161 .63966 .72506 .61869 .70427 Metetra .502 .706 .962 r .61869 .70427 .63966 .72506 .68245 .72508 .66185 .70396 Metetra .662 .767 .928 r .66185 .70396 .68245 .72508 .72543 .71207 .7053 .69071 Metetra .724 .785 .901 r .7053 .69071 .72543 .71207 .76851 .69138 .74891 .66984 Metetra .718 .786 .905 r .74891 .66984 .76851 .69138 .81209 .67126 .79303 .64951 Metetra .661 .769 .93 r .79303 .64951 .81209 .67126 .85679 .65786 .83825 .63579 Metetra .59 .741 .948 r .83825 .63579 .85679 .65786 .90292 .65127 .88487 .62873 Metetra .566 .729 .951 r .88487 .62873 .90292 .65127 .9502 .64672 .93264 .62367 Metetra .594 .74 .946 r .93264 .62367 .9502 .64672 .9981 .63962 .98107 .6161 Metetra .402 .66 .964 r .36044 .56472 .38308 .58438 .41969 .5888 .39728 .56901 Metetra 0 .319 .78 r .39728 .56901 .41969 .5888 .4568 .61113 .43452 .59112 Metetra 0 .193 .672 r .43452 .59112 .4568 .61113 .49516 .63949 .47298 .61922 Metetra 0 .223 .701 r .47298 .61922 .49516 .63949 .53498 .66792 .51294 .64732 Metetra 0 .366 .82 r .51294 .64732 .53498 .66792 .57625 .69116 .5544 .67018 Metetra .226 .564 .937 r .5544 .67018 .57625 .69116 .61869 .70427 .59712 .68289 Metetra .503 .707 .962 r .59712 .68289 .61869 .70427 .66185 .70396 .64066 .68224 Metetra .661 .766 .928 r .64066 .68224 .66185 .70396 .7053 .69071 .68459 .66874 Metetra .722 .785 .902 r .68459 .66874 .7053 .69071 .74891 .66984 .72874 .64771 Metetra .717 .786 .906 r .72874 .64771 .74891 .66984 .79303 .64951 .77341 .62719 Metetra .661 .77 .931 r .77341 .62719 .79303 .64951 .83825 .63579 .81916 .61312 Metetra .591 .743 .948 r .81916 .61312 .83825 .63579 .88487 .62873 .86628 .60557 Metetra .569 .731 .951 r .86628 .60557 .88487 .62873 .93264 .62367 .91455 .59995 Metetra .595 .741 .945 r .91455 .59995 .93264 .62367 .98107 .6161 .96353 .59187 Metetra .398 .66 .965 r .33719 .54478 .36044 .56472 .39728 .56901 .37425 .54892 Metetra 0 .321 .78 r .37425 .54892 .39728 .56901 .43452 .59112 .41161 .57077 Metetra 0 .198 .675 r .41161 .57077 .43452 .59112 .47298 .61922 .45018 .59854 Metetra 0 .23 .706 r .45018 .59854 .47298 .61922 .51294 .64732 .49028 .62624 Metetra 0 .374 .824 r .49028 .62624 .51294 .64732 .5544 .67018 .53193 .64865 Metetra .233 .569 .939 r .53193 .64865 .5544 .67018 .59712 .68289 .57493 .66091 Metetra .505 .707 .962 r .57493 .66091 .59712 .68289 .64066 .68224 .61886 .65988 Metetra .66 .766 .928 r .61886 .65988 .64066 .68224 .68459 .66874 .66327 .64614 Metetra .72 .785 .903 r .66327 .64614 .68459 .66874 .72874 .64771 .70797 .62496 Metetra .715 .786 .908 r .70797 .62496 .72874 .64771 .77341 .62719 .75321 .60426 Metetra .66 .77 .931 r .75321 .60426 .77341 .62719 .81916 .61312 .7995 .58985 Metetra .593 .744 .949 r .7995 .58985 .81916 .61312 .86628 .60557 .84713 .58176 Metetra .571 .733 .951 r .84713 .58176 .86628 .60557 .91455 .59995 .89591 .57554 Metetra .597 .742 .945 r .89591 .57554 .91455 .59995 .96353 .59187 .94544 .5669 Metetra .393 .662 .966 r .31327 .52461 .33719 .54478 .37425 .54892 .35056 .52858 Metetra 0 .324 .78 r .35056 .52858 .37425 .54892 .41161 .57077 .38805 .55011 Metetra 0 .203 .678 r .38805 .55011 .41161 .57077 .45018 .59854 .42673 .57747 Metetra 0 .238 .712 r .42673 .57747 .45018 .59854 .49028 .62624 .46696 .60468 Metetra 0 .382 .83 r .46696 .60468 .49028 .62624 .53193 .64865 .50881 .62656 Metetra .241 .574 .941 r .50881 .62656 .53193 .64865 .57493 .66091 .55209 .63831 Metetra .507 .709 .962 r .55209 .63831 .57493 .66091 .61886 .65988 .59641 .63687 Metetra .658 .766 .929 r .59641 .63687 .61886 .65988 .66327 .64614 .64131 .62288 Metetra .717 .784 .905 r .64131 .62288 .66327 .64614 .70797 .62496 .68657 .60158 Metetra .713 .786 .909 r .68657 .60158 .70797 .62496 .75321 .60426 .7324 .58073 Metetra .66 .771 .932 r .7324 .58073 .75321 .60426 .7995 .58985 .77924 .56595 Metetra .596 .746 .949 r .77924 .56595 .7995 .58985 .84713 .58176 .82739 .55728 Metetra .575 .735 .951 r .82739 .55728 .84713 .58176 .89591 .57554 .8767 .55039 Metetra .599 .743 .945 r .8767 .55039 .89591 .57554 .94544 .5669 .92679 .54116 Metetra .387 .663 .968 r .28866 .5043 .31327 .52461 .35056 .52858 .32618 .50808 Metetra 0 .328 .78 r .32618 .50808 .35056 .52858 .38805 .55011 .3638 .52921 Metetra 0 .21 .682 r .3638 .52921 .38805 .55011 .42673 .57747 .40259 .55605 Metetra 0 .247 .718 r .40259 .55605 .42673 .57747 .46696 .60468 .44295 .58267 Metetra 0 .392 .836 r .44295 .58267 .46696 .60468 .50881 .62656 .485 .60391 Metetra .249 .58 .944 r .485 .60391 .50881 .62656 .55209 .63831 .52857 .61505 Metetra .509 .71 .962 r .52857 .61505 .55209 .63831 .59641 .63687 .57328 .61316 Metetra .657 .765 .929 r .57328 .61316 .59641 .63687 .64131 .62288 .61869 .59894 Metetra .715 .784 .906 r .61869 .59894 .64131 .62288 .68657 .60158 .66453 .57756 Metetra .71 .785 .911 r .66453 .57756 .68657 .60158 .7324 .58073 .71094 .55657 Metetra .659 .772 .933 r .71094 .55657 .7324 .58073 .77924 .56595 .75835 .54142 Metetra .598 .748 .949 r .75835 .54142 .77924 .56595 .82739 .55728 .80704 .53211 Metetra .578 .737 .951 r .80704 .53211 .82739 .55728 .8767 .55039 .85688 .52449 Metetra .602 .744 .945 r .85688 .52449 .8767 .55039 .92679 .54116 .90754 .51461 Metetra .38 .666 .971 r .2633 .48397 .28866 .5043 .32618 .50808 .30105 .48754 Metetra 0 .332 .78 r .30105 .48754 .32618 .50808 .3638 .52921 .33881 .50816 Metetra 0 .219 .686 r .33881 .50816 .3638 .52921 .40259 .55605 .37772 .53435 Metetra 0 .259 .726 r .37772 .53435 .40259 .55605 .44295 .58267 .41823 .56022 Metetra 0 .404 .844 r .41823 .56022 .44295 .58267 .485 .60391 .46047 .58068 Metetra .26 .587 .946 r .46047 .58068 .485 .60391 .52857 .61505 .50434 .59113 Metetra .512 .711 .961 r .50434 .59113 .52857 .61505 .57328 .61316 .54945 .58874 Metetra .655 .765 .93 r .54945 .58874 .57328 .61316 .61869 .59894 .59537 .57429 Metetra .711 .783 .908 r .59537 .57429 .61869 .59894 .66453 .57756 .64179 .55287 Metetra .707 .785 .912 r .64179 .55287 .66453 .57756 .71094 .55657 .68882 .53178 Metetra .659 .773 .934 r .68882 .53178 .71094 .55657 .75835 .54142 .73681 .51624 Metetra .601 .75 .949 r .73681 .51624 .75835 .54142 .80704 .53211 .78604 .50623 Metetra .582 .74 .951 r .78604 .50623 .80704 .53211 .85688 .52449 .83643 .49781 Metetra .604 .746 .945 r .83643 .49781 .85688 .52449 .90754 .51461 .88768 .48722 Metetra .371 .67 .974 r .23712 .46384 .2633 .48397 .30105 .48754 .27512 .46714 Metetra 0 .339 .78 r .27512 .46714 .30105 .48754 .33881 .50816 .31303 .48709 Metetra 0 .231 .691 r .31303 .48709 .33881 .50816 .37772 .53435 .35207 .51245 Metetra 0 .274 .736 r .35207 .51245 .37772 .53435 .41823 .56022 .39274 .53739 Metetra 0 .419 .853 r .39274 .53739 .41823 .56022 .46047 .58068 .4352 .5569 Metetra .272 .595 .95 r .4352 .5569 .46047 .58068 .50434 .59113 .47936 .56651 Metetra .515 .713 .961 r .47936 .56651 .50434 .59113 .54945 .58874 .52489 .56355 Metetra .653 .764 .931 r .52489 .56355 .54945 .58874 .59537 .57429 .57132 .5489 Metetra .708 .782 .91 r .57132 .5489 .59537 .57429 .64179 .55287 .61834 .5275 Metetra .704 .785 .915 r .61834 .5275 .64179 .55287 .68882 .53178 .66599 .50636 Metetra .658 .774 .935 r .66599 .50636 .68882 .53178 .73681 .51624 .71458 .4904 Metetra .604 .753 .949 r .71458 .4904 .73681 .51624 .78604 .50623 .76437 .47963 Metetra .587 .743 .951 r .76437 .47963 .78604 .50623 .83643 .49781 .81531 .47031 Metetra .607 .748 .944 r .81531 .47031 .83643 .49781 .88768 .48722 .86716 .45894 Metetra .359 .676 .977 r .21003 .44424 .23712 .46384 .27512 .46714 .2483 .44719 Metetra 0 .348 .779 r .2483 .44719 .27512 .46714 .31303 .48709 .28639 .46626 Metetra 0 .245 .697 r .28639 .46626 .31303 .48709 .35207 .51245 .32559 .49052 Metetra 0 .293 .747 r .32559 .49052 .35207 .51245 .39274 .53739 .36644 .51426 Metetra .006 .438 .864 r .36644 .51426 .39274 .53739 .4352 .5569 .40913 .53257 Metetra .287 .606 .953 r .40913 .53257 .4352 .5569 .47936 .56651 .45361 .54117 Metetra .519 .715 .961 r .45361 .54117 .47936 .56651 .52489 .56355 .49955 .53758 Metetra .65 .763 .932 r .49955 .53758 .52489 .56355 .57132 .5489 .54651 .52275 Metetra .703 .781 .912 r .54651 .52275 .57132 .5489 .61834 .5275 .59414 .50146 Metetra .7 .785 .917 r .59414 .50146 .61834 .5275 .66599 .50636 .64242 .48032 Metetra .657 .775 .936 r .64242 .48032 .66599 .50636 .71458 .4904 .69162 .46393 Metetra .608 .757 .95 r .69162 .46393 .71458 .4904 .76437 .47963 .74198 .45229 Metetra .593 .747 .95 r .74198 .45229 .76437 .47963 .81531 .47031 .7935 .44197 Metetra .611 .75 .944 r .7935 .44197 .81531 .47031 .86716 .45894 .84596 .42974 Metetra .342 .685 .98 r .18192 .42575 .21003 .44424 .2483 .44719 .22047 .4282 Metetra 0 .36 .776 r .22047 .4282 .2483 .44719 .28639 .46626 .25879 .44606 Metetra 0 .266 .705 r .25879 .44606 .28639 .46626 .32559 .49052 .2982 .46883 Metetra 0 .319 .762 r .2982 .46883 .32559 .49052 .36644 .51426 .33927 .49098 Metetra .032 .462 .877 r .33927 .49098 .36644 .51426 .40913 .53257 .38222 .50776 Metetra .306 .619 .958 r .38222 .50776 .40913 .53257 .45361 .54117 .42703 .5151 Metetra .524 .718 .961 r .42703 .5151 .45361 .54117 .49955 .53758 .4734 .51078 Metetra .647 .762 .933 r .4734 .51078 .49955 .53758 .54651 .52275 .52089 .49581 Metetra .697 .78 .915 r .52089 .49581 .54651 .52275 .59414 .50146 .56913 .47473 Metetra .695 .785 .92 r .56913 .47473 .59414 .50146 .64242 .48032 .61806 .45368 Metetra .656 .778 .938 r .61806 .45368 .64242 .48032 .69162 .46393 .66789 .43684 Metetra .612 .761 .95 r .66789 .43684 .69162 .46393 .74198 .45229 .71885 .42423 Metetra .599 .751 .95 r .71885 .42423 .74198 .45229 .7935 .44197 .77096 .41277 Metetra .615 .753 .944 r .77096 .41277 .7935 .44197 .84596 .42974 .82404 .39957 Metetra .316 .698 .979 r .15255 .40936 .18192 .42575 .22047 .4282 .19147 .41107 Metetra 0 .378 .768 r .19147 .41107 .22047 .4282 .25879 .44606 .2301 .42716 Metetra 0 .295 .713 r .2301 .42716 .25879 .44606 .2982 .46883 .2698 .44783 Metetra 0 .354 .781 r .2698 .44783 .2982 .46883 .33927 .49098 .31115 .46779 Metetra .068 .494 .894 r .31115 .46779 .33927 .49098 .38222 .50776 .35442 .48254 Metetra .329 .635 .963 r .35442 .48254 .38222 .50776 .42703 .5151 .3996 .48829 Metetra .53 .721 .961 r .3996 .48829 .42703 .5151 .4734 .51078 .44641 .48311 Metetra .643 .761 .934 r .44641 .48311 .4734 .51078 .52089 .49581 .49443 .46807 Metetra .69 .779 .919 r .49443 .46807 .52089 .49581 .56913 .47473 .54329 .44736 Metetra .689 .786 .924 r .54329 .44736 .56913 .47473 .61806 .45368 .59288 .42649 Metetra .656 .78 .94 r .59288 .42649 .61806 .45368 .66789 .43684 .64336 .40918 Metetra .618 .766 .951 r .64336 .40918 .66789 .43684 .71885 .42423 .69494 .39545 Metetra .607 .756 .95 r .69494 .39545 .71885 .42423 .77096 .41277 .74764 .38268 Metetra .62 .756 .943 r .74764 .38268 .77096 .41277 .82404 .39957 .80136 .36839 Metetra .269 .712 .961 r .12157 .39697 .15255 .40936 .19147 .41107 .16096 .39744 Metetra 0 .403 .746 r .16096 .39744 .19147 .41107 .2301 .42716 .20008 .41081 Metetra 0 .338 .72 r .20008 .41081 .2301 .42716 .2698 .44783 .24022 .42836 Metetra 0 .406 .806 r .24022 .42836 .2698 .44783 .31115 .46779 .28199 .44518 Metetra .117 .538 .916 r .28199 .44518 .31115 .46779 .35442 .48254 .32566 .4571 Metetra .361 .657 .969 r .32566 .4571 .35442 .48254 .3996 .48829 .37125 .46075 Metetra .538 .726 .961 r .37125 .46075 .3996 .48829 .44641 .48311 .41853 .45453 Metetra .638 .759 .936 r .41853 .45453 .44641 .48311 .49443 .46807 .46709 .43951 Metetra .681 .777 .923 r .46709 .43951 .49443 .46807 .54329 .44736 .51656 .41939 Metetra .682 .786 .929 r .51656 .41939 .54329 .44736 .59288 .42649 .56683 .39885 Metetra .655 .784 .943 r .56683 .39885 .59288 .42649 .64336 .40918 .61797 .38103 Metetra .625 .773 .951 r .61797 .38103 .64336 .40918 .69494 .39545 .67019 .366 Metetra .616 .763 .95 r .67019 .366 .69494 .39545 .74764 .38268 .72351 .3517 Metetra .626 .76 .943 r .72351 .3517 .74764 .38268 .80136 .36839 .77789 .33616 Metetra .171 .698 .864 r .08825 .39239 .12157 .39697 .16096 .39744 .12836 .39061 Metetra .163 0 0 r .12836 .39061 .16096 .39744 .20008 .41081 .16829 .39951 Metetra 0 .403 .713 r .16829 .39951 .20008 .41081 .24022 .42836 .20919 .41206 Metetra 0 .485 .835 r .20919 .41206 .24022 .42836 .28199 .44518 .25161 .42404 Metetra .187 .602 .944 r .25161 .42404 .28199 .44518 .32566 .4571 .29585 .43181 Metetra .403 .688 .977 r .29585 .43181 .32566 .4571 .37125 .46075 .34195 .43253 Metetra .549 .732 .96 r .34195 .43253 .37125 .46075 .41853 .45453 .38971 .42499 Metetra .631 .757 .938 r .38971 .42499 .41853 .45453 .46709 .43951 .4388 .41015 Metetra .668 .774 .929 r .4388 .41015 .46709 .43951 .51656 .41939 .48889 .39094 Metetra .672 .787 .935 r .48889 .39094 .51656 .41939 .56683 .39885 .53983 .37094 Metetra .654 .789 .946 r .53983 .37094 .56683 .39885 .61797 .38103 .59168 .35254 Metetra .634 .782 .952 r .59168 .35254 .61797 .38103 .67019 .366 .64456 .33595 Metetra .627 .772 .949 r .64456 .33595 .67019 .366 .72351 .3517 .69853 .31983 Metetra .634 .765 .942 r .69853 .31983 .72351 .3517 .77789 .33616 .75358 .30284 Metetra .017 0 0 r .05101 .40398 .08825 .39239 .12836 .39061 .09238 .39778 Metetra .158 0 0 r .09238 .39778 .12836 .39061 .16829 .39951 .13383 .39867 Metetra .107 0 0 r .13383 .39867 .16829 .39951 .20919 .41206 .17612 .4025 Metetra .071 .61 .853 r .17612 .4025 .20919 .41206 .25161 .42404 .2197 .40633 Metetra .296 .701 .975 r .2197 .40633 .25161 .42404 .29585 .43181 .26484 .40743 Metetra .464 .732 .985 r .26484 .40743 .29585 .43181 .34195 .43253 .31161 .40376 Metetra .565 .742 .96 r .31161 .40376 .34195 .43253 .38971 .42499 .35991 .39444 Metetra .621 .753 .941 r .35991 .39444 .38971 .42499 .4388 .41015 .40952 .38004 Metetra .651 .771 .937 r .40952 .38004 .4388 .41015 .48889 .39094 .4602 .36223 Metetra .659 .788 .943 r .4602 .36223 .48889 .39094 .53983 .37094 .51183 .34304 Metetra .654 .797 .951 r .51183 .34304 .53983 .37094 .59168 .35254 .56441 .32397 Metetra .645 .794 .954 r .56441 .32397 .59168 .35254 .64456 .33595 .61799 .30545 Metetra .642 .782 .949 r .61799 .30545 .64456 .33595 .69853 .31983 .67264 .28709 Metetra .643 .771 .941 r .67264 .28709 .69853 .31983 .75358 .30284 .72838 .26837 Metetra .198 0 0 r .00636 .4495 .05101 .40398 .09238 .39778 .05008 .43518 Metetra .149 0 0 r .05008 .43518 .09238 .39778 .13383 .39867 .09451 .42136 Metetra .012 0 0 r .09451 .42136 .13383 .39867 .17612 .4025 .13964 .40857 Metetra .239 .754 .763 r .13964 .40857 .17612 .4025 .2197 .40633 .18553 .39689 Metetra .468 .851 .983 r .18553 .39689 .2197 .40633 .26484 .40743 .23233 .38587 Metetra .559 .801 .989 r .23233 .38587 .26484 .40743 .31161 .40376 .28015 .37478 Metetra .589 .757 .959 r .28015 .37478 .31161 .40376 .35991 .39444 .32908 .36283 Metetra .606 .748 .945 r .32908 .36283 .35991 .39444 .40952 .38004 .37917 .34932 Metetra .623 .765 .947 r .37917 .34932 .40952 .38004 .4602 .36223 .4304 .33369 Metetra .641 .792 .955 r .4304 .33369 .4602 .36223 .51183 .34304 .48273 .31573 Metetra .654 .81 .958 r .48273 .31573 .51183 .34304 .56441 .32397 .53608 .2958 Metetra .661 .811 .955 r .53608 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\[Alpha] depends on \[Theta]1 and \ t0; values for these in the numerical examples were chosen to give a Gaussian \ that starts reasonably localised and that spreads noticeably by in the time \ taken to reach the step. The value of \[Alpha] coming out of this is 41. So \ it is plausible to assume that values of \[Alpha] smaller than 1.5 is not of \ interest and we can just choose a representative value, say \[Alpha] = 4, for \ testing other dependencies. We now look at the time and \[CapitalDelta] \ dependence:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Plot3D[ b0[T] /. {\[Alpha] \[Rule] 4, \[Beta] \[Rule] 1}, {\[CapitalDelta], \(-0.8\), 0.8}, {T, 1.1, 5}\ ]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .81114 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% SurfaceGraphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 1.66533e-016 1.04977 -0.0679587 1.04977 [ [.13842 .20256 -23.9105 -9 ] [.13842 .20256 .0895 0 ] [.3345 .12542 -5.5867 -9 ] [.3345 .12542 .4133 0 ] [.54622 .04218 -15.5873 -9 ] [.54622 .04218 2.41266 0 ] [.77714 .11187 0 -6.14746 ] [.77714 .11187 6 2.85254 ] [.85293 .22792 0 -6.03643 ] [.85293 .22792 6 2.96357 ] [.92079 .33185 0 -5.93942 ] [.92079 .33185 6 3.06058 ] [.98191 .42546 0 -5.85393 ] [.98191 .42546 6 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.96935 .42924 L s .96935 .42924 m 1 .6535 L s 1 .6535 m .70298 .24544 L s .70298 .24544 m .68874 0 L s .03716 .25514 m 0 .48963 L s 0 .48963 m .70298 .24544 L s .70298 .24544 m .68874 0 L s .68874 0 m .03716 .25514 L s .40529 .59895 m .96935 .42924 L s .96935 .42924 m 1 .6535 L s 1 .6535 m .39787 .81114 L s .39787 .81114 m .40529 .59895 L s 0 0 m 1 0 L 1 .81114 L 0 .81114 L closepath clip newpath .5 Mabswid .589 .723 .936 r .37561 .76378 .39878 .78503 .43735 .77955 .41446 .75848 Metetra .594 .729 .938 r .41446 .75848 .43735 .77955 .47658 .77343 .454 .75253 Metetra .601 .735 .939 r .454 .75253 .47658 .77343 .51649 .76665 .49425 .74586 Metetra .608 .741 .94 r .49425 .74586 .51649 .76665 .55706 .75913 .53519 .73842 Metetra .616 .747 .939 r .53519 .73842 .55706 .75913 .59831 .7508 .57682 .73008 Metetra .626 .753 .938 r .57682 .73008 .59831 .7508 .6402 .7415 .61913 .72067 Metetra .638 .759 .936 r .61913 .72067 .6402 .7415 .68272 .73097 .66209 .70991 Metetra .655 .765 .931 r .66209 .70991 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.55473 .70872 .53201 .68671 Metetra .62 .75 .939 r .53201 .68671 .55473 .70872 .59746 .69923 .57516 .67716 Metetra .636 .758 .936 r .57516 .67716 .59746 .69923 .64086 .68823 .61902 .66593 Metetra .657 .766 .93 r .61902 .66593 .64086 .68823 .68488 .67519 .66352 .6524 Metetra .685 .774 .918 r .66352 .6524 .68488 .67519 .72941 .65924 .70853 .63559 Metetra .72 .78 .9 r .70853 .63559 .72941 .65924 .77426 .6389 .75385 .61393 Metetra .763 .784 .87 r .75385 .61393 .77426 .6389 .81905 .61179 .79911 .58496 Metetra .81 .784 .829 r .79911 .58496 .81905 .61179 .86317 .57429 .84367 .54518 Metetra .853 .781 .78 r .84367 .54518 .86317 .57429 .90567 .52193 .88666 .49081 Metetra .886 .776 .734 r .88666 .49081 .90567 .52193 .94558 .45217 .92729 .42059 Metetra .57 .706 .934 r .30253 .6945 .3275 .7186 .36688 .71384 .34216 .69008 Metetra .576 .714 .937 r .34216 .69008 .36688 .71384 .40702 .70836 .38258 .68495 Metetra .583 .722 .939 r .38258 .68495 .40702 .70836 .44792 .7021 .4238 .679 Metetra .592 .731 .94 r .4238 .679 .44792 .7021 .48959 .69494 .46582 .67208 Metetra .603 .74 .941 r .46582 .67208 .48959 .69494 .53201 .68671 .50863 .66398 Metetra .617 .749 .94 r .50863 .66398 .53201 .68671 .57516 .67716 .55222 .65443 Metetra .635 .758 .936 r .55222 .65443 .57516 .67716 .61902 .66593 .59653 .64296 Metetra .659 .767 .929 r .59653 .64296 .61902 .66593 .66352 .6524 .64151 .62889 Metetra .689 .775 .916 r .64151 .62889 .66352 .6524 .70853 .63559 .68702 .61114 Metetra .727 .781 .895 r .68702 .61114 .70853 .63559 .75385 .61393 .73284 .58799 Metetra .771 .784 .864 r .73284 .58799 .75385 .61393 .79911 .58496 .77857 .55696 Metetra .816 .783 .822 r .77857 .55696 .79911 .58496 .84367 .54518 .82359 .51478 Metetra .855 .778 .775 r .82359 .51478 .84367 .54518 .88666 .49081 .86711 .45847 Metetra .884 .774 .735 r .86711 .45847 .88666 .49081 .92729 .42059 .90849 .38806 Metetra .561 .699 .932 r .27695 .66925 .30253 .6945 .34216 .69008 .3168 .66525 Metetra .568 .707 .936 r .3168 .66525 .34216 .69008 .38258 .68495 .35748 .66054 Metetra .575 .716 .938 r .35748 .66054 .38258 .68495 .4238 .679 .39901 .65498 Metetra .585 .726 .941 r .39901 .65498 .4238 .679 .46582 .67208 .44137 .64838 Metetra .597 .736 .941 r .44137 .64838 .46582 .67208 .50863 .66398 .48457 .64051 Metetra .613 .746 .941 r .48457 .64051 .50863 .66398 .55222 .65443 .52858 .63099 Metetra .634 .757 .937 r .52858 .63099 .55222 .65443 .59653 .64296 .57337 .61931 Metetra .661 .767 .929 r .57337 .61931 .59653 .64296 .64151 .62889 .61884 .60465 Metetra .695 .776 .914 r .61884 .60465 .64151 .62889 .68702 .61114 .66486 .58582 Metetra .735 .782 .89 r .66486 .58582 .68702 .61114 .73284 .58799 .71119 .561 Metetra .779 .784 .857 r .71119 .561 .73284 .58799 .77857 .55696 .7574 .5277 Metetra .821 .781 .814 r .7574 .5277 .77857 .55696 .82359 .51478 .80291 .48302 Metetra .856 .775 .771 r .80291 .48302 .82359 .51478 .86711 .45847 .847 .42491 Metetra .882 .772 .736 r .847 .42491 .86711 .45847 .90849 .38806 .88919 .35457 Metetra .553 .69 .931 r .25074 .64271 .27695 .66925 .3168 .66525 .29079 .6392 Metetra .558 .699 .934 r .29079 .6392 .3168 .66525 .35748 .66054 .33172 .635 Metetra .566 .709 .938 r .33172 .635 .35748 .66054 .39901 .65498 .37353 .62994 Metetra .576 .72 .941 r .37353 .62994 .39901 .65498 .44137 .64838 .41622 .62378 Metetra .589 .731 .942 r .41622 .62378 .44137 .64838 .48457 .64051 .4598 .61622 Metetra .608 .744 .941 r .4598 .61622 .48457 .64051 .52858 .63099 .50424 .60681 Metetra .632 .756 .937 r .50424 .60681 .52858 .63099 .57337 .61931 .5495 .59493 Metetra .663 .768 .928 r .5495 .59493 .57337 .61931 .61884 .60465 .59548 .57963 Metetra .701 .777 .911 r .59548 .57963 .61884 .60465 .66486 .58582 .64202 .55957 Metetra .744 .783 .885 r .64202 .55957 .66486 .58582 .71119 .561 .68887 .53286 Metetra .787 .783 .849 r .68887 .53286 .71119 .561 .7574 .5277 .73559 .49708 Metetra .826 .778 .807 r .73559 .49708 .7574 .5277 .80291 .48302 .78162 .44984 Metetra .857 .772 .767 r .78162 .44984 .80291 .48302 .847 .42491 .82633 .39013 Metetra .879 .77 .738 r .82633 .39013 .847 .42491 .88919 .35457 .86936 .32015 Metetra .543 .68 .928 r .2239 .61476 .25074 .64271 .29079 .6392 .26413 .61181 Metetra .547 .689 .933 r .26413 .61181 .29079 .6392 .33172 .635 .30527 .60822 Metetra .555 .7 .937 r .30527 .60822 .33172 .635 .37353 .62994 .34734 .60378 Metetra .565 .712 .94 r .34734 .60378 .37353 .62994 .41622 .62378 .39035 .5982 Metetra .58 .726 .943 r .39035 .5982 .41622 .62378 .4598 .61622 .43429 .59107 Metetra .601 .74 .942 r .43429 .59107 .4598 .61622 .50424 .60681 .47916 .58186 Metetra .63 .755 .938 r .47916 .58186 .50424 .60681 .5495 .59493 .52489 .5698 Metetra .665 .769 .927 r .52489 .5698 .5495 .59493 .59548 .57963 .57139 .55379 Metetra .707 .779 .908 r .57139 .55379 .59548 .57963 .64202 .55957 .61848 .53233 Metetra .753 .784 .878 r .61848 .53233 .64202 .55957 .68887 .53286 .66587 .50347 Metetra .795 .782 .841 r .66587 .50347 .68887 .53286 .73559 .49708 .71312 .46502 Metetra .831 .776 .8 r .71312 .46502 .73559 .49708 .78162 .44984 .7597 .41523 Metetra .857 .77 .764 r .7597 .41523 .78162 .44984 .82633 .39013 .80507 .35417 Metetra .876 .769 .741 r .80507 .35417 .82633 .39013 .86936 .32015 .849 .28483 Metetra .531 .669 .925 r .19644 .58526 .2239 .61476 .26413 .61181 .23681 .58294 Metetra .535 .679 .93 r .23681 .58294 .26413 .61181 .30527 .60822 .27813 .58007 Metetra .542 .69 .936 r .27813 .58007 .30527 .60822 .34734 .60378 .32043 .57639 Metetra .553 .703 .94 r .32043 .57639 .34734 .60378 .39035 .5982 .36372 .57153 Metetra .569 .719 .943 r .36372 .57153 .39035 .5982 .43429 .59107 .40801 .56499 Metetra .594 .736 .943 r .40801 .56499 .43429 .59107 .47916 .58186 .45329 .55609 Metetra .627 .754 .938 r .45329 .55609 .47916 .58186 .52489 .5698 .4995 .54387 Metetra .668 .769 .926 r .4995 .54387 .52489 .5698 .57139 .55379 .54653 .52707 Metetra .715 .78 .903 r .54653 .52707 .57139 .55379 .61848 .53233 .59419 .50402 Metetra .762 .784 .871 r .59419 .50402 .61848 .53233 .66587 .50347 .64215 .47278 Metetra .803 .781 .832 r .64215 .47278 .66587 .50347 .71312 .46502 .68997 .4315 Metetra .835 .773 .794 r .68997 .4315 .71312 .46502 .7597 .41523 .73714 .37926 Metetra .857 .768 .763 r .73714 .37926 .7597 .41523 .80507 .35417 .78323 .31716 Metetra .872 .768 .744 r .78323 .31716 .80507 .35417 .849 .28483 .82809 .24872 Metetra .519 .657 .922 r .16836 .5541 .19644 .58526 .23681 .58294 .20884 .55247 Metetra .521 .666 .928 r .20884 .55247 .23681 .58294 .27813 .58007 .25029 .55043 Metetra .526 .678 .934 r .25029 .55043 .27813 .58007 .32043 .57639 .29277 .54765 Metetra .537 .693 .94 r .29277 .54765 .32043 .57639 .36372 .57153 .33631 .54369 Metetra .556 .711 .944 r .33631 .54369 .36372 .57153 .40801 .56499 .38093 .53791 Metetra .584 .731 .944 r .38093 .53791 .40801 .56499 .45329 .55609 .42661 .52944 Metetra .623 .752 .939 r .42661 .52944 .45329 .55609 .4995 .54387 .47331 .51712 Metetra .672 .77 .924 r .47331 .51712 .4995 .54387 .54653 .52707 .52089 .49944 Metetra .724 .782 .899 r .52089 .49944 .54653 .52707 .59419 .50402 .56913 .4746 Metetra .772 .785 .863 r .56913 .4746 .59419 .50402 .64215 .47278 .6177 .44077 Metetra .811 .78 .824 r .6177 .44077 .64215 .47278 .68997 .4315 .66613 .39662 Metetra .838 .772 .788 r .66613 .39662 .68997 .4315 .73714 .37926 .71395 .34217 Metetra .857 .767 .762 r .71395 .34217 .73714 .37926 .78323 .31716 .76081 .27938 Metetra .869 .769 .749 r .76081 .27938 .78323 .31716 .82809 .24872 .80662 .21201 Metetra .506 .645 .919 r .13965 .52131 .16836 .5541 .20884 .55247 .18019 .52039 Metetra .505 .653 .926 r .18019 .52039 .20884 .55247 .25029 .55043 .22174 .51926 Metetra .508 .665 .933 r .22174 .51926 .25029 .55043 .29277 .54765 .26436 .51753 Metetra .519 .681 .939 r .26436 .51753 .29277 .54765 .33631 .54369 .30811 .51464 Metetra .539 .701 .944 r .30811 .51464 .33631 .54369 .38093 .53791 .35302 .50979 Metetra .573 .725 .946 r .35302 .50979 .38093 .53791 .42661 .52944 .39908 .50189 Metetra .619 .75 .94 r .39908 .50189 .42661 .52944 .47331 .51712 .44626 .4895 Metetra .675 .772 .923 r .44626 .4895 .47331 .51712 .52089 .49944 .49441 .47086 Metetra .732 .784 .894 r .49441 .47086 .52089 .49944 .56913 .4746 .54328 .44409 Metetra .781 .786 .856 r .54328 .44409 .56913 .4746 .6177 .44077 .59249 .40762 Metetra .817 .78 .818 r .59249 .40762 .6177 .44077 .66613 .39662 .64159 .3608 Metetra .841 .773 .786 r .64159 .3608 .66613 .39662 .71395 .34217 .69013 .30456 Metetra .857 .77 .764 r .69013 .30456 .71395 .34217 .76081 .27938 .73783 .24148 Metetra .868 .773 .755 r .73783 .24148 .76081 .27938 .80662 .21201 .78462 .17517 Metetra .491 .635 .918 r .11027 .48719 .13965 .52131 .18019 .52039 .15083 .48702 Metetra .486 .642 .925 r .15083 .48702 .18019 .52039 .22174 .51926 .19244 .48685 Metetra .487 .653 .933 r .19244 .48685 .22174 .51926 .26436 .51753 .23515 .48625 Metetra .497 .669 .94 r .23515 .48625 .26436 .51753 .30811 .51464 .27907 .48454 Metetra .52 .692 .946 r .27907 .48454 .30811 .51464 .35302 .50979 .32423 .48072 Metetra .56 .72 .948 r .32423 .48072 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.45422 Metetra .5 .689 .952 r .24907 .45422 .27907 .48454 .32423 .48072 .29449 .45117 Metetra .549 .719 .952 r .29449 .45117 .32423 .48072 .37066 .47344 .34129 .4442 Metetra .612 .749 .942 r .34129 .4442 .37066 .47344 .41832 .46097 .38944 .43147 Metetra .68 .773 .921 r .38944 .43147 .41832 .46097 .46705 .44136 .43877 .41114 Metetra .741 .788 .89 r .43877 .41114 .46705 .44136 .51658 .41278 .48899 .38172 Metetra .79 .797 .857 r .48899 .38172 .51658 .41278 .5665 .37407 .5397 .34259 Metetra .828 .803 .828 r .5397 .34259 .5665 .37407 .61636 .32531 .59045 .29411 Metetra .856 .812 .805 r .59045 .29411 .61636 .32531 .66572 .26805 .64083 .23764 Metetra .879 .822 .788 r .64083 .23764 .66572 .26805 .71434 .20501 .69053 .17513 Metetra .894 .83 .776 r .69053 .17513 .71434 .20501 .76213 .13938 .73938 .10882 Metetra .409 .652 .958 r .04808 .42395 .08001 .45306 .12058 .45366 .08865 .4256 Metetra .399 .66 .964 r .08865 .4256 .12058 .45366 .16221 .45445 .13036 .42725 Metetra .407 .674 .97 r .13036 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.3872 .3862 .41494 .4307 .38987 .4044 .36147 Metetra .72 .764 .888 r .4044 .36147 .4307 .38987 .47563 .36657 .44986 .33788 Metetra .702 .761 .898 r .44986 .33788 .47563 .36657 .52107 .34521 .49583 .31665 Metetra .681 .76 .911 r .49583 .31665 .52107 .34521 .56714 .32604 .54245 .29792 Metetra .659 .76 .924 r .54245 .29792 .56714 .32604 .61395 .30914 .58985 .28141 Metetra .643 .761 .935 r .58985 .28141 .61395 .30914 .66163 .29418 .63813 .26637 Metetra .636 .762 .939 r .63813 .26637 .66163 .29418 .71026 .28033 .68734 .25169 Metetra .64 .761 .936 r .68734 .25169 .71026 .28033 .75983 .26644 .73747 .23636 Metetra .648 .757 .929 r .73747 .23636 .75983 .26644 .81029 .25153 .7885 .21992 Metetra .881 .791 .757 r .09985 .57032 .13181 .58938 .17997 .52259 .14948 .49897 Metetra .826 .79 .818 r .14948 .49897 .17997 .52259 .22532 .47978 .19577 .45364 Metetra .79 .78 .844 r .19577 .45364 .22532 .47978 .26999 .4454 .24117 .41751 Metetra .767 .771 .856 r .24117 .41751 .26999 .4454 .31458 .41504 .2864 .3859 Metetra .749 .764 .865 r .2864 .3859 .31458 .41504 .35934 .3872 .33172 .35731 Metetra .731 .759 .875 r .33172 .35731 .35934 .3872 .4044 .36147 .37731 .33141 Metetra .712 .757 .887 r .37731 .33141 .4044 .36147 .44986 .33788 .42328 .30828 Metetra .69 .758 .903 r .42328 .30828 .44986 .33788 .49583 .31665 .46977 .28803 Metetra .667 .762 .921 r .46977 .28803 .49583 .31665 .54245 .29792 .51695 .27037 Metetra .648 .768 .936 r .51695 .27037 .54245 .29792 .58985 .28141 .56495 .25445 Metetra .64 .774 .945 r .56495 .25445 .58985 .28141 .63813 .26637 .61385 .23902 Metetra .643 .776 .944 r .61385 .23902 .63813 .26637 .68734 .25169 .66367 .22294 Metetra .652 .773 .937 r .66367 .22294 .68734 .25169 .73747 .23636 .71438 .20567 Metetra .659 .764 .928 r .71438 .20567 .73747 .23636 .7885 .21992 .76597 .18738 Metetra .864 .766 .752 r .06762 .54427 .09985 .57032 .14948 .49897 .11808 .47391 Metetra .834 .784 .804 r .11808 .47391 .14948 .49897 .19577 .45364 .16558 .42403 Metetra .792 .764 .828 r .16558 .42403 .19577 .45364 .24117 .41751 .21182 .38512 Metetra .763 .748 .839 r .21182 .38512 .24117 .41751 .2864 .3859 .25767 .35205 Metetra .739 .738 .851 r .25767 .35205 .2864 .3859 .33172 .35731 .30348 .32332 Metetra .715 .735 .866 r .30348 .32332 .33172 .35731 .37731 .33141 .34948 .29859 Metetra .69 .738 .888 r .34948 .29859 .37731 .33141 .42328 .30828 .39584 .27774 Metetra .663 .75 .914 r .39584 .27774 .42328 .30828 .46977 .28803 .44279 .26023 Metetra .641 .767 .94 r .44279 .26023 .46977 .28803 .51695 .27037 .49053 .24483 Metetra .63 .786 .956 r .49053 .24483 .51695 .27037 .56495 .25445 .53917 .2298 Metetra .638 .801 .961 r .53917 .2298 .56495 .25445 .61385 .23902 .58876 .21355 Metetra .656 .805 .954 r .58876 .21355 .61385 .23902 .66367 .22294 .63923 .19529 Metetra .671 .797 .942 r .63923 .19529 .66367 .22294 .71438 .20567 .69054 .17524 Metetra .676 .781 .929 r .69054 .17524 .71438 .20567 .76597 .18738 .74271 .15434 Metetra .862 .767 .755 r .03435 .51739 .06762 .54427 .11808 .47391 .08572 .44742 Metetra .855 .775 .773 r .08572 .44742 .11808 .47391 .16558 .42403 .13544 .38539 Metetra .79 .722 .788 r .13544 .38539 .16558 .42403 .21182 .38512 .18285 .34024 Metetra .742 .686 .797 r .18285 .34024 .21182 .38512 .25767 .35205 .22923 .3055 Metetra .701 .667 .813 r .22923 .3055 .25767 .35205 .30348 .32332 .27517 .27913 Metetra .663 .666 .842 r .27517 .27913 .30348 .32332 .34948 .29859 .32112 .25993 Metetra .626 .685 .886 r .32112 .25993 .34948 .29859 .39584 .27774 .36751 .24621 Metetra .591 .723 .936 r .36751 .24621 .39584 .27774 .44279 .26023 .41472 .23553 Metetra .567 .774 .976 r .41472 .23553 .44279 .26023 .49053 .24483 .46298 .22514 Metetra .569 .821 .993 r .46298 .22514 .49053 .24483 .53917 .2298 .51237 .21266 Metetra .599 .859 .993 r .51237 .21266 .53917 .2298 .58876 .21355 .56281 .1966 Metetra .645 .88 .984 r .56281 .1966 .58876 .21355 .63923 .19529 .61415 .17644 Metetra .688 .884 .971 r .61415 .17644 .63923 .19529 .69054 .17524 .66625 .15251 Metetra .716 .865 .953 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.45381 .89736 .42578 Metetra .795 .736 .798 r .25607 .5753 .28164 .60311 .32278 .56495 .29777 .536 Metetra .677 .709 .872 r .29777 .536 .32278 .56495 .36282 .54956 .33811 .52068 Metetra .604 .688 .902 r .33811 .52068 .36282 .54956 .40311 .54158 .37864 .51329 Metetra .567 .681 .917 r .37864 .51329 .40311 .54158 .44407 .5364 .41984 .50895 Metetra .55 .684 .927 r .41984 .50895 .44407 .5364 .48583 .53211 .4619 .50558 Metetra .547 .694 .936 r .4619 .50558 .48583 .53211 .52847 .52776 .50487 .50206 Metetra .553 .706 .942 r .50487 .50206 .52847 .52776 .57201 .52275 .5488 .49764 Metetra .568 .721 .945 r .5488 .49764 .57201 .52275 .61643 .51661 .59365 .49165 Metetra .593 .735 .943 r .59365 .49165 .61643 .51661 .6617 .50877 .63939 .48337 Metetra .625 .747 .935 r .63939 .48337 .6617 .50877 .70774 .49845 .68589 .47194 Metetra .66 .755 .92 r .68589 .47194 .70774 .49845 .75441 .48466 .73303 .45663 Metetra .689 .756 .903 r .73303 .45663 .75441 .48466 .80155 .46683 .78065 .43756 Metetra .699 .755 .895 r .78065 .43756 .80155 .46683 .8491 .44608 .82881 .41665 Metetra .689 .753 .901 r .82881 .41665 .8491 .44608 .89736 .42578 .87778 .39693 Metetra .788 .722 .791 r .23008 .54417 .25607 .5753 .29777 .536 .27227 .50463 Metetra .668 .695 .866 r .27227 .50463 .29777 .536 .33811 .52068 .31285 .48987 Metetra .591 .674 .898 r .31285 .48987 .33811 .52068 .37864 .51329 .35357 .48345 Metetra .55 .668 .915 r .35357 .48345 .37864 .51329 .41984 .50895 .39498 .48028 Metetra .532 .672 .928 r .39498 .48028 .41984 .50895 .4619 .50558 .43729 .4781 Metetra .529 .684 .937 r .43729 .4781 .4619 .50558 .50487 .50206 .48058 .47559 Metetra .541 .701 .944 r .48058 .47559 .50487 .50206 .5488 .49764 .52489 .47177 Metetra .564 .719 .946 r .52489 .47177 .5488 .49764 .59365 .49165 .57017 .46576 Metetra .598 .736 .941 r .57017 .46576 .59365 .49165 .63939 .48337 .61637 .45667 Metetra .636 .747 .929 r .61637 .45667 .63939 .48337 .68589 .47194 .66333 .44375 Metetra .67 .751 .911 r .66333 .44375 .68589 .47194 .73303 .45663 .71093 .42679 Metetra .691 .75 .897 r .71093 .42679 .73303 .45663 .78065 .43756 .75909 .40681 Metetra .693 .748 .894 r .75909 .40681 .78065 .43756 .82881 .41665 .8079 .3862 Metetra .68 .749 .903 r .8079 .3862 .82881 .41665 .87778 .39693 .85759 .36721 Metetra .771 .699 .783 r .20389 .50801 .23008 .54417 .27227 .50463 .24637 .46996 Metetra .651 .673 .858 r .24637 .46996 .27227 .50463 .31285 .48987 .28708 .45669 Metetra .571 .655 .893 r .28708 .45669 .31285 .48987 .35357 .48345 .32791 .4519 Metetra .526 .65 .913 r .32791 .4519 .35357 .48345 .39498 .48028 .36947 .45042 Metetra .507 .658 .928 r .36947 .45042 .39498 .48028 .43729 .4781 .41198 .4498 Metetra .508 .675 .94 r .41198 .4498 .43729 .4781 .48058 .47559 .45557 .44845 Metetra .529 .697 .946 r .45557 .44845 .48058 .47559 .52489 .47177 .50025 .44511 Metetra .565 .72 .945 r .50025 .44511 .52489 .47177 .57017 .46576 .54596 .43864 Metetra .609 .737 .936 r .54596 .43864 .57017 .46576 .61637 .45667 .59261 .42814 Metetra .651 .745 .918 r .59261 .42814 .61637 .45667 .66333 .44375 .64003 .41326 Metetra .678 .743 .9 r .64003 .41326 .66333 .44375 .71093 .42679 .6881 .39472 Metetra .688 .739 .89 r .6881 .39472 .71093 .42679 .75909 .40681 .73682 .37444 Metetra .682 .739 .894 r .73682 .37444 .75909 .40681 .8079 .3862 .78632 .3547 Metetra .669 .743 .906 r .78632 .3547 .8079 .3862 .85759 .36721 .83676 .33659 Metetra .737 .66 .774 r .1779 .46375 .20389 .50801 .24637 .46996 .22026 .4305 Metetra .619 .638 .848 r .22026 .4305 .24637 .46996 .28708 .45669 .26087 .42051 Metetra .537 .625 .887 r .26087 .42051 .28708 .45669 .32791 .4519 .30166 .41858 Metetra .489 .626 .912 r .30166 .41858 .32791 .4519 .36947 .45042 .34326 .41968 Metetra .471 .641 .931 r .34326 .41968 .36947 .45042 .41198 .4498 .38593 .42113 Metetra .483 .667 .944 r .38593 .42113 .41198 .4498 .45557 .44845 .42979 .42099 Metetra .522 .698 .95 r .42979 .42099 .45557 .44845 .50025 .44511 .47484 .41762 Metetra .577 .725 .943 r .47484 .41762 .50025 .44511 .54596 .43864 .521 .40974 Metetra .631 .737 .925 r .521 .40974 .54596 .43864 .59261 .42814 .56809 .39681 Metetra .667 .736 .902 r .56809 .39681 .59261 .42814 .64003 .41326 .61595 .37951 Metetra .681 .728 .886 r .61595 .37951 .64003 .41326 .6881 .39472 .66451 .35985 Metetra .678 .723 .884 r .66451 .35985 .6881 .39472 .73682 .37444 .71384 .34027 Metetra .666 .726 .895 r .71384 .34027 .73682 .37444 .78632 .3547 .76406 .32212 Metetra .656 .737 .91 r .76406 .32212 .78632 .3547 .83676 .33659 .81526 .30502 Metetra .667 .587 .756 r .15303 .40518 .1779 .46375 .22026 .4305 .19432 .38335 Metetra .555 .574 .828 r .19432 .38335 .22026 .4305 .26087 .42051 .23435 .38033 Metetra .471 .574 .876 r .23435 .38033 .26087 .42051 .30166 .41858 .27479 .38382 Metetra .42 .589 .912 r .27479 .38382 .30166 .41858 .34326 .41968 .31626 .38919 Metetra .411 .621 .938 r .31626 .38919 .34326 .41968 .38593 .42113 .35902 .39346 Metetra .453 .669 .956 r .35902 .39346 .38593 .42113 .42979 .42099 .40316 .39415 Metetra .532 .715 .956 r .40316 .39415 .42979 .42099 .47484 .41762 .44864 .38927 Metetra .612 .738 .935 r .44864 .38927 .47484 .41762 .521 .40974 .49527 .3779 Metetra .663 .734 .903 r .49527 .3779 .521 .40974 .56809 .39681 .54281 .36088 Metetra .679 .716 .876 r .54281 .36088 .56809 .39681 .61595 .37951 .59109 .34092 Metetra .672 .7 .868 r .59109 .34092 .61595 .37951 .66451 .35985 .64014 .32135 Metetra .655 .697 .877 r .64014 .32135 .66451 .35985 .71384 .34027 .6901 .30403 Metetra .642 .709 .896 r .6901 .30403 .71384 .34027 .76406 .32212 .74108 .28831 Metetra .642 .73 .913 r .74108 .28831 .76406 .32212 .81526 .30502 .79304 .2724 Metetra .625 .55 .75 r .12864 .34057 .15303 .40518 .19432 .38335 .16907 .3257 Metetra .443 .471 .794 r .16907 .3257 .19432 .38335 .23435 .38033 .20773 .33457 Metetra .314 .459 .846 r .20773 .33457 .23435 .38033 .27479 .38382 .24713 .34944 Metetra .249 .506 .902 r .24713 .34944 .27479 .38382 .31626 .38919 .28812 .36276 Metetra .277 .594 .949 r .28812 .36276 .31626 .38919 .35902 .39346 .3309 .37089 Metetra .419 .707 .982 r .3309 .37089 .35902 .39346 .40316 .39415 .37548 .37055 Metetra .588 .773 .967 r .37548 .37055 .40316 .39415 .44864 .38927 .4216 .36001 Metetra .682 .761 .911 r .4216 .36001 .44864 .38927 .49527 .3779 .46884 .34055 Metetra .701 .718 .862 r .46884 .34055 .49527 .3779 .54281 .36088 .51683 .31679 Metetra .68 .676 .839 r .51683 .31679 .54281 .36088 .59109 .34092 .56547 .29463 Metetra .645 .653 .843 r .56547 .29463 .59109 .34092 .64014 .32135 .61497 .27764 Metetra .615 .655 .866 r .61497 .27764 .64014 .32135 .6901 .30403 .66558 .26495 Metetra .608 .682 .895 r .66558 .26495 .6901 .30403 .74108 .28831 .71733 .25291 Metetra .623 .718 .914 r .71733 .25291 .74108 .28831 .79304 .2724 .77007 .23858 Metetra .664 .741 .907 r .09904 .31293 .12864 .34057 .16907 .3257 .13983 .29762 Metetra .664 .741 .907 r .13983 .29762 .16907 .3257 .20773 .33457 .18129 .28206 Metetra 0 .093 .651 r .18129 .28206 .20773 .33457 .24713 .34944 .21774 .32439 Metetra 0 .121 .594 r .21774 .32439 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There is not a strong \ time dependence, except for very early times, although increasing from b0 to \ b1 to b2. For the latter, the asymmetry between up and down steps noticeably \ decreases with time.\ \>", "Text"], Cell["\<\ Finally we choose fixed values for T and \[Alpha] so that we can compare the \ \[CapitalDelta] dependence of the coefficients together on the same graph. \ \>", "Text"], Cell["\<\ ------------------------------------------------------------------------------\ -------\ \>", "Text"], Cell["\<\ Try to get the \[CapitalDelta] for which a chosen T will be larger than Tc:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[\(\[Alpha] \((1 + \[CapitalDelta])\)\)\/\(\[Alpha] + \ \[CapitalDelta]\) \[Equal] Tc, \[CapitalDelta]]\)], "Input"], Cell[BoxData[ \({{\[CapitalDelta] \[Rule] \(\[Alpha] - Tc\ \[Alpha]\)\/\(Tc - \ \[Alpha]\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[\(\[Alpha] \((1 - \[CapitalDelta])\)\)\/\(\[Alpha] - \ \[CapitalDelta]\) \[Equal] Tc, \[CapitalDelta]]\)], "Input"], Cell[BoxData[ \({{\[CapitalDelta] \[Rule] \(\(-\[Alpha]\) + Tc\ \[Alpha]\)\/\(Tc - \ \[Alpha]\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \({\[Alpha], \[CapitalDelta]} //. Join[Upstep, TGrep]\), "\[IndentingNewLine]", \({\[Alpha], \[CapitalDelta]} //. 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Ool2002?Ool008moo`03001oogoo08ioo`00Sgoo00<007ooOol0SWoo002?Ool00`00Oomoo`2>Ool0 08moo`03001oogoo08ioo`00Sgoo0P00Sgoo002?Ool00`00Oomoo`2>Ool008moo`03001oogoo08io o`00Sgoo00<007ooOol0SWoo002?Ool2002?Ool008moo`03001oogoo07Uoo`Il00moo`00Qgoo1@00 0goo00<007ooOol0KWoo2g`05Goo000>OolKO01POol00`00Oomoo`03Ool00`00Oomoo`13Ool[O00P Ool002Uoojal04]oo`00RGoo00<007ooOol0U7oo0029Ool00`00Oomoo`2DOol008Qoo`8009Ioo`00 ogoo8Goo0000\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.554507, -1.0536, \ 0.00386416, 0.0119092}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell["\<\ That confirms it, and we see that it is generally safe to assume that b0 >> \ b1 and b2, as was found for the 2 particular \[CapitalDelta] values in the \ numerical examples.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Reconstructing the concentration", "Subsection"], Cell["\<\ Define a2 and Z2 as functions of T, and the set of basis functions un:\ \>", "Text"], Cell[BoxData[ \(a2f[tt_] := a2 //. T \[Rule] tt\)], "Input"], Cell[BoxData[ \(Z2f[T_, Step_] := \(\((\(-\(1\/T\)\) + T)\)\ \((1 + \ \[CapitalDelta])\)\)\/\(\@2\ \((\[Beta] - \[Beta]\ \[CapitalDelta])\)\) /. Step\)], "Input"], Cell[BoxData[ \(u[x_, n_, \[Sigma]_] := \(\(\ \)\(2\^\(\(-n\)/2\)\ \@\[Pi]\ \((x\/\[Sigma])\ \)\^n\ g[x, \[Sigma]]\)\)\/Gamma[\(1 + n\)\/2]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(u[x, 0, \[Sigma]]\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\^\(-\(x\^2\/\(2\ \[Sigma]\^2\)\)\)\/\(\@\(2\ \[Pi]\)\ \ \[Sigma]\)\)], "Output"] }, Open ]], Cell[TextData[{ "First check how well the truncated series works for the example cases. \ Note that to transform back to the real coordinates, we need a factor ", Cell[BoxData[ \(TraditionalForm\`1/\@\(2 \(\( \[Gamma]\^2\)(t - t0)\)\)\)]], " in front because C is the concentration per unit length, and the length \ units of z and of x are related by this factor.:" }], "Text"], Cell[BoxData[ \(C2uxt[xx_, tt_, Step_] := \(1\/\@\(2 \( \[Gamma]\^2\) \((t - t0)\)\)\) \((b0[T] u[z - Z2, 0, 1\/\@\(2\ a2f[T]\)] + b1[T] u[z - Z2, 1, 1\/\@\(2\ a2f[T]\)] + b2[T] u[z - Z2, 2, 1\/\@\(2\ a2f[T]\)])\) //. 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ogoo009oo`03O01oogoo06Ioo`036n1oogoo03aoo`006goo00<007ooOol0EGoo00Ool001]oo`03001oogoo06]oo`036n1oog`0049o o`036n1oogoo04moo`006goo00<007ooOol0K7oo00"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.529838, -0.0326962, \ 0.0194713, 0.00252043}}], Cell[BoxData[ RowBox[{"{", RowBox[{\(15.90300000000002`\ Second\), ",", TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]}], "}"}]], "Output"] }, Open ]], Cell["\<\ It looks slightly worse than the original shown above with the numerical \ calculations, but only near the step. This is OK as the step has not been \ fully penetrated.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(T //. Join[TGrep, Tzrep, Upstep, {x \[Rule] 3, t \[Rule] 20}]\)], "Input"], Cell[BoxData[ \(1.5695268822630408`\)], "Output"] }, Open ]], Cell["\<\ But now revert to the scaled dimensions z and T. First, find the scaled \ parameters for the steps and use that to get new step definitions:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({\[Alpha], \[Beta], \[CapitalDelta]} //. Join[TGrep, Upstep]\)], "Input"], Cell[BoxData[ \({40.999999999999986`, 0.7808688094430304`, 0.19999999999999996`}\)], "Output"] }, Open ]], Cell[BoxData[ \(\(ups = {\[Alpha] \[Rule] 41. , \[Beta] \[Rule] 0.780869, \[CapitalDelta] \[Rule] 0.2};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \({\[Alpha], \[Beta], \[CapitalDelta]} //. Join[TGrep, Downstep]\)], "Input"], Cell[BoxData[ \({40.999999999999986`, 0.7808688094430304`, \(-0.3333333333333333`\)}\)], "Output"] }, Open ]], Cell[BoxData[ \(\(downs = {\[Alpha] \[Rule] 41. , \[Beta] \[Rule] 0.780869, \[CapitalDelta] \[Rule] \(-0.333333\)};\)\)], "Input"], Cell["\<\ Define the concentration as an expansion in the un functions:\ \>", "Text"], Cell[BoxData[ \(Cu[z_, T_, Step_] := \((b0[T] u[z, 0, 1\/\@\(2\ a2f[T]\)] + b1[T] u[z, 1, 1\/\@\(2\ a2f[T]\)] + b2[T] u[z, 2, 1\/\@\(2\ a2f[T]\)])\) /. Step\)], "Input"], Cell["\<\ The pure diffusive dispersion is, in the scaled coordinates, just a static \ gaussian:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Cdiff[z_] = u[z, 0, 1/\@2]\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\^\(-z\^2\)\/\@\[Pi]\)], "Output"] }, Open ]], Cell["\<\ Compare these two for a time shortly after the peak penetrates the step. Note \ that Z2 was taken as 0, i.e. we are actually plotting (z-Z2) here, which \ amounts to moving the \"camera\" with the peak. 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To look at that in more detail, use the approach \ leading up to equation 17.62. Choose a tolerance \[Epsilon] << 1 and \ determine the time parameter T\[Epsilon] when the parameter under \ consideration will have approached its large T asymptote to within \ \[Epsilon]. \nThis time scale T\[Epsilon] corresponds to a length scale L \ by\n", Cell[BoxData[ \(TraditionalForm\`L = \(V2\ \((t - \[Theta]1)\) = \(V2\ \((\[Theta]1 - t0)\) \((T\^2 - 1)\) = V2\ \((\[Sigma]1\^2/\[Gamma]\^2)\) \((T\[Epsilon])\)\^2\)\)\)]], " where \[Sigma]1 represents the variance of the plume when it reaches the \ step at x = x1, and in the last step the fact was used that since T\[Epsilon] \ represents the large time behaviour, only leading order terms in T are \ retained. So let us determine expressions for T\[Epsilon] from each of the \ critical variables. Start with a2:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[Series[a2, {T, \[Infinity], 2}]]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{ "1", "+", \(\(\[CapitalDelta]\ \((\[Alpha]\^2\ \((\(-4\) + \ \[CapitalDelta])\) + \[CapitalDelta] - 2\ \[Alpha]\ \[CapitalDelta])\)\ \((1\/T)\)\^2\)\/\((\ \[Alpha] + \[CapitalDelta])\)\^2\), "+", InterpretationBox[\(O[1\/T]\^3\), SeriesData[ T, DirectedInfinity[ 1], {}, 0, 3, 1]]}], SeriesData[ T, DirectedInfinity[ 1], {1, 0, Times[ \[CapitalDelta], Power[ Plus[ \[Alpha], \[CapitalDelta]], -2], Plus[ Times[ Power[ \[Alpha], 2], Plus[ -4, \[CapitalDelta]]], \[CapitalDelta], Times[ -2, \[Alpha], \[CapitalDelta]]]]}, 0, 3, 1]]], "Output"] }, Open ]], Cell[TextData[{ "The asymptote is 1, so the second term determines T\[Epsilon]. To \ determine its essential behaviour we may take \[Alpha] >>1 (that was the case \ for the examples, and anyway it was shown that the expressions are insenstive \ to \[Alpha] so we can just as well take a large value) and we also simplify \ to the case that |\[CapitalDelta]| << 1. Then the second term simplifies to \ ", Cell[BoxData[ \(TraditionalForm\`\(\(-4\) \[CapitalDelta]\)\/T\^2 = \[Epsilon]\)]], " so ", Cell[BoxData[ \(TraditionalForm\`\((T\[Epsilon])\)\^2 = \(4 \[CapitalDelta]\)\/\ \[Epsilon]\)]], " (the sign of \[Epsilon] is irrelevant) and hence \n\t\t\t", Cell[BoxData[ \(TraditionalForm\`L = 4 V2 \( \[Sigma]1\^2\/\[Gamma]\^2\) \ \((\[CapitalDelta]\/\[Epsilon])\)\)]], "\nas obtained previously (equation 17.62).\nThis, then is the length scale \ for the annealing of the Gaussian exponent." }], "Text"], Cell["\<\ Now do the same for b0; its difference from its asymptote represents the \ total symmetric distortions. Its series is:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[ Series[b0[T], {T, \[Infinity], 4}], {T > 0, \[Alpha] > 1, \[CapitalDelta] > \(-1\), \[CapitalDelta] < 1}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{ "1", "-", \(\(\((\[CapitalDelta]\^2\ \((1 - \[Alpha]\ \((\(-3\) + \ \[CapitalDelta])\) + \[CapitalDelta])\)\^2)\)\ \((1\/T)\)\^2\)\/\(2\ \ \((\((\(-1\) + \[CapitalDelta])\)\^2\ \((\[Alpha] + \ \[CapitalDelta])\)\^2)\)\)\), "+", InterpretationBox[\(O[1\/T]\^4\), SeriesData[ T, DirectedInfinity[ 1], {}, 0, 4, 1]]}], SeriesData[ T, DirectedInfinity[ 1], {1, 0, Times[ Rational[ -1, 2], Power[ Plus[ -1, \[CapitalDelta]], -2], Power[ \[CapitalDelta], 2], Power[ Plus[ \[Alpha], \[CapitalDelta]], -2], Power[ Plus[ 1, Times[ -1, \[Alpha], Plus[ -3, \[CapitalDelta]]], \[CapitalDelta]], 2]]}, 0, 4, 1]]], "Output"] }, Open ]], Cell[TextData[{ "Taking the same estimates for \[Alpha] and \[CapitalDelta] we now have ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon] = \(3 \[CapitalDelta]\^2\)\/\(2 \ T\^2\)\)]], " and that leads to \n\t\t", Cell[BoxData[ \(TraditionalForm\`L = \(\(3 V2\)\/2\) \(\[Sigma]1\^2\/\[Gamma]\^2\) \((\[CapitalDelta]\^2\ \/\[Epsilon])\)\)]], "\nThe significant difference here is that for small steps, the length \ scale is smaller by a factor \[CapitalDelta] for symmetric distortions to die \ away than that for the exponent.\n\nWe would expect to find the same result \ from the lowest order of the symmetric distortions, i.e. the b2 term. \ However, that does not happen:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[ Series[b2[T], {T, \[Infinity], 2}], {T > 0, \[Alpha] > 1, \[CapitalDelta] > \(-1\), \[CapitalDelta] < 1}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{\(\((\[Beta]\^2\ \[CapitalDelta]\ \((3\ \[CapitalDelta]\^3\ \ \((1 + 35\ \[CapitalDelta] + 35\ \[CapitalDelta]\^2 + \[CapitalDelta]\^3)\) + 4\ \[Alpha]\^3\ \[CapitalDelta]\ \((98 - 22\ \[CapitalDelta] - 97\ \[CapitalDelta]\^2 + 107\ \[CapitalDelta]\^3 - 49\ \[CapitalDelta]\^4 + 11\ \[CapitalDelta]\^5)\) + \[Alpha]\^4\ \((104 - 78\ \[CapitalDelta] - 94\ \[CapitalDelta]\^2 + 159\ \[CapitalDelta]\^3 - 91\ \[CapitalDelta]\^4 + 27\ \[CapitalDelta]\^5 - 3\ \[CapitalDelta]\^6)\) + 4\ \[Alpha]\ \[CapitalDelta]\^2\ \((3 + 92\ \[CapitalDelta] + 71\ \[CapitalDelta]\^2 - 37\ \[CapitalDelta]\^3 + 20\ \[CapitalDelta]\^4 - 7\ \[CapitalDelta]\^5 + 2\ \[CapitalDelta]\^6)\) + 2\ \[Alpha]\^2\ \[CapitalDelta]\ \((3 + 277\ \[CapitalDelta] + 89\ \[CapitalDelta]\^2 - 227\ \[CapitalDelta]\^3 + 170\ \[CapitalDelta]\^4 - 62\ \[CapitalDelta]\^5 + 14\ \[CapitalDelta]\^6)\))\)\ \((1\/T)\)\^2)\)/\((32\ \ \((\(-1\) + \[CapitalDelta])\)\^2\ \((1 + \[CapitalDelta])\)\ \((\[Alpha] + \ \[CapitalDelta])\)\^4)\)\), "+", InterpretationBox[\(O[1\/T]\^3\), SeriesData[ T, DirectedInfinity[ 1], {}, 2, 3, 1]]}], SeriesData[ T, DirectedInfinity[ 1], { Times[ Rational[ 1, 32], Power[ \[Beta], 2], Power[ Plus[ -1, \[CapitalDelta]], -2], \[CapitalDelta], Power[ Plus[ 1, \[CapitalDelta]], -1], Power[ Plus[ \[Alpha], \[CapitalDelta]], -4], Plus[ Times[ 3, Power[ \[CapitalDelta], 3], Plus[ 1, Times[ 35, \[CapitalDelta]], Times[ 35, Power[ \[CapitalDelta], 2]], Power[ \[CapitalDelta], 3]]], Times[ 4, Power[ \[Alpha], 3], \[CapitalDelta], Plus[ 98, Times[ -22, \[CapitalDelta]], Times[ -97, Power[ \[CapitalDelta], 2]], Times[ 107, Power[ \[CapitalDelta], 3]], Times[ -49, Power[ \[CapitalDelta], 4]], Times[ 11, Power[ \[CapitalDelta], 5]]]], Times[ Power[ \[Alpha], 4], Plus[ 104, Times[ -78, \[CapitalDelta]], Times[ -94, Power[ \[CapitalDelta], 2]], Times[ 159, Power[ \[CapitalDelta], 3]], Times[ -91, Power[ \[CapitalDelta], 4]], Times[ 27, Power[ \[CapitalDelta], 5]], Times[ -3, Power[ \[CapitalDelta], 6]]]], Times[ 4, \[Alpha], Power[ \[CapitalDelta], 2], Plus[ 3, Times[ 92, \[CapitalDelta]], Times[ 71, Power[ \[CapitalDelta], 2]], Times[ -37, Power[ \[CapitalDelta], 3]], Times[ 20, Power[ \[CapitalDelta], 4]], Times[ -7, Power[ \[CapitalDelta], 5]], Times[ 2, Power[ \[CapitalDelta], 6]]]], Times[ 2, Power[ \[Alpha], 2], \[CapitalDelta], Plus[ 3, Times[ 277, \[CapitalDelta]], Times[ 89, Power[ \[CapitalDelta], 2]], Times[ -227, Power[ \[CapitalDelta], 3]], Times[ 170, Power[ \[CapitalDelta], 4]], Times[ -62, Power[ \[CapitalDelta], 5]], Times[ 14, Power[ \[CapitalDelta], 6]]]]]]}, 2, 3, 1]]], "Output"] }, Open ]], Cell[TextData[{ "Clearly, this would put a factor ", Cell[BoxData[ \(TraditionalForm\`\[Beta]\^2\)]], " into the scale factor, and looking at the highest order term in \[Alpha], \ it has a ", Cell[BoxData[ \(TraditionalForm\`\[CapitalDelta]\)]], " rather than a ", Cell[BoxData[ \(TraditionalForm\`\[CapitalDelta]\^2\)]], " in it. That is the same discrepancy commented on before." }], "Text"], Cell["\<\ Finally, we look at the asymmetric distortions as represented by the lowest \ order one, b1. We have\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[ Series[b1[T], {T, \[Infinity], 2}], {T > 0, \[Alpha] > 1, \[CapitalDelta] > \(-1\), \[CapitalDelta] < 1}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{\(\((\[Beta]\ \[CapitalDelta]\ \((\[CapitalDelta]\ \((1 + 10\ \[CapitalDelta] + \[CapitalDelta]\^2)\) + 2\ \[Alpha]\ \[CapitalDelta]\ \((10 - \[CapitalDelta] - 2\ \[CapitalDelta]\^2 + \[CapitalDelta]\^3)\) + \ \[Alpha]\^2\ \((10 - 5\ \[CapitalDelta] - 4\ \[CapitalDelta]\^2 + 3\ \[CapitalDelta]\^3)\))\))\)/\((2\ \@\(2\ \[Pi]\)\ \ \((\[Alpha] + \[CapitalDelta])\)\^2\ \((\(-1\) + \[CapitalDelta]\^2)\)\ \ T)\)\), "+", InterpretationBox[\(O[1\/T]\^3\), SeriesData[ T, DirectedInfinity[ 1], {}, 1, 3, 1]]}], SeriesData[ T, DirectedInfinity[ 1], { Times[ Rational[ 1, 2], Power[ Times[ 2, Pi], Rational[ -1, 2]], \[Beta], \[CapitalDelta], Power[ Plus[ \[Alpha], \[CapitalDelta]], -2], Power[ Plus[ -1, Power[ \[CapitalDelta], 2]], -1], Plus[ Times[ \[CapitalDelta], Plus[ 1, Times[ 10, \[CapitalDelta]], Power[ \[CapitalDelta], 2]]], Times[ 2, \[Alpha], \[CapitalDelta], Plus[ 10, Times[ -1, \[CapitalDelta]], Times[ -2, Power[ \[CapitalDelta], 2]], Power[ \[CapitalDelta], 3]]], Times[ Power[ \[Alpha], 2], Plus[ 10, Times[ -5, \[CapitalDelta]], Times[ -4, Power[ \[CapitalDelta], 2]], Times[ 3, Power[ \[CapitalDelta], 3]]]]]]}, 1, 3, 1]]], "Output"] }, Open ]], Cell[TextData[{ "This time, the dominating term at large T behaves like ", Cell[BoxData[ \(TraditionalForm\`\(5 \[Beta]\ \[CapitalDelta]\)\/\(\(\@\(2 \ \[Pi]\)\) T\)\)]], " and so ", Cell[BoxData[ \(TraditionalForm\`T\[Epsilon]\^2 = \(\(25 \[Beta]\^2\)\/\(\(2\)\(\ \)\ \(\[Pi]\)\(\ \)\)\) \((\[CapitalDelta]\^2\/\[Epsilon]\^2)\)\)]], " which means that these distortions will die away over a longer scale than \ the exponent if the step ratio is larger than \[Epsilon], as seems a \ reasonable choice. But that also depends on \[Beta], ?!" }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Spacing of subsequent steps", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(\@\(1 + 2 \[Beta]\) /. downs\)], "Input"], Cell[BoxData[ \(1.6005430328485393`\)], "Output"] }, Open ]], Cell[TextData[{ "We have seen that the approximations for the bn are only satisfactory for \ times large enough that the gaussian plume has fully penetrated the first \ step. Full penetration can be reasonably defined by taking the half-width of \ a Gaussian as something between 2 and 3 standard deviations. Which is chosen, \ is a somewhat arbitrary choice. Here, to establish a lower limit on the step \ spacing, we make the least severe choice namely 2 standard deviations, i.e. \ the t penetration time parameter ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["Tp", FontSlant->"Italic"], "=", \(\@\(1 + 2 \[Beta]\)\)}], TraditionalForm]]], ". \nIn order to be able to calculate the concentration on the other side \ of a subsequent step, placed at a z-value \[Zeta] , it follows that the plume \ should not reach \[Zeta] at a time earlier than ", StyleBox["Tp. ", FontSlant->"Italic"], "This means that the peak value Z2 should still be within 2 standard \ deviations on the upstream side of \[Zeta], and if we take the standard \ deviation of the main b0 component of the peak as the measure, this has ", Cell[BoxData[ \(TraditionalForm\`\[Sigma] = 1/\@2\)]], ". So using the expression for Z2 we can calculate the smallest value of \ \[Zeta] by" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Zeta]min = Simplify[\(\((\(-\(1\/T\)\) + T)\)\ \((1 + \[CapitalDelta])\)\)\/\(\@2\ \ \((\[Beta] - \[Beta]\ \[CapitalDelta])\)\) + 2/\@\(2\ a2\) /. T \[Rule] \@\(1 + 2 \[Beta]\)]\)], "Input"], Cell[BoxData[ \(\(-\(\(1 + \[CapitalDelta]\)\/\(\@\(1\/2 + \[Beta]\)\ \((\(-1\) + \ \[CapitalDelta])\)\)\)\) + 2\/\@\(\(4\ \[Alpha]\^2\ \[CapitalDelta] + \((1 + 2\ \[Beta])\)\^2\ \((\ \[Alpha] + \[CapitalDelta])\)\^2 + \[Alpha]\ \((1 + 2\ \[Beta])\)\ \((\(-2\)\ \ \[CapitalDelta]\ \((1 + \[CapitalDelta])\) + \[Alpha]\ \((\(-1\) - 4\ \ \[CapitalDelta] + \[CapitalDelta]\^2)\))\)\)\/\(\[Beta]\ \((1 + 2\ \[Beta])\)\ \ \((\[Alpha] + \[CapitalDelta])\)\^2\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Zeta]min /. ups\)], "Input"], Cell[BoxData[ \(2.998001170610487`\)], "Output"] }, Open ]], Cell["\<\ Look at the time profile with which the main peak enters the second step, \ compared with the time variation of the bn coefficients :\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[{b0[T] //. ups, b1[T] //. ups, b2[T] //. ups, a2f[T] /. ups, u[\[Zeta]min - Z2f[T, ups], 0, 1/\@\(2\ a2\)] /. ups}, {T, 1, 5}, PlotRange 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- \ turquiose; a2 - blue; u0 function - purple.\ \>", "Text"], Cell["\<\ In both cases we see that in the time interval that is effectively selected \ in the boundary value problem time integration at the second step, the \ bn-coefficients are slowly varying compared to the peak function itself (the \ purple curve above). In particular, the dominating b0 coef is virtually \ constant; the other coefficients vary slightly, but make small contributions. \ This justifies taking the bn coefficients out of the integration at the \ second step, provided that the distance between steps 1 and 2 is not smaller \ than the value that corresponds to \[Zeta]min as calculated above.\ \>", "Text"], Cell["\<\ Notice that the calculated step spacing does not apply for \[CapitalDelta] = \ 0, as in that case there is no minimum time at which the bn become valid. \ Also, it is more specific than the length scale discussed previously, as it \ really is a limitation on the applicability of the approximations used rather \ than an intrinsic property.\ \>", "Text"], Cell[TextData[{ "To translate \[Zeta]min back to the actual physical variables, we write ", Cell[BoxData[ FormBox[ RowBox[{"\[Zeta]min", "=", RowBox[{\(\(x2 - x1\)\/\@\(2 \(\( \[Gamma]\^2\)(t - t0)\)\)\), "=", RowBox[{\(\(x2 - x1\)\/\@\(2 \(\( \[Gamma]\^2\)(\[Theta]1 - t0)\)\)\), FractionBox["1", StyleBox["Tp", FontSlant->"Italic"]]}]}]}], TraditionalForm]]], " where we used the fact that \[Zeta]min is to be evaluated at a time \ corresponding to ", StyleBox["T = Tp", FontSlant->"Italic"], ". Combining this with ", Cell[BoxData[ \(TraditionalForm\`\[Theta]2 = \[Theta]1 + \(x2 - x1\)\/V2\)]], " we can find that the corresponding limits on either \[Theta]2 or ", StyleBox["x2", FontSlant->"Italic"], " can be written as \n", Cell[BoxData[ \(TraditionalForm\`\(\[Theta]2 - \[Theta]1\)\/\(\[Theta]1 - t0\) \ \[GreaterEqual] \(\(1 - \[CapitalDelta]\)\/\(1 + \[CapitalDelta]\)\) 2 \[Beta]\/\@\(1 + 2 \[Beta]\) + \(2 \[Beta]\)\/\(1 + 2 \ \[Beta]\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(x2 - x1\)\/\(V1(\[Theta]1 - t0)\) \[GreaterEqual] 2 \[Beta] \@\( 1 + 2 \[Beta]\) + 2 \[Beta] 1 + \[CapitalDelta]\/\(1 - \[CapitalDelta]\)\)]], "\nTry these formulas out for the numerical examples:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\((\(\(1 - \[CapitalDelta]\)\/\(1 + \[CapitalDelta]\)\) \(2 \ \[Beta]\)\/\@\(1 + 2 \[Beta]\) + \(2 \[Beta]\)\/\(1 + 2 \[Beta]\))\) /. ups\)], "Input"], Cell[BoxData[ \(1.260143406555709`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\((\(\(1 - \[CapitalDelta]\)\/\(1 + \[CapitalDelta]\)\) \(2 \ \[Beta]\)\/\@\(1 + 2 \[Beta]\) + \(2 \[Beta]\)\/\(1 + 2 \[Beta]\))\) /. downs\)], "Input"], Cell[BoxData[ \(2.561148720371945`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\((2 \[Beta] \@\( 1 + 2 \[Beta]\) + 2 \[Beta] \( 1 + \[CapitalDelta]\)\/\(1 - \[CapitalDelta]\) /. ups)\)\)], "Input"], Cell[BoxData[ \(4.842235875034812`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\((2 \[Beta] \@\( 1 + 2 \[Beta]\) + 2 \[Beta] \( 1 + \[CapitalDelta]\)\/\(1 - \[CapitalDelta]\) /. downs)\)\)], "Input"], Cell[BoxData[ \(3.2804984606867085`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(V1 \((\[Theta]1 - t0)\) //. Upstep\)], "Input"], Cell[BoxData[ \(0.82`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(V1 \((\[Theta]1 - t0)\) //. Downstep\)], "Input"], Cell[BoxData[ \(0.82`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \( .82*3.2805\)], "Input"], Cell[BoxData[ \(2.69001`\)], "Output"] }, Open ]], Cell["\<\ Notice that the distance to the second step is specified completely in terms \ of the parameters for zone 1 and step 1. This is basically because \ penetrating step 1 can be determined by only looking at the input into step 1 \ from the upstream side. So the minimal step spacing for the next step beyond would be 0.82*4.84 = 4 \ units for the upstep, and 0.82*3.28 = 2.7 units for the downstep. In the \ calculations of stochasflow2.nb, the steps were placed at x1 = 1 and x2 = 4 \ i.e. at a spacing of 3 units in both cases. These values were merely chosen \ as seeming reasnable in comparison to the plume size, but are in quite good \ agreement with the formulas derived here. However, when studying the fluctuations in stochasflow2.nb, the 3rd step was \ placed at x3 = 7 for both the downupdown and updownup fluctuations. However, \ for the first case we now see that a much larger spacing for step 3 would \ have been appropriate to allow full penetration of step 2. That may well be \ why there was a substantial normalisation problem, and in general why the \ results for the two fluctuations differed.\ \>", "Text"], Cell["\<\ It is much harder to justify ignoring the variation of a2 with time. It \ certainly varies substantially over the T range discussed above - see the \ blue curves. However, it would be very hard to include this variation, as it \ would upset the basis of the approach used to solve the boundary value \ problem. In eq. 18.1, the two exponential factors would now depend on t' in \ different ways and that would invalidate the transformation to f used there, \ with severe consequences for everything done so far. The only tractable way seems to be to treat a2 in the same way as the bn's, \ i.e. take its value at a suitably chosen fixed time. An obvious candidate \ would be the (deterministic) time the peak reaches the next boundary. \ However, because of dispersion and indeed the time dependence of a2 that is \ not in fact the time at which the time profile at the step reaches its \ maximum; it would be even better to determine the timing of this peak value. \ If that is done, we can once more in effect make a series expansion of the \ time peak about this value. That justifies the neglect of time dependence \ because for time values either side of the peak, the exponential factor in \ the derivative will suppress any deviation. We now compare the performance of \ the two choices for the example steps.\ \>", "Text"], Cell["\<\ First, determine the arrival time at step 2 and the vallue of the \ corresponding T2, which is called star, and subsequently used as the starting \ guess for the actual peak time tpeak::\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\[Theta]2 := \[Theta]1 + \[Zeta]min \@\(\( 2 \( \[Gamma]\^2\) \((\ \[Theta]1 - t0)\) \((1 + 2 \[Beta])\)\)\/V2\^2\);\)\)], "Input"], Cell[BoxData[ \(General::"spell1" \(\(:\)\(\ \)\) "Possible spelling error: new symbol name \"\!\(\[Theta]2\)\" is \ similar to existing symbol \"\!\(\[Phi]2\)\"."\)], "Message"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(start = Simplify[\@\(\(\[Theta]2 - t0\)\/\(\[Theta]1 - t0\)\) //. 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