In classical statistical mechanics, the equipartition theorem relates the temperature of a system to 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 all of its various forms; for example, the average kinetic energy per degree of freedom in translational motion of a molecule should equal that in 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 atom in a monatomic ideal gas has an average kinetic energy of 3/2kBT in thermal equilibrium, where kB is the Boltzmann constant and T is the (thermodynamic) temperature. More generally, equipartition can be applied to any classical system in thermal equilibrium, no matter how complicated. It can be used to derive the ideal gas law, and the Dulong–Petit law for the specific heat capacities of solids. The equipartition theorem 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 accurate predictions in certain conditions, it is inaccurate when quantum effects are significant, such as at low 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 heat capacity 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 heat capacity were among the first signs to physicists of the 19th century that classical physics was incorrect and that a new, more subtle, scientific model was required. Along with other evidence, equipartition's failure to model black-body radiation—also known as the ultraviolet catastrophe—led Max Planck to suggest that energy in the oscillators in an object, which emit light, were quantized, a revolutionary hypothesis that spurred the development of quantum mechanics and quantum field theory.
Basic concept and simple examples
The name "equipartition" means "equal division," as derived from the Latin equi from the antecedent, æquus ("equal or even"), and partition from the noun, partitio ("division, portion"). 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 an inert noble gas, in thermal equilibrium at temperature T, has an average translational kinetic energy of 3/2kBT, where kB is the Boltzmann constant. As a consequence, since kinetic energy is equal to 1⁄2(mass)(velocity)2, 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 1⁄2kBT and therefore contributes 1⁄2kB 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 kin = 1 2 m | v | 2 = 1 2 m ( v x 2 + v y 2 + v z 2 ) , {\displaystyle H_{\text{kin}}={\tfrac {1}{2}}m|\mathbf {v} |^{2}={\tfrac {1}{2}}m\left(v_{x}^{2}+v_{y}^{2}+v_{z}^{2}\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 1⁄2kBT to the average kinetic energy in thermal equilibrium. Thus the average kinetic energy of the particle is 3/2kBT, as in the example of noble gases above. More generally, in a monatomic 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 total energy of an ideal gas of N particles is 3/2 N kB T. 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 such gas particles is 3/2NAkB = 3/2R, where NA is the Avogadro constant 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 when compared to monatomic gases. The mean kinetic energy also allows the root mean square speed vrms of the gas particles to be calculated:
v rms = ⟨ v 2 ⟩ = 3 k B T m = 3 R T M , {\displaystyle v_{\text{rms}}={\sqrt {\left\langle v^{2}\right\rangle }}={\sqrt {\frac {3k_{\text{B}}T}{m}}}={\sqrt {\frac {3RT}{M}}},}
where M = NAm 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.
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. According to classical mechanics, the rotational energy of such a molecule is given by
H r o t = 1 2 ( I 1 ω 1 2 + I 2 ω 2 2 + I 3 ω 3 2 ) , {\displaystyle 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 3/2kBT. Similarly, the equipartition theorem allows the average (more precisely, the root mean square) angular speed of the molecules to be calculated. The tumbling of rigid molecules—that is, the random rotations of molecules in solution—plays a key role in the relaxations observed by nuclear magnetic resonance, particularly protein NMR and residual dipolar couplings. 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 pot = 1 2 a q 2 , {\displaystyle H_{\text{pot}}={\tfrac {1}{2}}aq^{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 Hkin is
H kin = 1 2 m v 2 = p 2 2 m , {\displaystyle H_{\text{kin}}={\frac {1}{2}}mv^{2}={\frac {p^{2}}{2m}},}
where v and p = mv denote the velocity and momentum of the oscillator. Combining these terms yields the total energy
H = H kin + H pot = p 2 2 m + 1 2 a q 2 . {\displaystyle H=H_{\text{kin}}+H_{\text{pot}}={\frac {p^{2}}{2m}}+{\frac {1}{2}}aq^{2}.}
Equipartition therefore implies that in thermal equilibrium, the oscillator has average energy
⟨ H ⟩ = ⟨ H kin ⟩ + ⟨ H pot ⟩ = 1 2 k B T + 1 2 k B T = k B T , {\displaystyle \langle H\rangle =\langle H_{\text{kin}}\rangle +\langle H_{\text{pot}}\rangle ={\tfrac {1}{2}}k_{\text{B}}T+{\tfrac {1}{2}}k_{\text{B}}T=k_{\text{B}}T,}
where the angular brackets ⟨ … ⟩ {\displaystyle \left\langle \ldots \right\rangle } denote the average of the enclosed quantity, 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 kBT and hence contributes kB to the system's heat capacity. This can be used to derive the formula for Johnson–Nyquist noise and the Dulong–Petit law of solid heat capacities. The latter application was particularly significant in the history of equipartition.
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 kBT, the average total energy of the solid is 3N kBT, and its heat capacity is 3N kB. By taking N to be the Avogadro constant 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 specific heat capacities of solids, which stated that the specific heat capacity (per unit mass) of a solid element is inversely proportional to its atomic weight. A modern version is that the molar heat capacity of a solid 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. A more accurate theory, incorporating quantum effects, was developed by Albert Einstein (1907) and Peter Debye (1911). 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.
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 xs (for a fixed real number s) to the energy, then in thermal equilibrium the average energy of that part is kBT/s. There is a simple application of this extension to the sedimentation of particles under gravity. For example, the haze sometimes seen in beer can be caused by clumps of proteins that scatter light. 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 g r a v = m b g z {\displaystyle H^{\mathrm {grav} }=m_{\text{b}}gz}
where z is the height of the protein clump in the bottle and g is the acceleration due to 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
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 NekB to the heat capacity by the equipartition theorem, where Ne 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 xn, the following equipartition formula holds in thermal equilibrium for all indices m and n:
⟨ x m ∂ H ∂ x n ⟩ = δ m n k B T . {\displaystyle \left\langle x_{m}{\frac {\partial H}{\partial x_{n}}}\right\rangle =\delta _{mn}k_{\text{B}}T.}
Here δmn is the Kronecker delta, which is equal to one if m = n and is zero otherwise. The averaging brackets ⟨ … ⟩ {\displaystyle \left\langle \ldots \right\rangle } 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:
⟨ x n ∂ H ∂ x n ⟩ = k B T for all n {\displaystyle \left\langle x_{n}{\frac {\partial H}{\partial x_{n}}}\right\rangle =k_{\text{B}}T\quad {\text{for all }}n}
⟨ x m ∂ H ∂ x n ⟩ = 0 for all m ≠ n . {\displaystyle \left\langle x_{m}{\frac {\partial H}{\partial x_{n}}}\right\rangle =0\quad {\text{for all }}m\neq n.}
If a degree of freedom xn appears only as a quadratic term anxn2 in the Hamiltonian H, then the first of these formulae implies that
k B T = ⟨ x n ∂ H ∂ x n ⟩ = 2 ⟨ a n x n 2 ⟩ , {\displaystyle k_{\text{B}}T=\left\langle x_{n}{\frac {\partial H}{\partial x_{n}}}\right\rangle =2\left\langle a_{n}x_{n}^{2}\right\rangle ,}
which is twice the contribution that this degree of freedom makes to the average energy ⟨ H ⟩ {\displaystyle \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 anxns. The degrees of freedom xn are coordinates on the phase space of the system and are therefore commonly subdivided into generalized position coordinates qk and generalized momentum coordinates pk, where pk is the conjugate momentum to qk. In this situation, formula 1 means that for all k,
⟨ p k ∂ H ∂ p k ⟩ = ⟨ q k ∂ H ∂ q k ⟩ = k B T . {\displaystyle \left\langle p_{k}{\frac {\partial H}{\partial p_{k}}}\right\rangle =\left\langle q_{k}{\frac {\partial H}{\partial q_{k}}}\right\rangle =k_{\text{B}}T.}
Using the equations of Hamiltonian mechanics, these formulae may also be written
⟨ p k d q k d t ⟩ = − ⟨ q k d p k d t ⟩ = k B T . {\displaystyle \left\langle p_{k}{\frac {dq_{k}}{dt}}\right\rangle =-\left\langle q_{k}{\frac {dp_{k}}{dt}}\right\rangle =k_{\text{B}}T.}
Similarly, one can show using formula 2 that
⟨ p j ∂ H ∂ p k ⟩ = ⟨ q j ∂ H ∂ q k ⟩ = 0 for all j ≠ k . {\displaystyle \left\langle p_{j}{\frac {\partial H}{\partial p_{k}}}\right\rangle =\left\langle q_{j}{\frac {\partial H}{\partial q_{k}}}\right\rangle =0\quad {\text{ for all }}\,j\neq k.}
and
⟨ p j ∂ q k ∂ t ⟩ = − ⟨ q j ∂ p k ∂ t ⟩ = 0 for all j ≠ k . {\displaystyle \left\langle p_{j}{\frac {\partial q_{k}}{\partial t}}\right\rangle =-\left\langle q_{j}{\frac {\partial p_{k}}{\partial t}}\right\rangle =0\quad {\text{ for all }}\,j\neq k.}
Relation to the virial theorem
The general equipartition theorem is an extension of the virial theorem (proposed in 1870), which states that
⟨ ∑ k q k ∂ H ∂ q k ⟩ = ⟨ ∑ k p k ∂ H ∂ p k ⟩ = ⟨ ∑ k p k d q k d t ⟩ = − ⟨ ∑ k q k d p k d t ⟩ , {\displaystyle \left\langle \sum _{k}q_{k}{\frac {\partial H}{\partial q_{k}}}\right\rangle =\left\langle \sum _{k}p_{k}{\frac {\partial H}{\partial p_{k}}}\right\rangle =\left\langle \sum _{k}p_{k}{\frac {dq_{k}}{dt}}\right\rangle =-\left\langle \sum _{k}q_{k}{\frac {dp_{k}}{dt}}\right\rangle ,}
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
⟨ H k i n ⟩ = 1 2 m ⟨ p x 2 + p y 2 + p z 2 ⟩ = 1 2 ( ⟨ p x ∂ H k i n ∂ p x ⟩ + ⟨ p y ∂ H k i n ∂ p y ⟩ + ⟨ p z ∂ H k i n ∂ p z ⟩ ) = 3 2 k B T {\displaystyle {\begin{aligned}\langle H^{\mathrm {kin} }\rangle &={\frac {1}{2m}}\la