Equipartition theorem

History

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.

The history of the equipartition theorem is intertwined with that of specific 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 specific heat capacities of solid elements at room temperature were inversely proportional to the atomic weight of the element. Their law was used for many years as a technique for measuring atomic weights. However, subsequent studies by James Dewar and Heinrich Friedrich Weber showed that this Dulong–Petit law holds only at high temperatures; at lower temperatures, or for exceptionally hard solids such as diamond, the specific heat capacity was lower.

Experimental observations of the specific heat capacities 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/(mol·K), whereas that of diatomic gases should be roughly 7 cal/(mol·K). Experiments confirmed the former prediction, but found that molar heat capacities of diatomic gases were typically about 5 cal/(mol·K), and fell to about 3 cal/(mol·K) at very low temperatures. Maxwell noted in 1875 that the disagreement between experiment and the equipartition theorem was much worse than even these numbers suggest; 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. According to the classical Drude model, metallic electrons act as a nearly ideal gas, and so they should contribute ⁠3/2⁠ N_ek_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.

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. Lord Kelvin suggested that the derivation of the equipartition theorem must be incorrect, since it disagreed with experiment, but was unable to show how. In 1900 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. Albert Einstein provided that escape, by showing in 1906 that these anomalies in the specific heat were due to quantum effects, specifically the quantization of energy in the elastic modes of the solid. Einstein used the failure of equipartition to argue for the need of a new quantum theory of matter. Nernst's 1910 measurements of specific heats at low temperatures supported Einstein's theory, and led to the widespread acceptance of quantum theory among physicists.

General formulation of the equipartition theorem

The most general form of the equipartition theorem 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: $${\displaystyle ⟨x_m\frac{\mathrm{\partial }H}{\mathrm{\partial }{x}_n}⟩={\delta }_{mn}{k}_{\text{B}}T.}$$ Here δ_mn is the Kronecker delta, which is equal to one if m = n and is zero otherwise. The averaging brackets ${\displaystyle ⟨\dots ⟩}$ is assumed to be an ensemble average over phase space or, under an assumption of ergodicity, a time average of a single system.

The general equipartition theorem holds in both the microcanonical ensemble, when the total energy of the system is constant, and also in the canonical ensemble, when the system is coupled to a heat bath with which it can exchange energy. Derivations of the general formula are given later in the article.

The general formula is equivalent to the following two:

1. ${\displaystyle ⟨x_n\frac{\mathrm{\partial }H}{\mathrm{\partial }{x}_n}⟩=k_{\text{B}}T\text{for all }n}$
2. ${\displaystyle ⟨x_m\frac{\mathrm{\partial }H}{\mathrm{\partial }{x}_n}⟩=0\text{for all }m\ne n.}$

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 $${\displaystyle k_{\text{B}}T=⟨x_n\frac{\mathrm{\partial }H}{\mathrm{\partial }{x}_n}⟩=2⟨a_n{x}_n^2⟩,}$$ which is twice the contribution that this degree of freedom makes to the average energy ${\displaystyle ⟨H⟩}$. 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 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, $${\displaystyle ⟨p_k\frac{\mathrm{\partial }H}{\mathrm{\partial }{p}_k}⟩=⟨q_k\frac{\mathrm{\partial }H}{\mathrm{\partial }{q}_k}⟩=k_{\text{B}}T.}$$

Using the equations of Hamiltonian mechanics, these formulae may also be written $${\displaystyle ⟨p_k\frac{d{q}_k}{dt}⟩=-⟨q_k\frac{d{p}_k}{dt}⟩=k_{\text{B}}T.}$$

Similarly, one can show using formula 2 that $${\displaystyle ⟨p_j\frac{\mathrm{\partial }H}{\mathrm{\partial }{p}_k}⟩=⟨q_j\frac{\mathrm{\partial }H}{\mathrm{\partial }{q}_k}⟩=0\text{ for all }j\ne k.}$$ and $${\displaystyle ⟨p_j\frac{\mathrm{\partial }{q}_k}{\mathrm{\partial }t}⟩=-⟨q_j\frac{\mathrm{\partial }{p}_k}{\mathrm{\partial }t}⟩=0\text{ for all }j\ne k.}$$

Relation to the virial theorem

The general equipartition theorem is an extension of the virial theorem (proposed in 1870), which states that $${\displaystyle ⟨\sum _k{q}_k\frac{\mathrm{\partial }H}{\mathrm{\partial }{q}_k}⟩=⟨\sum _k{p}_k\frac{\mathrm{\partial }H}{\mathrm{\partial }{p}_k}⟩=⟨\sum _k{p}_k\frac{d{q}_k}{dt}⟩=-⟨\sum _k{q}_k\frac{d{p}_k}{dt}⟩,}$$ where t denotes time. 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.

Applications

Ideal gas law

Ideal gases provide an important application of the equipartition theorem. As well as providing the formula $${\displaystyle \begin{array}{cc}⟨H^{\mathrm{k}\mathrm{i}\mathrm{n}}⟩& =\frac{1}{2m}⟨p_x^2+p_y^2+p_z^2⟩\\ & =\frac{1}{2}\left(⟨p_x\frac{\mathrm{\partial }{H}^{\mathrm{k}\mathrm{i}\mathrm{n}}}{\mathrm{\partial }{p}_x}⟩+⟨p_y\frac{\mathrm{\partial }{H}^{\mathrm{k}\mathrm{i}\mathrm{n}}}{\mathrm{\partial }{p}_y}⟩+⟨p_z\frac{\mathrm{\partial }{H}^{\mathrm{k}\mathrm{i}\mathrm{n}}}{\mathrm{\partial }{p}_z}⟩\right)=\frac{3}{2}{k}_{\text{B}}T\end{array}}$$ for the average kinetic energy per particle, the equipartition theorem can be used to derive the ideal gas law from classical mechanics. 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 $${\displaystyle \begin{array}{cc}⟨\mathbf{q}\cdot \mathbf{F}⟩& =⟨q_x\frac{d{p}_x}{dt}⟩+⟨q_y\frac{d{p}_y}{dt}⟩+⟨q_z\frac{d{p}_z}{dt}⟩\\ & =-⟨q_x\frac{\mathrm{\partial }H}{\mathrm{\partial }{q}_x}⟩-⟨q_y\frac{\mathrm{\partial }H}{\mathrm{\partial }{q}_y}⟩-⟨q_z\frac{\mathrm{\partial }H}{\mathrm{\partial }{q}_z}⟩=-3k_{\text{B}}T,\end{array}}$$ 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 $${\displaystyle 3N{k}_{\text{B}}T=-⟨\sum _{k=1}^N{\mathbf{q}}_k\cdot {\mathbf{F}}_k⟩.}$$

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 $${\displaystyle -⟨\sum _{k=1}^N{\mathbf{q}}_k\cdot {\mathbf{F}}_k⟩=P{\oint }_{\text{surface}}\mathbf{q}\cdot d\mathbf{S},}$$ where dS is the infinitesimal area element along the walls of the container. Since the divergence of the position vector q is $${\displaystyle \mathbf{\nabla }\cdot \mathbf{q}=\frac{\mathrm{\partial }{q}_x}{\mathrm{\partial }{q}_x}+\frac{\mathrm{\partial }{q}_y}{\mathrm{\partial }{q}_y}+\frac{\mathrm{\partial }{q}_z}{\mathrm{\partial }{q}_z}=3,}$$ the divergence theorem implies that $${\displaystyle P{\oint }_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}}\mathbf{q}\cdot \mathbf{d}\mathbf{S}=P{\int }_{\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{m}\mathrm{e}}\left(\mathbf{\nabla }\cdot \mathbf{q}\right)dV=3PV,}$$ where dV is an infinitesimal volume within the container and V is the total volume of the container.

Putting these equalities together yields $${\displaystyle 3N{k}_{\text{B}}T=-⟨\sum _{k=1}^N{\mathbf{q}}_k\cdot {\mathbf{F}}_k⟩=3PV,}$$ which immediately implies the ideal gas law for N particles: $${\displaystyle PV=N{k}_{\text{B}}T=nRT,}$$ 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.

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. The classical energy of this system is $${\displaystyle H=\frac{{\left|{\mathbf{p}}_1\right|}^2}{2m_1}+\frac{{\left|{\mathbf{p}}_2\right|}^2}{2m_2}+\frac{1}{2}a{q}^2,}$$ 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 1⁄2k_BT to the total average energy, and 1⁄2k_B to the heat capacity. Therefore, the heat capacity of a gas of N diatomic molecules is predicted to be 7N·1⁄2k_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 ⁠7/2⁠N_Ak_B = ⁠7/2⁠R 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) and fall to 3 cal/(mol·K) at very low temperatures. 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. This discrepancy was a key piece of evidence showing the need for a quantum theory of matter.

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, 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. In such cases, the kinetic energy of a single particle is given by the formula $${\displaystyle H_{\mathrm{k}\mathrm{i}\mathrm{n}}\approx cp=c\sqrt{p_x^2+p_y^2+p_z^2}.}$$ Taking the derivative of H with respect to the p_x momentum component gives the formula $${\displaystyle p_x\frac{\mathrm{\partial }{H}_{\mathrm{k}\mathrm{i}\mathrm{n}}}{\mathrm{\partial }{p}_x}=c\frac{p_x^2}{\sqrt{p_x^2+p_y^2+p_z^2}}}$$ and similarly for the p_y and p_z components. Adding the three components together gives $${\displaystyle \begin{array}{cc}⟨H_{\mathrm{k}\mathrm{i}\mathrm{n}}⟩& =⟨c\frac{p_x^2+p_y^2+p_z^2}{\sqrt{p_x^2+p_y^2+p_z^2}}⟩\\ & =⟨p_x\frac{\mathrm{\partial }{H}^{\mathrm{k}\mathrm{i}\mathrm{n}}}{\mathrm{\partial }{p}_x}⟩+⟨p_y\frac{\mathrm{\partial }{H}^{\mathrm{k}\mathrm{i}\mathrm{n}}}{\mathrm{\partial }{p}_y}⟩+⟨p_z\frac{\mathrm{\partial }{H}^{\mathrm{k}\mathrm{i}\mathrm{n}}}{\mathrm{\partial }{p}_z}⟩\\ & =3k_{\text{B}}T\end{array}}$$ 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 Nk_BT.

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. 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 4πr²ρg(r), where ρ = N/V is the mean density of the gas. It follows that the mean potential energy associated to the interaction of the given particle with the rest of the gas is $${\displaystyle ⟨h_{\mathrm{p}\mathrm{o}\mathrm{t}}⟩={\int }_0^{\mathrm{\infty }}4\pi r^2\rho U(r)g(r)dr.}$$ The total mean potential energy of the gas is therefore ${\displaystyle ⟨H_{\text{pot}}⟩=\frac{1}{2}N⟨h_{\mathrm{p}\mathrm{o}\mathrm{t}}⟩}$, where N is the number of particles in the gas, and the factor 1⁄2 is needed because summation over all the particles counts each interaction twice. Adding kinetic and potential energies, then applying equipartition, yields the energy equation $${\displaystyle H=⟨H_{\mathrm{k}\mathrm{i}\mathrm{n}}⟩+⟨H_{\mathrm{p}\mathrm{o}\mathrm{t}}⟩=\frac{3}{2}N{k}_{\text{B}}T+2\pi N\rho {\int }_0^{\mathrm{\infty }}{r}^{2}U(r)g(r)dr.}$$ A similar argument, can be used to derive the pressure equation $${\displaystyle 3N{k}_{\text{B}}T=3PV+2\pi N\rho {\int }_0^{\mathrm{\infty }}{r}^3{U}^{\prime }(r)g(r)dr.}$$

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. Simple examples are provided by potential energy functions of the form $${\displaystyle H_{\mathrm{p}\mathrm{o}\mathrm{t}}=C{q}^s,}$$ where C and s are arbitrary real constants. In these cases, the law of equipartition predicts that $${\displaystyle k_{\text{B}}T=⟨q\frac{\mathrm{\partial }{H}_{\mathrm{p}\mathrm{o}\mathrm{t}}}{\mathrm{\partial }q}⟩=⟨q\cdot sC{q}^{s-1}⟩=⟨sC{q}^s⟩=s⟨H_{\mathrm{p}\mathrm{o}\mathrm{t}}⟩.}$$ 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: $${\displaystyle H_{\mathrm{p}\mathrm{o}\mathrm{t}}=\sum _{n=2}^{\mathrm{\infty }}{C}_n{q}^n}$$ 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 $${\displaystyle k_{\text{B}}T=⟨q\frac{\mathrm{\partial }{H}_{\mathrm{p}\mathrm{o}\mathrm{t}}}{\mathrm{\partial }q}⟩=\sum _{n=2}^{\mathrm{\infty }}⟨q\cdot n{C}_n{q}^{n-1}⟩=\sum _{n=2}^{\mathrm{\infty }}n{C}_n⟨q^n⟩.}$$ In contrast to the other examples cited here, the equipartition formula $${\displaystyle ⟨H_{\mathrm{p}\mathrm{o}\mathrm{t}}⟩=\frac{1}{2}{k}_{\text{B}}T-\sum _{n=3}^{\mathrm{\infty }}\left(\frac{n-2}{2}\right)C_n⟨q^n⟩}$$ does not allow the average potential energy to be written in terms of known constants.

Brownian motion

The equipartition theorem can be used to derive the Brownian motion of a particle from the Langevin equation. According to that equation, the motion of a particle of mass m with velocity v is governed by Newton's second law $${\displaystyle \frac{d\mathbf{v}}{dt}=\frac{1}{m}\mathbf{F}=-\frac{\mathbf{v}}{\tau }+\frac{1}{m}{\mathbf{F}}_{\mathrm{r}\mathrm{n}\mathrm{d}},}$$ 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 $${\displaystyle ⟨\mathbf{r}\cdot \frac{d\mathbf{v}}{dt}⟩+\frac{1}{\tau }⟨\mathbf{r}\cdot \mathbf{v}⟩=0}$$ for Brownian motion (since the random force F_rnd is uncorrelated with the position r). Using the mathematical identities $${\displaystyle \frac{d}{dt}\left(\mathbf{r}\cdot \mathbf{r}\right)=\frac{d}{dt}\left(r^2\right)=2\left(\mathbf{r}\cdot \mathbf{v}\right)}$$ and $${\displaystyle \frac{d}{dt}\left(\mathbf{r}\cdot \mathbf{v}\right)=v^2+\mathbf{r}\cdot \frac{d\mathbf{v}}{dt},}$$ the basic equation for Brownian motion can be transformed into $${\displaystyle \frac{d^2}{d{t}^2}⟨r^2⟩+\frac{1}{\tau }\frac{d}{dt}⟨r^2⟩=2⟨v^2⟩=\frac{6}{m}{k}_{\text{B}}T,}$$ where the last equality follows from the equipartition theorem for translational kinetic energy: $${\displaystyle ⟨H_{\mathrm{k}\mathrm{i}\mathrm{n}}⟩=⟨\frac{p^2}{2m}⟩=⟨\frac{1}{2}m{v}^2⟩=\frac{3}{2}{k}_{\text{B}}T.}$$ The above differential equation for ${\displaystyle ⟨r^2⟩}$ (with suitable initial conditions) may be solved exactly: $${\displaystyle ⟨r^2⟩=\frac{6k_{\text{B}}T{\tau }^2}{m}\left(e^{-t/\tau }-1+\frac{t}{\tau }\right).}$$ 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: $${\displaystyle ⟨r^2⟩\approx \frac{3k_{\text{B}}T}{m}{t}^2=⟨v^2⟩t^2.}$$ However, on long time scales, with t ≫ τ, the exponential and constant terms are negligible, and the squared distance grows only linearly: $${\displaystyle ⟨r^2⟩\approx \frac{6k_{\text{B}}T\tau }{m}t=\frac{6k_{\text{B}}Tt}{\gamma }.}$$ 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.

Stellar physics

The equipartition theorem and the related virial theorem have long been used as a tool in astrophysics. As examples, the virial theorem may be used to estimate stellar temperatures or the Chandrasekhar limit on the mass of white dwarf stars.

The average temperature of a star can be estimated from the equipartition theorem. Since most stars are spherically symmetric, the total gravitational potential energy can be estimated by integration $${\displaystyle H_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}=-{\int }_0^R\frac{4\pi r^{2}G}{r}M(r)\rho (r)dr,}$$ 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 $${\displaystyle H_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}=-\frac{3G{M}^2}{5R},}$$ where M is the star's total mass. Hence, the average potential energy of a single particle is $${\displaystyle ⟨H_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}⟩=\frac{H_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}}{N}=-\frac{3G{M}^2}{5RN},}$$ 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 $${\displaystyle ⟨r\frac{\mathrm{\partial }{H}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}}{\mathrm{\partial }r}⟩=⟨-H_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}⟩=k_{\text{B}}T=\frac{3G{M}^2}{5RN}.}$$ 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.

Star formation

The same formulae may be applied to determining the conditions for star formation in giant molecular clouds. 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 $${\displaystyle \frac{3G{M}^2}{5R}>3N{k}_{\text{B}}T.}$$ Assuming a constant density ρ for the cloud $${\displaystyle M=\frac{4}{3}\pi R^3\rho }$$ yields a minimum mass for stellar contraction, the Jeans mass M_J $${\displaystyle M_{\text{J}}^2={\left(\frac{5k_{\text{B}}T}{G{m}_p}\right)}^3\left(\frac{3}{4\pi \rho }\right).}$$ 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.

Derivations

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 translational kinetic energy, ⁠3/2⁠k_BT. However, this is true only for ideal gas, and the same result can be derived from the Maxwell–Boltzmann distribution. First, we choose to consider only the Maxwell–Boltzmann distribution of velocity of the z-component $${\displaystyle f(v_z)=\sqrt{{\displaystyle \frac{m}{2\pi k_{\text{B}}T}}}{e}^{\frac{-m{v_z}^2}{2k_{\text{B}}T}}}$$

with this equation, we can calculate the mean square velocity of the z-component $${\displaystyle ⟨{v_z}^2⟩={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(v_z){v_z}^{2}d{v}_z={\displaystyle \frac{k_{\text{B}}T}{m}}}$$

Since different components of velocity are independent of each other and share the same distribution function, the average translational kinetic energy is given by $${\displaystyle ⟨E_k⟩={\displaystyle \frac{3}{2}}m⟨{v_z}^2⟩={\displaystyle \frac{3}{2}}{k}_{\text{B}}T}$$

Notice, the Maxwell–Boltzmann distribution should not be confused with the Boltzmann distribution, which the former can be derived from the latter by assuming the energy of a particle is equal to its translational kinetic energy.

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.

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 1⁄2k_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. Integration over the variable x yields a factor $${\displaystyle Z_x={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}dx{e}^{-\beta A{x}^2}=\sqrt{\frac{\pi }{\beta A}},}$$ in the formula for Z. The mean energy associated with this factor is given by $${\displaystyle ⟨H_x⟩=-\frac{\mathrm{\partial }\log Z_x}{\mathrm{\partial }\beta }=\frac{1}{2\beta }=\frac{1}{2}{k}_{\text{B}}T}$$ as stated by the equipartition theorem.

General proofs

General derivations of the equipartition theorem can be found in many statistical mechanics textbooks, both for the microcanonical ensemble and for the canonical ensemble. They involve taking averages over the phase space of the system, which is a symplectic manifold.

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.

Secondly, the infinitesimal volume $${\displaystyle d\mathrm{\Gamma }=\prod _{i}d{q}_{i}d{p}_i}$$ of the phase space is introduced and used to define the volume Σ(E, ΔE) of the portion of phase space where the energy H of the system lies between two limits, E and E + ΔE: $${\displaystyle \mathrm{\Sigma }(E,\mathrm{\Delta }E)={\int }_{H\in \left[E,E+\mathrm{\Delta }E\right]}d\mathrm{\Gamma }.}$$ In this expression, ΔE is assumed to be very small, ΔE ≪ E. Similarly, Ω(E) is defined to be the total volume of phase space where the energy is less than E: $${\displaystyle \mathrm{\Omega }(E)={\int }_{H<E}d\mathrm{\Gamma }.}$$

Since ΔE is very small, the following integrations are equivalent $${\displaystyle {\int }_{H\in \left[E,E+\mathrm{\Delta }E\right]}\dots d\mathrm{\Gamma }=\mathrm{\Delta }E\frac{\mathrm{\partial }}{\mathrm{\partial }E}{\int }_{H<E}\dots d\mathrm{\Gamma },}$$ where the ellipses represent the integrand. From this, it follows that Σ is proportional to ΔE $${\displaystyle \mathrm{\Sigma }(E,\mathrm{\Delta }E)=\mathrm{\Delta }E\frac{\mathrm{\partial }\mathrm{\Omega }}{\mathrm{\partial }E}=\mathrm{\Delta }E\rho (E),}$$ where ρ(E) is the density of states. By the usual definitions of statistical mechanics, the entropy S equals k_B log Ω(E), and the temperature T is defined by $${\displaystyle \frac{1}{T}=\frac{\mathrm{\partial }S}{\mathrm{\partial }E}=k_{\text{B}}\frac{\mathrm{\partial }\log \mathrm{\Omega }}{\mathrm{\partial }E}=k_{\text{B}}\frac{1}{\mathrm{\Omega }}\frac{\mathrm{\partial }\mathrm{\Omega }}{\mathrm{\partial }E}.}$$

The canonical ensemble

In the canonical ensemble, the system is in thermal equilibrium with an infinite heat bath at temperature T (in kelvins). The probability of each state in phase space is given by its Boltzmann factor times a normalization factor ${\displaystyle \mathcal{N}}$, which is chosen so that the probabilities sum to one $${\displaystyle \mathcal{N}\int e^{-\beta H(p,q)}d\mathrm{\Gamma }=1,}$$ where β = 1/(k_BT). Using Integration by parts for a phase-space variable x_k the above can be written as $${\displaystyle \mathcal{N}\int e^{-\beta H(p,q)}d\mathrm{\Gamma }=\mathcal{N}\int d[x_k{e}^{-\beta H(p,q)}]d{\mathrm{\Gamma }}_k-\mathcal{N}\int x_k\frac{\mathrm{\partial }{e}^{-\beta H(p,q)}}{\mathrm{\partial }{x}_k}d\mathrm{\Gamma },}$$ where dΓ_k = dΓ/dx_k, i.e., the first integration is not carried out over x_k. Performing the first integral between two limits a and b and simplifying the second integral yields the equation $${\displaystyle \mathcal{N}\int {\left[e^{-\beta H(p,q)}{x}_k\right]}_{x_k=a}^{x_k=b}d{\mathrm{\Gamma }}_k+\mathcal{N}\int e^{-\beta H(p,q)}{x}_k\beta \frac{\mathrm{\partial }H}{\mathrm{\partial }{x}_k}d\mathrm{\Gamma }=1,}$$

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 $${\displaystyle \mathcal{N}\int e^{-\beta H(p,q)}{x}_k\frac{\mathrm{\partial }H}{\mathrm{\partial }{x}_k}d\mathrm{\Gamma }=⟨x_k\frac{\mathrm{\partial }H}{\mathrm{\partial }{x}_k}⟩=\frac{1}{\beta }=k_{\text{B}}T.}$$

Here, the averaging symbolized by ${\displaystyle ⟨\dots ⟩}$ is the ensemble average taken over the canonical ensemble.

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. Hence, its total energy is effectively constant; to be definite, we say that the total energy H is confined between E and E+dE. For a given energy E and spread dE, there is a region of phase space Σ 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_k or p_k) and x_n is given by

$${\displaystyle \begin{array}{cc}⟨x_m\frac{\mathrm{\partial }H}{\mathrm{\partial }{x}_n}⟩& =\frac{1}{\mathrm{\Sigma }}{\int }_{H\in \left[E,E+\mathrm{\Delta }E\right]}{x}_m\frac{\mathrm{\partial }H}{\mathrm{\partial }{x}_n}d\mathrm{\Gamma }\\ & =\frac{\mathrm{\Delta }E}{\mathrm{\Sigma }}\frac{\mathrm{\partial }}{\mathrm{\partial }E}{\int }_{H<E}{x}_m\frac{\mathrm{\partial }H}{\mathrm{\partial }{x}_n}d\mathrm{\Gamma }\\ & =\frac{1}{\rho }\frac{\mathrm{\partial }}{\mathrm{\partial }E}{\int }_{H<E}{x}_m\frac{\mathrm{\partial }\left(H-E\right)}{\mathrm{\partial }{x}_n}d\mathrm{\Gamma },\end{array}}$$

where the last equality follows because E is a constant that does not depend on x_n. Integrating by parts yields the relation $${\displaystyle \begin{array}{cc}{\int }_{H<E}{x}_m\frac{\mathrm{\partial }(H-E)}{\mathrm{\partial }{x}_n}d\mathrm{\Gamma }& ={\int }_{H<E}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_n}(x_m(H-E))d\mathrm{\Gamma }-{\int }_{H<E}{\delta }_{mn}(H-E)d\mathrm{\Gamma }\\ & ={\delta }_{mn}{\int }_{H<E}(E-H)d\mathrm{\Gamma },\end{array}}$$ 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).

Substitution of this result into the previous equation yields $${\displaystyle ⟨x_m\frac{\mathrm{\partial }H}{\mathrm{\partial }{x}_n}⟩={\delta }_{mn}\frac{1}{\rho }\frac{\mathrm{\partial }}{\mathrm{\partial }E}{\int }_{H<E}\left(E-H\right)d\mathrm{\Gamma }={\delta }_{mn}\frac{1}{\rho }{\int }_{H<E}d\mathrm{\Gamma }={\delta }_{mn}\frac{\mathrm{\Omega }}{\rho }.}$$

Since ${\displaystyle \rho =\frac{\mathrm{\partial }\mathrm{\Omega }}{\mathrm{\partial }E}}$ the equipartition theorem follows: $${\displaystyle ⟨x_m\frac{\mathrm{\partial }H}{\mathrm{\partial }{x}_n}⟩={\delta }_{mn}{\left(\frac{1}{\mathrm{\Omega }}\frac{\mathrm{\partial }\mathrm{\Omega }}{\mathrm{\partial }E}\right)}^{-1}={\delta }_{mn}{\left(\frac{\mathrm{\partial }\log \mathrm{\Omega }}{\mathrm{\partial }E}\right)}^{-1}={\delta }_{mn}{k}_{\text{B}}T.}$$

Thus, we have derived the general formulation of the equipartition theorem $${\displaystyle ⟨x_m\frac{\mathrm{\partial }H}{\mathrm{\partial }{x}_n}⟩={\delta }_{mn}{k}_{\text{B}}T,}$$ which was so useful in the applications described above.

Limitations

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. Consequently, it must be possible to exchange energy among all its various forms within the system, or with an external heat bath in the canonical ensemble. The number of physical systems that have been rigorously proven to be ergodic is small; a famous example is the hard-sphere system of Yakov Sinai. The requirements for isolated systems to ensure ergodicity—and, thus equipartition—have been studied, and provided motivation for the modern chaos theory of dynamical systems. A chaotic Hamiltonian system need not be ergodic, although that is usually a good assumption.

A commonly cited counter-example where energy is not shared among its various forms and where equipartition does not hold in the microcanonical ensemble is a system of coupled harmonic oscillators. If the system is isolated from the rest of the world, the energy in each normal mode is constant; energy is not transferred from one mode to another. Hence, equipartition does not hold for such a system; the amount of energy in each normal mode is fixed at its initial value. If sufficiently strong nonlinear terms are present in the energy function, energy may be transferred between the normal modes, leading to ergodicity and rendering the law of equipartition valid. However, the Kolmogorov–Arnold–Moser theorem states that energy will not be exchanged unless the nonlinear perturbations are strong enough; if they are too small, the energy will remain trapped in at least some of the modes.

Another simple example is an ideal gas of a finite number of colliding particles in a round vessel. Due to the vessel's symmetry, the angular momentum of such a gas is conserved. Therefore, not all states with the same energy are populated. This results in the mean particle energy being dependent on the mass of this particle, and also on the masses of all the other particles.

Another way ergodicity can be broken is by the existence of nonlinear soliton symmetries. In 1953, Fermi, Pasta, Ulam and Tsingou conducted computer simulations of a vibrating string that included a non-linear term (quadratic in one test, cubic in another, and a piecewise linear approximation to a cubic in a third). They found that the behavior of the system was quite different from what intuition based on equipartition would have led them to expect. Instead of the energies in the modes becoming equally shared, the system exhibited a very complicated quasi-periodic behavior. This puzzling result was eventually explained by Kruskal and Zabusky in 1965 in a paper which, by connecting the simulated system to the Korteweg–de Vries equation led to the development of soliton mathematics.

Failure due to quantum effects

The law of equipartition breaks down when the thermal energy k_BT is significantly smaller than the spacing between energy levels. Equipartition no longer holds because it is a poor approximation to assume that the energy levels form a smooth continuum, which is required in the derivations of the equipartition theorem above. Historically, the failures of the classical equipartition theorem to explain specific heats and black-body radiation were critical in showing the need for a new theory of matter and radiation, namely, quantum mechanics and quantum field theory.

To illustrate the breakdown of equipartition, consider the average energy in a single (quantum) harmonic oscillator, which was discussed above for the classical case. Neglecting the irrelevant zero-point energy term since it can be factored out of the exponential functions involved in the probability distribution, the quantum harmonic oscillator energy levels are given by E_n = nhν, where h is the Planck constant, ν is the fundamental frequency of the oscillator, and n is an integer. The probability of a given energy level being populated in the canonical ensemble is given by its Boltzmann factor $${\displaystyle P(E_n)=\frac{e^{-n\beta h\nu }}{Z},}$$ where β = 1/k_BT and the denominator Z is the partition function, here a geometric series $${\displaystyle Z=\sum _{n=0}^{\mathrm{\infty }}{e}^{-n\beta h\nu }=\frac{1}{1-e^{-\beta h\nu }}.}$$

Its average energy is given by $${\displaystyle ⟨H⟩=\sum _{n=0}^{\mathrm{\infty }}{E}_{n}P(E_n)=\frac{1}{Z}\sum _{n=0}^{\mathrm{\infty }}nh\nu e^{-n\beta h\nu }=-\frac{1}{Z}\frac{\mathrm{\partial }Z}{\mathrm{\partial }\beta }=-\frac{\mathrm{\partial }\log Z}{\mathrm{\partial }\beta }.}$$

Substituting the formula for Z gives the final result $${\displaystyle ⟨H⟩=h\nu \frac{e^{-\beta h\nu }}{1-e^{-\beta h\nu }}.}$$

At high temperatures, when the thermal energy k_BT is much greater than the spacing hν between energy levels, the exponential argument βhν is much less than one and the average energy becomes k_BT, in agreement with the equipartition theorem (Figure 10). However, at low temperatures, when hν ≫ k_BT, the average energy goes to zero—the higher-frequency energy levels are "frozen out" (Figure 10). As another example, the internal excited electronic states of a hydrogen atom do not contribute to its specific heat as a gas at room temperature, since the thermal energy k_BT (roughly 0.025 eV) is much smaller than the spacing between the lowest and next higher electronic energy levels (roughly 10 eV).

Similar considerations apply whenever the energy level spacing is much larger than the thermal energy. This reasoning was used by Max Planck and Albert Einstein, among others, to resolve the ultraviolet catastrophe of black-body radiation. The paradox arises because there are an infinite number of independent modes of the electromagnetic field in a closed container, each of which may be treated as a harmonic oscillator. If each electromagnetic mode were to have an average energy k_BT, there would be an infinite amount of energy in the container. However, by the reasoning above, the average energy in the higher-frequency modes goes to zero as ν goes to infinity; moreover, Planck's law of black-body radiation, which describes the experimental distribution of energy in the modes, follows from the same reasoning.

Other, more subtle quantum effects can lead to corrections to equipartition, such as identical particles and continuous symmetries. The effects of identical particles can be dominant at very high densities and low temperatures. For example, the valence electrons in a metal can have a mean kinetic energy of a few electronvolts, which would normally correspond to a temperature of tens of thousands of kelvins. Such a state, in which the density is high enough that the Pauli exclusion principle invalidates the classical approach, is called a degenerate fermion gas. Such gases are important for the structure of white dwarf and neutron stars. At low temperatures, a fermionic analogue of the Bose–Einstein condensate (in which a large number of identical particles occupy the lowest-energy state) can form; such superfluid electrons are responsible for superconductivity.