在数学 和理论物理 中,泛函导数 是方向导数 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 中导数 的扩展。数学里专门研究泛函导数的分支是泛函分析 。
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 :
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{\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
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/
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{\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 :
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{\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
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{\displaystyle \varphi }
的一次变分
δ
φ
{\displaystyle \delta \varphi }
代替
f
{\displaystyle f}
就得到
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{\displaystyle F}
的一次变分
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{\displaystyle \delta F}
;
在物理学中,通常用狄拉克δ函数
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{\displaystyle \delta (x-y)}
,而不是一般的测试函数
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{\displaystyle f(x)}
, 来求出点
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{\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):
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{\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
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{\displaystyle F[\varphi (x)+\varepsilon f(x)]}
可以展开成
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{\displaystyle \varepsilon }
的级数時 (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
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{\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導數 。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。
與函數的導數類似,泛函導數滿足下列的性質:(其中 F [ρ ] 和 G [ρ ] 為兩個泛函)
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{\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 皆為常數。
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{\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)))\,,}
若 F 和 G 為兩個泛函,則[3]
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{\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)))\ .}
若當中的 G 為一個普通的可導函數 g ,則上式化為[4]
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{\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
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{\displaystyle \varphi (x)}
发生微小变化时,泛函
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{\displaystyle F[\varphi (x)]}
如何变化。其中,
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{\displaystyle \varphi (x)}
的变化量的具体形式没有指明,
給定泛函
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{\displaystyle F[\rho ]=\int f({\boldsymbol {r)),\rho ({\boldsymbol {r))),\nabla \rho ({\boldsymbol {r))))\,d{\boldsymbol {r)),}
及在積分區域的邊界上恆為零的函數 ϕ (r ),由定義可得:
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{\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 為純量對向量的導數。[Note 1] 第三行則用到了散度的積法則 。第四行由高斯散度定理 及邊界上 ϕ =0 的條件得到。由於 ϕ 可以是任意的函數,由變分法基本引理 可知,所求泛函導數為
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{\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 ] 具有本節首段的形式,上述公式就適用。對於其他的泛函形式,可由定義出發,求出其泛函導數。(見库仑势能泛函 。)
以上公式可推廣到高維,並且有其他高階導數的情況。則泛函可寫成
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{\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 ) 為一個張量,其 ni 個分量分別為 i 階微分算子
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{\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] 與上面類似,由泛函導數的定義可知:
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{\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,張量
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{\displaystyle {\frac {\partial f}{\partial \left(\nabla ^{(i)}\rho \right)))}
具有 ni 個分量,各為 f 對 ρ 偏導數之偏導數,即:
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{\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))))\ ,}
並定義張量的純量積為
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{\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] 1927年的Thomas-Fermi模型 对于无相互作用的单一电子雲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
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{\displaystyle T_{\mathrm {TF} }[\rho ]=C_{\mathrm {F} }\int \rho ^{5/3}(\mathbf {r} )\,d\mathbf {r} .}
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{\displaystyle T_{\mathrm {TF} }[\rho ]}
只与电子密度有关
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{\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,
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{\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} ).}
托馬斯和費米利用了以下库仑 勢能泛函來描述電子與核之間的電勢
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{\displaystyle V[\rho ]=\int {\frac {\rho ({\boldsymbol {r)))}{|{\boldsymbol {r))|))\ d{\boldsymbol {r)).}
由泛函導數的定義,
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{\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
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{\displaystyle {\frac {\delta V}{\delta \rho ({\boldsymbol {r)))))={\frac {1}{|{\boldsymbol {r))|))\ .}
至於電子與電子間的相互作用,由以下庫侖勢能泛函描述:
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{\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} '\,.}
由定義,
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{\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))}
式末的兩個積分相等,因為可以交換第二個積分中 r 和 r′ 兩個變數,而不改變積分的值。因此,
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{\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))}
故電子-電子庫侖勢能泛函 J [ρ ] 的導數為[5]
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{\displaystyle {\frac {\delta J}{\delta \rho ({\boldsymbol {r)))))=\int {\frac {\rho ({\boldsymbol {r))')}{\vert {\boldsymbol {r))-{\boldsymbol {r))'\vert ))d{\boldsymbol {r))'\,.}
且其二階泛函導數為
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)
=
∂
∂
ρ
(
r
′
)
(
ρ
(
r
′
)
|
r
−
r
′
|
)
=
1
|
r
−
r
′
|
.
{\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 )).}
1935 年,魏茨泽克 提出,在托馬斯-費米動能泛函中添加一項梯度修正,使之能更準確描述分子的電子雲:
T
W
[
ρ
]
=
1
8
∫
∇
ρ
(
r
)
⋅
∇
ρ
(
r
)
ρ
(
r
)
d
r
=
∫
t
W
d
r
,
{\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
t
W
≡
1
8
∇
ρ
⋅
∇
ρ
ρ
and
ρ
=
ρ
(
r
)
.
{\displaystyle t_{\mathrm {W} }\equiv {\frac {1}{8)){\frac {\nabla \rho \cdot \nabla \rho }{\rho ))\qquad {\text{and))\ \ \rho =\rho ({\boldsymbol {r)))\ .}
由上節的公式可得
δ
T
W
δ
ρ
(
r
)
=
∂
t
W
∂
ρ
−
∇
⋅
∂
t
W
∂
∇
ρ
=
−
1
8
∇
ρ
⋅
∇
ρ
ρ
2
−
(
1
4
∇
2
ρ
ρ
−
1
4
∇
ρ
⋅
∇
ρ
ρ
2
)
where
∇
2
=
∇
⋅
∇
,
{\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))}
故所求泛函導數為[6]
δ
T
W
δ
ρ
(
r
)
=
1
8
∇
ρ
⋅
∇
ρ
ρ
2
−
1
4
∇
2
ρ
ρ
.
{\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 ))\ .}
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,
ρ
(
r
)
=
∫
ρ
(
r
′
)
δ
(
r
−
r
′
)
d
r
′
.
{\displaystyle \rho (\mathbf {r} )=\int \rho (\mathbf {r} ')\delta (\mathbf {r} -\mathbf {r} ')\,d\mathbf {r} '.}
这个泛函只依赖于
ρ
{\displaystyle \rho }
,像上面两个例子一样(就是说,它们都是“局部的”)。因此
δ
ρ
(
r
)
δ
ρ
(
r
′
)
=
∂
ρ
(
r
′
)
δ
(
r
−
r
′
)
∂
ρ
(
r
′
)
=
δ
(
r
−
r
′
)
.
{\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} ').}
离散随机变量 的熵 是概率质量函数 的一个泛函
H
[
p
(
x
)
]
=
−
∑
x
p
(
x
)
log
p
(
x
)
{\displaystyle {\begin{aligned}H[p(x)]=-\sum _{x}p(x)\log p(x)\end{aligned))}
于是
⟨
δ
H
δ
p
,
ϕ
⟩
=
∑
x
δ
H
[
p
(
x
)
]
δ
p
(
x
′
)
ϕ
(
x
′
)
=
d
d
ϵ
H
[
p
(
x
)
+
ϵ
ϕ
(
x
)
]
|
ϵ
=
0
=
−
d
d
ε
∑
x
[
p
(
x
)
+
ε
ϕ
(
x
)
]
log
[
p
(
x
)
+
ε
ϕ
(
x
)
]
|
ε
=
0
=
−
∑
x
[
1
+
log
p
(
x
)
]
ϕ
(
x
)
=
⟨
−
[
1
+
log
p
(
x
)
]
,
ϕ
⟩
.
{\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,
δ
H
δ
p
=
−
1
−
log
p
(
x
)
.
{\displaystyle {\frac {\delta H}{\delta p))=-1-\log p(x).}
令
F
[
φ
(
x
)
]
=
e
∫
φ
(
x
)
g
(
x
)
d
x
.
{\displaystyle F[\varphi (x)]=e^{\int \varphi (x)g(x)dx}.}
以
δ
{\displaystyle \delta }
函数作为测量函数
δ
F
[
φ
(
x
)
]
δ
φ
(
y
)
=
lim
ε
→
0
F
[
φ
(
x
)
+
ε
δ
(
x
−
y
)
]
−
F
[
φ
(
x
)
]
ε
=
lim
ε
→
0
e
∫
(
φ
(
x
)
+
ε
δ
(
x
−
y
)
)
g
(
x
)
d
x
−
e
∫
φ
(
x
)
g
(
x
)
d
x
ε
=
e
∫
φ
(
x
)
g
(
x
)
d
x
lim
ε
→
0
e
ε
∫
δ
(
x
−
y
)
g
(
x
)
d
x
−
1
ε
=
e
∫
φ
(
x
)
g
(
x
)
d
x
lim
ε
→
0
e
ε
g
(
y
)
−
1
ε
=
e
∫
φ
(
x
)
g
(
x
)
d
x
g
(
y
)
.
{\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
δ
F
[
φ
(
x
)
]
δ
φ
(
y
)
=
g
(
y
)
F
[
φ
(
x
)
]
.
{\displaystyle {\frac {\delta F[\varphi (x)]}{\delta \varphi (y)))=g(y)F[\varphi (x)].}
^ 在三維笛卡尔坐标系 中,
∂
f
∂
∇
ρ
=
∂
f
∂
ρ
x
i
^
+
∂
f
∂
ρ
y
j
^
+
∂
f
∂
ρ
z
k
^
,
where
ρ
x
=
∂
ρ
∂
x
,
ρ
y
=
∂
ρ
∂
y
,
ρ
z
=
∂
ρ
∂
z
and
i
^
,
j
^
,
k
^
are unit vectors along the x, y, z axes.
{\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))}
^ 例如,對於三維 (n = 3 ) 和二階 (i = 2 ) 導數,張量 ∇(2) 的分量為
[
∇
(
2
)
]
α
β
=
∂
2
∂
r
α
∂
r
β
where
α
,
β
=
1
,
2
,
3
.
{\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\,.}
^ 例如,當 n = 3 及 i = 2 時,張量的純量積為
∇
(
2
)
⋅
∂
f
∂
(
∇
(
2
)
ρ
)
=
∑
α
,
β
=
1
3
∂
2
∂
r
α
∂
r
β
∂
f
∂
ρ
α
β
where
ρ
α
β
≡
∂
2
ρ
∂
r
α
∂
r
β
.
{\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 ))}\ .}