| The renormalization group (RG) provides a powerful tool and concept in the study of dynamics of spatially extended systems: different microscopic dynamics may be related, via coarse graining, to the same macroscopic dynamics. This thesis makes two contributions in this direction: first, we obtain the correct description of the dynamics of a continuous system on a discrete lattice, and second, we study the macroscopic sharp interface limit of various phase field models of dendritic growth. We demonstrate the virtue of using coarse grained variables in simulations, as opposed to uniformly sampling the underlying continuous configuration. By coarse graining a system repeatedly, the 'perfect linear' operator is obtained as the fixed point of a RG flow. The basin of attraction defines the dynamic universality class with respect to detailed microscopic interactions. In linear problems, the perfect linear operator gives the exact result of physical quantities down to the chosen lattice grid size, subject to the inherent numerical error of a simulation. It is therefore more efficient than the conventional discretization, which requires smaller grid size. The direct application of the perfect linear operator to nonlinear Model A dynamics also leads to improvement. A modified perfect operator that has a short range of interaction is empirically determined and gives results comparable to that from using the perfect linear operator. In the dendritic growth problem, a variety of microscopic phase field models converge to the same macroscopic dynamics in the sharp interface limit. The boundary layer calculation, where the interface width is a small parameter, shows the importance of the phase field's exponential decaying behavior outside the interface region. Subject to this constraint, there is a universality class of phase field models regarding the form of potential functions in the model. The potential functions are defined in terms of a new set of functions, which ensures the existence of first order solutions and thus the universality of the model. The outer temperature field is shown to be discontinuous at the second order. |
