MaxwellBoltzmannin

MaxwellBoltzmannin is the Maxwell–Boltzmann distribution, a fundamental concept in statistical mechanics that describes the distribution of speeds of particles in an ideal classical gas at thermal equilibrium. It embodies the combined ideas of James Clerk Maxwell and Ludwig Boltzmann and underpins the connection between microscopic molecular motion and macroscopic properties such as temperature and pressure.

The probability density for a particle to have speed v is f(v) = 4π (m/(2πkT))^(3/2) v^2 exp(-mv^2/(2kT)), valid for v ≥ 0. This function gives the fraction of molecules with speeds in the range v to v + dv. The distribution concerns speeds rather than velocity components and assumes a single species of non-relativistic, distinguishable particles in a dilute, equilibrium gas.

Derived quantities include the most probable speed v_p = sqrt(2kT/m), the average speed v̄ = sqrt(8kT/(πm)), and the root-mean-square speed v_rms = sqrt(3kT/m). In molar terms, v_p = sqrt(2RT/M), v̄ = sqrt(8RT/(πM)), and v_rms = sqrt(3RT/M), where M is the molar mass and R is the gas constant.

Assumptions behind the distribution include classical, non-quantum statistics, elastic collisions, and thermal equilibrium in an ideal gas with indistinguishable translational motion. It is valid in the high-temperature, dilute limit, where quantum effects are negligible. At low temperatures or high densities, quantum statistics (Bose-Einstein or Fermi-Dirac) replace Maxwell–Boltzmann statistics, but the classical distribution emerges as a limiting case. The distribution is widely used in kinetic theory, chemistry, plasma physics, and astrophysics.