Equipartition theorem
Figure 1. Thermal motion of an α-helical
peptide. The jittery motion is random and
complex, and the energy of any particular
atom can fluctuate wildly. Nevertheless, the
equipartition theorem allows the average kin-
etic energy of each atom to be computed, as
well as the average potential energies of
many vibrational modes. The grey, red and
blue spheres represent atoms of carbon, oxy-
gen and nitrogen, respectively; the smaller
white spheres represent atoms of hydrogen.
In
classical
statistical mechanics,
the
equipartition theorem is a general formula
that relates the temperature of a system with
its average energies. The equipartition theor-
em is also known as the law of equiparti-
tion, equipartition of energy, or simply
equipartition. The original idea of equiparti-
tion was that, in thermal equilibrium, energy
is shared equally among all of its various
forms; for example, the average kinetic en-
ergy in the translational motion of a molecule
should equal the average kinetic energy in its
rotational motion.
The equipartition theorem makes quantit-
ative predictions. Like the virial theorem, it
gives the total average kinetic and potential
energies for a system at a given temperature,
from which the system’s heat capacity can be
computed. However, equipartition also gives
the average values of individual components
of the energy, such as the kinetic energy of a
particular particle or the potential energy of
a single spring. For example, it predicts that
every molecule in a monoatomic ideal gas
has an average kinetic energy of (3/2)kBT in
thermal
equilibrium, where kB
is
the
Boltzmann constant and T is the (thermody-
namic) temperature. More generally, it can
be applied to any classical system in thermal
equilibrium, no matter how complicated. The
equipartition theorem can be used to derive
the ideal gas law, and the Dulong–Petit law
for the specific heat capacities of solids. It
can also be used to predict the properties of
stars, even white dwarfs and neutron stars,
since it holds even