|Budget Amount *help
¥1,100,000 (Direct Cost : ¥1,100,000)
Fiscal Year 1999 : ¥200,000 (Direct Cost : ¥200,000)
Fiscal Year 1998 : ¥900,000 (Direct Cost : ¥900,000)
In pattern forming phenomena, growth models such as the Eden model and the diffusion-limited aggregation model, play an important role to study the mechanism producing rough surfaces and fractal surfaces under nonequiliburium conditions. Wide variety of patterns is well described by these growth models such as vapor deposited surfaces, bacterial colonies, and metal leaves in nature. Although many studies have been done to understand the geometrical and statistical properties of self-affine and fractal objects, the pattern selection mechanism of these patterns still remains unresolved. For example, the Eden growth process may happen to produce a sparse stringy or fractal surface by chance, while it is well known that Eden clusters are compact and have self-affine surfaces. How probable is the observed object in computer simulations or experiments ? To answer this question, the entropy spectrum of the probability of finding a growth path is studied and a discrete dynamical system associated with the perimeter sizes of Eden clusters is introduced to study how the clusters converge to the most probable pattern which is given by the ensemble average taken over possible clusters. The dynamical system with a few degrees of freedom effectively describes the time evolution of the perimeter size starting from a given initial cluster. The fixed point of the system corresponds to the steady state of the growth and the largest eigenvalue of the linearized operator at the fixed point is estimated numerically. We can investigate the stability of patterns and the pattern selection mechanism with making use of eigenvalues at the fixed point. The dynamical system introduced in this work proceeds the understanding of the dynamics of the self-affine surfaces producing by the growth models.