# Equipartition theorem

The equipartition theorem is a principle of classical (non-quantum) statistical mechanics which states that the internal energy of a system composed of a large number of particles at thermal equilibrium will distribute itself evenly among each of the quadratic degrees of freedom allowed to the particles of the system.

For example, in thermodynamics, the equipartition theorem says that the mean internal energy associated with each degree of freedom of a monatomic ideal gas is the same.

For a molecule of gas, each component of velocity has an associated kinetic energy. This kinetic energy is, on average,

${\displaystyle {\frac {1}{2}}k_{B}T}$

where ${\displaystyle k_{B}}$ is the Boltzmann constant, and T is the temperature of the molecule in kelvins. The components of velocity can be either linear or angular.

In general, for any system with a classical Hamiltonian of the form:

${\displaystyle H=\sum _{i}^{m}{a_{i}p_{i}^{2}}+\sum _{j}^{n}{b_{j}q_{j}^{2}}+U(p_{m+1},p_{m+2},...p_{M},q_{n+1},q_{n+2},...q_{N})}$
where ${\displaystyle a_{i}}$ and ${\displaystyle b_{i}}$ are constant with respect to all ${\displaystyle q_{i and ${\displaystyle p_{i,
${\displaystyle q_{j}}$ and ${\displaystyle p_{i}}$ are spatial coordinates and their conjugate momenta,

each degree of freedom ${\displaystyle q_{i}}$ and ${\displaystyle p_{j}}$ will contribute a total of ${\displaystyle {\frac {1}{2}}k_{B}T}$ to the system's total energy, resulting in a total of ${\displaystyle {\frac {1}{2}}(m+n)k_{B}T}$ equipartition energy.

The equipartition theorem is valid only in the classical limit of an energy continuum. The equipartition theorem breaks down in the limit of large gaps between quantum energy levels, because it becomes more difficult to excite degrees of freedom which are highly quantized, such as electronic excitations in non-metals, vibrational modes with a large ratio of force constant to reduced mass, or rotational degrees of freedom about an axis with a low moment of inertia.