User:Schenad/反应-扩散系统

电脑模拟两种不同化学物质在环面上的反应与扩散（采用Gray–Scott模型）。

反应-扩散系统（reaction–diffusion system）是一类数学模型的统称。这类模型和许多物理现象相关：最常见的是一种或多种化学物质的浓度在空间分布上的变化。在这个过程中，局部的化学反应使化学物质相互转变；扩散作用使物质向四周分散运动。反应-扩散系统模拟的是在反应和扩散这两种不同机制的竞争下，物质在空间分布上的具体行为。

反应-扩散系统不仅被应用到化學研究中，还可以描述没有化学反应参与的动力学过程，在生物学、地质学、物理学（中子扩散理论）和生态学领域中都有相应的例子。

在数学上，反应-扩散方程是半线性抛物偏微分方程，一般形式为：

$\partial _{t}{\boldsymbol {q}}={\underline {\underline {\boldsymbol {D}}}}\,\nabla ^{2}{\boldsymbol {q}}+{\boldsymbol {R}}({\boldsymbol {q}}),$

其中，q(x, t) 表示未知的向量函数，D是质量扩散率的對角矩陣，R 表示所有的局域性反应。

反应-扩散方程的解行为五花八门，其中包括行波的形成、类波现象，以及自组装图案（条纹、六边形或是更复杂的结构，例如耗散孤子）。

单组分反应-扩散方程

最简单的反应-扩散方程是一维的：

$\partial _{t}u=D\partial _{x}^{2}u+R(u)$

若去掉式子中的反应项 $R(u)$，此方程化为普通的扩散方程，即菲克第二定律。若 $R(u)=u(1-u)$，则方程化为费希尔方程，最初用于描述人口的流动。若 $R(u)=u(1-u^{2})$，则变为 Newell–Whitehead-Segel 方程，用于描述瑞利-贝纳德对流。更为一般的情况是泽尔多维奇方程：$R(u)=u(1-u)(u-\alpha )$ 且 $0<\alpha <1$，在燃烧理论中出现；它的特例 $R(u)=u^{2}-u^{3}$ 有时也被称作泽尔多维奇方程。

演化方程又可以写作变分形式：

$\partial _{t}u=-{\frac {\delta {\mathfrak {L}}}{\delta u}}$

于是，借助以下的泛函，给出了一个单调递减的自由能（free energy）${\mathfrak {L}}$ ：

${\mathfrak {L}}=\int _{-\infty }^{\infty }\left[{\tfrac {D}{2}}\left(\partial _{x}u\right)^{2}-V(u)\right]{\text{d}}x$

其中的势能 $V(u)$ 满足 $R(u)={\frac {dV(u)}{du}}$

费希尔方程的一个行波波前解。

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 u(x, t) = û(ξ) with ξ = x − ct, where c 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 c = 0, there is a simple proof for this statement: if u₀(x) is a stationary solution and u = u₀(x) + ũ(x, t) is an infinitesimally perturbed solution, linear stability analysis yields the equation

$\partial _{t}{\tilde {u}}=D\partial _{x}^{2}{\tilde {u}}-U(x){\tilde {u}},\qquad U(x)=-R^{\prime }(u)|_{u=u_{0}(x)}.$

With the ansatz ũ = ψ(x)exp(−λt) we arrive at the eigenvalue problem

${\hat {H}}\psi =\lambda \psi ,\qquad {\hat {H}}=-D\partial _{x}^{2}+U(x),$

of Schrödinger type where negative eigenvalues result in the instability of the solution. Due to translational invariance ψ = ∂_x u₀(x) is a neutral eigenfunction with the eigenvalue λ = 0, 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 ψ = ∂_x u₀(x) should have at least one zero, and for a non-monotonic stationary solution the corresponding eigenvalue λ = 0 cannot be the lowest one, thereby implying instability.

To determine the velocity c of a moving front, one may go to a moving coordinate system and look at stationary solutions:

$D\partial _{\xi }^{2}{\hat {u}}(\xi )+c\partial _{\xi }{\hat {u}}(\xi )+R({\hat {u}}(\xi ))=0.$

This equation has a nice mechanical analogue as the motion of a mass D with position û in the course of the "time" ξ under the force R with the damping coefficient c which allows for a rather illustrative access to the construction of different types of solutions and the determination of c.

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.

双组分反应-扩散方程

Two-component systems allow for a much larger range of possible phenomena than their one-component counterparts. An important idea that was first proposed by Alan Turing is that a state that is stable in the local system can become unstable in the presence of diffusion.

A linear stability analysis however shows that when linearizing the general two-component system

${\begin{pmatrix}\partial _{t}u\\\partial _{t}v\end{pmatrix}}={\begin{pmatrix}D_{u}&0\\0&D_{v}\end{pmatrix}}{\begin{pmatrix}\partial _{xx}u\\\partial _{xx}v\end{pmatrix}}+{\begin{pmatrix}F(u,v)\\G(u,v)\end{pmatrix}}$

${\tilde {\boldsymbol {q}}}_{\boldsymbol {k}}({\boldsymbol {x}},t)={\begin{pmatrix}{\tilde {u}}(t)\\{\tilde {v}}(t)\end{pmatrix}}e^{i{\boldsymbol {k}}\cdot {\boldsymbol {x}}}$

of the stationary homogeneous solution will satisfy

${\begin{pmatrix}\partial _{t}{\tilde {u}}_{\boldsymbol {k}}(t)\\\partial _{t}{\tilde {v}}_{\boldsymbol {k}}(t)\end{pmatrix}}=-k^{2}{\begin{pmatrix}D_{u}{\tilde {u}}_{\boldsymbol {k}}(t)\\D_{v}{\tilde {v}}_{\boldsymbol {k}}(t)\end{pmatrix}}+{\boldsymbol {R}}^{\prime }{\begin{pmatrix}{\tilde {u}}_{\boldsymbol {k}}(t)\\{\tilde {v}}_{\boldsymbol {k}}(t)\end{pmatrix}}.$

Turing's idea can only be realized in four equivalence classes of systems characterized by the signs of the Jacobian R′ of the reaction function. In particular, if a finite wave vector k is supposed to be the most unstable one, the Jacobian must have the signs

${\begin{pmatrix}+&-\\+&-\end{pmatrix}},\quad {\begin{pmatrix}+&+\\-&-\end{pmatrix}},\quad {\begin{pmatrix}-&+\\-&+\end{pmatrix}},\quad {\begin{pmatrix}-&-\\+&+\end{pmatrix}}.$

This class of systems is named activator-inhibitor system after its first representative: close to the ground state, one component stimulates the production of both components while the other one inhibits their growth. Its most prominent representative is the FitzHugh–Nagumo equation

${\begin{aligned}\partial _{t}u&=d_{u}^{2}\,\nabla ^{2}u+f(u)-\sigma v,\\\tau \partial _{t}v&=d_{v}^{2}\,\nabla ^{2}v+u-v\end{aligned}}$

with  f (u) = λu − u³ − κ which describes how an action potential travels through a nerve. Here, d_u, d_v, τ, σ and λ are positive constants.

When an activator-inhibitor system undergoes a change of parameters, one may pass from conditions under which a homogeneous ground state is stable to conditions under which it is linearly unstable. The corresponding bifurcation may be either a Hopf bifurcation to a globally oscillating homogeneous state with a dominant wave number k = 0 or a Turing bifurcation to a globally patterned state with a dominant finite wave number. The latter in two spatial dimensions typically leads to stripe or hexagonal patterns.

• Subcritical Turing bifurcation: formation of a hexagonal pattern from noisy initial conditions in the above two-component reaction-diffusion system of Fitzhugh-Nagumo type.
• 
  Noisy initial conditions at t = 0.
• 
  State of the system at t = 10.
• 
  Almost converged state at t = 100.

For the Fitzhugh-Nagumo example, the neutral stability curves marking the boundary of the linearly stable region for the Turing and Hopf bifurcation are given by

${\begin{aligned}q_{\text{n}}^{H}(k):&{}\quad {\frac {1}{\tau }}+\left(d_{u}^{2}+{\frac {1}{\tau }}d_{v}^{2}\right)k^{2}&=f^{\prime }(u_{h}),\\[6pt]q_{\text{n}}^{T}(k):&{}\quad {\frac {\kappa }{1+d_{v}^{2}k^{2}}}+d_{u}^{2}k^{2}&=f^{\prime }(u_{h}).\end{aligned}}$

If the bifurcation is subcritical, often localized structures (dissipative solitons) can be observed in the hysteretic region where the pattern coexists with the ground state. Other frequently encountered structures comprise pulse trains (also known as periodic travelling waves), spiral waves and target patterns. These three solution types are also generic features of two- (or more-) component reaction-diffusion equations in which the local dynamics have a stable limit cycle

• Other patterns found in the above two-component reaction-diffusion system of Fitzhugh-Nagumo type.
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  Rotating spiral.
• 
  Target pattern.
• 
  Stationary localized pulse (dissipative soliton).

三组分或多组分反应-扩散方程

For a variety of systems, reaction-diffusion equations with more than two components have been proposed, e.g. as models for the Belousov-Zhabotinsky reaction, for blood clotting or planar gas discharge systems.

It is known that systems with more components allow for a variety of phenomena not possible in systems with one or two components (e.g. stable running pulses in more than one spatial dimension without global feedback),. An introduction and systematic overview of the possible phenomena in dependence on the properties of the underlying system is given in.

实际应用和普遍性

In recent times, reaction–diffusion systems have attracted much interest as a prototype model for pattern formation. The above-mentioned patterns (fronts, spirals, targets, hexagons, stripes and dissipative solitons) can be found in various types of reaction-diffusion systems in spite of large discrepancies e.g. in the local reaction terms. It has also been argued that reaction-diffusion processes are an essential basis for processes connected to morphogenesis in biology and may even be related to animal coats and skin pigmentation. Other applications of reaction-diffusion equations include ecological invasions, spread of epidemics, tumour growth and wound healing. Another reason for the interest in reaction-diffusion systems is that although they are nonlinear partial differential equations, there are often possibilities for an analytical treatment.

实验验证

迄今为止，对于化学中的反应-扩散系统，控制良好的实验可由以下三种方式实现：

1. 在凝胶或毛细管中进行化学反应；
2. 研究催化转换器表面的温度脉冲；
3. 用反应-扩散系统作为神经脉冲传导的模型。

除了上述的例子之外，像等离子体之类的电子输运系统或是半导体在某种条件之下也可以用反应-扩散系统来描述。

数值处理

反应-扩散系统可以用数值分析求解。一些研究论文中可找到不少针对此系统采用的数值处理方法。同时，数值法也被用于相关的复几何学中。

另见

• 自激波
• 扩散控制反应
• 化学动力学
• 相空间方法
• 自催化反应
• 图式形成
• 自然中的图案
• 周期性行波
• 随机几何学
• MClone
• 形态发生的化学基础

反应-扩散方程的几个特例

• 费希尔方程
• 费希尔-柯尔莫哥洛夫方程
• 菲茨休 - 南云方程

外部链接

• Reaction-Diffusion by the Gray-Scott Model: Pearson's parameterization a visual map of the parameter space of Gray-Scott reaction diffusion.
• A Thesis on reaction-diffusion patterns with an overview of the field

[[Category:数学模型]] [[Category:拋物型偏微分方程]]