(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 753911, 13349] NotebookOptionsPosition[ 747572, 13150] NotebookOutlinePosition[ 748383, 13179] CellTagsIndexPosition[ 748251, 13173] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"2", "-", "2"}]], "Input", CellChangeTimes->{{3.523711971538747*^9, 3.523711983368647*^9}}], Cell[BoxData["0"], "Output", CellChangeTimes->{3.5237119853084*^9, 3.523723509831514*^9}] }, Open ]], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.523711989097267*^9}], Cell[CellGroupData[{ Cell["Random Walks on the Plane", "Title", CellChangeTimes->{{3.523711994222374*^9, 3.523712001185648*^9}, { 3.5237120562567253`*^9, 3.5237120683294697`*^9}}], Cell["\<\ Nicholas Wheeler 30 August 2011\ \>", "Text", CellChangeTimes->{{3.5237120750107183`*^9, 3.523712086664295*^9}}, FontSize->10], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.5237120978028507`*^9}], Cell[CellGroupData[{ Cell["Introduction", "Subsection", CellChangeTimes->{{3.523723529970912*^9, 3.523723533897232*^9}}], Cell["\<\ The following material is quoted from \"Simulated Random Walks\" (Notebook \ dated 4 April 2008)\ \>", "Text", CellChangeTimes->{{3.523723550199059*^9, 3.523723618196224*^9}, 3.52372366070949*^9}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.5237236360701857`*^9}] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Steps of unit length, random turn angles uniformly distributed\ \>", "Subsection", CellChangeTimes->{{3.523723698875144*^9, 3.523723724218607*^9}}], Cell["\<\ The following commands generate the steps that contribute to a population of \ four such 1000-step walks:\ \>", "Text", 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Let it, in particular, be \ determined by a distribution with a very \"long tail,\" so that occasional \ steps are exceptionally long. Such random walks (or \"flights\") are used to \ gain insight into the mechanism that underlies the departures from the \"law \ of diffusion\" that are encountered in a wide assortment of physical (+ \ biological, financial) applications. \ \>", "Text", CellChangeTimes->{{3.523737951318297*^9, 3.5237380145827208`*^9}, { 3.5237380720932817`*^9, 3.523738248452717*^9}}], Cell["To illustrate the point, I adopt the distribution law", "Text", CellChangeTimes->{{3.5237383245373096`*^9, 3.523738341766025*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Levy", "[", "x_", "]"}], ":=", RowBox[{"PDF", "[", RowBox[{ RowBox[{"FRatioDistribution", "[", RowBox[{"3", ",", "3.9999"}], "]"}], ",", "x"}], "]"}]}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"Levy", "[", "x", "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "5"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.523738373161018*^9, 3.5237383916741877`*^9}}], Cell[BoxData[ GraphicsBox[{{}, {}, {Hue[0.67, 0.6, 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