Computer Simulation and Computer Algebra

It is the aim of this chapter to enable the readers to implement solutions to problems in the physical sciences with a computer program, and carry out the ensuing computer studies. They will therefore be shown a few basic numerical methods, and the general spirit for mapping physics problems onto a computational algorithm. It is advisable to spend some time actually implementing the exercises proposed, since is only by so doing that one may learn about, and get a feel for, the spirit of scientific computing. Examples are given using the FORTRAN 77 language and the UNIX operating system. The graphics interface used is that of the SUN workstation.

In Statistical Physics one mostly deals with thermal motion of a system of particles at nonzero temperatures. For example, in a classical ideal gas of point-like molecules each particle has an average kinetic energy equal to dk _B T/2 in d dimensions. Here T is the absolute temperature and k _B = 1.6 × 10²³ Joule per Kelvin is Boltzmann’s constant. Statistical Physics is used try to explain such laws and to predict the properties of materials consisting of many such particles; therefore, in this example the specific heat is 3Nk _B /2 in three dimensions if the gas consists of N particles. In most applications, the number N of particles is very large, and they influence each other by their intermolecular forces. For example, a glass of beer contains about 10²⁵ water molecules, and if these molecules did not interact with each other, the beer would vanish by evaporation, not by drinking. These interactions are also unhealthy for theoretical physics since with interactions usually one cannot solve exactly the problem of how the molecules move and what their average energy is, because even on a computer it is not possible to store the positions and velocities of 10²⁵ point-like molecules. (The Cray-2 supercomputer has only two Gigabytes of main memory.) Instead, one is forced to work with a much smaller number of molecules, below 10⁶, and solve numerically the equations of motion arising from Newton’s law: force equals mass times acceleration. This method is called molecular dynamics and has already been used in the first chapter of this book by Zabolitzky. We will not deal with this technique here; readers who want to know more are referred to the book of D.W.Heermann [1].

If you calculate on a computer by means of “letters” rather than with numbers, say you want to expand (a + 27b ³ − 4C)⁵ or to integrate ∫ 5x ² sin ³ x dx, then you are applying “computer algebra” (CA). For that purpose you need: And last but not least, you should have an introduction on how to use the CA-System.

So far, this book is unbalanced: The authors who need the computer to think have introduced the reader to the language REDUCE they use; but the two other authors who can already think and who use the computer for simulations have assumed that the reader understands FORTRAN, historically the first higher programming language. A reader experienced in BASIC should be able to learn FORTRAN by simply reading the listed programs. True beginners may learn FORTRAN through the following remarks (the need for which was pointed out to us by J.Adler). Some statements are compared with the corresponding BASIC statements.