Boltzmann distribution

The Boltzmann distribution assigns each state a probability proportional to the exponential of its negative energy divided by thermal energy. It describes equilibrium populations of states at a fixed temperature.

Boltzmann factor: The exponential weight exp(−E/kT) assigned to a state of energy E at temperature T. It supplies the energy-dependent weight at the heart of the distribution.
Maxwell–Boltzmann distribution: The classical equilibrium distribution of particle velocities in an ideal gas. It follows from Boltzmann weighting when states are particle velocity states.
Ludwig Boltzmann: A nineteenth-century Austrian physicist who developed statistical mechanics and the kinetic theory of gases. His work established the statistical interpretation associated with this distribution.
Microcanonical ensemble: A statistical ensemble describing isolated systems with fixed energy, volume, and particle number. It assigns equal probabilities to accessible states rather than weighting them by energy.
Partition function: A sum of statistical weights that normalizes probabilities and encodes equilibrium thermodynamic properties. Its value turns Boltzmann weights into probabilities that sum to one.
Arrhenius equation: An equation relating a reaction rate constant to temperature and activation energy. Boltzmann statistics explain the temperature-dependent fraction of molecules able to cross an energy barrier.
James Clerk Maxwell: A Scottish physicist whose kinetic theory helped establish statistical descriptions of molecular motion. His molecular velocity law preceded Boltzmann's broader equilibrium framework.
Grand canonical ensemble: An ensemble for systems that exchange both energy and particles with a reservoir. Its state probabilities add particle-number weighting to thermal energy weighting.
Canonical ensemble: A statistical ensemble of systems held at fixed temperature, volume, and particle number. The distribution gives state probabilities for this ensemble.
Chemical potential: The change in a system's free energy when a particle is added at fixed conditions. It extends equilibrium state weighting to systems that exchange particles.

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