Scaling in nonlinear growth and diffusion-limited reaction system

Scaling has been a fascinating research area in statistical physics for decades since the pioneering work of L. Kadanoff. Recently the focus of research has shifted to complicated, nonlinear and far from equilibrium physical systems. With the aid of large scale computer simulations, we have studied scaling phenomenon in nonlinear growth models and in a diffusion-limited reaction system.; First we investigate a particular growth equation--the Kuramoto-Sivashinsky equation. We use the inverse method to study the renormalization group flow of that equation. We show how scaling occurs and how deterministic chaos at small scales develops into noisy dynamics at large scales, and how a small scale pattern becomes a large scale disordered fractal via an intermediate scaling regime. For the first time we obtain the dynamical renormalization flow of the system and find some interesting properties of the renormalized coefficients.; As a second example of scaling, we propose a point island model to describe a Molecular Beam Epitaxy (MBE) experiment. We study the cases of various critical island sizes i for nucleation as a function of initial coverage. A many body random walk theory is presented to explain the anomalous property of island size distribution. Using a version of mean-field theory we also obtain a closed form for the spatial correlation function. We also study the model using a mean field rate equation.; Lastly we study the diffusion-limited reactions {dollar}A + A to emptyset{dollar}, {dollar}A + B to emptyset{dollar} and the number of distinct sites visited by a random walker on a d dimensional tubular lattice: a square lattice of sizes {dollar}L times Wsp{lcub}d-l{rcub}{dollar} with {dollar}L gg W{dollar}. We are interested in the crossover time at which the system changes its behavior from that in high dimension to that in one dimension. We analytically solve the random walk problem on the tubular lattice and use it to explain the anomalous scaling of the crossover time for reaction {dollar}A + A to emptyset{dollar} on the tubular lattice.