# 擴散作用

(重新導向自 扩散)

## 唯象描述

### 菲克定律

${\displaystyle {\overrightarrow {J}}=-D\nabla c}$

${\displaystyle \iiint _{V}{\frac {\partial c}{\partial t}}dV=-\oint _{\mathbb {S} }{\overrightarrow {J}}\cdot \mathrm {d} \mathbf {a} =-\iiint _{V}\nabla \cdot {\overrightarrow {J}}dV=\iiint _{V}\nabla \cdot (D\nabla c)dV}$

${\displaystyle \iiint _{V}{\frac {\partial c}{\partial t}}dV=D\iiint _{V}\nabla ^{2}cdV}$

${\displaystyle {\frac {\partial c}{\partial t}}=D\nabla ^{2}c}$

### 熱力學分析

#### 熱動平衡條件

${\displaystyle dS={\Big (}{\frac {p_{A}}{T}}-{\frac {p_{B}}{T}}{\Big )}dV_{A}-{\Big (}{\frac {\mu _{A}}{T}}-{\frac {\mu _{B}}{T}}{\Big )}dN_{A}+{\Big (}{\frac {1}{T_{A}}}-{\frac {1}{T_{B}}}{\Big )}dU_{A}=0}$

${\displaystyle {\overrightarrow {J}}=-{\frac {Dc}{RT}}\nabla \mu }$

${\displaystyle {\overrightarrow {J}}=-D\nabla c}$

#### 昂薩格倒易關係

${\displaystyle {\overrightarrow {J}}_{i}=\sum _{j}L_{ij}X_{j}}$

${\displaystyle X_{k}=\nabla {\frac {\partial s(n)}{\partial n_{k}}}}$
${\displaystyle n_{k}}$被稱為熱力學廣義坐標，對於擴散過程取${\displaystyle n}$為濃度即可。