(* 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[ 6641299, 112775] NotebookOptionsPosition[ 6619075, 112115] NotebookOutlinePosition[ 6620203, 112152] CellTagsIndexPosition[ 6619975, 112144] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"2", "-", "2"}]], "Input", CellChangeTimes->{{3.53418106308436*^9, 3.534181063401561*^9}}], Cell[BoxData["0"], "Output", CellChangeTimes->{3.534181065013879*^9}] }, Open ]], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.534181069541072*^9}], Cell[CellGroupData[{ Cell[TextData[{ "Generalizations of the ", StyleBox["Nest", "Input"], " Command" }], "Title", CellChangeTimes->{{3.5341811007709427`*^9, 3.534181114122408*^9}}], Cell["\<\ Nicholas Wheeler 29 December 2011\ \>", "Text", CellChangeTimes->{{3.5341811388584423`*^9, 3.534181153707932*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.534182437726161*^9}], Cell[CellGroupData[{ Cell["Introduction", "Subsection", CellChangeTimes->{{3.53418244415221*^9, 3.534182449813504*^9}}], Cell["I have used these commands", "Text", CellChangeTimes->{{3.5341825753437443`*^9, 3.534182579920451*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Nest", "[", RowBox[{"f", ",", "x", ",", "4"}], "]"}]], "Input", CellChangeTimes->{{3.5341824994893017`*^9, 3.534182508418664*^9}}], Cell[BoxData[ RowBox[{"f", "[", RowBox[{"f", "[", RowBox[{"f", "[", RowBox[{"f", "[", "x", "]"}], "]"}], "]"}], "]"}]], "Output", CellChangeTimes->{3.5341825101769953`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"NestList", "[", RowBox[{"f", ",", "x", ",", "4"}], "]"}], "//", "TableForm"}]], "Input", CellChangeTimes->{{3.534182527148403*^9, 3.534182535647369*^9}}], Cell[BoxData[ TagBox[ TagBox[GridBox[{ {"x"}, { RowBox[{"f", "[", "x", "]"}]}, { RowBox[{"f", "[", RowBox[{"f", "[", "x", "]"}], "]"}]}, { RowBox[{"f", "[", RowBox[{"f", "[", RowBox[{"f", "[", "x", "]"}], "]"}], "]"}]}, { RowBox[{"f", "[", RowBox[{"f", "[", RowBox[{"f", "[", RowBox[{"f", "[", "x", "]"}], "]"}], "]"}], "]"}]} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.5599999999999999]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], Column], Function[BoxForm`e$, TableForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.5341825370496483`*^9}] }, Open ]], Cell["\<\ in some recent work involving random walksw on rectangular grids. But random \ walks on hexagonal grids (\"graphene\") have the property that the \ step-options alternate; the patterns is ABABABABA instead AAAAAAAAA. This \ circumstance motivates me to looks for a generalization of the nest command \ that produces composite functions of the form\ \>", "Text", CellChangeTimes->{{3.534182589971817*^9, 3.5341825981078873`*^9}, { 3.534182652331456*^9, 3.534182940902416*^9}}], Cell[BoxData[ RowBox[{"f", "[", RowBox[{"g", "[", RowBox[{"f", "[", RowBox[{"g", "[", RowBox[{"f", "[", "x", "]"}], "]"}], "]"}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.534182947066382*^9, 3.5341829747290697`*^9}}], Cell["and, more generally, ", "Text", CellChangeTimes->{{3.534182984178974*^9, 3.534182988266708*^9}}], Cell[BoxData[ RowBox[{ SubscriptBox["f", "5"], "[", RowBox[{ SubscriptBox["f", "4"], "[", RowBox[{ SubscriptBox["f", "3"], "[", RowBox[{ SubscriptBox["f", "2"], "[", RowBox[{ SubscriptBox["f", "1"], "[", "x", "]"}], "]"}], "]"}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.534183005159544*^9, 3.534183035118828*^9}}], Cell["\<\ This problem\[LongDash]the problem of producing such constructs \ automatically/efficiently\[LongDash]gave me so much difficulty that I was \ motivated to write for advice to Patrick T. Tam, author of \"A Physicist's \ Guide to Mathematica\" [I wrote on the 23rd, have as yet received no reply]. \ Here I describe several elementary solutions or the problem thus posed. \ \>", "Text", CellChangeTimes->{{3.53418305992236*^9, 3.534183139059044*^9}, { 3.5341831970222816`*^9, 3.5341832223231277`*^9}, {3.5341833456367493`*^9, 3.534183346809074*^9}, {3.534183979692309*^9, 3.534184129028103*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.5341841681540823`*^9}] }, Open ]], Cell[CellGroupData[{ Cell["Constructing composite functions", "Subsection", CellChangeTimes->{{3.5341841771826773`*^9, 3.534184183954753*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "Composition"}]], "Input", CellChangeTimes->{{3.5341798632951717`*^9, 3.534179866771001*^9}}], Cell[BoxData[ RowBox[{ StyleBox["\<\"\\!\\(\\*RowBox[{\\\"Composition\\\", \\\"[\\\", \ RowBox[{SubscriptBox[StyleBox[\\\"f\\\", \\\"TI\\\"], StyleBox[\\\"1\\\", \ \\\"TR\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"f\\\", \\\"TI\\\"], \ StyleBox[\\\"2\\\", \\\"TR\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"f\\\", \ \\\"TI\\\"], StyleBox[\\\"3\\\", \\\"TR\\\"]], \\\",\\\", StyleBox[\\\"\ \[Ellipsis]\\\", \\\"TR\\\"]}], \\\"]\\\"}]\\) represents a composition of \ the functions \\!\\(\\*SubscriptBox[StyleBox[\\\"f\\\", \\\"TI\\\"], \ StyleBox[\\\"1\\\", \\\"TR\\\"]]\\), \ \\!\\(\\*SubscriptBox[StyleBox[\\\"f\\\", \\\"TI\\\"], StyleBox[\\\"2\\\", \\\ \"TR\\\"]]\\), \\!\\(\\*SubscriptBox[StyleBox[\\\"f\\\", \\\"TI\\\"], \ StyleBox[\\\"3\\\", \\\"TR\\\"]]\\), \[Ellipsis] . \"\>", "MSG"], "\[NonBreakingSpace]", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/Composition"]}]], "Print", "PrintUsage", CellChangeTimes->{3.534184237020846*^9}, CellTags->"Info3534155436-6247818"] }, Open ]], Cell[TextData[{ "The problem here is that when the ", Cell[BoxData[ FormBox[ SubscriptBox["f", "k"], TraditionalForm]], FormatType->"TraditionalForm"], " are produced by some tabulated set of definitions they are enclosed in a \ collective { }, and so long as those remain in place the ", StyleBox["Composition", "Input"], " command does not work:" }], "Text", CellChangeTimes->{{3.534184351410083*^9, 3.53418439149546*^9}, { 3.534184426867091*^9, 3.534184493025635*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Table", "[", RowBox[{ SubscriptBox["f", "k"], ",", RowBox[{"{", RowBox[{"k", ",", "1", ",", "10"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.534184248908547*^9, 3.534184266627577*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ SubscriptBox["f", "1"], ",", SubscriptBox["f", "2"], ",", SubscriptBox["f", "3"], ",", SubscriptBox["f", "4"], ",", SubscriptBox["f", "5"], ",", SubscriptBox["f", "6"], ",", SubscriptBox["f", "7"], ",", SubscriptBox["f", "8"], ",", SubscriptBox["f", "9"], ",", SubscriptBox["f", "10"]}], "}"}]], "Output", CellChangeTimes->{3.5341842679150047`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Composition", "[", RowBox[{"{", RowBox[{ SubscriptBox["f", "1"], ",", SubscriptBox["f", "2"], ",", SubscriptBox["f", "3"], ",", SubscriptBox["f", "4"], ",", SubscriptBox["f", "5"], ",", SubscriptBox["f", "6"], ",", SubscriptBox["f", "7"], ",", SubscriptBox["f", "8"], ",", SubscriptBox["f", "9"], ",", SubscriptBox["f", "10"]}], "}"}], "]"}], "[", "x", "]"}]], "Input", CellChangeTimes->{{3.5341842935333347`*^9, 3.534184336048061*^9}, 3.534184529734025*^9}], Cell[BoxData[ RowBox[{ RowBox[{"{", RowBox[{ SubscriptBox["f", "1"], ",", SubscriptBox["f", "2"], ",", SubscriptBox["f", "3"], ",", SubscriptBox["f", "4"], ",", SubscriptBox["f", "5"], ",", SubscriptBox["f", "6"], ",", SubscriptBox["f", "7"], ",", SubscriptBox["f", "8"], ",", SubscriptBox["f", "9"], ",", SubscriptBox["f", "10"]}], "}"}], "[", "x", "]"}]], "Output", CellChangeTimes->{3.534184340690957*^9, 3.534184537110012*^9}] }, Open ]], Cell["\<\ I know of no command that serves to remove the offending brackets. By-hand \ removal works\ \>", "Text", CellChangeTimes->{{3.534185104917233*^9, 3.534185145416687*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Composition", "[", RowBox[{ SubscriptBox["f", "1"], ",", SubscriptBox["f", "2"], ",", SubscriptBox["f", "3"], ",", SubscriptBox["f", "4"], ",", SubscriptBox["f", "5"], ",", SubscriptBox["f", "6"], ",", SubscriptBox["f", "7"], ",", SubscriptBox["f", "8"], ",", SubscriptBox["f", "9"], ",", SubscriptBox["f", "10"]}], "]"}], "[", "x", "]"}]], "Input", CellChangeTimes->{{3.534184550428156*^9, 3.5341845540666513`*^9}}], Cell[BoxData[ RowBox[{ SubscriptBox["f", "1"], "[", RowBox[{ SubscriptBox["f", "2"], "[", RowBox[{ SubscriptBox["f", "3"], "[", RowBox[{ SubscriptBox["f", "4"], "[", RowBox[{ SubscriptBox["f", "5"], "[", RowBox[{ SubscriptBox["f", "6"], "[", RowBox[{ SubscriptBox["f", "7"], "[", RowBox[{ SubscriptBox["f", "8"], "[", RowBox[{ SubscriptBox["f", "9"], "[", RowBox[{ SubscriptBox["f", "10"], "[", "x", "]"}], "]"}], "]"}], "]"}], "]"}], "]"}], "]"}], "]"}], "]"}], "]"}]], "Output", CellChangeTimes->{3.5341845632251997`*^9}] }, Open ]], Cell[TextData[{ "but is a procedure that becomes unworkable when ", StyleBox["Composition", "Input"], " is folded into a compound command. 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It produces", "Text", CellChangeTimes->{{3.534185410333054*^9, 3.5341854119731483`*^9}, { 3.534185486311007*^9, 3.53418550491854*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"ComposeList", "[", RowBox[{ RowBox[{"{", RowBox[{ SubscriptBox["f", "1"], ",", SubscriptBox["f", "2"], ",", SubscriptBox["f", "3"], ",", SubscriptBox["f", "4"], ",", SubscriptBox["f", "5"], ",", SubscriptBox["f", "6"], ",", SubscriptBox["f", "7"], ",", SubscriptBox["f", "8"], ",", SubscriptBox["f", "9"], ",", SubscriptBox["f", "10"]}], "}"}], ",", "x"}], "]"}], "//", "TableForm"}]], "Input", CellChangeTimes->{{3.534185430723626*^9, 3.534185454401046*^9}}], Cell[BoxData[ TagBox[ TagBox[GridBox[{ {"x"}, { RowBox[{ SubscriptBox["f", "1"], "[", "x", "]"}]}, { RowBox[{ SubscriptBox["f", "2"], "[", RowBox[{ SubscriptBox["f", "1"], "[", "x", "]"}], "]"}]}, { RowBox[{ SubscriptBox["f", "3"], "[", RowBox[{ SubscriptBox["f", "2"], "[", RowBox[{ SubscriptBox["f", "1"], "[", "x", "]"}], "]"}], "]"}]}, { RowBox[{ 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procedure for constructing that result\ \>", "Text", CellChangeTimes->{{3.534185652526431*^9, 3.534185681871902*^9}, { 3.534185746215251*^9, 3.534185802564493*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"F", "[", "n_", "]"}], ":=", RowBox[{ SubscriptBox["f", "n"], "@", RowBox[{"F", "[", RowBox[{"n", "-", "1"}], "]"}]}]}], ";", RowBox[{ RowBox[{"F", "[", "0", "]"}], "=", "x"}], ";"}]], "Input", CellChangeTimes->{{3.534176656402876*^9, 3.53417666188018*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"F", "[", "10", "]"}]], "Input", CellChangeTimes->{{3.534185700529686*^9, 3.534185702951971*^9}}], Cell[BoxData[ RowBox[{ SubscriptBox["f", "10"], "[", RowBox[{ SubscriptBox["f", "9"], "[", RowBox[{ SubscriptBox["f", "8"], "[", RowBox[{ SubscriptBox["f", "7"], "[", RowBox[{ SubscriptBox["f", "6"], "[", RowBox[{ SubscriptBox["f", "5"], "[", RowBox[{ SubscriptBox["f", "4"], "[", RowBox[{ SubscriptBox["f", "3"], "[", RowBox[{ SubscriptBox["f", "2"], "[", RowBox[{ SubscriptBox["f", "1"], 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"Steps200", "]"}]], "Input", CellChangeTimes->{{3.534436481291992*^9, 3.534436488077406*^9}}], Cell[BoxData["201"], "Output", CellChangeTimes->{3.534436489333211*^9, 3.534440286930193*^9, 3.534440320724462*^9}] }, Open ]], Cell["and what a few of them look like:", "Text", CellChangeTimes->{{3.534437352637301*^9, 3.5344373635430117`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Steps200", "\[LeftDoubleBracket]", RowBox[{"1", ";;", "4"}], "\[RightDoubleBracket]"}]], "Input", CellChangeTimes->{{3.534436394573332*^9, 3.5344364174308434`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "0.8660254037844386`"}], ",", "0.5`"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "0.8660254037844386`"}], ",", RowBox[{"-", "0.5`"}]}], "}"}], ",", RowBox[{"{", RowBox[{"0.8660254037844386`", ",", "0.5`"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.534436418670985*^9, 3.534440296787099*^9, 3.5344403302446413`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"Steps200", "//", "First"}], "\[IndentingNewLine]", RowBox[{"Steps200", "//", "Last"}]}], "Input", CellChangeTimes->{{3.534436627640971*^9, 3.534436641189131*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"0", ",", "0"}], "}"}]], "Output", CellChangeTimes->{3.534436642854369*^9, 3.534440300839541*^9, 3.534440333663756*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{"0.8660254037844386`", ",", RowBox[{"-", "0.5`"}]}], "}"}]], "Output", CellChangeTimes->{3.534436642854369*^9, 3.534440300839541*^9, 3.534440333762375*^9}] }, Open ]], Cell["\<\ Here I use that data to check out the commands that will extract the x and \ y-components of those vectors:\ \>", "Text", CellChangeTimes->{{3.534437398076099*^9, 3.534437469730011*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ "Steps200", "\[LeftDoubleBracket]", "201", "\[RightDoubleBracket]"}], "\[LeftDoubleBracket]", StyleBox["1", FontColor->RGBColor[1, 0, 0]], "\[RightDoubleBracket]"}], 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I have never encountered the a boundary of this 30 \ \[Times] 30 display, though it is theoretically possible to achieve \ \>", "Text", CellChangeTimes->{{3.534442712724453*^9, 3.534442841265414*^9}, { 3.534442893950597*^9, 3.5344429629651613`*^9}, {3.5344433240904007`*^9, 3.534443326048128*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ SubscriptBox["x250", "max"], "=", RowBox[{ RowBox[{"250", RowBox[{"Cos", "[", FractionBox["\[Pi]", "12"], "]"}]}], "//", "N"}]}], "\[IndentingNewLine]", RowBox[{ SubscriptBox["y250", "max"], "=", RowBox[{ RowBox[{"125", RowBox[{"(", RowBox[{"1", "+", RowBox[{"Sin", "[", FractionBox["\[Pi]", "12"], "]"}]}], ")"}]}], "//", "N"}]}]}], "Input",\ CellChangeTimes->{{3.534443331383272*^9, 3.534443422868493*^9}}], Cell[BoxData["241.48145657226704`"], "Output", CellChangeTimes->{3.534443425482113*^9}], Cell[BoxData["157.3523806378151`"], "Output", CellChangeTimes->{3.534443425498982*^9}] }, Open ]], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.5344434630118303`*^9}] }, Open ]], Cell[CellGroupData[{ Cell["Preliminary remarks concerning the distribution of endpoints", \ "Subsection", CellChangeTimes->{{3.534443470686598*^9, 3.53444349747635*^9}}], Cell["\<\ We expect the endpoints of N-step random walks on a honycomb, as on a grid \ with square tiles or the unarticulated plane, to be normally distributed, \ with standard deviation proportional to N. \ \>", "Text", CellChangeTimes->{{3.5344443572423573`*^9, 3.534444470956751*^9}, { 3.534444678366527*^9, 3.5344447054353743`*^9}, {3.534445000241447*^9, 3.534445054117503*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Hyperlink", "[", RowBox[{ "\"\\"", ",", "\"\\""}], "]"}]], "Input", CellChangeTimes->{{3.534444724378882*^9, 3.5344447638596563`*^9}}], Cell[BoxData[ TagBox[ ButtonBox[ PaneSelectorBox[{False->"\<\"Random Walks\"\>", True-> StyleBox["\<\"Random Walks\"\>", "HyperlinkActive"]}, Dynamic[ CurrentValue["MouseOver"]], BaseStyle->{"Hyperlink"}, BaselinePosition->Baseline, FrameMargins->0, ImageSize->Automatic], BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Random_walk"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Random_walk"], Annotation[#, "http://en.wikipedia.org/wiki/Random_walk", "Hyperlink"]& ]], "Output", CellChangeTimes->{3.5344447727373743`*^9}] }, Open ]], Cell["\<\ We are in position now to demonstrate the point by Monte Carlo calculation: \ look to the endpoints of (say) 10,000 N-step honeycomb walks, increment N, \ then repeat the exercise. Etc. This, however, would be a very labor-intensive \ way to obtain approximate support of a theoretically anticipated result. A better way (less labor-intensive?) to obtain that information would be to \ (1) construct a hexagonal graph so large that it would be impossible for an \ N-step walker\[LongDash]departing from the origin\[LongDash]to reach a \ boudary of the graph, (2) build the site-specific step-probatilities into the \ design of correspondingly large Markov matrix \[DoubleStruckCapitalM], and \ (3) look to the evolving stochastic vector \ \>", "Text", CellChangeTimes->{{3.534445060010466*^9, 3.534445208550047*^9}, { 3.534445240904685*^9, 3.534445338085091*^9}, {3.53444543268547*^9, 3.534445436796813*^9}, {3.5344458493823137`*^9, 3.5344463306563883`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"MatrixPower", "[", RowBox[{"\[DoubleStruckCapitalM]", ",", "n"}], "]"}], ".", SubscriptBox["p", "0"]}]], "Output", CellChangeTimes->{{3.534446335562192*^9, 3.534446400680434*^9}}], Cell[TextData[{ "where ", Cell[BoxData[ SubscriptBox["p", "0"]], CellChangeTimes->{{3.534446335562192*^9, 3.534446400680434*^9}}], " describes known initial placement at the origin. But we confront at this \ point a nest of ", StyleBox["challenging technical problems", FontWeight->"Bold"], ":\n\n1) How most naturally to associated (say) 1000 nodes in hexagonal \ array with the elements of a stochastic vector " }], "Text", CellChangeTimes->{{3.534446415621646*^9, 3.534446480202812*^9}, 3.5345191202809143`*^9, {3.534536038117055*^9, 3.534536166928761*^9}}], Cell[BoxData[ RowBox[{ SubscriptBox["P", "0"], "=", SubscriptBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { SubscriptBox["p", "1"]}, { SubscriptBox["p", "2"]}, { SubscriptBox["p", "3"]}, {"\[VerticalEllipsis]"}, { SubscriptBox["p", "1000"]} }], "\[NoBreak]", ")"}], "0"]}]], "Input", CellChangeTimes->{{3.534536172477806*^9, 3.534536275595664*^9}}], Cell[TextData[{ "2) That done, how to construct the 1000 \[Times] 1000 adjacency matrix?\n\n\ 3) How to feed in the site-specific weights that enter into construction of \ the unbalanced matrix \[DoubleStruckCapitalM]?\n\n4) How to display ", Cell[BoxData[ SubscriptBox["P", "N"]], CellChangeTimes->{{3.534536459958075*^9, 3.534536462552328*^9}}], " as a density map on the hexagonal grid (which lives on the xy-plane)?" }], "Text", CellChangeTimes->{{3.534536282524406*^9, 3.5345364473546267`*^9}, { 3.534536478861724*^9, 3.534536519882852*^9}}], Cell[BoxData[""], "Input", CellChangeTimes->{{3.534536541948195*^9, 3.534536544085002*^9}}], Cell[TextData[{ "But so long as we restrict our attention (as above) to individual hexagonal \ walks we can avoid the full force of those problems, since such walks ", StyleBox["live", FontSlant->"Italic"], " on the tesselated xy-plane." }], "Text", CellChangeTimes->{{3.5345365752164097`*^9, 3.53453667223345*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.5345366815398006`*^9}] }, Open ]], Cell[CellGroupData[{ Cell["Construction of site-specific next-step probabilities", "Subsection", CellChangeTimes->{{3.534536722708886*^9, 3.534536740319912*^9}}], Cell[CellGroupData[{ Cell["Preparatory remarks", "Subsubsection", CellChangeTimes->{{3.534548621806378*^9, 3.534548629688212*^9}}], Cell[TextData[{ "In the unbiased hexagonal walks I at each step used ", StyleBox["RandomChoice", "Input"], " to instruct the walker which of three equally-weighted next-step options \ to select. But ", StyleBox["RancomChoice", "Input"], " is capable also of making arbitrarily weighted choices:" }], "Text", CellChangeTimes->{{3.53454431973178*^9, 3.534544520318261*^9}, { 3.534544554802747*^9, 3.534544636674478*^9}, 3.534544671805901*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "RandomChoice"}]], "Input", CellChangeTimes->{{3.534536758388094*^9, 3.534536762418067*^9}}], Cell[BoxData[ RowBox[{ StyleBox["\<\"\\!\\(\\*RowBox[{\\\"RandomChoice\\\", \\\"[\\\", \ RowBox[{\\\"{\\\", RowBox[{SubscriptBox[StyleBox[\\\"e\\\", \\\"TI\\\"], \ StyleBox[\\\"1\\\", \\\"TR\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"e\\\", \ \\\"TI\\\"], StyleBox[\\\"2\\\", \\\"TR\\\"]], \\\",\\\", StyleBox[\\\"\ \[Ellipsis]\\\", \\\"TR\\\"]}], \\\"}\\\"}], \\\"]\\\"}]\\) gives a \ pseudorandom choice of one of the \\!\\(\\*SubscriptBox[StyleBox[\\\"e\\\", \ \\\"TI\\\"], StyleBox[\\\"i\\\", \\\"TI\\\"]]\\). \ \\n\\!\\(\\*RowBox[{\\\"RandomChoice\\\", \\\"[\\\", \ RowBox[{StyleBox[\\\"list\\\", \\\"TI\\\"], \\\",\\\", StyleBox[\\\"n\\\", \\\ \"TI\\\"]}], \\\"]\\\"}]\\) gives a list of \\!\\(\\*StyleBox[\\\"n\\\", \ \\\"TI\\\"]\\) pseudorandom choices. \ \\n\\!\\(\\*RowBox[{\\\"RandomChoice\\\", \\\"[\\\", \ RowBox[{StyleBox[\\\"list\\\", \\\"TI\\\"], \\\",\\\", RowBox[{\\\"{\\\", \ RowBox[{SubscriptBox[StyleBox[\\\"n\\\", \\\"TI\\\"], StyleBox[\\\"1\\\", \ \\\"TR\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"n\\\", \\\"TI\\\"], \ StyleBox[\\\"2\\\", \\\"TR\\\"]], \\\",\\\", StyleBox[\\\"\[Ellipsis]\\\", \\\ \"TR\\\"]}], \\\"}\\\"}]}], \\\"]\\\"}]\\) gives an \ \\!\\(\\*RowBox[{SubscriptBox[StyleBox[\\\"n\\\", \\\"TI\\\"], StyleBox[\\\"1\ \\\", \\\"TR\\\"]], \\\"\[Cross]\\\", SubscriptBox[StyleBox[\\\"n\\\", \\\"TI\ \\\"], StyleBox[\\\"2\\\", \\\"TR\\\"]], \\\"\[Cross]\\\", \ \\\"\[Ellipsis]\\\", \\\" \\\"}]\\) array of pseudorandom choices. \ \\n\\!\\(\\*RowBox[{\\\"RandomChoice\\\", \\\"[\\\", \ RowBox[{RowBox[{\\\"{\\\", RowBox[{SubscriptBox[StyleBox[\\\"w\\\", \ \\\"TI\\\"], StyleBox[\\\"1\\\", \\\"TR\\\"]], \\\",\\\", \ SubscriptBox[StyleBox[\\\"w\\\", \\\"TI\\\"], StyleBox[\\\"2\\\", \ \\\"TR\\\"]], \\\",\\\", StyleBox[\\\"\[Ellipsis]\\\", \\\"TR\\\"]}], \\\"}\\\ \"}], \\\"->\\\", RowBox[{\\\"{\\\", RowBox[{SubscriptBox[StyleBox[\\\"e\\\", \ \\\"TI\\\"], StyleBox[\\\"1\\\", \\\"TR\\\"]], \\\",\\\", \ SubscriptBox[StyleBox[\\\"e\\\", \\\"TI\\\"], StyleBox[\\\"2\\\", \ \\\"TR\\\"]], \\\",\\\", StyleBox[\\\"\[Ellipsis]\\\", \\\"TR\\\"]}], \\\"}\\\ \"}]}], \\\"]\\\"}]\\) gives a pseudorandom choice weighted by the \ \\!\\(\\*SubscriptBox[StyleBox[\\\"w\\\", \\\"TI\\\"], StyleBox[\\\"i\\\", \\\ \"TI\\\"]]\\). \\n\\!\\(\\*RowBox[{\\\"RandomChoice\\\", \\\"[\\\", \ RowBox[{RowBox[{StyleBox[\\\"wlist\\\", \\\"TI\\\"], \\\"->\\\", \ StyleBox[\\\"elist\\\", \\\"TI\\\"]}], \\\",\\\", StyleBox[\\\"n\\\", \ \\\"TI\\\"]}], \\\"]\\\"}]\\) gives a list of \\!\\(\\*StyleBox[\\\"n\\\", \\\ \"TI\\\"]\\) weighted choices.\\n\\!\\(\\*RowBox[{\\\"RandomChoice\\\", \\\"[\ \\\", RowBox[{RowBox[{StyleBox[\\\"wlist\\\", \\\"TI\\\"], \\\"->\\\", \ StyleBox[\\\"elist\\\", \\\"TI\\\"]}], \\\",\\\", RowBox[{\\\"{\\\", \ RowBox[{SubscriptBox[StyleBox[\\\"n\\\", \\\"TI\\\"], StyleBox[\\\"1\\\", \ \\\"TR\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"n\\\", \\\"TI\\\"], \ StyleBox[\\\"2\\\", \\\"TR\\\"]], \\\",\\\", StyleBox[\\\"\[Ellipsis]\\\", \\\ \"TR\\\"]}], \\\"}\\\"}]}], \\\"]\\\"}]\\) gives an \ \\!\\(\\*RowBox[{SubscriptBox[StyleBox[\\\"n\\\", \\\"TI\\\"], StyleBox[\\\"1\ \\\", \\\"TR\\\"]], \\\"\[Cross]\\\", SubscriptBox[StyleBox[\\\"n\\\", \\\"TI\ \\\"], StyleBox[\\\"2\\\", \\\"TR\\\"]], \\\"\[Cross]\\\", StyleBox[\\\"\ \[Ellipsis]\\\", \\\"TR\\\"]}]\\) array of weighted choices.\"\>", "MSG"], "\[NonBreakingSpace]", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/RandomChoice"]}]], "Print", "PrintUsage", CellChangeTimes->{3.534536776763426*^9}, CellTags->"Info3534507965-2795975"] }, Open ]], Cell["\<\ as I demonstrate. Construct a triple of weights (elements of a 3-dimensional \ stochatic vector)\ \>", "Text", CellChangeTimes->{{3.534536836328095*^9, 3.534536860859644*^9}, { 3.5345446875432167`*^9, 3.534544757198604*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"V", "=", RowBox[{ RowBox[{"RandomInteger", "[", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "9"}], "}"}], ",", RowBox[{"{", RowBox[{"1", ",", "3"}], "}"}]}], "]"}], "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"W", "=", FractionBox["V", RowBox[{"Total", "[", "V", "]"}]]}], ";"}], "\[IndentingNewLine]", RowBox[{"Table", "[", RowBox[{ RowBox[{ SubscriptBox["w", "k"], "=", RowBox[{"W", "\[LeftDoubleBracket]", "k", "\[RightDoubleBracket]"}]}], ",", RowBox[{"{", RowBox[{"k", ",", "1", ",", "3"}], "}"}]}], "]"}]}], "Input", CellChangeTimes->{{3.534536872837668*^9, 3.534536922120873*^9}, { 3.5345369712787523`*^9, 3.5345369994942904`*^9}, {3.534537036487555*^9, 3.534537107169979*^9}, {3.534537253780171*^9, 3.53453725486559*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ FractionBox["1", "3"], ",", FractionBox["7", "24"], ",", FractionBox["3", "8"]}], "}"}]], "Output", CellChangeTimes->{3.5345371827678843`*^9, 3.5345372589040737`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Total", "[", "%", "]"}]], "Input", CellChangeTimes->{{3.5345372714007587`*^9, 3.534537274328279*^9}}], Cell[BoxData["1"], "Output", CellChangeTimes->{3.534537275542726*^9}] }, Open ]], Cell[TextData[{ "Our objective is to select \[DoubleStruckCapitalR] with probability (or \ \"weight\") ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["w", "1"], ",", " ", "\[DoubleStruckCapitalS]"}], TraditionalForm]], FormatType->"TraditionalForm"], " with weight ", Cell[BoxData[ FormBox[ SubscriptBox["w", "2"], TraditionalForm]], FormatType->"TraditionalForm"], ", \[DoubleStruckCapitalT] with weight ", Cell[BoxData[ FormBox[ SubscriptBox["w", "3"], TraditionalForm]], FormatType->"TraditionalForm"], ". 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", StyleBox["phase", FontSlant->"Italic"], " space in which the oscillator Hamiltonian ", StyleBox["is", FontSlant->"Italic"], " ", StyleBox["used to control step-length, ", FontSlant->"Italic"], "and randomness enters only into the decision whether the next step will be \ a \[Delta]x-step or a\n\[Delta]p-step. The resulting walks look typically \ like these:" }], "Text", CellChangeTimes->{{3.534690610849807*^9, 3.53469062056386*^9}, { 3.53469338424907*^9, 3.5346935376521873`*^9}, {3.534693579752316*^9, 3.534693691165964*^9}, {3.534693737490408*^9, 3.5346937668779984`*^9}, { 3.534694325882586*^9, 3.53469434508809*^9}}], Cell[BoxData[ GraphicsBox[ {Hue[0.67, 0.6, 0.6], PointSize[0.003], PointBox[CompressedData[" 1:eJyd3Hc4V/H/P34kihSSWaG0EFJKwrELFRqUTSllVsgohFQaZlb2zpasyHnJ Htl77z1e9n759rm+1/v5fP8+z+9fv/PPuW7X/Tr7ec7j+byuc84tC0NuIzIy MgoyMuqG/zv/vxNGJPv/TP9/jYXlqk29PZoLnPq/nEMjYRndRgNc+L/y/3hu he/pWMtj6Gn1lrrEWuhBF6kcZxGsWPC72Or1JtStYlV82g3QlYu3wy+bYeVk lxhqVr8AV9WXjEdFKaLOrexrOy4LHR9heoCODasNvaBWfHoDtd+eEEcvYazh fMxkgNQmcBNrWvL1GTvg5u1f5hRBq8Ctg6WCuQrWwO3ldfNmC4vAncmdmcfD n6F+NpTiVkkGrVJY+JG8FvpsUJ23WAjW/ZHAHMWaC9z7NNaey3sHNVXimEKT A9Z/PNh0BdsCHmAYPNx9cQ3xbOcamf9HKWyA1HlOK8QMeHCqUb6LfBB14ruG uit90PYs0dTTjtCK8VZSoUeg2S7J290oAJ4ZL2fOIGlhQw3ut9U5LICHc8T0 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But lessons learned from that computational experience are irrelevant to the \ problem now at hand.\ \>", "Text", CellChangeTimes->{{3.534694272594592*^9, 3.534694273529579*^9}, { 3.534694386747299*^9, 3.53469441474133*^9}}], Cell[BoxData[""], "Input", CellChangeTimes->{{3.5346944298893757`*^9, 3.5346944325712976`*^9}}], Cell[TextData[{ "My initial intention had been to lay down a scalar field ", StyleBox["\[Phi][x,y]", "Output"], " and to use its directional derivatives to assign local value to ", Cell[BoxData[ RowBox[{"{", RowBox[{ SubscriptBox["w", "1"], ",", SubscriptBox["w", "2"], ",", SubscriptBox["w", "3"]}], "}"}]], CellChangeTimes->{{3.5346948080596027`*^9, 3.5346948190853577`*^9}}], ". I look by way of preparation to the problem of \[Ellipsis] " }], "Text", CellChangeTimes->{{3.5346945793742332`*^9, 3.53469462603426*^9}, { 3.534694682614802*^9, 3.5346947621841307`*^9}, {3.534694796405753*^9, 3.534694801220245*^9}, {3.534694833029254*^9, 3.5346948674066257`*^9}, { 3.53469490403194*^9, 3.5346949084632587`*^9}, {3.534694942408702*^9, 3.534694948576762*^9}, {3.53469499848517*^9, 3.534694999707349*^9}, { 3.534695037753265*^9, 3.53469504680536*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Directional derivative construction", "Subsubsection", CellChangeTimes->{{3.5346950068787107`*^9, 3.53469501338516*^9}}], Cell["For illustrative purposes, let", "Text", CellChangeTimes->{{3.534702791439299*^9, 3.534702804314872*^9}}], Cell[BoxData[ RowBox[{"Clear", "[", RowBox[{"x", ",", "y", ",", "\[Phi]"}], "]"}]], "Input", CellChangeTimes->{{3.5347030711194897`*^9, 3.5347030769751596`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"\[Phi]", "[", RowBox[{"x_", ",", "y_"}], "]"}], ":=", RowBox[{ FractionBox["1", RowBox[{"6", "\[Pi]"}]], SuperscriptBox[ RowBox[{"(", RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]}], ")"}], "3"], SuperscriptBox["\[ExponentialE]", RowBox[{"-", RowBox[{"(", RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]}], ")"}]}]]}]}]], "Input", CellChangeTimes->{{3.5347028896813517`*^9, 3.534702970513805*^9}}], Cell["\<\ This scalar field is rotationally symmetric; in polar coordinates it becomes\ \>", "Text", CellChangeTimes->{{3.534703003374958*^9, 3.534703007303259*^9}, { 3.53470314937669*^9, 3.5347031602627497`*^9}, {3.534703199474533*^9, 3.534703228126539*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"\[Phi]", "[", RowBox[{ RowBox[{"r", " ", RowBox[{"Cos", "[", "\[Theta]", "]"}]}], ",", RowBox[{"r", " ", RowBox[{"Sin", "[", "\[Theta]", "]"}]}]}], "]"}], "//", "Simplify"}]], "Input", CellChangeTimes->{{3.534703015903965*^9, 3.534703042772291*^9}, { 3.5347030988671722`*^9, 3.534703101922936*^9}}], Cell[BoxData[ FractionBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"-", SuperscriptBox["r", "2"]}]], " ", SuperscriptBox["r", "6"]}], RowBox[{"6", " ", "\[Pi]"}]]], "Output", CellChangeTimes->{{3.534703089604006*^9, 3.534703103645919*^9}}] }, Open ]], Cell["\<\ and I have\[LongDash]for no good reason\[LongDash]arrannged for it to be \ normalized:\ \>", "Text", CellChangeTimes->{{3.534703250669488*^9, 3.5347032791464863`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Integrate", "[", RowBox[{ RowBox[{"2", "\[Pi]", " ", "r", " ", FractionBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"-", SuperscriptBox["r", "2"]}]], " ", SuperscriptBox["r", "6"]}], RowBox[{"6", " ", "\[Pi]"}]]}], ",", RowBox[{"{", RowBox[{"r", ",", "0", ",", "\[Infinity]"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.534695426512391*^9, 3.534695449354208*^9}, 3.5346954849843616`*^9, {3.534695590672709*^9, 3.534695594904621*^9}, { 3.534695637885292*^9, 3.534695649265793*^9}, 3.534695681280258*^9, 3.53469580510039*^9, {3.534703115607629*^9, 3.5347031320167437`*^9}}], Cell[BoxData["1"], "Output", CellChangeTimes->{ 3.534695454487988*^9, 3.534695486560017*^9, {3.53469565858366*^9, 3.5346956856679907`*^9}, 3.534695812220703*^9, 3.5347031374552393`*^9}] }, Open ]], Cell["\<\ By design, it depends very simply on its arguments. 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But the latter draws white \ curves where the ", StyleBox["UnitStep", "Input"], " discontinuity is encountered. 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additive 0.001 that was folded into the definitions of the ", Cell[BoxData[ SubscriptBox["f", "i"]], CellChangeTimes->{{3.534790617847481*^9, 3.534790620313408*^9}}], ")." }], "Text", CellChangeTimes->{{3.534790131925082*^9, 3.534790135260466*^9}, { 3.534790165959371*^9, 3.5347902198616257`*^9}, {3.534790253857379*^9, 3.5347902967888927`*^9}, {3.534790398483308*^9, 3.534790639615582*^9}}], Cell["\<\ As a final demonstration of where we now stand, I construct (which takes \ about 5 seconds)\ \>", "Text", CellChangeTimes->{{3.5347907885242*^9, 3.53479080291955*^9}, { 3.534790986026225*^9, 3.534790995139*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Table", "[", RowBox[{ RowBox[{"RandomChoice", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["f", "1"], "[", RowBox[{"1.53", ",", RowBox[{"-", "0.35"}]}], "]"}], ",", RowBox[{ SubscriptBox["f", "3"], "[", RowBox[{"1.53", ",", RowBox[{"-", "0.35"}]}], "]"}], ",", RowBox[{ SubscriptBox["f", "5"], "[", RowBox[{"1.53", ",", RowBox[{"-", 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\"random walks on corrugated \ graphine,\" but gave no indication of what he understood to be the physics of \ such processes or the source of his interest in them\[Ellipsis]which I find I \ am unable to guess.\n\nI do not expect the electrical/conductivity of a wire \ to depend measureably upon the ", StyleBox["shape", FontSlant->"Italic"], " of the wire, do not expect the effective resistance of a 2D sheet of \ conductive material (copper, graphene) to depend measureably upon the \ non-flatness/corregation of that sheet\[Ellipsis]unless, possibly, the \ characteristic length of the deformation is so small as to be comparable to \ the lattice constant, so small that is to say as to modulate the interatomic \ bonds.\n\nWhat am I to suppose is the physical embodiment of the \"walker\"? \ An electron, an \"exiton\" of some sort (spin?). I have no specific knowledge \ of the (random? stimulated?) migration of electrons/exitons on periodic \ structures, but suppose \"hopping models\" might come into play. I would not \ expect such hopping to occur at regularly spaced temporal intervals, but that \ is of no immediate concern: my effort has been to describe the geometry of \ trajectories, not how they develop in time.\n\nIssues of \ computational/practical concern arise. The \"corrugation length\" must be \ enough larger than step-length for its effect to become evident as a \ modulation written upon the designs of random walks, yet not so large as to \ require walks so long that they saturate memory or take too long to compute.\n\ \nIn short: I feel\[LongDash]as a modeler\[LongDash]handicapped by the fact \ that I have no clear sense of the underlying physics of the processes I am \ attempting to model, no sharp intuitive sense of the results I expect to \ emerge. Sina spoke vaguely of \"walks confined to channels,\" and it is for \ evidence of such (intuitively plausible) channeling that I will be looking\ \[Ellipsis]\n\n\[Ellipsis]which informs the design of what I will now take to \ be my \"scalar control field.\"" }], "Text", CellChangeTimes->{{3.5347987760724087`*^9, 3.5347987939292917`*^9}, { 3.53479883724334*^9, 3.534798965266642*^9}, {3.534799017798617*^9, 3.534799174959526*^9}, {3.53479921129913*^9, 3.534799296556498*^9}, { 3.534799340640277*^9, 3.5347994512191668`*^9}, {3.534799503744732*^9, 3.534799507642922*^9}, {3.534878734724506*^9, 3.5348787991054068`*^9}, { 3.534878836391532*^9, 3.534878903669998*^9}, {3.5348789344296913`*^9, 3.534878934673088*^9}, {3.534878969428014*^9, 3.534878981069109*^9}, { 3.5348790112970667`*^9, 3.534879201815508*^9}, {3.5348792391667433`*^9, 3.5348792509675207`*^9}, {3.534879283915584*^9, 3.534879411735869*^9}, { 3.534879462560651*^9, 3.534879570390871*^9}, {3.5348796180872993`*^9, 3.534879918987344*^9}, 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