Topics in complex systems

Abstract

Fundamental laws of physics, although successful in explaining many phenomena observed in nature and society, cannot account for the behaviour of complex, non-Hamiltonian systems. Much effort has been devoted to better understanding the topological properties of these systems. Neither ordered nor disordered, these systems of high variability are found in many areas of science. Studies on sandpiles, earthquakes and lattice gases have all yielded evidence of complexity in the form of power law distributions. This scalefree characteristic is believed to be the hall-mark of complexity known as self-organised criticality. Systems in the self-organised critical state regulate themselves and are resistant to error and attacks. The aim of this thesis is to further current knowledge of complex systems by proposing and analysing three models of real systems. Statistical mechanics and numerical simulations are used to analyse these models. The first model mimics herd behaviour in social groups and encompasses growth and addition. It has been found that when the growth rate is fast enough, the group size distribution conforms to a power law. When the growth rate is slow, the system runs out of free agents in finite time. The second model aims to capture the basic empirical measurements from hospital waiting lists. This model illustrates how the power law distributions found in empirical studies might arise, but also indicates that these distributions are unlikely to be caused by the preferential behaviour of patients or physicians. The third model is a salary comparison model; the salary distributions of most of its variants are power laws. Both mean field and 1-d versions of the model are analysed, and differences between the two versions are identified by looking at the mean absolute difference between the salaries in each version.