English

Noun

equipartition

1. the division of something into equal parts
2. (of a graph) the partition of its vertex set into sets whose sizes differ from each other by no more than 1

Derived terms

• equipartition theorem

In classical statistical mechanics, the equipartition theorem is a general formula that relates the temperature of a system with its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. The original idea of equipartition was that, in thermal equilibrium, energy is shared equally among its various forms; for example, the average kinetic energy in the translational motion of a molecule should equal the average kinetic energy in its rotational motion.

The equipartition theorem makes quantitative 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 an ideal gas has an average kinetic energy of (3/2)kBT in thermal equilibrium, where kB is the Boltzmann constant and T is the 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 classical 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 when relativistic effects are considered. Although the equipartition theorem makes very accurate predictions in certain conditions, it becomes inaccurate when quantum effects are significant, namely at low enough temperatures. When the thermal energy kBT is smaller than the quantum energy spacing in a particular degree of freedom, the average energy and heat capacity of this degree of freedom are less than the values predicted by equipartition. Such a degree of freedom is said to be "frozen out" when the thermal energy is much smaller than this spacing. For example, the specific heat of a solid decreases at low temperatures as various types of motion become frozen out, rather than remaining constant as predicted by equipartition. Such decreases in specific heat were the first sign to physicists of the 19th century that classical physics was incorrect and that new physics was needed. Along with other evidence, equipartition's failure for electromagnetic radiation — also known as the ultraviolet catastrophe — led Albert Einstein to suggest that light itself was quantized into photons, a revolutionary hypothesis that spurred the development of quantum mechanics and quantum field theory.

Basic concept and simple examples

The name "equipartition" means "share and share alike". The original concept of equipartition was that the total kinetic energy of a system is shared equally among all of its independent parts, on the average, once the system has reached thermal equilibrium. Equipartition also makes quantitative predictions for these energies. For example, it predicts that every atom of a noble gas, in thermal equilibrium at temperature T, has an average translational kinetic energy of (3/2)kBT, where kB is the Boltzmann constant. As a consequence, the heavier atoms of xenon have a lower average speed than do the lighter atoms of helium at the same temperature. Figure 2 shows the Maxwell–Boltzmann distribution for the speeds of the atoms in four noble gases.

In this example, the key point is that the kinetic energy is quadratic in the velocity. The equipartition theorem shows that in thermal equilibrium, any degree of freedom (such as a component of the position or velocity of a particle) which appears only quadratically in the energy has an average energy of ½kBT and therefore contributes ½kB to the system's heat capacity. This has many applications.

Translational energy and ideal gases

The (Newtonian) kinetic energy of a particle of mass m, velocity v is given by

H^ = \tfrac12 m |\mathbf|^2 = \tfrac m\left( v_^ + v_^ + v_^ \right),

where vx, vy and vz are the cartesian components of the velocity v. Here, H is short for Hamiltonian, and used henceforth as a symbol for energy because the Hamiltonian formalism plays a central role in the most general form of the equipartition theorem.

Since the kinetic energy is quadratic in the components of the velocity, by equipartition these three components each contribute ½kBT to the average kinetic energy in thermal equilibrium. Thus the average kinetic energy of the particle is (3/2)kBT, as in the example of noble gases above.

More generally, in an ideal gas, the total energy consists purely of (translational) kinetic energy: by assumption, the particles have no internal degrees of freedom and move independently of one another. Equipartition therefore predicts that the average total energy of an ideal gas of N particles is (3/2) N kBT.

It follows that the heat capacity of the gas is (3/2) N kB and hence, in particular, the heat capacity of a mole of a such gas particles is (3/2)NAkB=(3/2)R, where NA is Avogadro's number and R is the gas constant. Since R ≈ 2 cal/(mol·K), equipartition predicts that the molar heat capacity of an ideal gas is roughly 3 cal/(mol·K). This prediction is confirmed by experiment.

Rotational energy and molecular tumbling in solution

A similar example is provided by a rotating molecule with principal moments of inertia I1, I2 and I3. The rotational energy of such a molecule is given by

H^ = \tfrac ( I_ \omega_^ + I_ \omega_^ + I_ \omega_^ ),

where ω1, ω2, and ω3 are the principal components of the angular velocity. By exactly the same reasoning as in the translational case, equipartition implies that in thermal equilibrium the average rotational energy of each particle is (3/2)kBT. Similarly, the equipartition theorem allows the average (more precisely, the root mean square) angular speed of the molecules to be calculated. Rotational diffusion can also be observed by other biophysical probes such as fluorescence anisotropy, flow birefringence and dielectric spectroscopy.

Potential energy and harmonic oscillators

Equipartition applies to potential energies as well as kinetic energies: important examples include harmonic oscillators such as a spring, which has a quadratic potential energy

H^ = \tfrac 12 a q^,\,

where the constant a describes the stiffness of the spring and q is the deviation from equilibrium. If such a one dimensional system has mass m, then its kinetic energy Hkin is ½mv2 = p2/2m, where v and p = mv denote the velocity and momentum of the oscillator. Combining these terms yields the total energy and the Dulong–Petit law of solid molar heat capacities. The latter application was particularly significant in the history of equipartition.

Specific heat capacity of solids

For more details on the molar specific heat of solids, see Einstein solid and Debye model.

An important application of the equipartition theorem is to the specific heat capacity of a crystalline solid. Each atom in such a solid can oscillate in three independent directions, so the solid can be viewed as a system of 3N independent simple harmonic oscillators, where N denotes the number of atoms in the lattice. Since each harmonic oscillator has average energy kBT, the average total energy of the solid is 3NkBT, and its heat capacity is 3NkB.

By taking N to be Avogadro's number NA, and using the relation R = NAkB between the gas constant R and the Boltzmann constant kB, this provides an explanation for the Dulong–Petit law of molar heat capacities of solids, which states that the heat capacity per mole of atoms in the lattice is 3R ≈ 6 cal/(mol·K).

However, this law is inaccurate at lower temperatures, due to quantum effects; it is also inconsistent with the experimentally derived third law of thermodynamics, according to which the molar heat capacity of any substance must go to zero as the temperature goes to absolute zero. Over time, these clumps settle downwards under the influence of gravity, causing more haze near the bottom of a bottle than near its top. However, in a process working in the opposite direction, the particles also diffuse back up towards the top of the bottle. Once equilibrium has been reached, the equipartition theorem may be used to determine the average position of a particular clump of buoyant mass mb. For an infinitely tall bottle of beer, the gravitational potential energy is given by

H^ = m_ g z\,,

where z is the height of the protein clump in the bottle and g is the acceleration caused by gravity. Since s=1, the average potential energy of a protein clump equals kBT. Hence, a protein clump with a buoyant mass of 10 MDa (roughly the size of a virus) would produce a haze with an average height of about 2 cm at equilibrium. The process of such sedimentation to equilibrium is described by the Mason–Weaver equation.

History

This article uses the non-SI unit of cal/(mol·K) for molar specific heat, because it offers greater accuracy for single digits.For an approximate conversion to the corresponding SI unit of J/(mol·K), such values should be multiplied by 4.2 J/cal.

The equipartition of kinetic energy was proposed initially in 1843, and more correctly in 1845, by John James Waterston. In 1859, James Clerk Maxwell argued that the kinetic heat energy of a gas is equally divided between linear and rotational energy. In 1876, Ludwig Boltzmann expanded on this principle by showing that the average energy was divided equally among all the independent components of motion in a system. Boltzmann applied the equipartition theorem to provide a theoretical explanation of the Dulong–Petit law for the specific heat capacities of solids.

• Kinetic theory
• Statistical mechanics
• Quantum statistical mechanics

Notes and references

Further reading

• Statistical Mechanics

• Mathematical Foundations of Statistical Mechanics (G. Gamow, translator)

• Statistical Physics, Part 1

• Statistical Physics

• Statistical Mechanics: Methods and Applications

• Statistical Mechanics

• Pauli Lectures on Physics: Volume 4. Statistical Mechanics

• Statistical Mechanics, with Applications to Physics and Chemistry ASIN B00085D6OO

• The Principles of Statistical Mechanics

External links

• Applet demonstrating equipartition in real time for a mixture of monatomic and diatomic gases
• The equipartition theorem in stellar physics, written by Nir J. Shaviv, an associate professor at the Racah Institute of Physics in the Hebrew University of Jerusalem.