Random Numbers and Monte Carlo Methods

Abstract

Many-body problems often involve the calculation of integrals of very high dimension which cannot be treated by standard methods. For the calculation of thermodynamical averages Monte Carlo methods are very useful which sample the integration volume at randomly chosen points. This chapter begins with some basic statistics. Probability density and cumulative distribution are introduced. The construction of a histogram is described. The central limit theorem is discussed with some examples. Pseudo-random numbers with given distribution are generated. The principles of Monte Carlo integration and importance sampling are introduced. The Metropolis algorithm is studied in detail which is very helpful to calculate thermodynamic averages in configuration space. Computer experiments demonstrate the central limit theorem and nonlinear optimization with the Metropolis method.