统计熵 Statistical entropy — 公式推导、符号与局限

\[S = k_\mathrm{B}\ln W,\qquad S = -k_\mathrm{B}\sum_i p_i\ln p_i\]

含义Meaning

玻尔兹曼熵把熵与微观状态数联系起来；吉布斯形式适用于任意概率分布，与信息论的香农熵形式相同。

Boltzmann’s entropy links S to the number of microstates; Gibbs’s form applies to any distribution and matches Shannon’s information entropy.

符号Symbol	含义 Meaning	单位 Unit
\(W\)	与宏观态相容的微观状态数number of microstates	1
\(p_i\)	微观态 i 的概率probability of microstate i	1

熵是广延量，两个独立体系 \(W = W_1W_2\)，而 \(S = S_1 + S_2\)，只有对数函数满足 \(f(W_1W_2) = f(W_1) + f(W_2)\)。
S is extensive while W multiplies; only the logarithm turns products into sums.
等概率 \(p_i = 1/W\) 时吉布斯形式化为 \(k\ln W\)。
With equal probabilities 1/W the Gibbs form reduces to k ln W.
代入玻尔兹曼分布 \(p_i = e^{-\beta E_i}/Q\)：\(S = U/T + k_\mathrm{B}\ln Q\)。
Inserting Boltzmann probabilities gives \(S = U/T + k\ln Q\).
残余熵：CO 晶体中分子取向 CO/OC 随机，\(S = R\ln 2 = 5.76\ \mathrm{J/(mol\,K)}\)（实测约 4.6）。
Residual entropy: random CO/OC orientations give R ln 2 = 5.76 J mol⁻¹ K⁻¹ (observed ≈ 4.6).