含义Meaning
热平衡时分子在各能级上的最概然分布：能量越高的能级布居越少，按指数衰减；温度越高分布越平坦。统计热力学的核心。
The most probable distribution of molecules over energy levels at equilibrium: populations fall exponentially with energy, flattening as temperature rises. The core of statistical thermodynamics.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(N_i\) | 能级 i 上的分子数population of level i | 1 |
| \(g_i\) | 简并度degeneracy | 1 |
| \(\varepsilon_i\) | 能级能量level energy | J |
| \(q\) | 分子配分函数molecular partition function | 1 |
推导Derivation
N 个可区分粒子分配到各能级的方式数 \(W = N!/\prod_i n_i!\)（对简并能级再乘 \(g_i^{n_i}\)）。
The number of ways to distribute N particles is \(W = N!/\prod n_i!\) (times \(g_i^{n_i}\)).
在 \(\sum n_i = N\)、\(\sum n_i\varepsilon_i = E\) 约束下求 \(\ln W\) 的极大，用斯特林近似 \(\ln n! \approx n\ln n - n\) 和拉格朗日乘子 α、β。
Maximise \(\ln W\) subject to fixed N and E using Stirling’s approximation and Lagrange multipliers α, β.
得 \(n_i = g_ie^{-\alpha}e^{-\beta\varepsilon_i}\)；由归一化定 α，与热力学比较得 \(\beta = 1/k_\mathrm{B}T\)。
This gives \(n_i = g_ie^{-\alpha - \beta\varepsilon_i}\); normalisation fixes α and thermodynamics gives \(\beta = 1/k_\mathrm{B}T\).
两能级之比只取决于能量差与温度之比 \(\Delta\varepsilon/k_\mathrm{B}T\)；室温 \(k_\mathrm{B}T \approx 2.5\ \mathrm{kJ/mol} \approx 207\ \mathrm{cm^{-1}}\)。
Population ratios depend only on \(\Delta\varepsilon/k_\mathrm{B}T\); at room temperature \(k_\mathrm{B}T \approx 2.5\) kJ/mol ≈ 207 cm⁻¹.
适用条件与局限Assumptions & limitations
经典（玻尔兹曼）统计，要求每个量子态平均占据数 ≪ 1；低温、高密度下用玻色–爱因斯坦或费米–狄拉克统计。
Classical statistics need occupancy ≪ 1 per state; otherwise use Bose–Einstein or Fermi–Dirac statistics.
只描述平衡态；激光、等离子体等非平衡体系中可出现布居反转。
Equilibrium only; lasers and plasmas can show population inversion.
例题Worked example
振动能级间隔 1000 cm⁻¹ 时，298 K 下 N₁/N₀ = e^(−1000/207) = 0.008：几乎所有分子处于振动基态。
For a 1000 cm⁻¹ vibration, N₁/N₀ = e^(−4.83) = 0.008 at 298 K: nearly all molecules are in v = 0.