Monte Carlo Simulations Using Microsoft EXCEL®

This book ​offers step-by-step descriptions of various random systems and explores the world of computer simulations. In addition, this book offers a working introduction to those who want to learn how to create and run Monte Carlo simulations. Monte Carlo simulation has been a powerful computational tool for physics models, and when combined with the programming language Excel, this book is a valuable resource for readers who wish to acquire knowledge that can be applied to more complex systems. Visualization of the simulation results via the Visual Basic built in Microsoft EXCEL is presented as the first step towards the subject. Prior experience with the Excel add-in VBA is kept to a minimum. In addition, a chapter on quantum optimization simulation utilizing Python is added to explore the quantum computation. Readers will gain a fundamental knowledge and techniques of simulation physics, which can be extended to STEM projects and other research projects. Preface Contents 1 Probability Distribution Functions 1.1 Electron Spins in Magnetic Field—Binomial Distribution 1.1.1 Configuration of Spin Array 1.1.2 Simulation of Binominal Distribution 1.2 Radioactive Decay—Poisson Distribution 1.2.1 Decay Equation 1.2.2 Binominal Distribution to Poisson Distribution 1.3 Gaussian Distribution 1.3.1 Poisson to Gaussian 1.3.2 Binominal to Gaussian 1.4 White Noise—Uniform Distribution to Gaussian Distribution 1.5 Central Limit Theorem References 2 Idea of Monte Carlo Simulations 2.1 Calculation of π 2.2 Calculation of Definite Integrals 2.3 Radioactive Decay 2.4 Random Walk 2.4.1 One-Dimensional Random Walk 2.4.2 Two-Dimensional Random Walk 2.5 Percolation References 3 Brownian Motion and Diffusion Equation 3.1 Motion of a Particle Driven by Collisions with Surrounding Particles 3.1.1 One-Dimensional Collision 3.1.2 Two-Dimensional Collision 3.2 Langevin Equation 3.3 Smoluchowski Equation to Diffusion Equation 3.3.1 Smoluchowski Equation to Fokker-Plank Equation 3.3.2 Fokker Plank Equation to Diffusion Equation 3.4 Diffusion Process by Random Walk 3.4.1 One-Dimensional Diffusion 3.4.2 Two-Dimensional Diffusion 3.5 Analytical Solution of One-Dimensional Diffusion Equation 3.5.1 Trial Function Method 3.5.2 Spectral Method 3.6 Numerical Analysis of One-Dimensional Diffusion Equation 3.6.1 Particle Diffusion 3.6.2 Heat Conduction 3.6.3 Analytical Solution of Heat Equation References 4 Quantum Diffusion Monte Carlo Method 4.1 One-Dimensional Infinite Potential Well 4.1.1 Imaginary Time Schrödinger Equation 4.1.2 A Particle in One Dimensional Potential Box 4.2 Quantum Diffusion Monte Carlo Method 4.2.1 Basic Idea of Quantum Diffusion Monte Carlo Method 4.2.2 Harmonic Oscillator 4.2.3 Three-Dimensional Harmonic Oscillator 4.2.4 Hydrogen Atom 4.2.5 Helium Atom 4.2.6 Hydrogen Molecule 4.3 Variational Monte Carlo and Path Integral Monte Carlo Methods 4.3.1 Variational Monte Carlo (VMC) Method 4.3.2 Path Integral Monte Carlo (PIMC) Method References 5 Metropolis–Hastings Algorithm for Ising Model 5.1 Algorithm of Metropolis and Hastings 5.2 Application to Ising Model 5.3 One-Dimensional Ising Model 5.3.1 Exact Solution 5.3.2 Monte Carlo Simulation 5.4 Two-Dimensional Ising Model 5.5 Quantum Optimization Using Ising Model 5.5.1 Optimization by Quantum Annealing 5.5.2 Addition of Horizontal Field 5.5.3 Traveling Salesman References 6 Chaos and Fractal 6.1 Chaos 6.1.1 Lorentz Attractor 6.1.2 Logistic Function 6.1.3 Nonlinear Pendulum 6.1.4 Nonlinear Double Pendulum 6.2 Fractal 6.2.1 Triadic Koch Curve 6.2.2 Sierpinski Triangle 6.2.3 Determination of Fractal Dimensions 6.2.4 Note on Chaos and Fractal 6.2.5 Mandelbrot Figure References Appendix A1 EXCEL Options A1.1 Enabling VBA Macro A1.2 Adding “Data Analysis” A2.3 Autofill A2 VBA Codes A2.1 Two-Dimensional Random Walk on a Square Lattice A2.2 Ground State of Three-Dimensional Harmonic Oscillator by QDMC Method A2.3 Ground State of Hydrogen Atom by QDMC Method A2.4 Ground State of Helium Atom by QDMC Method A2.5 Ground State of Hydrogen Molecule by QDMC Method References Another book on the Monte Carlo simulation? Yes, but this is a kind of book I wanted to have when I was a student. It guides you to explore the fantastic world of Monte Carlo simulations and acquire basic computational knowledge to create and run your own simulation programs. The intended readers are undergraduate to graduate students who are interested in engaging in simulation projects. There are several frequently cited simulation examples from probability distribution functions, computation of π-value, nuclear decay, and random walks, which are extended to classical diffusion problems, quantum diffusion Monte Carlo method, and Ising models along with descriptions of their procedures. A brief introduction of quantum annealing for optimization utilizing Ising models, and descriptions of chaos and fractals are also included with actual examples.