Monte Carlo Simulations in Statistical Physics

Abstract

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].