(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 13.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 1294862, 24574] NotebookOptionsPosition[ 1260953, 23942] NotebookOutlinePosition[ 1261478, 23960] CellTagsIndexPosition[ 1261435, 23957] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["3.029 Spring 2022\[LineSeparator]Lectures 09 & 10 - 02/28/2022, \ 03/02/2022", "Subtitle", CellChangeTimes->{{3.8525512993398438`*^9, 3.8525513206118402`*^9}, { 3.852652054138073*^9, 3.8526520591301193`*^9}, {3.853194369726288*^9, 3.8531943739664793`*^9}, 3.8531971130005827`*^9, {3.853361889945813*^9, 3.853361893353859*^9}, {3.854455749568426*^9, 3.8544557604776297`*^9}, { 3.8545621484661007`*^9, 3.8545621501697083`*^9}, {3.85499650158072*^9, 3.854996504060568*^9}, 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CellChangeTimes->{{3.85499720266166*^9, 3.854997211629056*^9}},ExpressionUUID->"24e936e0-2a54-4f4c-9544-\ cefbca8db01f"], Cell[CellGroupData[{ Cell["Lattice Walk", "Subsubsection", CellChangeTimes->{{3.854997960905527*^9, 3.854997966466023*^9}},ExpressionUUID->"dbd57cd1-b13e-4f99-8d3e-\ 5586e859b372"], Cell[CellGroupData[{ Cell["\<\ Imagine we have some random walkers (molecules, vacancies, drunken sailors..)\ \ \>", "Item", CellChangeTimes->{{3.8549972292672*^9, 3.854997311725926*^9}, { 3.8549974904133167`*^9, 3.854997511507317*^9}},ExpressionUUID->"f15b584c-5efb-45b2-a824-\ 58cf9b8c4820"], Cell[CellGroupData[{ Cell["\<\ Which take steps on a square lattice using equal probabilities\ \>", "Subitem", CellChangeTimes->{{3.8549972292672*^9, 3.854997311725926*^9}, { 3.8549974904133167`*^9, 3.854997551065209*^9}, {3.8549976405652657`*^9, 3.854997640566683*^9}},ExpressionUUID->"e38f13d5-7af2-4058-a3ab-\ d8a5df37237d"], Cell[CellGroupData[{ Cell["\<\ First, let\[CloseCurlyQuote]s get a list of the possible steps (including \ staying still)\ \>", "Subsubitem", CellChangeTimes->{{3.8549972292672*^9, 3.854997311725926*^9}, { 3.8549974904133167`*^9, 3.854997551065209*^9}, {3.854997641945588*^9, 3.8549976751697617`*^9}},ExpressionUUID->"a36a1bbb-deb6-4c45-af2c-\ 2969bc1671b8"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"steps", "=", RowBox[{"Select", "[", RowBox[{ RowBox[{"Tuples", "[", RowBox[{ RowBox[{"Range", "[", RowBox[{ RowBox[{"-", "1"}], ",", "1"}], "]"}], ",", "2"}], "]"}], ",", RowBox[{ RowBox[{ RowBox[{"Norm", "[", "#", "]"}], "<=", "1"}], "&"}]}], "]"}]}]], "Input",\ CellChangeTimes->{{3.854997566628237*^9, 3.854997570921715*^9}, { 3.8549976298834057`*^9, 3.85499763187112*^9}}, CellLabel->"In[2]:=",ExpressionUUID->"c5fdac06-0aa9-4f7e-8441-995ffbc5ffe9"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", RowBox[{"-", "1"}]}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0"}], "}"}], 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Open ]], Cell[CellGroupData[{ Cell["\<\ We can now use RandomPoint to sample N points on this surface\ \>", "Item", CellChangeTimes->{{3.8550448822007008`*^9, 3.855044893824998*^9}, { 3.855045327030426*^9, 3.8550453458706017`*^9}},ExpressionUUID->"9a39aaa6-3c9e-4a1f-a701-\ f3b55d641e00"], Cell[BoxData[ RowBox[{"Manipulate", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"With", "[", RowBox[{ RowBox[{"{", RowBox[{"reg", "=", RowBox[{"implicitSphericalCap", "[", "\[Phi]", "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"Graphics3D", "[", RowBox[{"Point", "[", RowBox[{"RandomPoint", "[", RowBox[{"reg", ",", "numberOfPoints"}], "]"}], "]"}], "]"}]}], "]"}], ",", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"\[Phi]", ",", "\[Pi]"}], "}"}], ",", "0.1", ",", RowBox[{"4", "\[Pi]"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"numberOfPoints", ",", "1000"}], "}"}], ",", "100", ",", "10000"}], "}"}], ",", RowBox[{"Paneled", "\[Rule]", "False"}]}], "]"}]], "Input", 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CellLabel->"In[68]:=",ExpressionUUID->"265f610b-12d8-487d-b8cb-c6fcd0a9f7bb"], Cell[BoxData[ RowBox[{"{", RowBox[{"0.0013297265625`", ",", "Null"}], "}"}]], "Output", CellChangeTimes->{{3.8550455002320766`*^9, 3.855045504704117*^9}, 3.855046437443842*^9}, CellLabel->"Out[68]=",ExpressionUUID->"3f87e41e-dae1-42fa-b3bf-b5f38bb8e14c"] }, Open ]] }, Open ]], Cell["\<\ Instead, we can construct a symbolic formula for sampling uniformly over a \ spherical cap\ \>", "Item", CellChangeTimes->{{3.855045476988516*^9, 3.8550454777642593`*^9}, { 3.855045518264275*^9, 3.855045605610331*^9}, {3.855045660042913*^9, 3.855045711231154*^9}},ExpressionUUID->"e7442ea4-d955-4f41-b330-\ 3ed04256ca95"], Cell[CellGroupData[{ Cell["\<\ We start by noticing that the probability density of a random point on the \ sphere lying on the red circle intersection above, is proportional to the \ circumference\ \>", "Item", CellChangeTimes->{{3.855045476988516*^9, 3.8550454777642593`*^9}, { 3.855045518264275*^9, 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3.855052906995029*^9}},ExpressionUUID->"2e2c2c1b-deee-4502-af78-\ fe59a85139df"], Cell[CellGroupData[{ Cell[TextData[{ "What AnglePath3D does is equivalent to a \[OpenCurlyDoubleQuote]Folding\ \[CloseCurlyDoubleQuote] a dot-product over the list of rotations:\ \[LineSeparator] ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{"{", RowBox[{ SubscriptBox[ UnderscriptBox[ StyleBox["R", "TI"], "_"], "1"], "\[CenterDot]", RowBox[{"(", RowBox[{"1", ",", "0", ",", "0"}], ")"}], ",", SubscriptBox[ UnderscriptBox[ StyleBox["R", "TI"], "_"], "2"], "\[CenterDot]", SubscriptBox[ UnderscriptBox[ StyleBox["R", "TI"], "_"], "1"], "\[CenterDot]", "\[CenterDot]", RowBox[{"(", RowBox[{"1", ",", "0", ",", "0"}], ")"}], ",", SubscriptBox[ UnderscriptBox[ StyleBox["R", "TI"], "_"], "3"], "\[CenterDot]", SubscriptBox[ UnderscriptBox[ StyleBox["R", "TI"], "_"], "2"], "\[CenterDot]", SubscriptBox[ UnderscriptBox[ StyleBox["R", "TI"], "_"], "1"], "\[CenterDot]", RowBox[{"(", RowBox[{"1", ",", 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