泛函导数

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数学和理论物理中,泛函导数方向导数的推广。后者对一个有限维向量求微分,而前者则对一个连续函数(可视为无穷维向量)求微分。它们都可以认为是简单的一元微积分导数的扩展。数学里专门研究泛函导数的分支是泛函分析

定义[编辑]

设有流形 M representing (连续/光滑/有某些边界条件等的)函数 φ 以及泛函 F

F\colon M \rightarrow \mathbb{R} \quad \mbox{or} \quad F\colon M \rightarrow \mathbb{C} ,

F泛函导数,记为{\delta F}/{\delta\varphi},是一个满足以下条件的分布

对任何测量函数 f:



\begin{align}
\left\langle \frac{\delta F[\varphi(x)]}{\delta\varphi(x)}, f(x) \right\rangle 
&= \int \frac{\delta F[\varphi(x)]}{\delta\varphi(x')} f(x')dx' \\
&= \lim_{\varepsilon\to 0}\frac{F[\varphi(x)+\varepsilon f(x)]-F[\varphi(x)]}{\varepsilon} \\
&= \left.\frac{d}{d\epsilon}F[\varphi+\epsilon f]\right|_{\epsilon=0}.
\end{align}

\varphi第一变分\delta\varphi代替f就得到F的第一变分\delta F;this is similar to how the differential is obtained from the gradient. Using a function f with unit norm yields the directional derivative along that function.

在物理学中,通常用狄拉克δ函数 \delta(x-y),而不是一般的测试函数 f(x), 来求出点y处的泛函导数(this is a point of the whole functional derivative as 偏导数梯度的一个分量):

\frac{\delta F[\varphi(x)]}{\delta \varphi(y)}=\lim_{\varepsilon\to 0}\frac{F[\varphi(x)+\varepsilon\delta(x-y)]-F[\varphi(x)]}{\varepsilon}.

This works in cases when F[\varphi(x)+\varepsilon f(x)] formally can be expanded as a series (or at least up to first order) in \varepsilon. The formula is however not mathematically rigorous, since F[\varphi(x)+\varepsilon\delta(x-y)] is usually not even defined.

正式表述[编辑]

通过更仔细地定义函数空间,泛函导数的定义可以更准确、正式。例如,当函数空间是一个巴拿赫空间时, 泛函导数 becomes known as the Fréchet导数, while one 在更一般的局部凸空间上使用Gâteaux 导数。注意,著名的希尔伯特空间巴拿赫空间的特例。更正式的处理允许将普通微积分数学分析的定理推广为泛函分析中对应的定理,以及大量的新定理。

δ函数作为测量函数[编辑]

上面给出的定义是基于一种对所有测量函数 f都成立的关系,因此有人可能会想,它在 f是一个指定的函数(比如说狄拉克δ函数)时也应该成立。但是,δ函数不是一个合理的测量函数。

在定义中,泛函导数描述了整个函数\varphi(x)发生微小变化时,泛函F[\varphi(x)]如何变化。其中,\varphi(x)的变化量的具体形式没有指明,

例子[编辑]

托马斯-费米动能泛函[编辑]

1927年的托马斯-费米模型 used a kinetic energy functional for a noninteracting uniform electron gas in a first attempt of density-functional theory of electronic structure:

T_\mathrm{TF}[\rho] = C_\mathrm{F} \int \rho^{5/3}(\mathbf{r}) \, d\mathbf{r}.

T_\mathrm{TF}[\rho] depends only on the charge density \rho(\mathbf{r}) and does not depend on its gradient, Laplacian, or other higher-order derivatives (functionals like this are called “local”). Therefore,

\frac{\delta T_\mathrm{TF}[\rho]}{\delta \rho} = C_\mathrm{F} \frac{\partial \rho^{5/3}(\mathbf{r})}{\partial \rho(\mathbf{r})}  = \frac{5}{3} C_\mathrm{F}  \rho^{2/3}(\mathbf{r}).

Coulomb势能泛函[编辑]

Weizsäcker 动能泛函[编辑]

将函数表示成泛函[编辑]

最后,注意到任何函数都可以以积分的形式表示成一个泛函。例如,

\rho(\mathbf{r}) = \int \rho(\mathbf{r}') \delta(\mathbf{r}-\mathbf{r}')\, d\mathbf{r}'.

这个泛函只依赖于\rho,像上面两个例子一样(就是说,它们都是“local”)。因此

\frac{\delta \rho(\mathbf{r})}{\delta\rho(\mathbf{r}')}=\frac{\partial \rho(\mathbf{r}') \delta(\mathbf{r}-\mathbf{r}')}{\partial \rho(\mathbf{r}')} = \delta(\mathbf{r}-\mathbf{r}').

[编辑]

离散随机变量概率质量函数的一个泛函


\begin{align}
H[p(x)] = -\sum_x p(x) \log p(x)
\end{align}

于是


\begin{align}
\left\langle \frac{\delta H}{\delta p}, \phi \right\rangle 
& {} = \sum_x \frac{\delta H[p(x)]}{\delta p(x')} \, \phi(x') \\
& {} = \left. \frac{d}{d\epsilon} H[p(x) + \epsilon\phi(x)] \right|_{\epsilon=0}\\
& {} = -\frac{d}{d\varepsilon} \left. \sum_x [p(x) + \varepsilon\phi(x)] \log [p(x) + \varepsilon\phi(x)] \right|_{\varepsilon=0} \\
& {} = \displaystyle -\sum_x [1+\log p(x)]\phi(x)\\
& {} = \left\langle -[1+\log p(x)], \phi \right\rangle.
\end{align}

最后,


\frac{\delta H}{\delta p} = -1-\log p(x).

指数[编辑]

 F[\varphi(x)]= e^{\int \varphi(x) g(x)dx}.

\delta函数作为测量函数


\begin{align}
\frac{\delta F[\varphi(x)]}{\delta \varphi(y)} 
& {} = \lim_{\varepsilon\to 0}\frac{F[\varphi(x)+\varepsilon\delta(x-y)]-F[\varphi(x)]}{\varepsilon}\\
& {} = \lim_{\varepsilon\to 0}\frac{e^{\int (\varphi(x)+\varepsilon\delta(x-y)) g(x)dx}-e^{\int \varphi(x) g(x)dx}}{\varepsilon}\\
& {} = e^{\int \varphi(x) g(x)dx}\lim_{\varepsilon\to 0}\frac{e^{\varepsilon \int \delta(x-y) g(x)dx}-1}{\varepsilon}\\
& {} = e^{\int \varphi(x) g(x)dx}\lim_{\varepsilon\to 0}\frac{e^{\varepsilon g(y)}-1}{\varepsilon}\\
& {} = e^{\int \varphi(x) g(x)dx}g(y).
\end{align}

因此

 \frac{\delta F[\varphi(x)]}{\delta \varphi(y)} = g(y) F[\varphi(x)].

参考来源[编辑]