MaxwellBoltzmannverdeling

The Maxwell–Boltzmann distribution is a probability distribution that describes the distribution of speeds of particles in an ideal gas at thermodynamic equilibrium under classical mechanics. It arises from statistical considerations of many particles and the Boltzmann distribution, assuming non-interacting particles and a large number of particles in a three-dimensional space.

The velocity distribution can be written for each component as f(vx, vy, vz) = (m/(2πkT))^(3/2) exp(-m(vx^2 + vy^2 + vz^2)/(2kT)), where m is the particle mass, k is the Boltzmann constant, and T is the absolute temperature. The distribution of speeds, obtained by integrating over directions, is f(v) = 4π (m/(2πkT))^(3/2) v^2 exp(-mv^2/(2kT)).

Key properties include the most probable speed vp = sqrt(2kT/m), the mean speed ⟨v⟩ = sqrt(8kT/πm), and the root-mean-square speed vrms = sqrt(3kT/m). The average kinetic energy per molecule is (3/2)kT.

Applications and limitations: the Maxwell–Boltzmann distribution underpins kinetic theory of gases and is used in astrophysics, atmospheric science, and engineering to model dilute gases. It is valid when quantum effects are negligible (high temperature or low density) and interactions between particles are minimal. At low temperatures or high densities, quantum statistics (Bose–Einstein or Fermi–Dirac) or non-ideal interactions must be considered.

Historical context: the distribution is named for James Clerk Maxwell and Ludwig Boltzmann, who developed the foundational ideas in the 1860s and 1870s.