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Cell[CellGroupData[{
Cell["Understanding Statistical Physics", "Title"],
Cell["\<\
Project Suggestions for APAM1601
Due: April 4, 2012\
\>", "Subsubtitle",
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Cell["\<\
All projects must have\[Ellipsis]
Name:____________
Email:_____@columbia.edu\
\>", "Subsubtitle"],
Cell[CellGroupData[{
Cell["Introduction", "Section"],
Cell[TextData[{
"During the past few lectures, we've explored simple models for the \
evolution of complex systems interacting with random motions. Although these \
models are very simple, they can be used to develop insights to real physical \
systems, such as magnets and polymers (and proteins).\n\nFor your next \
project, I ask that you prepare a notebook that reports your investigations \
related to a problem or question in statistical physics and/or random \
numbers. I listed below two possible subjects each phrased like a homework \
problem, but I encourage you to look beyond the simple answer to the \
question. Like before, your notebook should contain ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" expressions and graphics which illustrate your solution. Try not to \
include a large number of repeated expressions. Instead, generate a table or \
graphic of your results. Also, consider collecting ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" expressions that are used repeatedly into a single ",
StyleBox["Module[...]",
FontWeight->"Bold"],
" that can be executed within a ",
StyleBox["Table[ ]",
FontWeight->"Bold"],
" statement or by mapping to elements of a list. \n\nIn all cases, format \
your notebook and include textual comments and descriptions. Your notebook \
need not be long. It should begin with an Introduction describing your \
problem, approach, and a brief abstract of your results. It should also be \
interesting and arrive at a definite conclusion.\n\nI've included two \
articles from ",
StyleBox["American Scientist",
FontSlant->"Italic"],
" (written by former editor Brian Hayes) that provide some background for \
your investigations. One is entitled \"The World in a Spin\" descibing the \
history and application of the Ising model for magnetism. The other is \
entitled \"How to Avoid Yourself\" and describes counting of possible \
self-avoiding random walks. Feel free to use an idea contained in these \
articles to motivate questions for you project. "
}], "Text"]
}, Open ]],
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Cell["\<\
Behavior of Materials at Critical Temperatures
(and Second-Order Phase Transitions)\
\>", "Section"],
Cell[TextData[{
"One characteristic of a second-order phase transitions is the divergence of \
thermodynamic and statistical properties at the critical temperature. For the \
Ising model for a magnetic material,these include specific heat, ",
StyleBox["C, ",
FontSlant->"Italic"],
"the magnetic susceptibility, \[Chi], and the characteristic length, \[Xi], \
of the correlation function ",
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"(",
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") = \[LeftAngleBracket]",
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" ",
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"\[RightAngleBracket]. These three parameters diverge with some power of 1/|",
StyleBox["T - ",
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Cell[BoxData[
FormBox[
SubscriptBox["T", "c"], TraditionalForm]]],
Cell[BoxData[
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". (An interesting quantitative puzzle is to find the value of \[Alpha] for \
the divergent quantities.)\n\nIn this problem, you are to explore the \
temperature dependences of the thermodynamic and statistical quantities of \
the Ising model near the critical temperature, ",
Cell[BoxData[
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". \n\nSome thermodynamic properties can be found simply by calculating the \
fluctuations of averages of the energy, ",
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", and of the magnetization, ",
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". For example, the ",
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" is related to energy fluctuations",
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"/ ",
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". The fluctuations in the magnetization of the lattice are proportional to \
the ",
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"\[Chi]",
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"."
}], "Text"],
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"There are many possible problems that you can explore using the ",
StyleBox["Ising[...]",
FontWeight->"Bold"],
" function that we defined in class. You could also modify this function to \
explore behvior described in Brian Hayes's article.\n\nBelow, I have \
suggested two directions to get you started. You may explore either or both, \
or nether, as you wish."
}], "Text"],
Cell[TextData[{
StyleBox["Suggestion 1",
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": Calculate the specific heat,",
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", as a function of ",
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" by estimating the derivative ",
StyleBox["C",
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" = ",
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". Compare this estimate with the calculations for ",
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" using calculations of the ",
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" in the lattice energy, ",
StyleBox[" C = ",
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"/ ",
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". Compare with your estimate using the derivative.\n\nNotice: The ",
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" of the fluctuations near the critical temperature, ",
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" \[Rule] ",
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", become stronger and also slower. (This is because they are physically \
larger.) This influences your estimations of \[LeftAngleBracket]",
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"\[RightAngleBracket] and \[LeftAngleBracket]",
Cell[BoxData[
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"\[RightAngleBracket].\n\nBe aware that the size an dimension of your \
simulations will significantly change your results. For a ",
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FontSlant->"Italic"],
" lattice, the size is enormous and the dimensions are three. With a \
computer, we can easily perform one dimensional studies; it is possible to do \
large two-dimensional studies; while 3D Ising studies are difficult but \
possible with small dimensions along the side. (You can even try a 4D \
example! However, so far, neither myself or any other FFSEAS student has \
submitted a 3D Ising simulation.) To detect a second order phase transition, \
you need at least two dimensions with a large lattice dimension. (Be sure you \
fully debug your expressions before executing large statistical computer \
runs!) The size of the lattice effects the size of your estimate for \
thermodynamical quantities. For the 2D Ising lattice, the maximum value of \
the specific heat scales like ",
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"~ ",
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"log ",
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"(or the specific heat per spin site, scales like log ",
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".) As you make your lattice larger, the divergence of the specific heat \
becomes easier to detect."
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Cell[TextData[{
StyleBox["Suggestion 2",
FontWeight->"Bold"],
": I really like Figure 1 in Brian Hayes's \"The World in a Spin\". It shows \
a snap-shot of a 2D Ising lattice with a temperature gradient. At the bottom, \
",
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" = ",
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"and at the top, ",
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" = ",
Cell[BoxData[
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". This allows a single lattice to perform a temperature scan! For this \
suggestion, re-write the ",
StyleBox["Ising[...]",
FontWeight->"Bold"],
" command with a linearly-varying temperature along one lattice dimension \
(instead of a constant). Perform studies like that listed in the suggestion \
above\[LongDash]except perform all of your temperature scans simultaneously. \
Of course, you will also need to re-write the ",
StyleBox["energy[...]",
FontWeight->"Bold"],
" command so that it returns a list of lattice energy along the axis of \
temperature variation. "
}], "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell["Random and Self-Avoiding Walks", "Section"],
Cell["\<\
Using our in-class notebook describing the \"pivot\" method for generating \
self-avoiding walks and Brian Hayes's article \"How to Avoid Yourself\", \
compare the statistics of (i) random walks, (ii) non-reversing walks, and \
(iii) self-avoiding walks. \
\>", "Text"],
Cell[TextData[{
"Your analyses will have several steps. First, you must verify that your \
three \"stepper\" functions work as designed and that you are able to \
construct ensembles of walks using the three techniques. Secondly, compute \
statistical averages and variations of your walks as a function of the length \
of the walks. The average position is the \"center of mass\", {",
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",",
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"}. The root-mean-square size of the walkers is \[CapitalDelta] \[Congruent] \
",
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TraditionalForm]}], "|"}], ")"}], "2"],
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". \[CapitalDelta] is a measure of the average \"radius\" of random walk \
process. The end-to-end length is ",
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" - ",
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"| is a measure of how far one can walk in ",
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"alculate averages of \[CapitalDelta] and ",
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" as a function of the number of steps, ",
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", for a large number of random walkers and for each of the three types of \
walks. We know that \[LeftAngleBracket]",
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" scale like ",
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" for an unbiased random walk. How do these quantities vary with ",
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" for the other two types of walks?"
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