Complexity : a study of fractals and self-organized criticality
Huynh, Hoai Nguyen
Date of Issue2013
School of Physical and Mathematical Sciences
Imperial College London
Over the past few decades, Complex Systems or Complexity has emerged as a new field of Science to study abundant complicated behaviours of systems with nonlinear interactions among many degrees of freedom. These systems can range from a very simple system like one-dimensional map (May R., 1976 Nature 261 459) or a collective system with many (spatial) degrees of freedom like cellular model of sandpile (Bak P., Tang C., and Wiesenfeld K., 1987 Phys. Rev. Lett. 59 381) in theoretical study to complicated natural systems like the Atmosphere (Peters O., Hertlein C., and Christensen K., 2001 Phys. Rev. Lett. 88 018701) or the Earth’s crust (Gutenberg B., and Richter C. F., 1955 Nature 176 795). The emergent feature of these systems is the ubiquitous scale-invariance in temporal as well as spatial observables. In this thesis, Complexity is looked at from two perspectives: Fractals and Self-Organized Criticality. They both share the same path from simplicity to complexity: The repeated application of simple microscopic interacting rules among elements of a physical system, as time evolves, gives rise to very complicated macroscopic structures observed. This thesis comprises of two parts: first part is a study of Fractals, and second part is a study of Self-Organized Criticality. In the first part, an idea of creating fractals by using the geometric arc as the basic element is presented. This approach of generating fractals, through the tuning of just three parameters, gives a universal way to obtain many different fractals including the classic ones. The fractals generated using this arc-fractal system are shown to possess a number of features, one of which is the ability to tile the space. Furthermore, by assuming that coastline formation is based purely on the processes of erosion and deposition, the arc-fractal system can also serve as a dynamical model of coastal morphology, with each level of its construction corresponding to the time evolution of the shape of the coastal features. Remarkably, the results indicate that the arc-fractal system can provide an explanation on the origin of fractality in real coastline. In the second part, high-accuracy moment analysis is performed to analyse the avalanche size, duration and area distribution of the Abelian Manna model. The model is studied on a vast number of lattices in different dimensions ranging from one to three, including the noninteger ones, with various detailed structures.
DRNTU::Science::Physics::Atomic physics::Statistical physics