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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Reaction–diffusion equation ： ウィキペディア英語版 Reaction–diffusion system Reaction–diffusion systems are mathematical models which explain how the concentration of one or more substances distributed in space changes under the influence of two processes: local chemical reactions in which the substances are transformed into each other, and diffusion which causes the substances to spread out over a surface in space. Reaction–diffusion systems are naturally applied in chemistry. However, the system can also describe dynamical processes of non-chemical nature. Examples are found in biology, geology and physics and ecology. Mathematically, reaction–diffusion systems take the form of semi-linear parabolic partial differential equations. They can be represented in the general form :$\backslash partial\_t\; \backslash boldsymbol\; =\; \backslash underline\; \backslash ,\backslash nabla^2\; \backslash boldsymbol\; +\; \backslash boldsymbol(\backslash boldsymbol),$ where each component of the vector represents the concentration of one substance, is a diagonal matrix of diffusion coefficients, and accounts for all local reactions. The solutions of reaction–diffusion equations display a wide range of behaviours, including the formation of travelling waves and wave-like phenomena as well as other self-organized patterns like stripes, hexagons or more intricate structure like dissipative solitons. == One-component reaction–diffusion equations == The simplest reaction–diffusion equation concerning the concentration of a single substance in one spatial dimension, :$\backslash partial\_t\; u\; =\; D\; \backslash partial^2\_x\; u\; +\; R(u),$ is also referred to as the KPP (Kolmogorov-Petrovsky-Piskounov) equation.〔A. Kolmogorov et al., Moscow Univ. Bull. Math. A 1 (1937): 1〕 If the reaction term vanishes, then the equation represents a pure diffusion process. The corresponding equation is Fick's second law. The choice yields Fisher's equation that was originally used to describe the spreading of biological populations,〔R. A. Fisher, Ann. Eug. 7 (1937): 355〕 the Newell-Whitehead-Segel equation with to describe Rayleigh-Benard convection,〔A. C. Newell and J. A. Whitehead, J. Fluid Mech. 38 (1969): 279〕〔L. A. Segel, J. Fluid Mech. 38 (1969): 203〕 the more general Zeldovich equation with and that arises in combustion theory,〔Y. B. Zeldovich and D. A. Frank-Kamenetsky, Acta Physicochim. 9 (1938): 341〕 and its particular degenerate case with that is sometimes referred to as the Zeldovich equation as well.〔B. H. Gilding and R. Kersner, Travelling Waves in Nonlinear Diffusion Convection Reaction, Birkhäuser (2004)〕 The dynamics of one-component systems is subject to certain restrictions as the evolution equation can also be written in the variational form :$\backslash partial\_t\; u=-\backslash frac$ and therefore describes a permanent decrease of the "free energy" $\backslash mathfrak\; L$ given by the functional :$\backslash mathfrak\; L=\backslash int\_^\backslash infty\; \backslash left(\backslash left\; (\backslash partial\_xu\; \backslash right\; )^2-V(u)\backslash right)\backslash textx$ with a potential such that In systems with more than one stationary homogeneous solution, a typical solution is given by travelling fronts connecting the homogeneous states. These solutions move with constant speed without changing their shape and are of the form with , where is the speed of the travelling wave. Note that while travelling waves are generically stable structures, all non-monotonous stationary solutions (e.g. localized domains composed of a front-antifront pair) are unstable. For , there is a simple proof for this statement:〔P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Springer (1979)〕 if is a stationary solution and is an infinitesimally perturbed solution, linear stability analysis yields the equation :$\backslash partial\_t\; \backslash tilde=D\backslash partial\_x^2\; \backslash tilde-U(x)\backslash tilde,\backslash qquad\; U(x)\; =\; -R^(u)|\_.$ With the ansatz we arrive at the eigenvalue problem :$\backslash hat\; H\backslash psi=\backslash lambda\backslash psi,\; \backslash qquad\; \backslash hat\; H=-D\backslash partial\_x^2+U(x),$ of Schrödinger type where negative eigenvalues result in the instability of the solution. Due to translational invariance is a neutral eigenfunction with the eigenvalue , and all other eigenfunctions can be sorted according to an increasing number of knots with the magnitude of the corresponding real eigenvalue increases monotonically with the number of zeros. The eigenfunction should have at least one zero, and for a non-monotonic stationary solution the corresponding eigenvalue cannot be the lowest one, thereby implying instability. To determine the velocity of a moving front, one may go to a moving coordinate system and look at stationary solutions: :$D\; \backslash partial^2\_\backslash hat(\backslash xi)+\; c\backslash partial\_\; \backslash hat(\backslash xi)+R(\backslash hat(\backslash xi))=0.$ This equation has a nice mechanical analogue as the motion of a mass with position in the course of the "time" under the force with the damping coefficient c which allows for a rather illustrative access to the construction of different types of solutions and the determination of . When going from one to more space dimensions, a number of statements from one-dimensional systems can still be applied. Planar or curved wave fronts are typical structures, and a new effect arises as the local velocity of a curved front becomes dependent on the local radius of curvature (this can be seen by going to polar coordinates). This phenomenon leads to the so-called curvature-driven instability.〔A. S. Mikhailov, Foundations of Synergetics I. Distributed Active Systems, Springer (1990)〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Reaction–diffusion system」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.