爱因斯坦–斯莫卢霍夫斯基关系（均方位移） Einstein–Smoluchowski relation (mean-square displacement) — 公式推导、符号与局限

\[\langle x^2\rangle = 2Dt\ (\text{一维}),\qquad \langle r^2\rangle = 6Dt\ (\text{三维}),\qquad D = \frac{\lambda^2}{2\tau}\]\[\langle x^2\rangle = 2Dt\ (\text{1D}),\qquad \langle r^2\rangle = 6Dt\ (\text{3D}),\qquad D = \frac{\lambda^2}{2\tau}\]

含义Meaning

布朗运动粒子的均方位移随时间线性增长（1905–1906）。这一关系被佩兰的实验验证，确立了原子、分子的真实存在并测出阿伏加德罗常数。

The mean-square displacement of a Brownian particle grows linearly with time (1905–06). Perrin’s experiments confirmed it, establishing the reality of molecules and measuring Avogadro’s number.

符号说明Symbols

符号Symbol	含义 Meaning	单位 Unit
\(\langle x^2\rangle\)	均方位移mean-square displacement	m²
\(\lambda, \tau\)	每步步长与每步时间（随机行走）step length and step time (random walk)	m, s

推导Derivation

一维随机行走，每步 ±λ 概率相等，N = t/τ 步后 \(\langle x^2\rangle = N\lambda^2\)。
A 1D random walk of N = t/τ steps ±λ gives \(\langle x^2\rangle = N\lambda^2\).
与扩散方程高斯解的方差 2Dt 比较得 \(D = \lambda^2/2\tau\)。
Matching the Gaussian variance 2Dt gives \(D = \lambda^2/2\tau\).
三维各方向独立，方差相加得 6Dt。
In 3D the variances add to 6Dt.

适用条件与局限Assumptions & limitations

适用于正常扩散；拥挤环境（细胞内、聚合物网络）中可出现反常扩散 \(\langle x^2\rangle \propto t^\alpha\)。
Normal diffusion; crowded media can show anomalous diffusion.

例题Worked example

蔗糖在水中 D = 5.2×10⁻¹⁰ m²/s：1 s 扩散约 √(6Dt) = 56 μm；扩散 1 cm 则需约 9 小时。

Sucrose in water travels ~56 μm in 1 s but needs ~9 h for 1 cm.