Maxwell–Boltzmann distribution

The classical probability distribution of particle speeds or energies in thermal equilibrium. Its shape depends on temperature and particle mass.

Boltzmann factor: The exponential factor giving the relative thermal weight of a state with a specified energy. It supplies the energy-dependent weighting underlying the equilibrium distribution.
Statistical mechanics: The framework connecting the microscopic states of many particles to macroscopic physical properties. The distribution is a classical statistical-mechanical result for equilibrium particles.
Fermi–Dirac statistics: Quantum statistics describing indistinguishable fermions, which obey the Pauli exclusion principle. Fermions at low temperatures need this distribution rather than the classical one.
Maxwell–Boltzmann plot: A graphing method that linearizes a Maxwell–Boltzmann population relation to estimate temperature or energy differences. It turns thermal population ratios into a line whose slope reveals temperature.
James Clerk Maxwell: A Scottish physicist whose work established classical electromagnetic theory and advanced kinetic theory. Maxwell derived the molecular velocity distribution for gases in 1860.
Equipartition theorem: A classical result assigning an average energy of one-half kT to each independent quadratic degree of freedom. It links the distribution’s average kinetic energy to temperature.
Probability density function: A function whose integral over an interval gives the probability of a continuous random variable falling there. The curve gives probabilities over speed or energy intervals, not at a single exact value.
Bose–Einstein statistics: Quantum statistics describing indistinguishable bosons, which can occupy the same quantum state. Bosons can accumulate in states in ways the classical distribution cannot describe.
Arrhenius equation: An equation relating a chemical reaction’s rate constant to temperature and activation energy. The distribution’s high-energy tail helps explain why reaction rates rise with temperature.
Ludwig Boltzmann: An Austrian physicist who developed statistical mechanics and explained thermodynamics through molecular behavior. Boltzmann extended the kinetic theory and connected molecular statistics with thermodynamics.

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