泛函导数

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

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

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

对任何测量函数 f:

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

用 ${\displaystyle \varphi }$  的一次变分 ${\displaystyle \delta \varphi }$  代替 ${\displaystyle f}$  就得到 ${\displaystyle F}$  的一次变分 ${\displaystyle \delta F}$ ；

在物理学中，通常用狄拉克δ函数 ${\displaystyle \delta (x-y)}$ ，而不是一般的测试函数 ${\displaystyle f(x)}$ , 来求出点${\displaystyle y}$ 处的泛函导数(这是整个泛函变分的关键点，就像偏导数是梯度的一个分量):

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

这适用于${\displaystyle F[\varphi (x)+\varepsilon f(x)]}$  可以展开成${\displaystyle \varepsilon }$ 的级数时 (或者至少能展为1阶). 但是这一表达在数学上并不严格,因为 ${\displaystyle F[\varphi (x)+\varepsilon \delta (x-y)]}$ 一般而言并未定义。

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, 泛函导数就是著名的Fréchet导数, 而这在更一般的局部凸空间上使用加托导数。注意，著名的希尔伯特空间是巴拿赫空间的特例。更正式的处理允许将普通微积分和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。

与函数的导数类似，泛函导数满足下列的性质：（其中 F[ρ] 和 G[ρ] 为两个泛函）

• 线性：^[1]

${\displaystyle {\frac {\delta (\lambda F+\mu G)[\rho ]}{\delta \rho (x)}}=\lambda {\frac {\delta F[\rho ]}{\delta \rho (x)}}+\mu {\frac {\delta G[\rho ]}{\delta \rho (x)}},}$ 

其中 λ, μ 皆为常数。

• 积法则：^[2]

${\displaystyle {\frac {\delta (FG)[\rho ]}{\delta \rho (x)}}={\frac {\delta F[\rho ]}{\delta \rho (x)}}G[\rho ]+F[\rho ]{\frac {\delta G[\rho ]}{\delta \rho (x)}}\,,}$ 

• 链式法则：

若 F 和 G 为两个泛函，则^[3]

${\displaystyle \displaystyle {\frac {\delta F[G[\rho ]]}{\delta \rho (y)}}=\int dx{\frac {\delta F[G(\rho )]}{\delta G[\rho (x)]}}\ {\frac {\delta G[\rho ]}{\delta \rho (y)}}\ .}$ 

若当中的 G 为一个普通的可导函数 g，则上式化为^[4]

${\displaystyle \displaystyle {\frac {\delta F[g(\rho )]}{\delta \rho (y)}}={\frac {\delta F[g(\rho )]}{\delta g[\rho (x)]}}\ {\frac {dg(\rho )}{d\rho (y)}}\ .}$ 

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${\displaystyle \varphi (x)}$ 发生微小变化时，泛函${\displaystyle F[\varphi (x)]}$ 如何变化。其中，${\displaystyle \varphi (x)}$ 的变化量的具体形式没有指明，

公式

给定泛函

${\displaystyle F[\rho ]=\int f({\boldsymbol {r}},\rho ({\boldsymbol {r}}),\nabla \rho ({\boldsymbol {r}}))\,d{\boldsymbol {r}},}$ 

及在积分区域的边界上恒为零的函数 ϕ(r)，由定义可得：

${\displaystyle {\begin{aligned}\int {\frac {\delta F}{\delta \rho ({\boldsymbol {r}})}}\,\phi ({\boldsymbol {r}})\,d{\boldsymbol {r}}&=\left[{\frac {d}{d\varepsilon }}\int f({\boldsymbol {r}},\rho +\varepsilon \phi ,\nabla \rho +\varepsilon \nabla \phi )\,d{\boldsymbol {r}}\right]_{\varepsilon =0}\\&=\int \left({\frac {\partial f}{\partial \rho }}\,\phi +{\frac {\partial f}{\partial \nabla \rho }}\cdot \nabla \phi \right)d{\boldsymbol {r}}\\&=\int \left[{\frac {\partial f}{\partial \rho }}\,\phi +\nabla \cdot \left({\frac {\partial f}{\partial \nabla \rho }}\,\phi \right)-\left(\nabla \cdot {\frac {\partial f}{\partial \nabla \rho }}\right)\phi \right]d{\boldsymbol {r}}\\&=\int \left[{\frac {\partial f}{\partial \rho }}\,\phi -\left(\nabla \cdot {\frac {\partial f}{\partial \nabla \rho }}\right)\phi \right]d{\boldsymbol {r}}\\&=\int \left({\frac {\partial f}{\partial \rho }}-\nabla \cdot {\frac {\partial f}{\partial \nabla \rho }}\right)\phi ({\boldsymbol {r}})\ d{\boldsymbol {r}}\,.\end{aligned}}}$ 

其中第二行用到了 f 的全微分， ∂f /∂∇ρ 为标量对向量的导数。^{[Note 1]} 第三行则用到了散度的积法则。第四行由高斯散度定理及边界上 ϕ=0 的条件得到。由于 ϕ 可以是任意的函数，由变分法基本引理可知，所求泛函导数为

${\displaystyle {\frac {\delta F}{\delta \rho ({\boldsymbol {r}})}}={\frac {\partial f}{\partial \rho }}-\nabla \cdot {\frac {\partial f}{\partial \nabla \rho }}}$ 

其中 ρ = ρ(r) 且 f = f (r, ρ, ∇ρ)。只要 F[ρ] 具有本节首段的形式，上述公式就适用。对于其他的泛函形式，可由定义出发，求出其泛函导数。（见库仑势能泛函。)

以上公式可推广到高维，并且有其他高阶导数的情况。则泛函可写成

${\displaystyle F[\rho ({\boldsymbol {r}})]=\int f({\boldsymbol {r}},\rho ({\boldsymbol {r}}),\nabla \rho ({\boldsymbol {r}}),\nabla ^{(2)}\rho ({\boldsymbol {r}}),\dots ,\nabla ^{(N)}\rho ({\boldsymbol {r}}))\,d{\boldsymbol {r}},}$ 

其中向量 r ∈ ℝⁿ，而 ∇⁽ⁱ⁾ 为一个张量，其 nⁱ 个分量分别为 i 阶微分算子

${\displaystyle \left[\nabla ^{(i)}\right]_{\alpha _{1}\alpha _{2}\cdots \alpha _{i}}={\frac {\partial ^{\,i}}{\partial r_{\alpha _{1}}\partial r_{\alpha _{2}}\cdots \partial r_{\alpha _{i}}}}\qquad \qquad {\text{where}}\quad \alpha _{1},\alpha _{2},\cdots ,\alpha _{i}=1,2,\cdots ,n\ .}$ ^{[Note 2]}

与上面类似，由泛函导数的定义可知：

${\displaystyle {\begin{aligned}{\frac {\delta F[\rho ]}{\delta \rho }}&{}={\frac {\partial f}{\partial \rho }}-\nabla \cdot {\frac {\partial f}{\partial (\nabla \rho )}}+\nabla ^{(2)}\cdot {\frac {\partial f}{\partial \left(\nabla ^{(2)}\rho \right)}}+\dots +(-1)^{N}\nabla ^{(N)}\cdot {\frac {\partial f}{\partial \left(\nabla ^{(N)}\rho \right)}}\\&{}={\frac {\partial f}{\partial \rho }}+\sum _{i=1}^{N}(-1)^{i}\nabla ^{(i)}\cdot {\frac {\partial f}{\partial \left(\nabla ^{(i)}\rho \right)}}\ .\end{aligned}}}$ 

式中，张量 ${\displaystyle {\frac {\partial f}{\partial \left(\nabla ^{(i)}\rho \right)}}}$  具有 nⁱ 个分量，各为 f 对 ρ 偏导数之偏导数，即：

${\displaystyle \left[{\frac {\partial f}{\partial \left(\nabla ^{(i)}\rho \right)}}\right]_{\alpha _{1}\alpha _{2}\cdots \alpha _{i}}={\frac {\partial f}{\partial \rho _{\alpha _{1}\alpha _{2}\cdots \alpha _{i}}}}\qquad \qquad {\text{where}}\quad \rho _{\alpha _{1}\alpha _{2}\cdots \alpha _{i}}\equiv {\frac {\partial ^{\,i}\rho }{\partial r_{\alpha _{1}}\,\partial r_{\alpha _{2}}\cdots \partial r_{\alpha _{i}}}}\ ,}$ 

并定义张量的标量积为

${\displaystyle \nabla ^{(i)}\cdot {\frac {\partial f}{\partial \left(\nabla ^{(i)}\rho \right)}}=\sum _{\alpha _{1},\alpha _{2},\cdots ,\alpha _{i}=1}^{n}\ {\frac {\partial ^{\,i}}{\partial r_{\alpha _{1}}\,\partial r_{\alpha _{2}}\cdots \partial r_{\alpha _{i}}}}\ {\frac {\partial f}{\partial \rho _{\alpha _{1}\alpha _{2}\cdots \alpha _{i}}}}\ .}$  ^{[Note 3]}

例子

托马斯-费米动能泛函

1927年的托马斯-费米模型对于无相互作用的单一电子云使用了动能泛函是密度泛函理论关于电子结构的第一次尝试

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

${\displaystyle T_{\mathrm {TF} }[\rho ]}$  只与电子密度有关 ${\displaystyle \rho (\mathbf {r} )}$  并且不依赖于其梯度, Laplacian, 或者其他更高阶的微分 (像这样的泛函被称为是“局部的”). 因此，

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

库仑势能泛函

托马斯和费米利用了以下库仑势能泛函来描述电子与核之间的电势

${\displaystyle V[\rho ]=\int {\frac {\rho ({\boldsymbol {r}})}{|{\boldsymbol {r}}|}}\ d{\boldsymbol {r}}.}$ 

由泛函导数的定义，

${\displaystyle {\begin{aligned}\int {\frac {\delta V}{\delta \rho ({\boldsymbol {r}})}}\ \phi ({\boldsymbol {r}})\ d{\boldsymbol {r}}&{}=\left[{\frac {d}{d\varepsilon }}\int {\frac {\rho ({\boldsymbol {r}})+\varepsilon \phi ({\boldsymbol {r}})}{|{\boldsymbol {r}}|}}\ d{\boldsymbol {r}}\right]_{\varepsilon =0}\\&{}=\int {\frac {1}{|{\boldsymbol {r}}|}}\,\phi ({\boldsymbol {r}})\ d{\boldsymbol {r}}\,.\end{aligned}}}$ 

故

${\displaystyle {\frac {\delta V}{\delta \rho ({\boldsymbol {r}})}}={\frac {1}{|{\boldsymbol {r}}|}}\ .}$ 

至于电子与电子间的相互作用，由以下库仑势能泛函描述：

${\displaystyle J[\rho ]={\frac {1}{2}}\iint {\frac {\rho (\mathbf {r} )\rho (\mathbf {r} ')}{\vert \mathbf {r} -\mathbf {r} '\vert }}\,d\mathbf {r} d\mathbf {r} '\,.}$ 

由定义，

${\displaystyle {\begin{aligned}\int {\frac {\delta J}{\delta \rho ({\boldsymbol {r}})}}\phi ({\boldsymbol {r}})d{\boldsymbol {r}}&{}=\left[{\frac {d\ }{d\epsilon }}\,J[\rho +\epsilon \phi ]\right]_{\epsilon =0}\\&{}=\left[{\frac {d\ }{d\epsilon }}\,\left({\frac {1}{2}}\iint {\frac {[\rho ({\boldsymbol {r}})+\epsilon \phi ({\boldsymbol {r}})]\,[\rho ({\boldsymbol {r}}')+\epsilon \phi ({\boldsymbol {r}}')]}{\vert {\boldsymbol {r}}-{\boldsymbol {r}}'\vert }}\,d{\boldsymbol {r}}d{\boldsymbol {r}}'\right)\right]_{\epsilon =0}\\&{}={\frac {1}{2}}\iint {\frac {\rho ({\boldsymbol {r}}')\phi ({\boldsymbol {r}})}{\vert {\boldsymbol {r}}-{\boldsymbol {r}}'\vert }}\,d{\boldsymbol {r}}d{\boldsymbol {r}}'+{\frac {1}{2}}\iint {\frac {\rho ({\boldsymbol {r}})\phi ({\boldsymbol {r}}')}{\vert {\boldsymbol {r}}-{\boldsymbol {r}}'\vert }}\,d{\boldsymbol {r}}d{\boldsymbol {r}}'\\\end{aligned}}}$ 

式末的两个积分相等，因为可以交换第二个积分中 r 和 r′ 两个变数，而不改变积分的值。因此，

${\displaystyle \int {\frac {\delta J}{\delta \rho ({\boldsymbol {r}})}}\phi ({\boldsymbol {r}})d{\boldsymbol {r}}=\int \left(\int {\frac {\rho ({\boldsymbol {r}}')}{\vert {\boldsymbol {r}}-{\boldsymbol {r}}'\vert }}d{\boldsymbol {r}}'\right)\phi ({\boldsymbol {r}})d{\boldsymbol {r}}}$ 

故电子－电子库仑势能泛函 J[ρ] 的导数为^[5]

${\displaystyle {\frac {\delta J}{\delta \rho ({\boldsymbol {r}})}}=\int {\frac {\rho ({\boldsymbol {r}}')}{\vert {\boldsymbol {r}}-{\boldsymbol {r}}'\vert }}d{\boldsymbol {r}}'\,.}$ 

且其二阶泛函导数为

${\displaystyle {\frac {\delta ^{2}J[\rho ]}{\delta \rho (\mathbf {r} ')\delta \rho (\mathbf {r} )}}={\frac {\partial }{\partial \rho (\mathbf {r} ')}}\left({\frac {\rho (\mathbf {r} ')}{\vert \mathbf {r} -\mathbf {r} '\vert }}\right)={\frac {1}{\vert \mathbf {r} -\mathbf {r} '\vert }}.}$ 

魏茨泽克动能泛函

1935 年，魏茨泽克提出，在托马斯－费米动能泛函中添加一项梯度修正，使之能更准确描述分子的电子云：

${\displaystyle T_{\mathrm {W} }[\rho ]={\frac {1}{8}}\int {\frac {\nabla \rho (\mathbf {r} )\cdot \nabla \rho (\mathbf {r} )}{\rho (\mathbf {r} )}}d\mathbf {r} =\int t_{\mathrm {W} }\ d\mathbf {r} \,,}$ 

其中

${\displaystyle t_{\mathrm {W} }\equiv {\frac {1}{8}}{\frac {\nabla \rho \cdot \nabla \rho }{\rho }}\qquad {\text{and}}\ \ \rho =\rho ({\boldsymbol {r}})\ .}$ 

由上节的公式可得

${\displaystyle {\begin{aligned}{\frac {\delta T_{\mathrm {W} }}{\delta \rho ({\boldsymbol {r}})}}&={\frac {\partial t_{\mathrm {W} }}{\partial \rho }}-\nabla \cdot {\frac {\partial t_{\mathrm {W} }}{\partial \nabla \rho }}\\&=-{\frac {1}{8}}{\frac {\nabla \rho \cdot \nabla \rho }{\rho ^{2}}}-\left({\frac {1}{4}}{\frac {\nabla ^{2}\rho }{\rho }}-{\frac {1}{4}}{\frac {\nabla \rho \cdot \nabla \rho }{\rho ^{2}}}\right)\qquad {\text{where}}\ \ \nabla ^{2}=\nabla \cdot \nabla \ ,\end{aligned}}}$ 

故所求泛函导数为^[6]

${\displaystyle {\frac {\delta T_{\mathrm {W} }}{\delta \rho ({\boldsymbol {r}})}}=\ \ \,{\frac {1}{8}}{\frac {\nabla \rho \cdot \nabla \rho }{\rho ^{2}}}-{\frac {1}{4}}{\frac {\nabla ^{2}\rho }{\rho }}\ .}$ 

将函数表示成泛函

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，

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

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

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

熵

离散随机变量的熵是概率质量函数的一个泛函

${\displaystyle {\begin{aligned}H[p(x)]=-\sum _{x}p(x)\log p(x)\end{aligned}}}$ 

于是

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

最后，

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

指数

令

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

以${\displaystyle \delta }$ 函数作为测量函数

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

因此

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

1. ^ 在三维笛卡尔坐标系中，

   ${\displaystyle {\begin{aligned}{\frac {\partial f}{\partial \nabla \rho }}={\frac {\partial f}{\partial \rho _{x}}}\mathbf {\hat {i}} +{\frac {\partial f}{\partial \rho _{y}}}\mathbf {\hat {j}} +{\frac {\partial f}{\partial \rho _{z}}}\mathbf {\hat {k}} \,,\qquad &{\text{where}}\ \rho _{x}={\frac {\partial \rho }{\partial x}}\,,\ \rho _{y}={\frac {\partial \rho }{\partial y}}\,,\ \rho _{z}={\frac {\partial \rho }{\partial z}}\,\\&{\text{and}}\ \ \mathbf {\hat {i}} ,\ \mathbf {\hat {j}} ,\ \mathbf {\hat {k}} \ \ {\text{are unit vectors along the x, y, z axes.}}\end{aligned}}}$ 

2. ^ 例如，对于三维 (n = 3) 和二阶 (i = 2) 导数，张量 ∇⁽²⁾ 的分量为

   ${\displaystyle \left[\nabla ^{(2)}\right]_{\alpha \beta }={\frac {\partial ^{\,2}}{\partial r_{\alpha }\,\partial r_{\beta }}}\qquad \qquad {\text{where}}\quad \alpha ,\beta =1,2,3\,.}$ 

3. ^ 例如，当 n = 3 及 i = 2时，张量的标量积为

   ${\displaystyle \nabla ^{(2)}\cdot {\frac {\partial f}{\partial \left(\nabla ^{(2)}\rho \right)}}=\sum _{\alpha ,\beta =1}^{3}\ {\frac {\partial ^{\,2}}{\partial r_{\alpha }\,\partial r_{\beta }}}\ {\frac {\partial f}{\partial \rho _{\alpha \beta }}}\qquad {\text{where}}\ \ \rho _{\alpha \beta }\equiv {\frac {\partial ^{\,2}\rho }{\partial r_{\alpha }\,\partial r_{\beta }}}\ .}$ 

• R. G. Parr, W. Yang, “Density-Functional Theory of Atoms and Molecules”, Oxford University Press, Oxford 1989.
• B. A. Frigyik, S. Srivastava and M. R. Gupta, Introduction to Functional Derivatives, UWEE Tech Report 2008-0001. https://web.archive.org/web/20120207192424/http://www.ee.washington.edu/research/guptalab/publications/functionalDerivativesIntroduction.pdf