The Maxwell-Boltzmann distribution is a continuous probability distribution for the speed of particles in an idealised gas at thermal equilibrium. It is the three-dimensional analogue of the Rayleigh Distribution: the magnitude of a vector with independent, zero-mean normal components.
A random variable \(X\) follows a Maxwell-Boltzmann distribution with scale parameter \(a > 0\) if
\[ X \sim \text{Maxwell}(a). \]
The probability density function is
\[ f(x) = \sqrt{\frac{2}{\pi}}\,\frac{x^2}{a^3} \exp\!\left(-\frac{x^2}{2a^2}\right), \quad x \geq 0. \]
Interpretation: The factor \(x^2\) suppresses small speeds and the Gaussian factor suppresses large ones, giving a single peak at the most probable speed \(x = a\sqrt{2}\).
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
from scipy.stats import maxwell
sns.set_style('whitegrid')
sns.set_palette('Set2')
x = np.linspace(0, 10, 400)
fig, ax = plt.subplots(figsize=(7, 4))
for a in (1, 2, 3):
ax.plot(x, maxwell.pdf(x, scale=a), lw=2, label=rf'$a = {a}$')
ax.set_xlabel('$x$')
ax.set_ylabel('$f(x)$')
ax.set_title('Maxwell-Boltzmann density')
ax.legend()
plt.show()
For \(X \sim \text{Maxwell}(a)\),
\[ \mathbb{E}[X] = 2a\sqrt{\frac{2}{\pi}} \quad \& \quad \operatorname{Var}(X) = \frac{3\pi - 8}{\pi}\,a^2. \]
Proof. With the substitution \(u = x^2/(2a^2)\) and the gamma integral \(\int_0^\infty u^{s-1} e^{-u}\,du = \Gamma(s)\),
\[ \mathbb{E}[X] = \sqrt{\frac{2}{\pi}}\,\frac{1}{a^3} \int_0^\infty x^3 e^{-x^2/(2a^2)} \, dx = 2a\sqrt{\frac{2}{\pi}}. \]
Since \(\mathbb{E}[X^2] = 3a^2\), the variance is \(3a^2 - \frac{8}{\pi}a^2 = \frac{3\pi - 8}{\pi}a^2\).
The speeds of molecules in an ideal gas at temperature \(T\) follow a Maxwell-Boltzmann distribution with scale \(a = \sqrt{k_B T / m}\).
Mean speed, most probable speed, and root-mean-square speed of gas molecules are read directly from the distribution’s moments.
The fraction of molecules exceeding an activation energy — governing chemical reaction rates — is obtained from the tail of the distribution.