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Notebook[{
Cell["Homework 1 \[LongDash] Solution", "Title"],

Cell["\<\
3.016 Mathematical Methods for Materials Scientists and Engineers
S. M. Allen
September 30, 2005\
\>", "Subsubtitle"],

Cell[CellGroupData[{

Cell["Individual Exercise I-1: ", "Subsection"],

Cell["\<\
First, recognize that you will need to use ListPlot to make the \
final plot of your data.  ListPlot produces a plot from (x,y) pairs that are \
stored in a Table.  The program you need to write has to produce output that \
will be written to a Table with the proper format.
The function Random[Integer] is used here; it returns either 0 or 1 with \
probability 1/2.
Quoting Charles Cantrell: \"All of the problem is contained in one line of \
input.  First I create a table of data members that contains the trial number \
and the number of required steps to reach the endpoint.  Then I plot the data \
using the ListPlot command.  The While loop runs until the position is either \
100 or -100, and the Table command is run 100 times--giving us 100 data \
points.\"\
\>", "Text"],

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            While[Abs[location] < 100, 
              If[Random[Integer] \[Equal] 0, location = location + 1, 
                location = location - 1]; timestep = timestep + 1]; 
            timestep}, {i, 100}];\)\), "\n", 
    \(ListPlot[data, 
      AxesLabel \[Rule] {"\<Trial number\>", "\<TimeSteps\>"}]\)}], "Input"],

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\>"],
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0.291263, 245.816}}],

Cell[BoxData[
    TagBox[\(\[SkeletonIndicator]  Graphics  \[SkeletonIndicator]\),
      False,
      Editable->False]], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Individual Exercise I-2:", "Subsection"],

Cell["\<\
Note: Some features of Omar Fabian's work were incorporated in the \
solution to this exercise.\
\>", "Text"],

Cell[CellGroupData[{

Cell["1.", "Subsubsection"],

Cell[TextData[StyleBox["Input the LJ potential, differentiate and set \
derivative == 0 to find minima.", "Text"]], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(LJ = a\/r^12 - b\/r^6\)], "Input"],

Cell[BoxData[
    \(a\/r\^12 - b\/r\^6\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(MinimaOfLJ\  = \ Solve[D[LJ, r] == \ 0, r]\)], "Input"],

Cell[BoxData[
    \({{r \[Rule] \(-\(\(2\^\(1/6\)\ a\^\(1/6\)\)\/b\^\(1/6\)\)\)}, {r \
\[Rule] \(2\^\(1/6\)\ a\^\(1/6\)\)\/b\^\(1/6\)}, {r \[Rule] \(-\(\(\((\(-1\))\
\)\^\(1/3\)\ 2\^\(1/6\)\ a\^\(1/6\)\)\/b\^\(1/6\)\)\)}, {r \[Rule] \
\(\((\(-1\))\)\^\(1/3\)\ 2\^\(1/6\)\ a\^\(1/6\)\)\/b\^\(1/6\)}, {r \[Rule] \
\(-\(\(\((\(-1\))\)\^\(2/3\)\ 2\^\(1/6\)\ a\^\(1/6\)\)\/b\^\(1/6\)\)\)}, {r \
\[Rule] \(\((\(-1\))\)\^\(2/3\)\ 2\^\(1/6\)\ a\^\(1/6\)\)\/b\^\(1/6\)}}\)], \
"Output"]
}, Open  ]],

Cell["\<\
Note that there are six roots but only one of them is \"physical\" \
\[LongDash] i.e., real and positive (the second root). Use this to set rmin. \
Note how Replace is used to extract the value of the second root.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Minr = r /. \ MinimaOfLJ[\([2]\)]\)], "Input"],

Cell[BoxData[
    \(\(2\^\(1/6\)\ a\^\(1/6\)\)\/b\^\(1/6\)\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["2. ", "Subsubsection"],

Cell["Replace r with rmin in LJ to get EMin = LJ(rmin).", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(MinE = \ \ LJ /. r -> Minr\)], "Input"],

Cell[BoxData[
    \(\(-\(b\^2\/\(4\ a\)\)\)\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["3.", "Subsubsection"],

Cell["\<\
The Solve function can be used to eliminate specific variables from \
a set of equations.  Here we have the equation for rMin in terms of (a,b) and \
the equation for EMin in terms of (a,b).  Solve is used to find expressions \
for a and b in terms of rMin and EMin.\
\>", "Text"],

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
    \(NewVars = 
      Solve[{rMin \[Equal] Minr, EMin \[Equal] MinE}, {a, b}]\)], "Input"],

Cell[BoxData[
    \({{a \[Rule] \(-EMin\)\ rMin\^12, 
        b \[Rule] \(-2\)\ EMin\ rMin\^6}}\)], "Output"]
}, Open  ]],

Cell["\<\
All that remains is to Replace a and b with these new \
variables.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(LJModified = LJ /. NewVars[\([1]\)]\)], "Input"],

Cell[BoxData[
    \(\(2\ EMin\ rMin\^6\)\/r\^6 - \(EMin\ rMin\^12\)\/r\^12\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["4.", "Subsubsection"],

Cell["The force is obtained by differentiation", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(Force\  := \ \(-\[PartialD]\_r\ LJModified\)\), "\[IndentingNewLine]", 
    \(Force\)}], "Input"],

Cell[BoxData[
    \(\(12\ EMin\ rMin\^6\)\/r\^7 - \(12\ EMin\ rMin\^12\)\/r\^13\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["5.", "Subsubsection"],

Cell["\<\
Define new \"normalized\" variables rNorm = r / rMin and LJNorm = \
LJ / EMin and use them to define a normalized LJ function LJNorm(rNorm). \
Recongnizing that I will eventually want to plot these functions, I define \
them with delayed assignment :=\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(LJNorm := \ 
      Simplify[\(-\((LJModified/EMin)\)\) /. rMin \[Rule] r/rNorm, 
        Assumptions \[Rule] {a > 0\  && b > 0}]\), "\[IndentingNewLine]", 
    \(LJNorm\)}], "Input"],

Cell[BoxData[
    \(1\/rNorm\^12 - 2\/rNorm\^6\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[{
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      Simplify[\(-\((Force\ rMin/EMin)\)\) /. rMin \[Rule] r/rNorm, 
        Assumptions \[Rule] {a > 0\  && b > 0}]\), "\[IndentingNewLine]", 
    \(ForceNorm\)}], "Input"],

Cell[BoxData[
    \(\(-\(\(12\ \((\(-1\) + rNorm\^6)\)\)\/rNorm\^13\)\)\)], "Output"]
}, Open  ]],

Cell["To demonstrate that LJNorm and ForceNorm are dimensionless:", "Text"],

Cell[BoxData[{
    \(\(EMinUnits = Mass\ Length^2/Time^2;\)\), "\[IndentingNewLine]", 
    \(\(ForceUnits = 
        Mass\ Length^2/\((Time^2\ Length)\);\)\), "\[IndentingNewLine]", 
    \(\(rUnits\  = Length;\)\), "\[IndentingNewLine]", 
    \(\(rMinUnits\  = \ Length;\)\)}], "Input"],

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Cell[BoxData[{
    \(\(rBarUnits\  = \ rUnits/rMinUnits;\)\), "\[IndentingNewLine]", 
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Cell[BoxData[
    \(True\)], "Output"]
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Cell[CellGroupData[{

Cell[BoxData[{
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        ForceUnits\ rMinUnits/EMinUnits;\)\), "\[IndentingNewLine]", 
    \(FBarUnits\  \[Equal] 1\)}], "Input"],

Cell[BoxData[
    \(True\)], "Output"]
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Cell[TextData[{
  "So, the variables ",
  Cell[BoxData[
      \(TraditionalForm\`r\&_\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`F\&_\)]],
  " are dimensionless."
}], "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["6.", "Subsubsection"],

Cell["\<\
Now for the plot. Note in the use of the DisplayFunction option to \
control whether or not the graphics are rendered to the screen.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(NewPlot\  = \ 
      Plot[{LJNorm, ForceNorm}, \ {rNorm, \ 0, \ 2}, \ 
        PlotRange\  \[Rule] \ {\(-4\), 
            4}, \ \ \ PlotStyle\  \[Rule] \ {{Thickness[ .01], \ 
              Hue[0.3]}, {Thickness[ .01], \ Hue[1]}}, 
        AxesLabel -> {"\<r/rMin\>", "\<Normalized LJ and F\>"}, 
        DisplayFunction \[Rule] Identity]\)], "Input"],

Cell[BoxData[
    TagBox[\(\[SkeletonIndicator]  Graphics  \[SkeletonIndicator]\),
      False,
      Editable->False]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

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    \(Show[NewPlot, Graphics[Text["\<Force\>", {1.5\ , \(-2\)}]], 
      Graphics[Text["\<Potential\>", { .6, 2}]], 
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I will use the same approach as above, using as my constants the \
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Use function Series to get the Taylor series representation, \
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Note that this expression is a parabolic fit to the bottom of the \
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The coefficient of the second-order term of HarmonicApprox is equal \
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\>", "Text"],

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