Katsuya Honda

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.

On the fractal structure and statistics of contour lines on a self-affine surface

Contour lines of a self-affine surface with a specified value of Hurst exponent H show certain special characteristics in fractal structure and statistics. The first argument is that the fractal dimension D e of an entire pattern formed by contour lines of the same altitude, similar to coastlines, should be distinguished from the fractal dimension D c of single contour lines. It is then confirmed that D e =2- H . The exponent ζ characterizing a power-law form of the size distribution of closed contour lines, similar to islands, is found to be equal to D e . Self-avoiding fractional Brownian motion is newly introduced to derive a new scaling law D c =2/(1+ H ).

Reconsideration of the renormalization-group theory on intermittent chaos

Abstract The renormalization-group theory has made us believe that the average laminar length for type III intermittency is inversely proportional to the control parameter. However this does not agree with the result of numerical computations. This contradiction is removed through studying the scaling relation for the length of each laminar.

Theory on Formation of the Ripple Phase in Bilayer Membranes

To discuss the phase transition from the L α or L β phase to the periodic ripple ( P β ) phase, we propose a model free energy, which is a functional of two kinds of fields. One denotes the displacement of the polar head groups of lipid molecules and the other the order parameter describing the configuration of hydrocarbon chains. The free energy is noted to include no intrinsic lengths to generate the periodic pattern spontaneously. The ripple phase can be found to occur intermediately between two flat phases due to large fluctuations of the order parameter. The temperature dependence of the period and a qualitative phase diagram are also given.

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.

On a Hard-Sphere Crystal

On the basis of the effective potential method, the single-particle distribution-function in the three-dimensional hard-sphere crystal is investigated. The effective potential in a unit cell which is determined self-consistently depends not only on the displacement of a particle from the lattice point but also on the configuration of its neighboring lattice points. The effective potential is expanded in terms of the above displacement up to the quadratic terms including a linear term and its coefficients are determined by the use of the self-consistent equations. The well-localized distribution of particles around their own lattice sites is obtained below a threshold specific volume and the hard-sphere crystal cannot exist stably above that volume. The correlations between displacements of a pair of particles in the directions parallel and perpendicular to the line connecting the nearest-neighboring sites are discussed. The pressure and the isothermal compressibility as a function of volume are also studied. The effective two-body interaction potential is introduced in an intuitive way. By the use of it, the dispersion relations of the lattice vibration are investigated.

Dynamic Renormalization Group Approach to the Kardar-Parisi-Zhang Equation and Its Results Derived

By removing a trick hidden in the original KPZ paper, a correction in the application of the dynamic renormalization group technique to the KPZ problem is proposed. It is stressed that a fourth derivative term plays an essential role in formulating a consistent scenario. In a one-loop approximation, α=2- z =8(4- d )/(48+4 d - d 2 ), is obtained for the roughness exponent α and the dynamic exponent z , which gives reasonable numbers. For 2< d <4, a smooth surface (α=0, z =2) can also appear depending on the bare coefficients, whereas for d ≥4, only the smooth surface is apparent.

Is intermittent motion of outer flow in the turbulent boundary layer deterministic chaos

Intermittent phenomena observed in the outer regions of the turbulent boundary layer are studied. A new idea is proposed in which the dynamics generating the intermittent phenomena can be described by a one‐dimensional map; Xi+1=(1+e)Xi+uXzi, for 0≤Xi≤Xc 1, u>0). The binary sequence {si} is constructed from carefully processed data of instantaneous streamwise velocities, and also from the map. The encoding is made in such a way that the turbulent state is referred to as si=1 and the nonturbulent state as si=0 at a discretized time i. Both sequences are found to have the common essential properties of intermittent chaos; the probability P(l) of finding nonturbulent states with length l exhibits the power law with an exponent r for l<lc, lc being a cutoff. From the exponent r, one can assign z almost equal to 4, while the control parameter e is closely related to the distance from the wall. The scaling function of P(l) agrees very well with the one predicted from the map. The cutoff length lc i...

On the possibility of a liquid ferromagnet

Abstract The melting phenomenon and the magnetic property of a Ising spin lattice are investigated by making use of a model recently used by Nakano. It is shown that the appearance of a ferromagnetic liquid state is rather difficult.