| In this dissertation, we study various complex phenomena occurring in heterogeneous materials. Heterogeneous media, i.e., materials that are non-uniform, are in fact abundant both in nature and man-made systems. We first address how we can efficiently characterize complex data obtained from measuring various physical properties in such materials. We later extend this concept to the curved space and relate this to various deformed structures in heterogeneous materials. We also show how we can efficiently coarsen heterogeneous media using wavelet transformation and describe various finite-difference and finite-element algorithms to solve coarsened-transport equations in such systems.;In the second part of this dissertation, we mainly focus on ionic systems and their physical properties. At first, we study two-dimensional ionic fluids and their phase equilibria and investigate the presence of various ionic species, such as dipoles and larger clusters at various temperatures and densities. We also investigate the insulating-to-conducting transition (Kosterlitz-Thouless) and study the changes that happen during this phase transition. Later, we extend this formalism, to ionic fluids in presence of immobile ions (quench ionic disorder) and investigate how the distribution of immobile ions affects various cluster species present in such systems. We finally report the results of diffusion of ions in a quench medium and study how immobile ions affect the motion of mobile ions. |
