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Cell[CellGroupData[{

Cell[BoxData[{\(Clear[r, x, rd, xd, r0, rdot0, x0, xdot0, b, z, epsilon, 
      smax, BC, Udot, Vdot, f]\), "\[IndentingNewLine]", 
    RowBox[{\(Off[General::"\<spell1\>"]\), "\[IndentingNewLine]", 
      "\[IndentingNewLine]", 
      StyleBox[\( (*Define\ the\ constants\ in\ the\ potential\ and\ the\ \
boundary\ conditions . \ 
              The\ parameter\ b\ controls\ the\ relative\ height\ of\ the\ \
potential\ wells\ and\ the\ parameter\ z\ controls\ the\ height\ of\ the\ \
false\ vacuum\ well . \ For\ a\ given\ potential, \ 
          there\ is\ a\ unique\ set\ of\ boundary\ conditions\ corresponding\ \
to\ the\ instanton . \ 
              Samples\ are\ given\ in\ the\ next\ cell\ below\ and\ can\ be\ \
copied\ and\ pasted\ between\ the\ lines\ \(\(below\)\(.\)\)*) \),
        FontColor->RGBColor[1, 0, 0]], "\[IndentingNewLine]", 
      "\[IndentingNewLine]", 
      StyleBox[\( (*To\ begin, \ 
          copy\ some\ sample\ data\ and\ paste\ between\ the\ \
\(\(lines\)\(.\)\)*) \),
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      RowBox[{
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        StyleBox[\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(\
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(*\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(---\)\)\)\
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      StyleBox[\( (*Define\ the\ auxiliary\ \(\(function\)\(.\)\)*) \),
        FontColor->RGBColor[1, 0, 0]], 
      "\[IndentingNewLine]"}], "\[IndentingNewLine]", 
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      "\[IndentingNewLine]", "\[IndentingNewLine]", 
      StyleBox[\( (*Find\ the\ extrema\ of\ the\ \(potential : \ 
              s1 < s2 < \(\(s3\)\(.\)\)\)*) \),
        FontColor->RGBColor[1, 0, 0]], 
      "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(Udot[
          x_] := \(-\((\(-x\) - b\ x\^2 + x\^3)\)\);\), "\[IndentingNewLine]",
   
    RowBox[{\(Uddot[x_] := \(-\((\(-1\) - 2\ b\ x + 3\ x\^2)\)\);\), 
      "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(s1 = \(Sort[
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            Less]\)[\([1]\)];\), "\[IndentingNewLine]", \(s2 = \(Sort[
            p /. Solve[Udot[p] \[Equal] 0, p], 
            Less]\)[\([2]\)];\), "\[IndentingNewLine]", 
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      "\[IndentingNewLine]", 
      StyleBox[\( (*Plot\ the\ potential*) \),
        FontColor->RGBColor[1, 0, 0]], 
      "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(V[x_] := 
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        x_] := \(-V[x]\)\), "\[IndentingNewLine]", 
    RowBox[{\(Plot[V[x], {x, \(-2\), 2}, 
          AxesLabel \[Rule] {"\<x\>", "\<V[x]\>"}];\), "\[IndentingNewLine]", 
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      StyleBox[\( (*The\ remaining\ initial\ conditions\ are\ r[s = 0] = 0, \ 
          xdot[s = 0] = 0, \ 
          and\ rdot[
                s = 0] = \(1. \ Unfortunately\ the\ numerics\ cannot\ deal\ \
with\ these\ \((the\ friction\ term\ is\ undefined)\) . \ 
                  We\ define\ an\ interval\ in\ x\ of\ 10^\((\(-16\))\)\ \
around\ the\ initial\ position\ \((limited\ by\ machine\ precision)\)\ where\ \
a\ Taylor\ expansion\ to\ first\ order\ can\ be\ used . \ 
                  The\ equation\ of\ motion\ for\ r\ is\ r[s] = s\), \ 
          and\ the\ equation\ of\ motion\ for\ x\ is\ x[s]\  = \ 
            x0\  - \ Udot[x0]\/8\ s\^2 . \ 
                  We\ use\ these\ equations\ to\ solve\ for\ the\ initial\ \
conditions\ NEAR\ \(\(x0\)\(.\)\)*) \),
        FontColor->RGBColor[1, 0, 0]], 
      "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(x0 = 
        x0 - 10\^\(-16\);\), "\[IndentingNewLine]", \(r0 := 
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        1;\), "\[IndentingNewLine]", 
    RowBox[{\(xdot0 := N[\(-\(Udot[x0]\/4\)\)\ r0];\), "\[IndentingNewLine]", 
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      StyleBox[\( (*These\ are\ the\ boundary\ conditions\ we\ will\ use\ for\
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      StyleBox[\( (*We\ define\ a\ system\ of\ 4\ 1  
            st\ order\ equations\ from\ the\ 2  
            nd\ order\ equations\ for\ r \((s)\)\ and\ x \(\((s)\)\(.\)\)*) \
\),
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                3\ rd[s]\/r[s]\ xd[s] + Udot[x[s]] == 0, 
            rd[s] \[Equal] \(r'\)[s], xd[s] \[Equal] \(x'\)[s]};\), 
      "\[IndentingNewLine]", "\[IndentingNewLine]", "\[IndentingNewLine]", 
      StyleBox[\( (*This\ function\ numerically\ integrates\ the\ equations\ \
of\ motion\ to\ find\ r \((s)\) . \ The\ inputs\ are\ smax, \ 
          which\ is\ the\ maximum\ time\ to\ integrate\ to, \ and\ BC, \ 
          which\ is\ a\ list\ containing\ the\ boundary\ \
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            BC_] := \((solution = 
              NDSolve[Join[Eqns, BC], {r, x, rd, xd}, {s, 0, smax}, 
                MaxSteps \[Rule] 
                  200000]; \(\(Evaluate[{r[t]} /. 
                    solution]\)[\([1]\)]\)[\([1]\)])\);\), 
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      StyleBox[\( (*This\ function\ numerically\ integrates\ the\ equations\ \
of\ motion\ to\ find\ x \((s)\) . \ The\ inputs\ are\ smax, \ 
          which\ is\ the\ maximum\ time\ to\ integrate\ to, \ and\ BC, \ 
          which\ is\ a\ list\ containing\ the\ boundary\ \
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                    solution]\)[\([1]\)]\)[\([1]\)])\);\), 
      "\[IndentingNewLine]", "\[IndentingNewLine]", "\[IndentingNewLine]", 
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          Plot[r[s, smax, BC], {s, 0, smax}, 
            PlotStyle \[Rule] {{Thickness[ .009], Hue[0. ]}}, 
            PlotRange \[Rule] All, 
            AxesLabel \[Rule] {"\<s\>", "\<r[s]\>"}];\), 
      "\[IndentingNewLine]", "\[IndentingNewLine]", 
      StyleBox[\( (*Plot\ x \((s)\)\ between\ its\ two\ \(\(zeros\)\(.\)\)*) \
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        FontColor->RGBColor[1, 0, 0]], 
      "\[IndentingNewLine]"}], "\[IndentingNewLine]", 
    RowBox[{\(p3 = 
          Show[Plot[x[s, smax, BC], {s, 0, smax}, 
              PlotStyle \[Rule] {{Thickness[ .009], Hue[0.6]}}, 
              PlotRange \[Rule] All, AxesLabel \[Rule] {"\<s\>", "\<x[s]\>"}, 
              DisplayFunction \[Rule] Identity\ ], 
            Graphics[{{Dashing[{ .005,  .005}], 
                  Line[{{0, s1}, {smax, s1}}]}, {Dashing[{ .005,  .005}], 
                  Line[{{0, s2}, {smax, s2}}]}, {Dashing[{ .005,  .005}], 
                  Line[{{0, s3}, {smax, s3}}]}}], \ 
            PlotRange \[Rule] {s1 -  .1, s3 +  .1}, 
            DisplayFunction \[Rule] $DisplayFunction];\), 
      "\[IndentingNewLine]", "\[IndentingNewLine]", 
      StyleBox[\( (*Calculate\ the\ action\ of\ the\ \(\(instanton\)\(.\)\)*) \
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    RowBox[{\(SI = \(-4\)\ Pi\^2\ NIntegrate[\((r[s, smax, BC])\)\^3\ U[
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                smax}];\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", 
    RowBox[{\(SBG = \(8\ Pi\^2\)\/\(3\ epsilon\^4\)\ 1\/U[s1];\), 
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Cell[BoxData[
    \("The instanton action is:"\)], "Print"],

Cell[BoxData[
    \(\(-2485.833841014521`\)\)], "Print"],

Cell[BoxData[
    \("The total Euclidean action is:"\)], "Print"],

Cell[BoxData[
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Cell[BoxData[
    StyleBox[\(Some\ example\ initial\ conditions\ for\ single - 
        pass\ instantons . \ 
            Copy\ and\ paste\ in\ the\ cell\ \(\(above\)\(.\)\)\),
      "Section",
      FontColor->RGBColor[0, 1, 0]]], "Input"],

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      "\[IndentingNewLine]", 
      RowBox[{\(b :=  .3;\), "\[IndentingNewLine]", \(z :=  .1;\), 
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        "\[IndentingNewLine]", \(b :=  .3;\), 
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        "\[IndentingNewLine]", \(b :=  .3;\), 
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        "\[IndentingNewLine]", \(b :=  .3;\), 
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        StyleBox["\[IndentingNewLine]",
          FontColor->RGBColor[1, 0, 0]], "\[IndentingNewLine]", \(b :=  .3;\),
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        "\[IndentingNewLine]", \(epsilon :=  .85;\), 
        "\[IndentingNewLine]", \(x0 := 0.9161368121027716;\), 
        "\[IndentingNewLine]", \(smax = 8.711289175379738;\), 
        "\[IndentingNewLine]", 
        StyleBox["\[IndentingNewLine]",
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--\(--\(--\(---\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)*) \),
          FontColor->RGBColor[1, 0, 0]], "\[IndentingNewLine]", 
        "\[IndentingNewLine]", \(b :=  .3;\), 
        "\[IndentingNewLine]", \(z :=  .0001;\), 
        "\[IndentingNewLine]", \(epsilon :=  .85;\), 
        "\[IndentingNewLine]", \(x0 := 0.9166596628532317;\), 
        "\[IndentingNewLine]", \(smax = 8.713066498859538;\), 
        "\[IndentingNewLine]", "\[IndentingNewLine]", 
        StyleBox[\( (*\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(\
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        StyleBox[\( (*\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(\
--\(--\(--\(---\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)*) \),
          FontColor->RGBColor[1, 0, 0]], "\[IndentingNewLine]", 
        StyleBox["\[IndentingNewLine]",
          FontColor->RGBColor[1, 0, 0]], \(b :=  .3;\), 
        "\[IndentingNewLine]", \(z :=  .00001;\), 
        "\[IndentingNewLine]", \(epsilon :=  .85;\), 
        "\[IndentingNewLine]", \(x0 := 0.91670796518813;\), 
        "\[IndentingNewLine]", \(smax = 8.715448496063078;\), 
        "\[IndentingNewLine]", 
        StyleBox["\[IndentingNewLine]",
          FontColor->RGBColor[1, 0, 0]], 
        StyleBox[\( (*\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(\
--\(--\(--\(---\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)*) \),
          FontColor->RGBColor[1, 0, 0]], "\[IndentingNewLine]", 
        StyleBox[\( (*\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(\
--\(--\(--\(---\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)*) \),
          FontColor->RGBColor[1, 0, 0]], "\[IndentingNewLine]", 
        StyleBox["\[IndentingNewLine]",
          FontColor->RGBColor[1, 0, 0]], \(b :=  .05;\), 
        "\[IndentingNewLine]", \(z :=  .1;\), 
        "\[IndentingNewLine]", \(epsilon :=  .7;\), 
        "\[IndentingNewLine]", \(x0 := 0.871002797708976;\), 
        "\[IndentingNewLine]", \(smax = 8.599999999999999;\), 
        "\[IndentingNewLine]", "\[IndentingNewLine]", 
        StyleBox[\( (*\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(\
--\(--\(--\(---\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)*) \),
          FontColor->RGBColor[1, 0, 0]], 
        StyleBox["\[IndentingNewLine]",
          FontColor->RGBColor[1, 0, 0]], 
        StyleBox[\( (*\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(\
--\(--\(--\(---\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)*) \),
          FontColor->RGBColor[1, 0, 0]], "\[IndentingNewLine]", 
        "\[IndentingNewLine]", \(b :=  .05;\), 
        "\[IndentingNewLine]", \(z :=  .01;\), 
        "\[IndentingNewLine]", \(epsilon :=  .7;\), 
        "\[IndentingNewLine]", \(x0 := 0.9060392763395042;\), 
        "\[IndentingNewLine]", \(smax = 9.099999999999999;\), 
        "\[IndentingNewLine]", "\[IndentingNewLine]", 
        StyleBox[\( (*\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(\
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          FontColor->RGBColor[1, 0, 0]], 
        StyleBox["\[IndentingNewLine]",
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        StyleBox[\( (*\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(\
--\(--\(--\(---\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)*) \),
          FontColor->RGBColor[1, 0, 0]], "\[IndentingNewLine]", 
        "\[IndentingNewLine]", \(b :=  .05;\), 
        "\[IndentingNewLine]", \(z :=  .001;\), 
        "\[IndentingNewLine]", \(epsilon :=  .7;\), 
        "\[IndentingNewLine]", \(x0 := 0.909402656808811;\), 
        "\[IndentingNewLine]", \(smax = 9.099999999999999;\), 
        "\[IndentingNewLine]", "\[IndentingNewLine]", 
        StyleBox[\( (*\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(\
--\(--\(--\(---\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)*) \),
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--\(--\(--\(---\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)*) \),
          FontColor->RGBColor[1, 0, 0]], "\[IndentingNewLine]", 
        "\[IndentingNewLine]", \(b :=  .05;\), 
        "\[IndentingNewLine]", \(z :=  .0001;\), 
        "\[IndentingNewLine]", \(epsilon :=  .7;\), 
        "\[IndentingNewLine]", \(x0 := 0.9097361611585042;\), 
        "\[IndentingNewLine]", \(smax = 9.15;\), 
        StyleBox["\[IndentingNewLine]",
          FontColor->RGBColor[1, 0, 0]], "\[IndentingNewLine]", 
        StyleBox[\( (*\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(\
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        StyleBox[\( (*\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(\
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        StyleBox["\[IndentingNewLine]",
          FontColor->RGBColor[1, 0, 0]], \(b :=  .05;\), 
        "\[IndentingNewLine]", \(z :=  .1;\), 
        "\[IndentingNewLine]", \(epsilon :=  .9;\), 
        "\[IndentingNewLine]", \(x0 := 0.4505732202523388;\), 
        "\[IndentingNewLine]", \(smax = 6.4;\), 
        StyleBox["\[IndentingNewLine]",
          FontColor->RGBColor[1, 0, 0]], 
        StyleBox["\[IndentingNewLine]",
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        StyleBox["\[IndentingNewLine]",
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        "\[IndentingNewLine]", \(b :=  .05;\), 
        "\[IndentingNewLine]", \(z :=  .01;\), 
        "\[IndentingNewLine]", \(epsilon :=  .9;\), 
        "\[IndentingNewLine]", \(x0 := 0.5406508779247492;\), 
        "\[IndentingNewLine]", \(smax = 6.8;\), 
        StyleBox["\[IndentingNewLine]",
          FontColor->RGBColor[1, 0, 0]], "\[IndentingNewLine]", 
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          FontColor->RGBColor[1, 0, 0]], 
        StyleBox["\[IndentingNewLine]",
          FontColor->RGBColor[1, 0, 0]], 
        "\[IndentingNewLine]", \(b :=  .05;\), 
        "\[IndentingNewLine]", \(z :=  .001;\), 
        "\[IndentingNewLine]", \(epsilon :=  .9;\), 
        "\[IndentingNewLine]", \(x0 := 0.5490273355737542;\), 
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        StyleBox["\[IndentingNewLine]",
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        StyleBox["\[IndentingNewLine]",
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