"Tópicos em mecânica estatística de sistemas complexos: uma abordagem mecânico-estatística de dois tópicos de interesse em finanças, economia e sociologia".
Instituto de Física, Universidade de São Paulo.
In the present work we explore two topics of interest in finance, economics and social anthropology through the application of techniques from information theory and statistical mechanics. In the first topic we study the connexion between statistical dependency theory, information theory and copula theory. The concept of copula distribution is reviewed and applied to the reformulation of the definition of dependency measures given by Rényi. It is then shown that the mutual information satisfy all the requirements to be a good dependency measure. We derive an identity between mutual information and the entropy of the copula distribution and a more specific decomposition of the mutual information of an elliptical distribution into its linear and non-linear parts. We evaluate the risk of using naive measures as statistical dependency measures by showing that linear correlation can grossly underestimate dependency. Those results are used to develop a method to detect deviation from gaussian dependence in pairs of random variables and apply it to financial time series. Finally, we discuss a method to adjust t-copulas to empirical data through the determination of the mutual information and Kendalls tau. In the second topic we develop a model for the emergence of authority in pre-agricultural human societies. We discuss empirical motivations with roots in neuroscience, primatology and anthropology for a mathematical model able to explain the ample variability of forms of human social organization in the egalitarian-hierarchical axis. The model results from the application of information theory on a hypothesis about the evolutive costs involved in social life. It generates a rich phase diagram, with different regimes which can be interpreted as different types of societal organization, from egalitarian to hierarchical. The control parameters of the model are connected to the cognitive capacity of the species in question, the size of the group and ecological and social pressures.