玻尔兹曼分布Boltzmann distribution

两个能级的粒子数之比 N₂/N₁ = (g₂/g₁)·exp(−ΔE/kT)。室温下 kT ≈ 2.5 kJ/mol（0.0257 eV）：转动能级间隔远小于 kT，许多能级被占据；振动能级间隔通常大于 kT，绝大多数分子处于基态；电子激发能远大于 kT，几乎无热激发。气体分子速率的麦克斯韦-玻尔兹曼分布显示，升温使分布展宽、峰值右移，高能分子比例迅速增加。

The population ratio of two levels is N₂/N₁ = (g₂/g₁)·exp(−ΔE/kT). At room temperature kT ≈ 2.5 kJ/mol (0.0257 eV): rotational spacings are much smaller, so many levels are populated; vibrational spacings are usually larger, so most molecules sit in the ground level; electronic excitation energies are far larger, so there is almost no thermal excitation. The Maxwell–Boltzmann speed distribution broadens and shifts to higher speeds on heating, and the fraction of energetic molecules rises sharply.

1860 年麦克斯韦推导出理想气体分子的速率分布。1868–1877 年奥地利物理学家路德维希·玻尔兹曼把它推广到任意能量形式，并提出熵的统计解释 S = k·ln W（刻在他的墓碑上）。吉布斯在 1902 年用系综理论给出了更普遍的表述。

In 1860 Maxwell derived the speed distribution of ideal-gas molecules. Between 1868 and 1877 the Austrian physicist Ludwig Boltzmann generalised it to any form of energy and gave the statistical interpretation of entropy, S = k·ln W, engraved on his tombstone. Gibbs gave a more general formulation with ensemble theory in 1902.

玻尔兹曼因子 e^(−Ea/RT) 出现在阿伦尼乌斯方程中，解释了反应速率随温度的指数增长；它决定了 NMR 信号强度（核自旋能级粒子数差极小，约 10⁻⁵）、构象异构体的比例、平衡常数与标准自由能的关系 K = e^(−ΔG°/RT)，以及激光需要的粒子数反转为何必须打破玻尔兹曼分布。

The Boltzmann factor e^(−Ea/RT) appears in the Arrhenius equation, explaining the exponential rise of reaction rate with temperature; it sets NMR signal strength (nuclear-spin population differences are tiny, about 10⁻⁵), conformer ratios and the relation between equilibrium constant and standard free energy, K = e^(−ΔG°/RT), and it explains why lasers need a population inversion that defeats the Boltzmann distribution.