Hiroyasu Toyoki

Generalization and the Fractal Dimensionality of Diffusion-Limited Aggregation

Diffusion-limited aggregation (DLA) model is generalized to incorporate the dielectric breakdown model proposed by Niemeyer et al. , and the new simulation method is proposed. While a growing cluster is still in the diffusion (Laplace) field, the local growth probability at a perimeter site P ps of the cluster is now given by p g ( P ps )∼ | ∇φ( P ps ) | n , where φ( P ) is the probability of finding at a point P a random walker launched far away from the cluster. Ordinary DLA corresponds to η=1. Based on the theory of DLA proposed by Honda et al. , the fractal dimension d f for this generalized DLA is derived as d f ={ d s 2 +η( d w -1)} / { d s +η( d w -1)}, where d s is the dimension of space in which aggregation processes take place and d w is the fractal dimension of random walker trajectory. Both d s and d w are allowed to take any number larger than or equal to one. This formula is also applicable to Eden model (η=0) correctly, which means that the generalized DLA model naturally bridges a gap betw...

A theory of fractal dimensionality for generalized diffusion-limited aggregation

A dimensional consideration is presented to obtain the fractal dimension D of the diffusion-limited aggregates which grow in a space with general dimensionality d s (any number ≧1) through deposition of diffusing particles following trajectories with a fractal dimension d w (Levy flight). We find rigorously D =( d s 2 + d w -1)/( d s + d w -1), which is in excellent agreement with many computer simulation results.

Motion of Defects in the Phase Ordering Process of Nematic Liquid Crystals

Dislinations and monopoles in the phase ordering of three-dimensional nematic liquid crystals follwing a quench are numerically investigated. The total length of dislinations decreases as L ∼ t -0.95±0.07 . Monopoles are not detected except in the early transient regime. It is found that monopoles change to a ring disclination, and rings reconnect to shrink to nothing.

Vortex Dynamics in the Ordering Process of the Three-Dimensional Planar System

A three-dimensional cell-dynamical-system with planar symmetry is used to study the ordering process following a quench. The configuration of vortex strings is investigated and their total length is found to decrease with time as t -0.91 . It is shown that the form factor obeys dynamical scaling with a characteristic length increasing as t 0.45 . The results are consistent with the previous theoretical result.

Cell Dynamical Approach to the Ordering Process of the Three-Dimensional Heisenberg System

The ordering process of the quenched time-dependent Ginzburg-Landau model for the three-component nonconserved order-parameter in the three-dimensional system is numerically investigated. The form factor is found to obey a dynamical scaling law with a characteristic length growing as t 1/2 . The density of topological defects and the energy density exhibit a power decay that is consistent with the scaling of the form factor. The interaction of the defects is briefly discussed.

The motion of interfaces and the similarity dimension

Abstract A discrete model of interfaces with self-similarity is proposed to show the power-law decay of the total area. In the case that the similarity dimension is equal to the spatial dimension a t − 1 2 law appears.

Fractal Dimensionality of the Reversible Diffusion-Limited Aggregation : New Aspect of Equilibrium Patterns

We consider loopless clusters which are restructured repeatedly by diffusive particles released from the cluster itself, being essentially in equilibrium. The resulting clusters are self-similar with fractal dimension d f . By means of the dimensional analysis we give d f ={ d s 2 +2( d w -1)}/{ d s +2( d w -1)}, where d w and d s . denote the dimensions of the trajectory of the diffusive particles and of the space in which the cluster is embedded, respectively. The agreement with the numerical result in the case of d s = d w =2 obtained by Botet and Jullien is very good. We further make sure our theoretical prediction by simulating the limiting case of d w =1 (rectilinear trajectory). These constitute a breakthrough that the structure of the cluster in equilibrium can depend on the particle motion d w , which seems to be irrelevant to pattern formation.

Commensurate, incommensurate and chaotic phase in DNA double helices

Abstract It is shown numerically that in DNA double helix model there exist metastable states other than B-form. Depending on the strength of base pair coupling in double helices, it is found that the phase diagram consists of regimes with commensurate, incommensurate and chaotic phase, indicating the existence of complex open state in DNA double helices.

Boundary Effects for Macroscopic Patterns in Nematic Thin Layers under a Hybrid Boundary Condition

Macroscopic director patterns in the phase ordering process and equilibrium state of nematic liquid crystals under a hybrid boundary conditions are numerically studied. A new cell-dynamics model for nematic liquid crystals including anisotropic elastic constants is used. Competition between bulk and surface elastic energies yields periodic modulated equilibrium structures. String-like deformation between boojums is found when uniaxial anisotropy is present on a boundary.

Jamming Transitions in a Two-Dimensional Cellular Automation Model of Cars with Finite Trip Length

Abstract Dynamical phase transitions in two-dimensional traffic flow on the decorated square lattice are studied numerically. The square lattice point and the decorated site denote crossing and road respectively. In the present model, a car has the origin, the destination and the deterministic path between them, which are given at the birth of the car. The average velocity as a function of the car density has two discontinuous points, which divide the free flow state, the partial jam state and the frozen jam state. The phase transition is confirmed by the temporal change of the maximum cluster size, whose power spectrum exhibits the power-law decay. The exponent changes discontinuously at the transition between the free flow state and the partial jam state.