law of statistical mechanics stating that, in a system in thermal equilibrium, on the average, an equal amount of energy will be associated with each independent energy state. Based on the work of physicists James Clerk Maxwell of Scotland and Ludwig Boltzmann of Germany, this law states specifically that a system of particles in equilibrium at absolute temperature T will have an average energy of ¹/₂kT associated with each degree of freedom (see freedom, degree of), in which k is the Boltzmann constant. In addition, any degree of freedom contributing potential energy will have another ¹/₂kT associated with it. For a system of s degrees of freedom, of which t have associated potential energies, the total average energy of the system is ¹/₂(s + t)kT.

For example, an atom of a gas has three degrees of freedom (the three spatial, or position, coordinates of the atom) and will, therefore, have an average total energy of ³/₂kT. For an atom in a solid, vibratory motion involves potential energy as well as kinetic energy, and both modes will contribute a term ¹/₂kT, resulting in an average total energy of 3kT.