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 kinetic 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, oxygen 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 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)k_BT in thermal equilibrium, where k_B 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 k_BT 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.

[edit] Basic concept and simple examples

Figure 2. Probability density functions of the molecular speed for four noble gases at a temperature of 298.15 K (25 °C). The four gases are helium (4He), neon (20Ne), argon (40Ar) and xenon (132Xe); the superscripts indicate their mass numbers. These probability density functions have dimensions of probability times inverse speed; since probability is dimensionless, they can be expressed in units of seconds per meter.

Figure 2. Probability density functions of the molecular speed for four noble gases at a temperature of 298.15 K (25 °C). The four gases are helium (⁴He), neon (²⁰Ne), argon (⁴⁰Ar) and xenon (¹³²Xe); the superscripts indicate their mass numbers. These probability density functions have dimensions of probability times inverse speed; since probability is dimensionless, they can be expressed in units of seconds per meter.

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)k_BT, where k_B 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 ½k_BT and therefore contributes ½k_B to the system's heat capacity. This has many applications.

[edit] Translational energy and ideal gases

please see math/README to configure.): H^{\mathrm{kin}} = \tfrac12 m |\mathbf{v}|^2 = \tfrac{1}{2} m\left( v_{x}^{2} + v_{y}^{2} + v_{z}^{2} \right),

where v_x, v_y and v_z 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 ½k_BT to the average kinetic energy in thermal equilibrium. Thus the average kinetic energy of the particle is (3/2)k_BT, 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 k_BT.

It follows that the heat capacity of the gas is (3/2) N k_B and hence, in particular, the heat capacity of a mole of a such gas particles is (3/2)N_Ak_B=(3/2)R, where N_A 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.^[1]

The mean kinetic energy also allows the root mean square speed v_rms of the gas particles to be calculated:

please see math/README to configure.): v_{\mathrm{rms}} = \sqrt{\langle v^{2} \rangle} = \sqrt{\frac{3 k_{B} T}{m}} = \sqrt{\frac{3 R T}{M}},

where M = N_Am is the mass of a mole of gas particles. This result is useful for many applications such as Graham's law of effusion, which provides a method for enriching uranium.^[2]

[edit] 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

please see math/README to configure.): H^{\mathrm{rot}} = \tfrac{1}{2} ( I_{1} \omega_{1}^{2} + I_{2} \omega_{2}^{2} + I_{3} \omega_{3}^{2} ),

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)k_BT. Similarly, the equipartition theorem allows the average (more precisely, the root mean square) angular speed of the molecules to be calculated.^[3]

[edit] 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

please see math/README to configure.): H^{\mathrm{pot}} = \tfrac 12 a q^{2},\,

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 H^kin is ½mv² = p²/2m, where v and p = mv denote the velocity and momentum of the oscillator. Combining these terms yields the total energy^[6]

please see math/README to configure.): H = H^{\mathrm{kin}} + H^{\mathrm{pot}} = \frac{p^{2}}{2m} + \frac{1}{2} a q^{2}.

Equipartition therefore implies that in thermal equilibrium, the oscillator has average energy

please see math/README to configure.): \langle H \rangle = \langle H^{\mathrm{kin}} \rangle + \langle H^{\mathrm{pot}} \rangle = \tfrac{1}{2} k_{B} T + \tfrac{1}{2} k_{B} T = k_{B} T,

where the angular brackets Failed to parse (Missing texvc executable; please see math/README to configure.): \left\langle \ldots \right\rangle

This result is valid for any type of harmonic oscillator, such as a pendulum, a vibrating molecule or a passive electronic oscillator. Systems of such oscillators arise in many situations; by equipartition, each such oscillator receives an average total energy k_BT and hence contributes k_B to the system's heat capacity. This can be used to derive the formula for Johnson–Nyquist noise^[8] and the Dulong–Petit law of solid molar heat capacities. The latter application was particularly significant in the history of equipartition.

Figure 3. Atoms in a crystal can vibrate about their equilibrium positions in the lattice. Such vibrations account largely for the heat capacity of crystalline dielectrics; with metals, electrons also contribute to the heat capacity.

[edit] Specific heat capacity of solids

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 k_BT, the average total energy of the solid is 3Nk_BT, and its heat capacity is 3Nk_B.

By taking N to be Avogadro's number N_A, and using the relation R = N_Ak_B between the gas constant R and the Boltzmann constant k_B, 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.^[8] A more accurate theory, incorporating quantum effects, was developed by Albert Einstein (1907) and Peter Debye (1911).^[9]

Many other physical systems can be modeled as sets of coupled oscillators. The motions of such oscillators can be decomposed into normal modes, like the vibrational modes of a piano string or the resonances of an organ pipe. On the other hand, equipartition often breaks down for such systems, because there is no exchange of energy between the normal modes. In an extreme situation, the modes are independent and so their energies are independently conserved. This shows that some sort of mixing of energies, formally called ergodicity, is important for the law of equipartition to hold.

[edit] Sedimentation of particles

Potential energies are not always quadratic in the position. However, the equipartition theorem also shows that if a degree of freedom x contributes only a multiple of x^s (for a fixed real number s) to the energy, then in thermal equilibrium the average energy of that part is k_BT/s.

There is a simple application of this extension to the sedimentation of particles under gravity.^[10] For example, the haze sometimes seen in beer can be caused by clumps of proteins that scatter light.^[11] 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 m_b. For an infinitely tall bottle of beer, the gravitational potential energy is given by

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 k_BT. 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.^[12]

[edit] History

The equipartition of kinetic energy was proposed initially in 1843, and more correctly in 1845, by John James Waterston.^[13] In 1859, James Clerk Maxwell argued that the kinetic heat energy of a gas is equally divided between linear and rotational energy.^[14] 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.^[15]^[16] Boltzmann applied the equipartition theorem to provide a theoretical explanation of the Dulong–Petit law for the specific heat capacities of solids.

Figure 4. Idealized plot of the molar specific heat of a diatomic gas against temperature. It agrees with the value (7/2)R predicted by equipartition at high temperatures (where R is the gas constant), but decreases to (5/2)R and then (3/2)R at lower temperatures, as the vibrational and rotational modes of motion are "frozen out". The failure of the equipartition theorem led to a paradox that was only resolved by quantum mechanics. For most molecules, the transitional temperature T_rot is much less than room temperature, whereas T_vib can be ten times larger or more. A typical example is carbon monoxide, CO, for which T_rot ≈ 2.8 K and T_vib ≈ 3103 K. For molecules with very large or weakly bound atoms, T_vib can be close to room temperature (about 300 K); for example, T_vib ≈ 308 K for iodine gas, I₂.^[17]

The history of the equipartition theorem is intertwined with that of molar heat capacity, both of which were studied in the 19th century. In 1819, the French physicists Pierre Louis Dulong and Alexis Thérèse Petit discovered that the molar-specific heats of solids at room temperature were almost all identical, roughly 6 cal/(mol·K).^[18] Their law was used for many years as a technique for measuring atomic weights.^[9] However, subsequent studies by James Dewar and Heinrich Friedrich Weber showed that this Dulong–Petit law holds only at high temperatures;^[19] at lower temperatures, or for exceptionally hard solids such as diamond, the specific heat was lower.^[20]

Experimental observations of the specific heat of gases also raised concerns about the validity of the equipartition theorem. The theorem predicts that the molar heat capacity of simple monatomic gases should be roughly 3 cal/(mole·K), whereas that of diatomic gases should be roughly 7 cal/(mol·K). Experiments confirmed the former prediction,^[1] but found that molar heat capacities of diatomic gases were typically about 5 cal/(mol·K),^[21] and fell to about 3 cal/(mol·K) at very low temperatures.^[22] Maxwell noted in 1875 that the disagreement between experiment and the equipartition theorem was much worse than even these numbers suggest;^[23] since atoms have internal parts, heat energy should go into the motion of these internal parts, making the predicted specific heats of monatomic and diatomic gases much higher than 3 cal/(mol·K) and 7 cal/(mol·K), respectively.

A third discrepancy concerned the specific heat of metals.^[24] According to the classical Drude model, metallic electrons act as a nearly ideal gas, and so they should contribute (3/2) N_e k_B to the heat capacity by the equipartition theorem, where N_e is the number of electrons. Experimentally, however, electrons contribute little to the heat capacity: the molar heat capacities of many conductors and insulators are nearly the same.^[24]

Several explanations of equipartition's failure to account for molar heat capacities were proposed. Boltzmann defended the derivation of his equipartition theorem as correct, but suggested that gases might not be in thermal equilibrium because of their interactions with the aether.^[25] Lord Kelvin suggested that the derivation of the equipartition theorem must be incorrect, since it disagreed with experiment, but was unable to show how.^[26] Lord Rayleigh instead put forward a more radical view that the equipartition theorem and the experimental assumption of thermal equilibrium were both correct; to reconcile them, he noted the need for a new principle that would provide an "escape from the destructive simplicity" of the equipartition theorem.^[27] Albert Einstein provided that escape, by showing in 1907 that these anomalies in the specific heat were due to quantum effects, specifically the quantization of energy in the elastic modes of the solid.^[28] Einstein used the failure of equipartition to argue for the need of a new quantum theory of matter.^[9] Nernst's 1910 measurements of specific heats at low temperatures^[29] supported Einstein's theory, and led to the widespread acceptance of quantum theory among physicists.^[30]

[edit] General formulation of the equipartition theorem

The most general form of the equipartition theorem^[10]^[7]^[3] states that under suitable assumptions (discussed below), for a physical system with Hamiltonian energy function H and degrees of freedom x_n, the following equipartition formula holds in thermal equilibrium for all indices m and n:

Failed to parse (Missing texvc executable;

please see math/README to configure.): \! \Bigl\langle x_{m} \frac{\partial H}{\partial x_{n}} \Bigr\rangle = \delta_{mn} k_{B} T.

Here δ_mn is the Kronecker delta, which is equal to one if m=n and is zero otherwise. The averaging brackets Failed to parse (Missing texvc executable; please see math/README to configure.): \left\langle \ldots \right\rangle

please see math/README to configure.): \Bigl\langle x_{n} \frac{\partial H}{\partial x_{n}} \Bigr\rangle = k_{B} T for all n.

please see math/README to configure.): \Bigl\langle x_{m} \frac{\partial H}{\partial x_{n}} \Bigr\rangle = 0 for all m≠n.

If a degree of freedom x_n appears only as a quadratic term a_nx_n² in the Hamiltonian H, then the first of these formulae implies that

please see math/README to configure.): k_{B} T = \Bigl\langle x_{n} \frac{\partial H}{\partial x_{n}}\Bigr\rangle = 2\langle a_n x_n^2 \rangle,

which is twice the contribution that this degree of freedom makes to the average energy Failed to parse (Missing texvc executable; please see math/README to configure.): \langle H\rangle . Thus the equipartition theorem for systems with quadratic energies follows easily from the general formula. A similar argument, with 2 replaced by s, applies to energies of the form a_nx_n^s.

The degrees of freedom x_{n</sup> are coordinates on the phase space of the system and are therefore commonly subdivided into generalized position coordinates q_k and generalized momentum coordinates p_k, where p_k is the conjugate momentum to q_k. In this situation, formula 1 means that for all k,}

please see math/README to configure.): \Bigl\langle p_{k} \frac{\partial H}{\partial p_{k}} \Bigr\rangle = \Bigl\langle q_{k} \frac{\partial H}{\partial q_{k}} \Bigr\rangle = k_{B} T.

please see math/README to configure.): \Bigl\langle p_{k} \frac{dq_{k}}{dt} \Bigr\rangle = -\Bigl\langle q_{k} \frac{dp_{k}}{dt} \Bigr\rangle = k_{B} T.

please see math/README to configure.): \Bigl\langle q_{j} \frac{\partial H}{\partial q_{k}} \Bigr\rangle, \quad \Bigl\langle q_{j} \frac{\partial H}{\partial p_{k}} \Bigr\rangle, \quad \Bigl\langle p_{j} \frac{\partial H}{\partial p_{k}} \Bigr\rangle, \quad \Bigl\langle p_{j} \frac{\partial H}{\partial q_{k}} \Bigr\rangle, \quad \Bigl\langle q_{k} \frac{\partial H}{\partial p_{k}} \Bigr\rangle, and Failed to parse (Missing texvc executable; please see math/README to configure.): \Bigl\langle p_{k} \frac{\partial H}{\partial q_{k}} \Bigr\rangle

[edit] Relation to the virial theorem

The general equipartition theorem is an extension of the virial theorem (proposed in 1870^[32]), which states that

please see math/README to configure.): \Bigl\langle \sum_{k} q_{k} \frac{\partial H}{\partial q_{k}} \Bigr\rangle = \Bigl\langle \sum_{k} p_{k} \frac{\partial H}{\partial p_{k}} \Bigr\rangle = \Bigl\langle \sum_{k} p_{k} \frac{dq_{k}}{dt} \Bigr\rangle = -\Bigl\langle \sum_{k} q_{k} \frac{dp_{k}}{dt} \Bigr\rangle,

where t denotes time.^[6] Two key differences are that the virial theorem relates summed rather than individual averages to each other, and it does not connect them to the temperature T. Another difference is that traditional derivations of the virial theorem use averages over time, whereas those of the equipartition theorem use averages over phase space.

[edit] Applications

[edit] Ideal gas law

Ideal gases provide an important application of the equipartition theorem. As well as providing the formula

please see math/README to configure.): \begin{align} \langle H^{\mathrm{kin}} \rangle &= \frac{1}{2m} \langle p_{x}^{2} + p_{y}^{2} + p_{z}^{2} \rangle\\ &= \frac{1}{2} \biggl( \Bigl\langle p_{x} \frac{\partial H^{\mathrm{kin}}}{\partial p_{x}} \Bigr\rangle + \Bigl\langle p_{y} \frac{\partial H^{\mathrm{kin}}}{\partial p_{y}} \Bigr\rangle + \Bigl\langle p_{z} \frac{\partial H^{\mathrm{kin}}}{\partial p_{z}} \Bigr\rangle \biggr) = \frac{3}{2} k_{B} T \end{align}

for the average kinetic energy per particle, the equipartition theorem can be used to derive the ideal gas law from classical mechanics.^[3] If q = (q_x, q_y, q_z) and p = (p_x, p_y, p_z) denote the position vector and momentum of a particle in the gas, and F is the net force on that particle, then

please see math/README to configure.): \begin{align} \langle \mathbf{q} \cdot \mathbf{F} \rangle &= \Bigl\langle q_{x} \frac{dp_{x}}{dt} \Bigr\rangle + \Bigl\langle q_{y} \frac{dp_{y}}{dt} \Bigr\rangle + \Bigl\langle q_{z} \frac{dp_{z}}{dt} \Bigr\rangle\\ &=-\Bigl\langle q_{x} \frac{\partial H}{\partial q_x} \Bigr\rangle - \Bigl\langle q_{y} \frac{\partial H}{\partial q_y} \Bigr\rangle - \Bigl\langle q_{z} \frac{\partial H}{\partial q_z} \Bigr\rangle = -3k_{B} T, \end{align}

where the first equality is Newton's second law, and the second line uses Hamilton's equations and the equipartition formula. Summing over a system of N particles yields

please see math/README to configure.): 3Nk_{B} T = - \biggl\langle \sum_{k=1}^{N} \mathbf{q}_{k} \cdot \mathbf{F}_{k} \biggr\rangle.

Figure 5. The kinetic energy of a particular molecule can fluctuate wildly, but the equipartition theorem allows its average energy to be calculated at any temperature. Equipartition also provides a derivation of the ideal gas law, an equation that relates the pressure, volume and temperature of the gas. (In this diagram five of the molecules have been colored red to track their motion; this coloration has no other significance.)

By Newton's third law and the ideal gas assumption, the net force on the system is the force applied by the walls of their container, and this force is given by the pressure P of the gas. Hence

please see math/README to configure.): -\biggl\langle\sum_{k=1}^{N} \mathbf{q}_{k} \cdot \mathbf{F}_{k}\biggr\rangle = P \oint_{\mathrm{surface}} \mathbf{q} \cdot \mathbf{dS},

where dS is the infinitesimal area element along the walls of the container. Since the divergence of the position vector q is

please see math/README to configure.): \boldsymbol\nabla \cdot \mathbf{q} = \frac{\partial q_{x}}{\partial q_{x}} + \frac{\partial q_{y}}{\partial q_{y}} + \frac{\partial q_{z}}{\partial q_{z}} = 3,

please see math/README to configure.): P \oint_{\mathrm{surface}} \mathbf{q} \cdot \mathbf{dS} = P \int_{\mathrm{volume}} \left( \boldsymbol\nabla \cdot \mathbf{q} \right) dV = 3PV,

where dV is an infinitesimal volume within the container and V is the total volume of the container.

please see math/README to configure.): 3Nk_{B} T = -\biggl\langle \sum_{k=1}^{N} \mathbf{q}_{k} \cdot \mathbf{F}_{k} \biggr\rangle = 3PV,

where n=N/N_A is the number of moles of gas and R=N_Ak_B is the gas constant. Although equipartition provides a simple derivation of the ideal-gas law and the internal energy, the same results can be obtained by an alternative method using the partition function^[33].

[edit] Diatomic gases

A diatomic gas can be modelled as two masses, m₁ and m₂, joined by a spring of stiffness a, which is called the rigid rotor-harmonic oscillator approximation.^[17] The classical energy of this system is

please see math/README to configure.): H = \frac{\left| \mathbf{p}_{1} \right|^{2}}{2m_{1}} + \frac{\left| \mathbf{p}_{2} \right|^{2}}{2m_{2}} + \frac{1}{2} a q^{2},

where p₁ and p₂ are the momenta of the two atoms, and q is the deviation of the inter-atomic separation from its equilibrium value. Every degree of freedom in the energy is quadratic and, thus, should contribute ½k_BT to the total average energy, and ½k_B to the heat capacity. Therefore, the heat capacity of a gas of N diatomic molecules is predicted to be 7N · ½k_B: the momenta p₁ and p₂ contribute three degrees of freedom each, and the extension q contributes the seventh. It follows that the heat capacity of a mole of diatomic molecules with no other degrees of freedom should be (7/2)N_Ak_B=(7/2)R and, thus, the predicted molar heat capacity should be roughly 7 cal/(mol·K). However, the experimental values for molar heat capacities of diatomic gases are typically about 5 cal/(mol·K)^[21] and fall to 3 cal/(mol·K) at very low temperatures.^[22] This disagreement between the equipartition prediction and the experimental value of the molar heat capacity cannot be explained by using a more complex model of the molecule, since adding more degrees of freedom can only increase the predicted specific heat, not decrease it.^[23] This discrepancy was a key piece of evidence showing the need for a quantum theory of matter.

Figure 6. A combined X-ray and optical image of the Crab Nebula. At the heart of this nebula there is a rapidly rotating neutron star which has about one and a half times the mass of the Sun but is only 25 km across (roughly the size of Madrid). The equipartition theorem is useful in predicting the properties of such neutron stars.

[edit] Extreme relativistic ideal gases

Equipartition was used above to derive the classical ideal gas law from Newtonian mechanics. However, relativistic effects become dominant in some systems, such as white dwarfs and neutron stars,^[7] and the ideal gas equations must be modified. The equipartition theorem provides a convenient way to derive the corresponding laws for an extreme relativistic ideal gas.^[3] In such cases, the kinetic energy of a single particle is given by the formula

please see math/README to configure.): H^{\mathrm{kin}} \approx cp = c \sqrt{p_{x}^{2} + p_{y}^{2} + p_{z}^{2}}.

Taking the derivative of H with respect to the p_x momentum component gives the formula

please see math/README to configure.): p_{x} \frac{\partial H^{\mathrm{kin}}}{\partial p_{x}} = c \frac{p_{x}^{2}}{\sqrt{p_{x}^{2} + p_{y}^{2} + p_{z}^{2}}}

and similarly for the p_y and p_z components. Adding the three components together gives

please see math/README to configure.): \begin{align} \langle H^{\mathrm{kin}} \rangle &= \biggl\langle c \frac{p_{x}^{2} + p_{y}^{2} + p_{z}^{2}}{\sqrt{p_{x}^{2} + p_{y}^{2} + p_{z}^{2}}} \biggr\rangle\\ &= \Bigl\langle p_{x} \frac{\partial H^{\mathrm{kin}}}{\partial p_{x}} \Bigr\rangle + \Bigl\langle p_{y} \frac{\partial H^{\mathrm{kin}}}{\partial p_{y}} \Bigr\rangle + \Bigl\langle p_{z} \frac{\partial H^{\mathrm{kin}}}{\partial p_{z}} \Bigr\rangle\\ &= 3 k_{B} T \end{align}

where the last equality follows from the equipartition formula. Thus, the average total energy of an extreme relativistic gas is twice that of the non-relativistic case: for N particles, it is 3 N k_BT.

[edit] Non-ideal gases

In an ideal gas the particles are assumed to interact only through collisions. The equipartition theorem may also be used to derive the energy and pressure of "non-ideal gases" in which the particles also interact with one another through conservative forces whose potential U(r) depends only on the distance r between the particles.^[3] This situation can be described by first restricting attention to a single gas particle, and approximating the rest of the gas by a spherically symmetric distribution. It is then customary to introduce a radial distribution function g(r) such that the probability density of finding another particle at a distance r from the given particle is equal to 4πr²ρ g(r), where ρ=N/V is the mean density of the gas.^[34] It follows that the mean potential energy associated to the interaction of the given particle with the rest of the gas is

please see math/README to configure.): \langle h^{\mathrm{pot}} \rangle = \int_{0}^{\infty} 4\pi r^{2} \rho U(r) g(r)\, dr.

The total mean potential energy of the gas is therefore Failed to parse (Missing texvc executable; please see math/README to configure.): \langle H^{pot} \rangle = \tfrac12 N \langle h^{\mathrm{pot}} \rangle , where N is the number of particles in the gas, and the factor ½ is needed because summation over all the particles counts each interaction twice. Adding kinetic and potential energies, then applying equipartition, yields the energy equation

please see math/README to configure.): H = \langle H^{\mathrm{kin}} \rangle + \langle H^{\mathrm{pot}} \rangle = \frac{3}{2} Nk_{B}T + 2\pi N \rho \int_{0}^{\infty} r^{2} U(r) g(r) \, dr.

please see math/README to configure.): 3Nk_{B}T = 3PV + 2\pi N \rho \int_{0}^{\infty} r^{3} U'(r) g(r)\, dr.

[edit] Anharmonic oscillators

An anharmonic oscillator (in contrast to a simple harmonic oscillator) is one in which the potential energy is not quadratic in the extension q (the generalized position which measures the deviation of the system from equilibrium). Such oscillators provide a complementary point of view on the equipartition theorem.^[35]^[36] Simple examples are provided by potential energy functions of the form

where C and s are arbitrary real constants. In these cases, the law of equipartition predicts that

please see math/README to configure.): k_{B} T = \Bigl\langle q \frac{\partial H^{\mathrm{pot}}}{\partial q} \Bigr\rangle = \langle q \cdot s C q^{s-1} \rangle = \langle s C q^{s} \rangle = s \langle H^{\mathrm{pot}} \rangle.

Thus, the average potential energy equals k_BT/s, not k_BT/2 as for the quadratic harmonic oscillator (where s=2).

More generally, a typical energy function of a one-dimensional system has a Taylor expansion in the extension q:

please see math/README to configure.): H^{\mathrm{pot}} = \sum_{n=2}^{\infty} C_{n} q^{n}

for non-negative integers n. There is no n=1 term, because at the equilibrium point, there is no net force and so the first derivative of the energy is zero. The n=0 term need not be included, since the energy at the equilibrium position may be set to zero by convention. In this case, the law of equipartition predicts that^[35]

please see math/README to configure.): k_{B} T = \Bigl\langle q \frac{\partial H^{\mathrm{pot}}}{\partial q} \Bigr\rangle = \sum_{n=2}^{\infty} \langle q \cdot n C_{n} q^{n-1} \rangle = \sum_{n=2}^{\infty} n C_{n} \langle q^{n} \rangle.

please see math/README to configure.): \langle H^{\mathrm{pot}} \rangle = \frac{1}{2} k_{B} T - \sum_{n=3}^{\infty} \left( \frac{n - 2}{2} \right) C_{n} \langle q^{n} \rangle

does not allow the average potential energy to be written in terms of known constants.

[edit] Brownian motion

Figure 7. Typical Brownian motion of a particle in three dimensions.

The equipartition theorem can be used to derive the Brownian motion of a particle from the Langevin equation.^[3] According to that equation, the motion of a particle of mass m with velocity v is governed by Newton's second law

please see math/README to configure.): \frac{d\mathbf{v}}{dt} = \frac{1}{m} \mathbf{F} = -\frac{\mathbf{v}}{\tau} + \frac{1}{m} \mathbf{F}^{\mathrm{rnd}},

where F^rnd is a random force representing the random collisions of the particle and the surrounding molecules, and where the time constant τ reflects the drag force that opposes the particle's motion through the solution. The drag force is often written F^drag = - γv; therefore, the time constant τ equals m/γ.

The dot product of this equation with the position vector r, after averaging, yields the equation

please see math/README to configure.): \Bigl\langle \mathbf{r} \cdot \frac{d\mathbf{v}}{dt} \Bigr\rangle + \frac{1}{\tau} \langle \mathbf{r} \cdot \mathbf{v} \rangle = 0

for Brownian motion (since the random force F^rnd is uncorrelated with the position r). Using the mathematical identities

please see math/README to configure.): \frac{d}{dt} \left( \mathbf{r} \cdot \mathbf{r} \right) = \frac{d}{dt} \left( r^{2} \right) = 2 \left( \mathbf{r} \cdot \mathbf{v} \right)

please see math/README to configure.): \frac{d}{dt} \left( \mathbf{r} \cdot \mathbf{v} \right) = v^{2} + \mathbf{r} \cdot \frac{d\mathbf{v}}{dt},

please see math/README to configure.): \frac{d^{2}}{dt^{2}} \langle r^{2} \rangle + \frac{1}{\tau} \frac{d}{dt} \langle r^{2} \rangle = 2 \langle v^{2} \rangle = \frac{6}{m} k_{B} T,

where the last equality follows from the equipartition theorem for translational kinetic energy:

please see math/README to configure.): \langle H^{\mathrm{kin}} \rangle = \Bigl\langle \frac{p^{2}}{2m} \Bigr\rangle = \langle \tfrac{1}{2} m v^{2} \rangle = \tfrac{3}{2} k_{B} T.

The above differential equation for Failed to parse (Missing texvc executable; please see math/README to configure.): \langle r^2\rangle

please see math/README to configure.): \langle r^{2} \rangle = \frac{6k_{B} T \tau^{2}}{m} \left( e^{-t/\tau} - 1 + \frac{t}{\tau} \right).

On small time scales, with t << τ, the particle acts as a freely moving particle: by the Taylor series of the exponential function, the squared distance grows approximately quadratically:

please see math/README to configure.): \langle r^{2} \rangle \approx \frac{3k_{B} T}{m} t^{2} = \langle v^{2} \rangle t^{2}.

However, on long time scales, with t >> τ, the exponential and constant terms are negligible, and the squared distance grows only linearly:

please see math/README to configure.): \langle r^{2} \rangle \approx \frac{6k_{B} T\tau}{m} t = 6\gamma k_{B} T t.

This describes the diffusion of the particle over time. An analogous equation for the rotational diffusion of a rigid molecule can be derived in a similar way.

Figure 8. Equipartition gives an accurate estimate of the temperature at the Sun's core.

[edit] Stellar physics

The average temperature of a star can be estimated from the equipartition theorem.^[40] Since most stars are spherically symmetric, the total gravitational potential energy can be estimated by integration

please see math/README to configure.): H^{\mathrm{grav}}_{\mathrm{tot}} = -\int_{0}^{R} \frac{4\pi r^{2} G}{r} M(r)\, \rho(r)\, dr,

where M(r) is the mass within a radius r and ρ(r) is the stellar density at radius r; G represents the gravitational constant and R the total radius of the star. Assuming a constant density throughout the star, this integration yields the formula

please see math/README to configure.): H^{\mathrm{grav}}_{\mathrm{tot}} = - \frac{3G M^{2}}{5R},

where M is the star's total mass. Hence, the average potential energy of a single particle is

please see math/README to configure.): \langle H^{\mathrm{grav}} \rangle = \frac{H^{\mathrm{grav}}_{\mathrm{tot}}}{N} = - \frac{3G M^{2}}{5RN},

where N is the number of particles in the star. Since most stars are composed mainly of ionized hydrogen, N equals roughly (M/m_p), where m_p is the mass of one proton. Application of the equipartition theorem gives an estimate of the star's temperature

please see math/README to configure.): \Bigl\langle r \frac{\partial H^{\mathrm{grav}}}{\partial r} \Bigr\rangle = \langle -H^{\mathrm{grav}} \rangle = k_{B} T = \frac{3G M^{2}}{5RN}.

Substitution of the mass and radius of the Sun yields an estimated solar temperature of T = 14 million kelvins, very close to its core temperature of 15 million kelvins. However, the Sun is much more complex than assumed by this model — both its temperature and density vary strongly with radius — and such excellent agreement (≈7% relative error) is partly fortuitous.^[41]

[edit] Star formation

The same formulae may be applied to determining the conditions for star formation in giant molecular clouds.^[42] A local fluctuation in the density of such a cloud can lead to a runaway condition in which the cloud collapses inwards under its own gravity. Such a collapse occurs when the equipartition theorem — or, equivalently, the virial theorem — is no longer valid, i.e., when the gravitational potential energy exceeds twice the kinetic energy

please see math/README to configure.): M_{J}^{2} = \left( \frac{5k_{B}T}{G m_{p}} \right)^{3} \left( \frac{3}{4\pi \rho} \right)

Substituting the values typically observed in such clouds (T=150 K, ρ = 2×10^-16 g/cm³) gives an estimated minimum mass of 17 solar masses, which is consistent with observed star formation. This effect is also known as the Jeans instability, after the British physicist James Hopwood Jeans who published it in 1902.^[43]

[edit] Derivations

[edit] Kinetic energies and the Maxwell–Boltzmann distribution

The original formulation of the equipartition theorem states that, in any physical system in thermal equilibrium, every particle has exactly the same average kinetic energy, (3/2)k_BT.^[44] This may be shown using the Maxwell–Boltzmann distribution (see Figure 2), which is the probability distribution

please see math/README to configure.): f (v) = 4 \pi \left( \frac{m}{2 \pi k_B T}\right)^{3/2}\!\!v^2 \exp \Bigl( \frac{-mv^2}{2k_B T} \Bigr)

for the speed of a particle of mass m in the system, where the speed v is the magnitude Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{v_x^2 + v_y^2 + v_z^2}

The Maxwell–Boltzmann distribution applies to any system composed of atoms, and assumes only a canonical ensemble, specifically, that the kinetic energies are distributed according to their Boltzmann factor at a temperature T.^[44] The average kinetic energy for a particle of mass m is then given by the integral formula

please see math/README to configure.): \langle H^{\mathrm{kin}} \rangle = \langle \tfrac{1}{2} m v^{2} \rangle = \int _{0}^{\infty} \tfrac{1}{2} m v^{2}\ f(v)\ dv = \tfrac{3}{2} k_{B} T,

as stated by the equipartition theorem. The same result can also be obtained by averaging the particle energy using the probability of finding the particle in certain quantum energy state^[33].

[edit] Quadratic energies and the partition function

More generally, the equipartition theorem states that any degree of freedom x which appears in the total energy H only as a simple quadratic term Ax², where A is a constant, has an average energy of ½k_BT in thermal equilibrium. In this case the equipartition theorem may be derived from the partition function Z(β), where β=1/(k_BT) is the canonical inverse temperature.^[45] Integration over the variable x yields a factor

please see math/README to configure.): Z_{x} = \int_{-\infty}^{\infty} dx \ e^{-\beta A x^{2}} = \sqrt{\frac{\pi}{\beta A}},

please see math/README to configure.): \langle H_{x} \rangle = - \frac{\partial \log Z_{x}}{\partial \beta} = \frac{1}{2\beta} = \frac{1}{2} k_{B} T

[edit] General proofs

To explain these derivations, the following notation is introduced. First, the phase space is described in terms of generalized position coordinates q_j together with their conjugate momenta p_j. The quantities q_j completely describe the configuration of the system, while the quantities (q_j,p_j) together completely describe its state.

of the phase space is introduced and used to define the volume Γ(E, ΔE) of the portion of phase space where the energy H of the system lies between two limits, E and E+ΔE:

please see math/README to configure.): \Gamma (E, \Delta E) = \int_{H \in \left[E, E+\Delta E \right]} d\Gamma .

In this expression, ΔE is assumed to be very small, ΔE<<E. Similarly, Σ(E) is defined to be the total volume of phase space where the energy is less than E:

please see math/README to configure.): \int_{H \in \left[ E, E+\Delta E \right]} \ldots d\Gamma = \Delta E \frac{\partial}{\partial E} \int_{H < E} \ldots d\Gamma,

where the ellipses represent the integrand. From this, it follows that Γ is proportional to ΔE

please see math/README to configure.): \Gamma = \Delta E \ \frac{\partial \Sigma}{\partial E} = \Delta E \ \rho(E),

please see math/README to configure.): \frac{1}{T} = \frac{\partial S}{\partial E} = k_{b} \frac{\partial \log \Sigma}{\partial E} = k_{b} \frac{1}{\Sigma}\,\frac{\partial \Sigma}{\partial E} .

[edit] The canonical ensemble

please see math/README to configure.): \mathcal{N} \int e^{-\beta H(p, q)} d\Gamma = 1,

where β = 1/k_BT. Integration by parts for a phase-space variable x_k (which could be either q_k or p_k) between two limits a and b yields the equation

please see math/README to configure.): \mathcal{N} \int \left[ e^{-\beta H(p, q)} x_{k} \right]_{x_{k}=a}^{x_{k}=b} d\Gamma_{k}+ \mathcal{N} \int e^{-\beta H(p, q)} x_{k} \beta \frac{\partial H}{\partial x_{k}} d\Gamma = 1,

where dΓ_k = dΓ/dx_k, i.e., the first integration is not carried out over x_k. The first term is usually zero, either because x_k is zero at the limits, or because the energy goes to infinity at those limits. In that case, the equipartition theorem for the canonical ensemble follows immediately

please see math/README to configure.): \mathcal{N} \int e^{-\beta H(p, q)} x_{k} \frac{\partial H}{\partial x_{k}} \,d\Gamma = \Bigl\langle x_{k} \frac{\partial H}{\partial x_{k}} \Bigr\rangle = \frac{1}{\beta} = k_{B} T.

Here, the averaging symbolized by Failed to parse (Missing texvc executable; please see math/README to configure.): \langle \ldots \rangle

[edit] The microcanonical ensemble

In the microcanonical ensemble, the system is isolated from the rest of the world, or at least very weakly coupled to it.^[7] Hence, its total energy is effectively constant; to be definite, we say that the total energy H is confined between E and E+ΔE. For a given energy E and spread ΔE, there is a region of phase space Γ in which the system has that energy, and the probability of each state in that region of phase space is equal, by the definition of the microcanonical ensemble. Given these definitions, the equipartition average of phase-space variables x_m (which could be either q_kor p_k) and x_n is given by

please see math/README to configure.): \begin{align} \Bigl\langle x_{m} \frac{\partial H}{\partial x_{n}} \Bigr \rangle &= \frac{1}{\Gamma} \, \int_{H \in \left[ E, E+\Delta E \right]} x_{m} \frac{\partial H}{\partial x_{n}} \,d\Gamma\\ &=\frac{\Delta E}{\Gamma}\, \frac{\partial}{\partial E} \int_{H < E} x_{m} \frac{\partial H}{\partial x_{n}} \,d\Gamma\\ &= \frac{1}{\rho} \,\frac{\partial}{\partial E} \int_{H < E} x_{m} \frac{\partial \left( H - E \right)}{\partial x_{n}} \,d\Gamma, \end{align}

where the last equality follows because E is a constant that does not depend on x_n. Integrating by parts yields the relation

please see math/README to configure.): \begin{align} \int_{H < E} x_{m} \frac{\partial ( H - E )}{\partial x_{n}} \,d\Gamma &= \int_{H < E} \frac{\partial}{\partial x_{n}} \bigl( x_{m} ( H - E ) \bigr) \,d\Gamma - \int_{H < E} \delta_{mn} ( H - E ) d\Gamma\\ &= \delta_{mn} \int_{H < E} ( E - H ) \,d\Gamma, \end{align}

since the first term on the right hand side of the first line is zero (it can be rewritten as an integral of H - E on the hypersurface where H = E).

please see math/README to configure.): \Bigl\langle x_{m} \frac{\partial H}{\partial x_{n}} \Bigr\rangle = \delta_{mn} \frac{1}{\rho} \, \frac{\partial}{\partial E} \int_{H < E}\left( E - H \right)\,d\Gamma = \delta_{mn} \frac{1}{\rho} \, \int_{H < E} \,d\Gamma = \delta_{mn} \frac{\Sigma}{\rho}.

Since Failed to parse (Missing texvc executable; please see math/README to configure.): \rho = \frac{\partial \Sigma}{\partial E}

please see math/README to configure.): \Bigl\langle x_{m} \frac{\partial H}{\partial x_{n}} \Bigr\rangle = \delta_{mn} \Bigl(\frac{1}{\Sigma} \frac{\partial \Sigma}{\partial E}\Bigr)^{-1} = \delta_{mn} \Bigl(\frac{\partial \log \Sigma} {\partial E}\Bigr)^{-1} = \delta_{mn} k_{B} T.

Failed to parse (Missing texvc executable;

please see math/README to configure.): \! \Bigl\langle x_{m} \frac{\partial H}{\partial x_{n}} \Bigr\rangle = \delta_{mn} k_{B} T,

[edit] Limitations

Figure 9. Energy is not shared among the various normal modes in an isolated system of ideal coupled oscillators; the energy in each mode is constant and independent of the energy in the other modes. Hence, the equipartition theorem does not hold for such a system in the microcanonical ensemble (when isolated), although it does hold in the canonical ensemble (when coupled to a heat bath). However, by adding a sufficiently strong nonlinear coupling between the modes, energy will be shared and equipartition holds in both ensembles.

[edit] Requirement of ergodicity

The law of equipartition holds only for ergodic systems in thermal equilibrium, which implies that all states with the same energy must be equally likely to be populated.^[7] Consequently, it must be possible to exchange energy among all its various forms within the syst

Contents

[edit] Basic concept and simple examples

[edit] Translational energy and ideal gases

[edit] Rotational energy and molecular tumbling in solution

[edit] Potential energy and harmonic oscillators

[edit] Specific heat capacity of solids

[edit] Sedimentation of particles

[edit] History

[edit] General formulation of the equipartition theorem

[edit] Relation to the virial theorem

[edit] Applications

[edit] Ideal gas law

[edit] Diatomic gases

[edit] Extreme relativistic ideal gases

[edit] Non-ideal gases

[edit] Anharmonic oscillators

[edit] Brownian motion

[edit] Stellar physics

[edit] Star formation

[edit] Derivations

[edit] Kinetic energies and the Maxwell–Boltzmann distribution

[edit] Quadratic energies and the partition function

[edit] General proofs

[edit] The canonical ensemble

[edit] The microcanonical ensemble

[edit] Limitations

[edit] Requirement of ergodicity