(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 40672, 1343] NotebookOptionsPosition[ 36503, 1214] NotebookOutlinePosition[ 36862, 1230] CellTagsIndexPosition[ 36819, 1227] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Atomic Clusters", "Title"], Cell["\<\ APAM 1601 Columbia University\ \>", "Subsubtitle"], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell[TextData[{ "In this notebook, we continue to explore the use of random walks and the \ Rosenbluth-Teller-Metropolis algorithm to understand complicated physical \ systems in \"thermal equilibrium\". For this analysis, we will explore the \ three-dimensional structure of clusters of \"atoms\" interacting through the \ Lennard-Jones potential. ", ButtonBox["Sir John Edward Lennard-Jones (1894-1954) ", BaseStyle->"Hyperlink", ButtonData:>{ URL["http://www.quantum-chemistry-history.com/Le-Jo1Ue.htm"], None}], "was a leading theoretical chemist of his time. His simple (yet highly \ nonlinear) model for the potential energy between two atoms or molecules has \ become a much studied system of molecular configuration. The potential energy \ between two atoms is a combination of an ", StyleBox["attractive", FontSlant->"Italic"], " van der Waals force (due to the electric polarization of atoms) and a ", StyleBox["repulsive", FontSlant->"Italic"], " term resulting from the overlap of electron orbits. Today's research \ trying to understand the configurations found in inorganic nanoscience and in \ protein folding continue to make these sorts of models an active area of \ research. 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When the atoms move closer together, a strong potential gradient \ separates them." }], "Text"], Cell[CellGroupData[{ Cell["Monte Carlo Simulation Rules", "Subsection"], Cell[TextData[{ "The \"rules\" for our numerical simulation are similar to those used for \ many problems in statistical physics (like an ideal ", StyleBox["non-interacting", FontSlant->"Italic"], " atmosphere)\[Dash]with one difference. For particles interacting by way of \ the Lennard-Jones potential, all particles are strongly ", StyleBox["interacting with each other", FontSlant->"Italic"], ". Many large problems in statistical physics assume that particles ", StyleBox["only interact with nearest neighbors. ", FontSlant->"Italic"], "When a particle moves, we need to compute the change in system energy by \ summing over all interactions." }], "Text"], Cell[TextData[{ "We will follow the following set of rules:\nStep 1) Initialize the \ positions of the particles within a sphere of given radius..\nStep 2) Select \ a random particle. Make a potential random step ", StyleBox["in any direction. However, ", FontSlant->"Italic"], "if the step causes the particle to escapee \nStep 3) Calculate the energy \ ", StyleBox["change", FontSlant->"Italic"], " to the entire system if the particle makes the step.\nStep 4a) If the \ energy change is negative, accept the step.\nStep 4b) If the energy change is \ positive, pick a random number between 0 and 1. If the random number is less \ than ", StyleBox["Exp[ - (energy change)/kT]", FontWeight->"Bold"], ", then accept the step.\nStep 4c) Otherwise, reject the step and leave the \ position (and energy) unchanged.\nStep 5) Go to Step 2 and repeat (for a very \ long time....)" }], "Text"], Cell[TextData[{ "These rules comprise the \"Metropolis\" method for computational \ statistical physics. It is problably the most often used computational method \ in physics! See online biography for ", ButtonBox["Nick Metropolis", BaseStyle->"Hyperlink", ButtonData:>{ URL["http://scienceworld.wolfram.com/biography/Metropolis.html"], None}], ". The real insight behind the algorythm was developed by late ", ButtonBox["Marshall Rosenbluth", BaseStyle->"Hyperlink", ButtonData:>{ URL["http://www.utexas.edu/faculty/council/2004-2005/memorials/rosenbluth/\ rosenbluth.html"], None}], ", the father of modern plasma physics. 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