Equipartition theorem

In classical thermal equilibrium, each independent quadratic term in a system’s energy contributes an average energy of one-half the Boltzmann constant times the temperature.

Boltzmann distribution: The equilibrium probability distribution that assigns states weights proportional to the exponential of negative energy divided by thermal energy. Its energy weighting makes the average of each quadratic term equal to half of kBT.
Quadratic form: A homogeneous polynomial of degree two in a set of variables. The theorem assigns its standard contribution to independent quadratic energy terms.
Heat capacity: The change in a system’s thermal energy per unit change in temperature. Summing equipartition contributions gives classical heat-capacity predictions.
Quantum harmonic oscillator: A quantum system with equally spaced energy levels and a quadratic potential. Its discrete levels suppress the classical half-kBT contributions at low temperature.
Canonical ensemble: A statistical ensemble describing systems in thermal equilibrium with a heat reservoir at fixed temperature. Equipartition is derived by averaging energy over this fixed-temperature ensemble.
Thermal energy: The characteristic energy scale of a system at temperature T, given by the Boltzmann constant times T. Equipartition measures each contribution against this thermal energy scale.
Ideal gas: A gas model whose particles have negligible volume and interact only through elastic collisions. Translational quadratic terms yield the ideal gas’s classical internal energy.
Virial theorem: A relation between time-averaged kinetic and potential energies in bound systems. It generalizes energy-sharing relations beyond purely quadratic classical terms.
Gaussian integral: An integral of an exponential quadratic function, central to probability theory and statistical mechanics. Integrating a quadratic energy term against the Boltzmann weight yields its half-kBT average.
Classical mechanics: The theory describing motion using Newtonian laws or equivalent classical formulations. Equipartition assumes classical states rather than quantized energy levels.

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