Basic Concepts in Computational Physics

Abstract

Traditionally physics is divided into two fields of activities: theoretical and experimental. As a consequence of the stunning increase in computer power and of the development of more powerful numerical techniques, a new branch of physics was established over the last decades: Computational Physics. This new branch was introduced as a spin-off of what nowadays is commonly called computer simulations. They play an increasingly important role in physics and in related sciences as well as in industrial applications and serve two purposes, namely: • Direct simulation of physical processes such as ı Molecular dynamics or ı Monte Carlo simulation of physical processes • Solution of complex mathematical problems such as ı Differential equations ı Minimization problems ı High-dimensional integrals or sums This book addresses all these scenarios on a very basic level. It is addressed to lecturers who will have to teach a basic course/basic courses in Computational Physics or numerical methods and to students as a companion in their first steps into the realm of this fascinating field of modern research. Following these intentions this book was divided into two parts. Part I deals with deterministic methods in Computational Physics. We discuss, in particular, numerical differentiation and integration, the treatment of ordinary differential equations, and we present some notes on the numerics of partial differential equations. Each section within this part of the book is complemented by numerous applications. Part II of this book provides an introduction to stochastic methods in Computational Physics. In particular, we will examine how to generate random numbers following a given distribution, summarize the basics of stochastics in order to establish the necessary background to understand techniques like MARKOV-Chain Monte Carlo. Finally, algorithms of stochastic optimization are discussed. Again, numerous examples out of physics like diffusion processes or the POTTS model are investigated exhaustively. Finally, this book contains an appendix that augments the main parts of the book with a detailed discussion of supplementary topics. This book is not meant to be just a collection of algorithms which can immediately be applied to various problems which may arise in Computational Physics. On the contrary, the scope of this book is to provide the reader with a mathematically well-founded glance behind the scene of Computational Physics. Thus, particular emphasis is on a clear analysis of the various topics and to even provide in some cases the necessary means to understand the very background of these methods. Although there is a barely comprehensible amount of excellent literature on Computational Physics, most of these books seem to concentrate either on deterministic methods or on stochastic methods. It is not our goal to competewith these rather specific works. On the contrary, it is the particular focus of this book to discuss deterministic methods on par with stochastic methods and to motivate these methods by concrete examples out of physics and/or engineering.