Monte Carlo Simulations using Microsoft Excel

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