麦克斯韦–玻尔兹曼速率分布 Maxwell–Boltzmann speed distribution — 公式推导、符号与局限

\[f(v) = 4\pi \left(\frac{m}{2\pi k_\mathrm{B}T}\right)^{3/2} v^2 \exp\!\left(-\frac{mv^2}{2k_\mathrm{B}T}\right)\]

含义Meaning

给出理想气体分子速率在 \(v\) 到 \(v+dv\) 之间的概率 \(f(v)\,dv\)。温度升高时分布变宽、峰右移、峰值降低。

\(f(v)\,dv\) is the probability that a molecule’s speed lies between \(v\) and \(v+dv\). Heating broadens the curve and moves the peak right.

符号Symbol	含义 Meaning	单位 Unit
\(f(v)\)	速率概率密度speed probability density	s·m⁻¹
\(m\)	分子质量molecular mass	kg
\(k_\mathrm{B}\)	玻尔兹曼常数Boltzmann constant	J·K⁻¹

每个速度分量独立且服从玻尔兹曼分布：\(g(v_x) \propto \exp(-mv_x^2/2k_\mathrm{B}T)\)，归一化得 \(\sqrt{m/2\pi k_\mathrm{B}T}\) 前因子。
Each velocity component is independent and Boltzmann-distributed: \(g(v_x) \propto \exp(-mv_x^2/2k_\mathrm{B}T)\), normalised by \(\sqrt{m/2\pi k_\mathrm{B}T}\).
三维速度分布为三者乘积，只依赖 \(v^2 = v_x^2+v_y^2+v_z^2\)。
The 3D velocity distribution is the product, depending only on \(v^2\).
对方向积分：速率在 \(v\) 到 \(v+dv\) 的速度点落在半径 v、厚 dv 的球壳内，体积 \(4\pi v^2dv\)。
Integrate over directions: the shell of radius v and thickness dv has volume \(4\pi v^2dv\).
\[f(v)\,dv = \left(\frac{m}{2\pi k_\mathrm{B}T}\right)^{3/2} e^{-mv^2/2k_\mathrm{B}T}\, 4\pi v^2\,dv\]
\(v^2\) 因子使低速概率趋于 0，指数因子使高速概率迅速衰减，因此出现极大值。
The \(v^2\) factor kills low speeds and the exponential kills high ones, giving a maximum.

经典统计，适用于理想气体平衡态；极低温或高密度时需量子统计（玻色–爱因斯坦、费米–狄拉克）。
Classical statistics for ideal gases at equilibrium; quantum statistics are needed at very low T or high density.
非平衡（如激波、刚混合）时不成立。
Not valid away from equilibrium.