Invariância de escala e difusão anômala em sistemas complexos urbanos e biológicos

Abstract

Statistical physics has proved to be fruitful in the study of systems far from traditional physics. Physicists have applied techniques and concepts from statistical mechanics in the study of data from complex systems of the most diverse areas of knowledge. Several studies focus on time series of financial markets, social and biological systems. Many of these approaches use concepts derived from phase transition physics and anomalous diffusion. In this thesis, we apply the concepts of scale invariance and anomalous diffusion in the study of urban and biological complex systems. In Chapter 1, we investigate a metric that takes into account the nonlinearites in the relationship between urban indicators and population size, where we show that this scale-ajusted metric can be used to quantify and predict urban indicators. In Chapter 2, we show that the spatial correlation in the number of per capita homicides decays exponentially and that this correlation is independent of the population dynamics. We also show that this correlation leads to clusters of cities that can be modelled as a percolation-like transition. In Chapter 3, we make a complete characterization of the diffusion patterns of four species of protozoa. We show that the spread of these protozoan is superdiffusive and that there are long-range correlations in the radial velocities. In Chapter 4, we use a similar approach showing that C. Elegans is also characterized by superdiffusion and long-range correlations in the velocities. We further show that the exponents characterizing their dynamics change with ageing and diseases, similar to what was previously found in human physiology. Finally, in Chapter 5, we propose an extension for the comb-model via Langevin-like equations driven by fractional Gaussian noises (long-range correlated). We show that the correlations can affects the diffusive behavior in a non-trivial fashion, resulting in a quite rich diffusive scenario, that can be applied in the context of complex systems such as living cells.