A new field opened when modern computers offered the possibility of performing extensive simulations of large systems. This allows known behaviours to be checked and provides an exploratory guide in circumstances where theoretical tools are absent. Measurements of observables can be compared both to existing theoretical expectations – providing a crucial test for their validity – and to experiments – checking the modelling of a physical system –. This chapter presents the background material needed to design simulations on a (relatively) large scale. Some numerical examples have already been presented in previous chapters, and we only give a few further illustrations, pertaining mainly to lattice gauge theory, the usefulness of which relies extensively on this method. We also describe a practical implementation of real space renormalization, known as the Monte Carlo Renormalization Group method (Ma, Swendsen, Wilson). Finally, we discuss specific issues relevant to the extension of the simulations to fermionic systems.

Algorithms

Generalities

Systems with up to 106 to 107 variables can be handled by computers, and these numbers may soon be significantly increased. The measurements can be sufficiently numerous to allow statistical accuracy. Although simulated systems still have a modest size as compared to macroscopic systems, collective effects already clearly appear and accurate results about critical phenomena emerge from the numerical simulations. It turns out, when investigating more closely the available numerical methods, that one gets a better insight into the foundations of equilibrium statistical physics, the ergodicity problems, the meaning of probabilities and, last but not least, ways to approach equilibrium.