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as ", ButtonBox["Kinetic Monte Carlo", BaseStyle->"Hyperlink", ButtonData->{ URL["https://en.wikipedia.org/wiki/Kinetic_Monte_Carlo"], None}, ButtonNote->"https://en.wikipedia.org/wiki/Kinetic_Monte_Carlo"], " (KMC) in the Materials Science vernacular" }], "Subitem", CellChangeTimes->{{3.859446539456017*^9, 3.85944662785948*^9}, { 3.85944668405542*^9, 3.8594466933567333`*^9}, {3.85944682188491*^9, 3.859446822759149*^9}},ExpressionUUID->"4486e144-4456-4848-9997-\ f9a61b0f902c"], Cell[TextData[{ "It also goes by other popular names, such as ", ButtonBox["Markov chain Monte Carlo", BaseStyle->"Hyperlink", ButtonData->{ URL["https://en.wikipedia.org/wiki/Markov_chain_Monte_Carlo"], None}, ButtonNote->"https://en.wikipedia.org/wiki/Markov_chain_Monte_Carlo"], " (MCMC)" }], "Subitem", CellChangeTimes->{{3.859446539456017*^9, 3.8594466737438927`*^9}},ExpressionUUID->"d65385ef-4fb5-49f2-9b57-\ 5ecf88fdfa1f"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "The are many flavours of KMC, but we will focus on the most popular \ \[OpenCurlyDoubleQuote]rejection-KMC\[CloseCurlyDoubleQuote] algorithm, given \ by the ", ButtonBox["Metropolis-Hastings", BaseStyle->"Hyperlink", ButtonData->{ URL["https://en.wikibooks.org/wiki/Molecular_Simulation/Monte_Carlo_\ Methods#Metropolis_Monte_Carlo_Simulation"], None}, ButtonNote-> "https://en.wikibooks.org/wiki/Molecular_Simulation/Monte_Carlo_Methods#\ Metropolis_Monte_Carlo_Simulation"], " algorithm" }], "Item", CellChangeTimes->{{3.859446810780512*^9, 3.8594469013245564`*^9}},ExpressionUUID->"d11b1af6-fc23-4772-bd21-\ 723a059d5491"], Cell["In-fact, you saw this last semester in 3.019", "Subitem", CellChangeTimes->{{3.859446810780512*^9, 3.859446929321035*^9}},ExpressionUUID->"b1c3e4f2-9a20-49ed-895f-\ bcd6ea50fcb6"], Cell[CellGroupData[{ Cell["\<\ We will briefly review the method in 1D, and then address one of its \ shortcomings\ \>", "Subitem", CellChangeTimes->{{3.859446810780512*^9, 3.859446945457416*^9}, { 3.859446979268395*^9, 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Cell[TextData[{ StyleBox["Important Note: ", FontWeight->"Bold"], "The above slider gives the impression of \[OpenCurlyDoubleQuote]time\ \[CloseCurlyDoubleQuote]. However, the moves are chosen at random and have no \ physical meaning. They should instead be interpreted as plausible equilibrium \ states the system can access in that configuration at the specified \ temperature" }], "Item", CellChangeTimes->{{3.859448334727644*^9, 3.859448439626802*^9}},ExpressionUUID->"7ce58297-8b60-44da-ae0d-\ e60a7ed5807e"], Cell["\<\ This has important implications. Namely since we have no temporal \ information, we can only describe equilibrium properties and not dynamical \ properties\ \>", "Subitem", CellChangeTimes->{{3.859448334727644*^9, 3.859448505034751*^9}, 3.859448727643618*^9},ExpressionUUID->"609d9caf-c2db-4f77-8798-\ b5bd07969605"] }, Open ]], Cell[CellGroupData[{ Cell["That\[CloseCurlyQuote]s it folks!", "Item", CellChangeTimes->{{3.859448601773905*^9, 3.8594486356013393`*^9}},ExpressionUUID->"f64d3f60-41a2-46ef-8658-\ 507c7c37e109"], Cell[CellGroupData[{ Cell["\<\ Despite its simplicity, the MCMC method is very useful in modeling many \ thermally-activated phenomena, such as:\ \>", "Subitem", CellChangeTimes->{{3.859448601773905*^9, 3.859448667637973*^9}},ExpressionUUID->"6d22df2c-8578-4a1d-88e9-\ 470e52909b83"], Cell["Sintering", "Subsubitem", CellChangeTimes->{{3.859448601773905*^9, 3.8594486709520884`*^9}},ExpressionUUID->"eed05882-bae5-436d-a831-\ 659d68101296"], Cell["Surface Roughening", "Subsubitem", CellChangeTimes->{{3.859448601773905*^9, 3.859448677778018*^9}},ExpressionUUID->"504c74e0-e53b-468b-b66d-\ 4f6983f44b3a"], Cell["Vacancy Diffusion", "Subsubitem", CellChangeTimes->{{3.859448601773905*^9, 3.859448688548462*^9}, { 3.859448750889439*^9, 3.859448750891129*^9}},ExpressionUUID->"64b3f588-26ee-46ed-b105-\ 972a3dadd671"], Cell[" Coarsening", "Subsubitem", CellChangeTimes->{{3.859448601773905*^9, 3.859448688548462*^9}, { 3.859448751181061*^9, 3.859448753514517*^9}},ExpressionUUID->"f849c596-2fd1-4ada-a00d-\ 6bed3e7ba46e"], Cell["Dislocation Mobility", "Subsubitem", CellChangeTimes->{{3.859448601773905*^9, 3.859448688548462*^9}, { 3.859448751181061*^9, 3.859448769359763*^9}},ExpressionUUID->"8e269264-8b21-47f8-9f38-\ 4320ce59ee62"], Cell["etc..", "Subsubitem", CellChangeTimes->{{3.859448601773905*^9, 3.859448688548462*^9}, { 3.859448751181061*^9, 3.859448770542015*^9}},ExpressionUUID->"7e61d91e-cb4c-46e6-a70a-\ 794abdb224d4"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Meta-Dynamics", "Section", CellChangeTimes->{{3.859448782999133*^9, 3.859448786143858*^9}},ExpressionUUID->"7ed8b50b-8a6e-4e08-9c5d-\ 4a40036429a5"], Cell["\<\ As we saw in our KMC simulation above, the MCMC simulation is likely to get \ \[OpenCurlyDoubleQuote]stuck\[CloseCurlyDoubleQuote] in a local energy minimum\ \>", "Item", CellChangeTimes->{{3.859448808656392*^9, 3.859448863299155*^9}},ExpressionUUID->"8425439d-b607-48b5-a4b8-\ 276320d3cf32"], Cell[CellGroupData[{ Cell["\<\ There are many possible remedies for this, the simplest being increasing the \ temperature, however most are computationally expensive\ \>", "Item", CellChangeTimes->{{3.859448808656392*^9, 3.859448932428295*^9}},ExpressionUUID->"0d433497-4fe7-4fce-85c4-\ 0bfa9d0100fa"], Cell["\<\ This is because the particle spends most of its time in these local minima \ and very rarely (even at high temperatures) explores metastable configurations\ \>", "Subitem", CellChangeTimes->{{3.859448808656392*^9, 3.859448966236546*^9}},ExpressionUUID->"3f94566e-d82f-441d-a14b-\ 034d759af344"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Exploring these metastable states can be important when we wish to \ \[OpenCurlyDoubleQuote]reconstruct\[CloseCurlyDoubleQuote] an unknown \ potential energy surface\ \>", "Item", CellChangeTimes->{{3.859449656411772*^9, 3.8594497145097446`*^9}},ExpressionUUID->"233a924a-3e68-46da-9663-\ 8cb0ef735470"], Cell["\<\ E.g. in the Ammonia molecule example above, we could simulate its trajectory \ using molecular dynamics, and from that information wish to recover the \ energy landscape\ \>", "Subitem", CellChangeTimes->{{3.859449656411772*^9, 3.859449734947268*^9}, { 3.859449785196889*^9, 3.8594498391171427`*^9}},ExpressionUUID->"73cb0b96-9fea-497b-9fc4-\ 608dbc92f8b8"] }, Open ]], Cell["\<\ Metadynamics attempts to solve this by \[OpenCurlyDoubleQuote]filling\ \[CloseCurlyDoubleQuote] in the free energy minima using a history-dependent \ biasing potential\ \>", "Item", CellChangeTimes->{{3.859449594738253*^9, 3.859449650905863*^9}, { 3.859449904800469*^9, 3.859449924528421*^9}},ExpressionUUID->"9fa82f13-269b-4a09-ad1c-\ 882bf7eeeba1"], Cell[CellGroupData[{ Cell["Kernel Estimation", "Subsection", CellChangeTimes->{{3.859450503422021*^9, 3.859450506282073*^9}},ExpressionUUID->"2d60b760-5f25-41e6-b524-\ 3510a253b544"], Cell[CellGroupData[{ Cell["\<\ First, let\[CloseCurlyQuote]s investigate the locations visited by our \ unbiased MCMC simulation above\ \>", "Item", CellChangeTimes->{{3.859449933044674*^9, 3.859449962805443*^9}},ExpressionUUID->"06cf4abe-1f52-4916-bfa8-\ 8a17fdef5c05"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Histogram", "[", RowBox[{"unbiasedSimulation", ",", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"-", "7"}], "/", "4"}], ",", RowBox[{"3", "/", "2"}], ",", RowBox[{"1", "/", "16"}]}], "}"}], ",", RowBox[{"Frame", "\[Rule]", "True"}], ",", RowBox[{"ChartStyle", "\[Rule]", 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