# 泛函导数

## 定义

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

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

\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.

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

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.

## 例子

### 托马斯-费米动能泛函

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,

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

### 将函数表示成泛函

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

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

### 熵

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

\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}

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

### 指数

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

$\delta$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}

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