1 Introduction

1.1 Goal, contents, and outline

Nowadays, (theoretical) physicists are mainly concerned with two subjects: one is the devising of models, normally in the form of differential equations, ordinary or partial, while the other is their approximate solution, normally with the help of larger or smaller computers. In a certain way, computational physics can be seen as the link between theory and experiment. The often involved equations need a numerical treatment, usually in connection with certain simplifications and approximations. Once the code is written, the experimental part begins. A computer on which, for instance, a turbulent pipe flow is simulated can to some extent be regarded as a test station with easy access to output values such as flow rates, velocities, temperatures and also to input parameters like material constants or geometric boundary conditions.

The present text tries to show, by means of examples coming from different corners of physics, how physical and mathematical questions can be answered using a computer. Problems from classical mechanics normally rely on ordinary differential equations, while those of electrodynamics, quantum mechanics, continuum mechanics or hydrodynamics are formulated by partial differential equations, which are often nonlinear.

This is neither a numeric book, nor a tutorial for a special computer language, for which we refer to the literature [1]. On the other hand, most of the discussed algorithms are realized in FORTRAN. FORTRAN is an acronym composed of FORmula TRANslator. This already indicates that it is more appropriate and convenient for solving problems originating from theoretical physics or applied mathematics, like managing complex numbers or solving differential equations.

During the past 20 years or so, the software package MATLAB, also based on FORTRAN routines, has enjoyed more and more popularity in research as well as in teaching [2]. The notion is formed as another acronym from MATrix LABoratory. Thus, MATLAB shows its strengths in the interactive treatment of problems resulting from vector or matrix calculus, but other problems related to differential equations can also be tackled. Programming is similar to FORTRAN and for beginners many things are easier to implement and handle. In addition, MATLAB captivates with its comfortable and convenient graphic output features.

MATLAB is devised as a standalone system. It runs under Windows and Linux. No additional text editor, compiler or software libraries are needed. However, contrary to FORTRAN, once a MATLAB code is written, it is not compiled to an effective machine code, but rather interpreted line by line. Processes may thus become incredibly slow. Almost everything said here about MATLAB is also valid for PYTHON that came into vogue during the last 10 years.

MATLAB (and PYTHON) may have some advantages in program development and testing, but if in the end a fast and effective code is desired (which should usually be the case), a compiler language is inevitably preferred. Nevertheless, we shall give some examples also in MATLAB, mainly those based on matrix calculation and linear algebra.

In the meantime, highly developed computer algebra systems exist [3] and can be used to check the formulas derived in the book, but we do not go into this matter.

The motto of the book could read 'Learning by Doing'. Abstract formulations are avoided as much as possible. Sections where theory dominates are followed by an example or an application.

The chapters, of course, can be read in sequence, albeit not necessarily. If someone is interested, for instance, in partial differential equations, it is possible to start with Chapter 7 and consult Chapters 3-6 for potential issues. The same is true for the statistical part, Chapter 9. It should be understandable also without knowledge of the preceding chapters. Depending on the subject, certain basic knowledge in (theoretical) physics is required. For Chapters 4, 5, and 8, classical mechanics and a little bit of quantum mechanics should be at hand, while for Chapter 8 additional basics in hydrodynamics won't hurt. In Chapter 9 we presume a certain familiarity with the foundations of statistical mechanics.

In any case, it is recommended to have a computer or notebook running nearby and, according to the motto, experiment with and check the presented routines immediately while reading.

Figure 1.1 A simple neuronal network where five input neurons are connected to three output neurons by dynamic connections (from Chapter 2).

Figure 1.2 Time evolution of low-dimensional systems from Chapter 4. Left: two-planet problem with gravitational interaction. The orbit of the inner planet (red) is unstable and the planet is thrown out of the system. Right: deterministic chaos for the driven pendulum, Poincaré section.

The topics up to Chapter 5, and to some extent also in Chapter 6, can be categorized under the notion low-dimensional systems, problems that can be described with only a few variables. In Chapter 2, we start with iterative maps as a pre-stage to differential equations. Here, notions like stability, chaos, or fractals are introduced and related to each other. Neural networks are considered as special maps mapping input neurons to output neurons by dynamic connections (Figure 1.1). Although going back to the 50s, neural nets nowadays have regained huge attention because they provide the basic principle of artificial intelligence devices.

Besides iterative maps, applications from Newton's classical mechanics come to mind, for instance, the computations of planetary orbits or the motion of a pendulum (Figure 1.2). Naturally for this case, ordinary differential equations play a central role. For three dimensions and more (or better: degrees of freedom), chaotic behavior becomes possible, if not the rule. Large parts of Chapters 2-4 are dedicated to the so-called deterministic chaos.

Figure 1.3 Epidemic compartment models consist of normally three or more compartments where the individuals are located according to their state of health, see Chapter 6.

Figure 1.4 A microscopic traffic flow model describes a number of interacting cars by their positions and velocities, see Chapter 6.

Chapter 6 leaves physical subjects for a while and turns to models where time-delay, memory, and stochastic terms are on the main focus. Two systems are studied in detail. The first is an epidemic model, also known as SIR or compartment model, Figure 1.3, based on ordinary differential equations but including memory terms; the second treats road traffic flow by means of microscopic considerations that allow the prediction of self-organized congestion, Figure 1.4.

Starting with Chapter 6 high-dimensional (or infinite-dimensional) systems come into the focus, normally described by field theories in the form of partial differential equations (Chapter 7), but also by ordinary differential equations with memory or time-delay terms (Chapter 6). These can be linear, as is very often seen in quantum mechanics or electrodynamics, but also nonlinear, as in hydrodynamics (Figure 1.5) or in diffusion-driven macroscopic pattern formation (Figure 1.6). Caused by nonlinearities, patterns may emerge on most diverse spatio-temporal scales (Chapter 8) , the dynamics becoming chaotic or turbulent (Figure 1.7).

Figure 1.5 Numerical solutions of field equations, see Chapter 8. Left: time-dependent Schrödinger equation, probability density and center of mass of an electron in a constant magnetic field. Right: Hydrodynamic vortex for the driven cavity.

Figure 1.6 Macroscopic pattern formation for the example of a reaction-diffusion system, time series. Out of a large-scale oscillating solution, nuclei emerge spontaneously followed by the generation of small-scaled Turing structures. The numerical integration of the Brusselator equations is shown in Chapter 8.

Figure 1.7 Two-dimensional flow field showing turbulence, time series. Due to a vertical temperature gradient, an initially stable layer becomes unstable (top). Warmer air (red) rises in the form of 'plumes' in the colder air (blue) (from Chapter 8).

Finally, Chapter 9 is devoted to Monte Carlo methods. Here, randomness plays a key role in the determination of equilibrium states. Using Monte Carlo simulations, phase transitions can be studied without resorting to differential equations (Figure 1.8). The application on the Ising model of spatially arranged spins is a prominent example.

Figure 1.8 Monte Carlo simulation of a gas comprised of 1000 particles with hardcore rejection and...