Maxwell-Boltzmann Distribution
A probability distribution describing the speeds of molecules in a gas at thermal equilibrium, crucial for kinetic theory calculations.
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The statement of the theorem
Let be the velocity vector of a particle of mass in a gas at temperature . The probability density function for this velocity, derived from the canonical ensemble, is given by:\\begin{equation} f(\textbf{v}) = \left(\frac{m}{2\pi k_B T}\right)^{\frac{3}{2}} e^{-\frac{m(v_x^2 + v_y^2 + v_z^2)}{2k_B T}} \end{equation}\.\\text{The normalization condition requires that the integral over all velocity space equals unity:}\ \nobreak \int_{\mathbb{R}^3} f(\textbf{v}) \, d^3\textbf{v} = 1.\\text{The distribution of the speed magnitude } v = |\textbf{v}| \text{ is given by the radial probability density function } P(v) = 4\pi v^2 f(\textbf{v}) \text{, which yields:}\ \nobreak P(v) = \left(\frac{m}{2\pi k_B T}\right)^{\frac{3}{2}} 4\pi v^2 e^{-\frac{m v^2}{2k_B T}} \text{ for } v \ge 0.
Source: Wikipedia