(* 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[ 386762, 7870] NotebookOptionsPosition[ 369435, 7562] NotebookOutlinePosition[ 369954, 7580] CellTagsIndexPosition[ 369911, 7577] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["3.029 Spring 2022\[LineSeparator]Lecture 11 - 03/07/2022", "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}, {3.855176893175762*^9, 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chains with correlations in \ 2D\ \>", "Item", CellChangeTimes->{{3.855055477266987*^9, 3.855055477585175*^9}, { 3.8551698050037327`*^9, 3.8551698485482492`*^9}, {3.8556050347688847`*^9, 3.855605040459993*^9}},ExpressionUUID->"2efb080c-6960-42ef-be3d-\ 19201bc63efa"], Cell["We\[CloseCurlyQuote]ll then try assembling a number of them together", \ "Item", CellChangeTimes->{{3.855055477266987*^9, 3.855055477585175*^9}, { 3.8551698050037327`*^9, 3.8551698822979393`*^9}},ExpressionUUID->"b96b53d9-b234-43d0-8b8b-\ 2a08472b36f4"], Cell[CellGroupData[{ Cell["\<\ First - let\[CloseCurlyQuote]s write a simple function to return a polymer \ with N molecular units and some correlation\ \>", "Item", CellChangeTimes->{{3.8551699344312077`*^9, 3.855169948725903*^9}, { 3.8551699969141073`*^9, 3.855170022324267*^9}, {3.8551700534317493`*^9, 3.855170053433975*^9}},ExpressionUUID->"03b414f1-ffd0-4184-b4f8-\ 3b471682a2a8"], Cell[CellGroupData[{ Cell["It will be convenient to define a \[OpenCurlyDoubleQuote]correlation\ \[CloseCurlyDoubleQuote] variable", "Subitem", CellChangeTimes->{{3.8551699344312077`*^9, 3.855169948725903*^9}, { 3.8551699969141073`*^9, 3.855170022324267*^9}, {3.85517005504324*^9, 3.855170075994033*^9}},ExpressionUUID->"563a07e7-86c5-4830-bfec-\ f3dbf49abfea"], Cell["\<\ Which has a value of 1 when the polymer is \[OpenCurlyDoubleQuote]maximally\ \[CloseCurlyDoubleQuote] correlated (i.e. the next bond angle is chosen from \ a range {0,0})\ \>", "Subsubitem", CellChangeTimes->{{3.8551699344312077`*^9, 3.855169948725903*^9}, { 3.8551699969141073`*^9, 3.855170022324267*^9}, {3.85517005504324*^9, 3.855170128407077*^9}},ExpressionUUID->"119d7a78-170c-4fda-9357-\ 72c55c6262ae"], Cell["\<\ and a value of 0 when the polymer is uncorrelated (i.e. the next bond angle \ is chosen from the full range of angles {-\[Pi],\[Pi]})\ \>", "Subsubitem", CellChangeTimes->{{3.8551699344312077`*^9, 3.855169948725903*^9}, { 3.8551699969141073`*^9, 3.855170022324267*^9}, 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Cell[CellGroupData[{ Cell["Our first step is to write a collision-detection function", "Item", CellChangeTimes->{{3.855171116646644*^9, 3.8551711280666723`*^9}, { 3.855171313048789*^9, 3.855171313050551*^9}},ExpressionUUID->"d509b70b-7e59-4731-aa57-\ 827d0697e7b1"], Cell["\<\ The simplest idea is to compute the distance b/w all the particles\ \>", "Subitem", CellChangeTimes->{{3.855171116646644*^9, 3.8551711280666723`*^9}, { 3.855171314143138*^9, 3.855171341669619*^9}, {3.855171545684564*^9, 3.855171549554865*^9}},ExpressionUUID->"8bbe50b0-91aa-43a4-ad40-\ 0fbfecb3ffff"], Cell[CellGroupData[{ Cell["And check no-distances are less the bond length", "Subitem", CellChangeTimes->{{3.855171116646644*^9, 3.8551711280666723`*^9}, { 3.855171314143138*^9, 3.855171341669619*^9}, {3.855171545684564*^9, 3.855171559041312*^9}, {3.855649298193787*^9, 3.8556493119200363`*^9}},ExpressionUUID->"83b9dd9a-9a01-433a-88bf-\ 4cec2c2edde0"], Cell[CellGroupData[{ Cell["Distance between two particles", 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CellChangeTimes->{{3.855171413951977*^9, 3.855171501778103*^9}},ExpressionUUID->"8a9a4e64-1f81-42f8-aac9-\ d0f76da551f8"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ If you\[CloseCurlyQuote]re curious about all the ways to calculate pairwise \ distances in the WL - there\[CloseCurlyQuote]s a nice SE post about it here:\ \>", "Subitem", CellChangeTimes->{{3.855171533708517*^9, 3.8551715406524353`*^9}, { 3.855171615302339*^9, 3.8551716390462008`*^9}},ExpressionUUID->"e4bfdeb7-bb77-4442-8258-\ 1564b1135cf4"], Cell["\<\ https://mathematica.stackexchange.com/questions/21861/fastest-way-to-\ calculate-matrix-of-pairwise-distances\ \>", "Subsubitem", CellChangeTimes->{{3.855171533708517*^9, 3.8551715406524353`*^9}, { 3.855171615302339*^9, 3.855171639589957*^9}},ExpressionUUID->"59ef05d3-8d20-453e-8e48-\ 9adeaaaf2b87"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ The built-in `DistanceMatrix` function is actually reasonably fast (on \ numeric points using EuclideanDistance)\ \>", "Subitem", 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"0"} }, GridBoxAlignment->{"Columns" -> {{Center}}, "Rows" -> {{Baseline}}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{{3.855171699828109*^9, 3.855171711337804*^9}}, CellLabel-> "Out[55]//MatrixForm=",ExpressionUUID->"0610cf9f-6eb8-4cfd-93c6-\ 585edc13ada0"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ One quick common trick on Distance matrices is that often times if we want to \ compare distances we can just use SquaredEuclideanDistance and compare \ against bond_length^2\ \>", "Item", CellChangeTimes->{{3.855172998398219*^9, 3.855173054378327*^9}},ExpressionUUID->"74fb40b8-3ec6-4825-b14f-\ 994d0dd7efa9"], Cell["\<\ This saves us having to compute the outermost square root in all the entries \ above!\ \>", "Subitem", CellChangeTimes->{{3.855172998398219*^9, 3.855173071160063*^9}},ExpressionUUID->"c8bb78a8-f9b2-4736-bea2-\ a2642c43802d"] }, Open ]], Cell[CellGroupData[{ Cell["However, this still does more work than we need it to", "Item", CellChangeTimes->{{3.855172964439636*^9, 3.855172981715353*^9}},ExpressionUUID->"a7f9105e-695b-4d91-a1d3-\ 3b736d2be275"], Cell["\<\ Remember we\[CloseCurlyQuote]ll actually be building this polymer chain \ incrementally\ \>", "Subitem", CellChangeTimes->{{3.855172964439636*^9, 3.855172996687158*^9}, { 3.855173075432047*^9, 3.855173076391027*^9}},ExpressionUUID->"ab8b38bd-5fc3-4e2b-b740-\ 9d36f6ccc402"], Cell[CellGroupData[{ Cell["\<\ I.e. we only need to check if the point we\[CloseCurlyQuote]re trying to add \ collides with any of the existing points\ \>", "Subitem", CellChangeTimes->{{3.855172964439636*^9, 3.855172996687158*^9}, { 3.855173075432047*^9, 3.855173097915821*^9}},ExpressionUUID->"9963ab39-0fc3-412c-9f53-\ 4e134ca8cebe"], Cell["Since the previous points have already been checked", "Subsubitem", CellChangeTimes->{{3.855172964439636*^9, 3.855172996687158*^9}, { 3.855173075432047*^9, 3.855173107152231*^9}},ExpressionUUID->"11fcbd68-45dc-42aa-9476-\ 0289bcf2eb4a"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["We can do this efficiently using `Nearest`", "Item", CellChangeTimes->{{3.8551731146295424`*^9, 3.855173124111429*^9}, { 3.855175294465577*^9, 3.855175294468445*^9}},ExpressionUUID->"ffb4a1a2-80c9-4ae6-8f46-\ ecd8c453b33c"], Cell["\<\ This uses a KDtree implementation - ask me about it if \ there\[CloseCurlyQuote]s time!\ \>", "Subitem", CellChangeTimes->{{3.8551731146295424`*^9, 3.855173124111429*^9}, { 3.8551754135161037`*^9, 3.855175448899599*^9}},ExpressionUUID->"e65dfc59-4a73-48e1-8c25-\ 9335a5139171"], Cell[CellGroupData[{ Cell["\<\ Essentially we\[CloseCurlyQuote]ll build a fast look-up table of the existing \ particles - and then find the nearest particle to our candidate\ \>", "Subitem", 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