Department of Applied Science for Materials and Electronics,
Interdisciplinary Graduate of Engineering Sciences, Kyushu University
Kasuga, Fukuoka 816-8580, Japan
Overview of our laboratory
We conduct experiments and computer simulations to elucidate the physical mechanisms of dissipative structures in nonlinear-nonequilibrium systems and then apply these results to materials science. Our research area is related to both nonlinear science and materials science, and covers the topics of fractal, chaos, pattern formation, and complex systems.
||Yuka Kawasaki, Yue Zhao|
||Yoko Hamasaki, Kazuki Yamada|
Disspipative structures formed in diffusion fields
- Nature comes in various forms and structures. Our research focuses on elucidating physical mechanisms of pattern formation in nonlinear/nonequilibrium systems.
Temperature fields, concentration fields, electrostatic fields, and velocity fields are examples of diffusion fields and produce various dissipative structures. For example, dendrites have constant branch spacing and a tip curvature radius for a certain degree of nonequilibrium (Fig. 1). However, fractal patterns do not have such a characteristic length, but do have an irregular branching self-similar structure (Fig. 2). Dense-branching morphologies (DBMs) have an intermediate form (Fig. 3).
One important problem in nonlinear physics is how the characteristic length of a dendrite is determined. Theoretically, the interface growth is described by a certain integrodifferential equation. The solvability condition of this equation predicts that the tip curvature is proportional to the growth speed. Our experimental results support this prediction. The symmetry of microscopic constituent molecules determines the overall pattern symmetry. However, this connection of symmetries is not trivial. As can be seen in Fig. 1, it appears that a certain law is hidden behind the overall pattern, which consists of branches with various lengths. In addition to regular growth, the dendritic patterns are also subjected to oscillatory or chaotic growth in the region where the anisotropy of growth is mixed.
A self-similar (fractal) pattern is also known as a diffusion-limited aggregation(DLA), which occurs when the anisotropy of the aggregate vanishes. Although it looks like a random branched pattern, our measurements indicate that it has self-similarity with a fractal dimension of 1.67. This pattern can be divided into subsets of various fractal dimensions as determined by the probability measure of the interface.
As a DBM grows, the branching is frequently repeated and the envelope is stably preserved. Although it appears that protruding interfaces, which are due to fluctuations, screen other interfaces, this is not the case. According to our experiments, this stability mechanism is due to the negative feedback of the branching phenomena, which lead to a reduction in speed.
These patterns are individually studied as patterns of nonlinear-nonequilibrium systems. In addition, a transition occurs between the patterns when the anisotropy or the degree of nonequilibrium of the aggregates is varied. This fact recalls the argument of statistical mechanics in relation to phase transitions. The pattern transition is also related to the existence of any physical quantity characterizing these patterns. In order to study these problems, we are measuring the distribution of the branch lengths in these patterns.
In addition, with regard to mathematical, branching self-similar fractal patterns, we are studying the topological entropy spectrum that exhibits the dynamical nature of the branches, which is also related to the foregoing topics.
Furthermore, with regard to anisotropic, self-affine fractal patterns, we are studying the topological entropy spectra of mathematical models and mountain ranges, and dynamical scaling exponents for experimental systems of crystal growth.
Nonlinear Dynamics and pattern formation
- 1. Convection and pattern formation in liquid crystals
Consider a nematic liquid crystal sandwiched between electrodes. When an alternating electric field is applied, the liquid crystal initially stays at rest. However, as the strength of the electric field increases, a roll-like convection, called the Williams domain, appears. When the electric field is further increased, the roll structure begins to be disturbed; the rolls reconnect and a complicated space-time pattern appears. This state is called the fluctuating Williams domain or defect chaos state. It seems that the factors responsible for the disturbance are not fully understood. As a model, a differential equation of the Ginzburg-Landau type equation, which represents the temporal variation of the roll structure amplitude, is combined with an equation, which represents the flow field of long-wavelength components. Then a numerical calculation for the combined equation is conducted. In a certain parameter region, the zigzag instability of the rolls dominates and the rolls begin to reconnect. The reconnections can occur on a regular or chaotic basis. Fig. 4 illustrates a pattern formed by chaotic reconnections in which creation and annihilation of defects are ongoing.
2. Coupled oscillators and excitatory systems
A system of combined limit cycle oscillators exhibits a phenomenon called mutual pulling where the frequencies of the oscillators are aligned. We are using computer simulations to study the phenomenon of collective mutual pulling, which occurs in a combined system of many limit cycle oscillators. As an application, we are studying the generation mechanism of brain waves and cardiac arrhythmia. Figs. 5a-c illustrate a simulation process to eliminate severe arrhythmia, called ventricular fibrillation, by an external alternating force. Fig. 5a shows an oscillating state that corresponds to ventricular fibrillation, which is disturbed and spatially not synchronized. As the external force increases, it is gradually synchronized with the external force. Eventually, the system pulsates in synch with the external force and the ventricular fibrillation is eliminated.
3. Growth patterns
We are calculating the growth patterns of diffusion fields using coupled map lattices and phase field models. The following figures fig.6 illustrate a dendrite, which resembles a snow crystal, obtained by a coupled map lattice and a dendrite with periodic side branches obtained by a phase field model.
Solitons and vortices of matter waves have been found in Bose-Einstein condensates and studied in detail. We are conducting simulations of the Gross-Pitaevskii equation and are studying the dynamics of solitons and vortices. Figs. 7a and 7b illustrate simulations of a collision between a potential wall and a soliton. While the shape is preserved, the soliton is reflected when the wall is high (Fig. 7a) and is transmitted when the wall is low (Fig. 7b). The threshold is considerably lower than that of a classical mechanics system, which can be attributed to the tunneling phenomenon.
- List of publications
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