Equipartition theorem

In classical statistical mechanics, the equipartition theorem (also known as the law of equipartition) expresses the idea that, at thermal equilibrium, energy tends to be distributed equally among all of its possible forms. In principle, the equipartition theorem pertains to every type of physical system, ranging from organ pipes to electrical networks to stars. However, it holds only when energy can be freely exchanged among its forms, i.e., when ergodicity is satisfied.

The equipartition theorem is generally used to determine the average kinetic and potential energies of a system, even when the system is too complicated to be solved exactly. For example, it allows the ideal gas law to be derived from classical mechanics, as well as the universal specific heat of solids at high temperatures (the Dulong-Petit law). The equipartition theorem holds even when when relativistic effects are considered; for example, it may be used to derive the Chandrasekhar limit on the mass of a white dwarf star.

As an important special case, each quadratic form of energy receives an average thermal energy of ½k_BT, where k_B is the Boltzmann constant and T is the temperature in Kelvin. As a consequence, each degree of freedom that appears quadratically in the energy contributes ½k_B to the system's heat capacity. For example, the average kinetic energy of a particle in a system — which depends quadratically on the particle momentum — at thermal equilibrium is (3/2) k_BT, no matter what its mass is or what forces within the system act on it. The factor of (3/2) corresponds to the three independent forms of energy, namely, the motions along the x, y and z directions; knowing this energy and the mass of the particle allows the root mean square speed of the particle to be computed. Similarly, the average potential energy of a spring — which depends quadratically on the the spring's extension — in thermal equilibrium is ½k_BT, no matter what its spring constant is; knowing this and the spring constant allows the mean deviation from equilibrium to be computed.

The equipartition theorem is a classical result, and is not strictly true when quantum physics is considered; equipartition is valid only in the classical limit of large quantum numbers. Indeed, the failure of the equipartition theorem for electromagnetic radiation — the so-called ultraviolet catastrophe — led Albert Einstein to suggest that light was quantized into photons, a pivotal hypothesis that spurred the development of quantum mechanics and quantum field theory. Likewise, its failure to account for the specific heats of solids and diatomic gases at low temperatures also promoted the acceptance of quantum physics, which did account for them.

History

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

The history of the equipartition theorem is intertwined with that of specific heats, 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 heats of solids at room temperature were almost all identical, roughly 6 cal/(mole·K). Their law was used for many years as a technique of measuring atomic weights. However, subsequent studies by James Dewar and Heinrich Friedrich Weber shows that this Dulong-Petit law holds only at high temperatures; at lower temperatures, or for exceptionally stiff solids such as diamond, the specific heat was lower.

Experimental observations of the specific heat of gases also raised concerns about the validity of the equipartition theorem. As described below, the theorem predicts that the specific heat of simple monatomic gases should be roughly 3 cal/(mole·K), whereas that of diatomic gases should be roughly 7 cal/(mole·K). Experiments confirmed the former prediction, but found that the latter was instead 5 cal/(mole·K), which falls to 3 cal/(mole·K) at very low temperatures. More generally, molecules were believed to be composed of parts (atoms) already in the 19th century, and should have much higher specific heats than observed, as noted first by Maxwell in 1875.

The failure of the equipartition theorem to account for the specific heats of solids and gases was addressed in several ways. 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. Finally, Lord Rayleigh defended both the derivation and the experimental assumption of thermal equilibrium, and 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 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. Einstein used the failure of the equipartition theorem to argue for the need of a new quantum theory of matter, years ahead of other physicists. A refinement of Einstein's theory by Peter Debye is still the principal theory of specific heats in solids at all temperatures, and led to the prediction of phonons.

Mathematical formulation

Quantitatively, the generalized equipartition theorem states that

$\langle q_{k}{\frac {\partial H}{\partial q_{k}}}\rangle =\langle p_{k}{\frac {\partial H}{\partial p_{k}}}\rangle =k_{B}T=\langle p_{k}{\frac {dq_{k}}{dt}}\rangle =-\langle q_{k}{\frac {dp_{k}}{dt}}\rangle$

where H is the Hamiltonian energy function of a physical system, q_k and p_k are its k^th generalized coordinate and generalized momentum, respectively, and the latter two equations follow from Hamiltonian mechanics. The angular brackets $\left\langle \ldots \right\rangle$ symbolize taking the average of the enclosed quantity in thermal equilibrium, either the ensemble average over phase space (the usual meaning) or the time average of a single system over a long time. The generalized 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.

In many cases, H varies quadratically in x_k, e.g., as a kinetic energy term h_k = p_k²/2m or as a potential energy h_k = C q_k². In such cases, the average energy in that degree of freedom is ½k_BT. Expressed another way, each quadratic degree of freedom contributes ½k_B to the system's heat capacity.

The equipartition theorem also states that the corresponding averages for different variables are always zero

$\langle q_{m}{\frac {\partial H}{\partial q_{n}}}\rangle =\langle q_{m}{\frac {\partial H}{\partial p_{n}}}\rangle =\langle p_{m}{\frac {\partial H}{\partial p_{n}}}\rangle =\langle p_{m}{\frac {\partial H}{\partial q_{n}}}\rangle =0$

when m≠n, as is the average for any generalized coordinate q_k and its conjugate momentum p_k

$\langle q_{k}{\frac {\partial H}{\partial p_{k}}}\rangle =\langle p_{k}{\frac {\partial H}{\partial q_{k}}}\rangle =0$

These various results can be summarized in the formula

$\langle x_{m}{\frac {\partial H}{\partial x_{n}}}\rangle =\delta _{mn}k_{B}T$

where δ_mn is the Kronecker delta and x_m and x_n are any two variables in phase space, either generalized coordinates or generalized momenta.

Requirement of ergodicity

The equipartition holds only if energy can be freely exchanged among its various forms in the system, or with an external heat bath in the canonical ensemble. This is equivalent to requiring that the system be in thermal equilibrium, or that the system is ergodic.

A commonly cited counter-example is that of a harmonic system with multiple normal modes. If the oscillator is isolated from the rest of the world, the energy in each mode is fixed and cannot be exchanged; hence, the equipartition theorem does not hold for such a system. These energies usually begin exchanging if sufficiently strong nonlinear terms are present in the energy function. However, the Kolmogorov–Arnold–Moser theorem states that energy will not be exchanged if the nonlinear perturbations are too small. Studies of the requirements for isolated systems to ensure ergodicity — and, thus, that the equipartition theorem holds — led to the modern chaos theory.

Relation to the virial theorem

The equipartition theorem is an extension of the virial theorem (proposed in 1870), which relates the averages of the sums

$\langle \sum _{k}q_{k}{\frac {\partial H}{\partial q_{k}}}\rangle =\langle \sum _{k}p_{k}{\frac {\partial H}{\partial p_{k}}}\rangle =\langle \sum _{k}p_{k}{\frac {dq_{k}}{dt}}\rangle =-\langle \sum _{k}q_{k}{\frac {dp_{k}}{dt}}\rangle$

where t represent time and the summation is over all the degrees of freedom k. Two key differences are that the virial theorem relates the summed averages to each other, rather than individual averages, and also 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. However, these two types of averages should yield the same result, assuming ergodicity, and both have been used to estimate the total internal energy of complex physical systems.

Application to simple harmonic oscillators

The simplest application of the equipartition theorem is to a simple harmonic oscillator. A typical oscillator is a particle of mass m attached to a spring of stiffness k, for which the energy is the sum of the kinetic and potential energies

$H=H^{kin}+H^{pot}={\frac {p^{2}}{2m}}+{\frac {1}{2}}kq^{2}$

where q is the extension of the spring from equilibrium, and p is the momentum of the particle. The energy function H is quadratic in both q and p, indicating that they each contribute ½k_BT to the total average energy. Since

$\langle p{\frac {\partial H}{\partial p}}\rangle =\langle p\cdot {\frac {p}{m}}\rangle =\langle {\frac {p^{2}}{m}}\rangle =2\langle H^{kin}\rangle =k_{B}T$

and

$\langle q{\frac {\partial H}{\partial q}}\rangle =\langle q\cdot kq\rangle =\langle kq^{2}\rangle ==2\langle H^{pot}\rangle =k_{B}T$

the total average energy in this system is k_BT

$\langle H\rangle =\langle H^{kin}\rangle +\langle H^{pot}\rangle ={\frac {1}{2}}k_{B}T+{\frac {1}{2}}k_{B}T=k_{B}T$

This result is valid for any type of harmonic oscillator, such as a pendulum, a vibrating molecule or a passive electronic oscillator; for instance, the equipartition theorem can be used to derive the formula for Johnson-Nyquist noise.

The equipartition theorem also holds for sets of independent oscillators, such as the normal modes of coupled oscillators, e.g., the modes of a piano string, the resonances of an organ pipe, or the lattice vibrations in a solid. If there are N independent oscillators, then their average total energy is Nk_BT, and their specific heat at constant volume is Nk_B. In particular, a mole of such oscillators would have a specific heat of N_Ak_B=R where N_A is Avogadro's number and R is the universal gas constant, roughly 2 cal/(mole·K). This is the explanation for the Dulong-Petit law of molar specific heats of solids; each mole of atoms in the lattice contributes 3R≈6 cal/(mole·K) to the specific heat, since each atom can oscillate in three independent directions.

Application to anharmonic oscillators

It is instructive to calculate the equipartition of energy for an anharmonic oscillator, one in which the potential energy is not quadratic in the extension q. For example, if the energy function has the form

$H==H^{kin}+H^{pot}={\frac {p^{2}}{2m}}+Cq^{m}$

where C and m are arbitrary constants, the equipartition theorem predicts that

$\langle q{\frac {\partial H}{\partial q}}\rangle =\langle q\cdot mCq^{m-1}\rangle =\langle mCq^{m}\rangle =m\langle H^{pot}\rangle =k_{B}T$

More generally, if the energy function is expanded in a Taylor series in its extension q

$H={\frac {p^{2}}{2m}}+\sum _{m=2}^{\infty }C_{m}q^{m}$

the equipartition theorem predicts that

$\langle q{\frac {\partial H}{\partial q}}\rangle =\sum _{m=2}^{\infty }\langle q\cdot mC_{m}q^{m-1}\rangle =\sum _{m=2}^{\infty }mC_{m}\langle q^{m}\rangle =k_{B}T$

Contrary to the other examples cited here, this latter equipartition result does not allow the average potential energy to be written in simple form

$\langle H^{pot}\rangle ={\frac {1}{2}}k_{B}T-\sum _{m=3}^{\infty }\left({\frac {m-2}{2}}\right)C_{m}\langle q^{m}\rangle$

Application to non-relativistic ideal gases

Ideal gases provide another good illustration of the equipartition theorem. The classical kinetic energy of a single particle of mass m is given by

$H^{kin}={\frac {1}{2m}}\left|\mathbf {p} \right|^{2}={\frac {1}{2m}}\left(p_{x}^{2}+p_{y}^{2}+p_{z}^{2}\right)$

where (p_x, p_y, p_z) are the Cartesian components of the momentum p of the particle. For the p_x component, the equipartition theorem equals

$\langle p_{x}{\frac {\partial H^{kin}}{\partial p_{x}}}\rangle =\langle p_{x}{\frac {p_{x}}{m}}\rangle =k_{B}T$

and similarly for the p_y and p_z components. Adding them together and dividing by two gives the average kinetic energy of a particle in three dimensions

${\frac {1}{2}}\left[\langle p_{x}{\frac {\partial H^{kin}}{\partial p_{x}}}\rangle +\langle p_{y}{\frac {\partial H^{kin}}{\partial p_{y}}}\rangle +\langle p_{z}{\frac {\partial H^{kin}}{\partial p_{z}}}\rangle \right]={\frac {1}{2m}}\left[p_{x}^{2}+p_{y}^{2}+p_{z}^{2}\right]=\langle H^{kin}\rangle ={\frac {3}{2}}k_{B}T$

In an ideal gas, there is no potential energy; by assumption, the particles have no internal degrees of freedom and move independently of one another. Therefore, the total energy consists only of their kinetic energies; quantitatively, the average total energy U of an ideal gas of N particles is 3/2 N k_BT and its specific heat is 3/2 N k_B. Thus, a mole of a monatomic gas should have a specific heat at constant volume of (3/2)N_Ak_B=(3/2)R where again N_A is Avogadro's number and R is the universal gas constant, roughly 2 cal/(mole·K). Therefore, the predicted molar specific heat should be roughly 3 cal/(mole·K), which was confirmed by experiment. From the mean kinetic energy, the root mean square speed of the molecules can be calculated.

The equipartition theorem also allows us to derive the ideal gas law. Starting from the equation

$\langle q_{x}{\frac {\partial H^{kin}}{\partial q_{x}}}\rangle =-\langle q_{x}{\frac {dp_{x}}{dt}}\rangle =k_{B}T$

and the counterparts for the q_y and q_z components, we obtain the formula

$-\langle q_{x}{\frac {dp_{x}}{dt}}\rangle -\langle q_{y}{\frac {dp_{y}}{dt}}\rangle -\langle q_{z}{\frac {dp_{z}}{dt}}\rangle =-\langle \mathbf {q} \cdot \mathbf {F} \rangle =3k_{B}T$

where q is the position vector of the particle and F is the net force on that particle. Summing over the N particles yields

$-\langle \sum _{k=1}^{N}\mathbf {q} _{k}\cdot \mathbf {F} _{k}\rangle =3Nk_{B}T$

Since the particles don't interact, the only force on them is the inwards pressure P applied by the walls of their container. Therefore, quantity to be averaged is

$\sum _{k=1}^{N}\mathbf {q} _{k}\cdot \mathbf {F} _{k}=-P\oint _{\mathrm {surface} }\mathbf {q} _{k}\cdot \mathbf {dS}$

where dS is an infinitesimal area on the walls of the container. Using the divergence theorem, this integral can be shown to be constant

$\sum _{k=1}^{N}\mathbf {q} _{k}\cdot \mathbf {F} _{k}=-P\int _{\mathrm {volume} }\left({\boldsymbol {\nabla }}\cdot \mathbf {q} _{k}\right)dV=-3PV$

where dV is an infinitesimal volume within the container and V is the total volume of the container. This follows because the divergence of the position vector q in three dimensions is three

${\boldsymbol {\nabla }}\cdot \mathbf {q} _{k}={\frac {\partial q_{x}}{\partial q_{x}}}+{\frac {\partial q_{y}}{\partial q_{y}}}+{\frac {\partial q_{z}}{\partial q_{z}}}=3$

Putting this together

$-\langle \sum _{k=1}^{N}\mathbf {q} _{k}\cdot \mathbf {F} _{k}\rangle =3PV=3Nk_{B}T$

yields the ideal gas law for N particles

$PV=Nk_{B}T$

Application to extreme relativistic ideal gases

An instructive counterpoint to this derivation is an extreme relativistic ideal gas, such as occur in white dwarf and neutron stars. In such cases, the kinetic energy of a single particle is given by a different formula

$H=pc=c{\sqrt {p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}}$

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

$\langle p_{x}{\frac {\partial H^{kin}}{\partial p_{x}}}\rangle =\langle c{\frac {p_{x}^{2}}{\sqrt {p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}}}\rangle =k_{B}T$

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

$\langle p_{x}{\frac {\partial H^{kin}}{\partial p_{x}}}\rangle +\langle p_{y}{\frac {\partial H^{kin}}{\partial p_{y}}}\rangle +\langle p_{z}{\frac {\partial H^{kin}}{\partial p_{z}}}\rangle =\langle c{\frac {p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}{\sqrt {p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}}}\rangle =\langle H\rangle =3k_{B}T$

Thus, the average energy in the extreme relativistic case is twice that of the non-relativistic case; the average total energy U of an extreme relativistic ideal gas of N particles is 3 N k_BT.

Application to non-ideal gases

The equipartition theorem may also be used to derive the energy and pressure of non-ideal gases, which are gases in which the particles interact with one another. To illustrate this, we assume for simplicity that the gas particles are spherically symmetric (isotropic) and that they interact only by a pair potential V(r), where r is the distance between the two particles. It is customary to introduce a radial distribution function g(r) such that the probability density of finding another particle at a distance r is given by 4πr²ρ g(r), where ρ=N/V is the mean density of the gas. In this case, the mean potential energy for one particle equals the integral

$\langle h^{pot}\rangle ={\frac {1}{2}}\int _{0}^{\infty }dr\ 4\pi r^{2}\rho g(r)V(r)$

where the factor of two is introduced to avoid double counting the interactions; in other words, each particle gets half of its interaction with another particle. Therefore, the total energy for N particles equals

$U=\langle H^{kin}\rangle +N\langle h^{pot}\rangle ={\frac {3}{2}}Nk_{B}T+2\pi N\rho \int _{0}^{\infty }dr\ r^{2}g(r)V(r):$

By an analogous argument, it is possible to derive the non-ideal pressure equation

$3Nk_{B}T=3PV+2\pi N\rho \int _{0}^{\infty }dr\ r^{2}g(r)r{\frac {dV}{dr}}$

Application to molecular tumbling

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. Such rotational diffusion can be observed by several other biophysical probes such as fluorescence anisotropy, flow birefringence and dielectric spectroscopy.

The equipartition theorem allows the mean angular speeds of the molecule to be calculated. The rotational energy of the molecule is given by

$H^{rot}={\frac {1}{2}}\left[I_{1}\omega _{1}^{2}+I_{2}\omega _{2}^{2}+I_{3}\omega _{3}^{2}\right]$

where I_1-3 and ω_1-3 are the moments of inertia and angular velocities about the molecule's three principal axes, respectively. This energy can also be written in terms of the corresponding angular momenta L_1-3

$H^{rot}={\frac {L_{1}^{2}}{2I_{1}}}+{\frac {L_{2}^{2}}{2I_{2}}}+{\frac {L_{3}^{2}}{2I_{3}}}$

where L_k = I_kω_k. Application of the equipartition theorem yields the relation

$\langle L_{1}{\frac {\partial H^{rot}}{\partial L_{1}}}\rangle =\langle L_{1}{\frac {L_{1}}{I_{1}}}\rangle =k_{B}T$

and similarly for L₂ and L₃. Thus, each rotational degree of freedom possesses a mean rotational kinetic energy of ½ k_BT and contributes ½ k_B to the specific heat, from which the mean rotational speeds about each principal axis can be calculated. As another consequence, the mean rotational energy of the molecule equals the mean translational energy of the molecule

$\langle H^{rot}\rangle ={\frac {3}{2}}k_{B}T$

Application to 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 is governed by Newton's second law

${\frac {d\mathbf {v} }{dt}}={\frac {1}{m}}\mathbf {F} _{tot}=-{\frac {\mathbf {v} }{\tau }}+{\frac {1}{m}}\mathbf {F} _{rnd}$

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.

Taking the dot product of this equation with the position vector r and averaging yields the basic equation for Brownian motion

$\langle \mathbf {r} \cdot {\frac {d\mathbf {v} }{dt}}\rangle +{\frac {1}{\tau }}\langle \mathbf {r} \cdot \mathbf {v} \rangle =0$

since the random force F_rnd is uncorrelated with the position r. Using the mathematical identities

${\frac {d}{dt}}\left(\mathbf {r} \cdot \mathbf {r} \right)={\frac {d}{dt}}\left(r^{2}\right)=2\left(\mathbf {r} \cdot \mathbf {v} \right)$

and

${\frac {d}{dt}}\left(\mathbf {r} \cdot \mathbf {v} \right)=v^{2}+\mathbf {r} \cdot {\frac {d\mathbf {v} }{dt}}$

the basic equation for Brownian motion can be transformed into

${\frac {d^{2}}{dt^{2}}}\langle r^{2}\rangle +{\frac {1}{\tau }}{\frac {d}{dt}}\langle r^{2}\rangle =2\langle v^{2}\rangle ={\frac {6}{m}}k_{B}T$

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

$\langle H^{kin}\rangle =\langle {\frac {p^{2}}{2m}}\rangle ==\langle {\frac {1}{2}}mv^{2}\rangle ={\frac {3}{2}}k_{B}T$

This equation may be solved exactly

$\langle r^{2}\rangle ={\frac {6k_{B}T\tau ^{2}}{m}}\left[{\frac {t}{\tau }}-1+e^{-t/\tau }\right]$

For small times, t << τ, the particle acts as a freely moving particle; the squared distance grows quadratically

$\langle r^{2}\rangle \approx {\frac {3k_{B}T}{m}}t^{2}=\langle v^{2}\rangle t^{2}$

However, for long times, t >> τ, the squared distance grows only linearly with the time

$\langle r^{2}\rangle \approx {\frac {6k_{B}T\tau }{m}}t=6\gamma k_{B}Tt$

which 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.

Application to diatomic gases

A diatomic gas can be modelled as two masses, m₁ and m₂, joined by a spring of stiffness k. The classical energy of this system is

$H={\frac {\left|\mathbf {p} _{1}\right|^{2}}{2m_{1}}}+{\frac {\left|\mathbf {p} _{2}\right|^{2}}{2m_{2}}}+{\frac {1}{2}}kq^{2}$

where q is the deviation of the inter-atomic separation from its equilibrium value. Each degree of freedom in the Hamiltonian energy function is quadratic and, thus, should contribute ½k_BT to the total average energy, and ½k_B. Therefore, the specific heat of a gas of N diatomic molecules is predicted to be 7N · ½k_B; for each molecule, p₁ and p₂ each contribute three degrees of freedom, whereas q contributes the seventh. Therefore, the specific heat of a mole of diatomic molecules with no other degrees of freedom should be (7/2)N_Ak_B=(7/2)R where again N_A is Avogadro's number and R is the universal gas constant, roughly 2 cal/(mole·K). Therefore, the predicted molar specific heat should be roughly 7 cal/(mole·K); however, the measured value is roughly 5 cal/(mole·K) and falls to 3 cal/(mole·K) at very low temperatures. This disagreement cannot be explained by using a more complex model of the molecule, since adding more degrees of freedom can only increase the classical specific heat, not decrease it. This discrepancy was a key piece of evidence showing the need for a quantum theory of matter.

Failure in the quantum regime

The equipartition theorem breaks down when the thermal energy k_BT is significantly smaller than the spacing between energy levels. In such cases, it is a poor approximation to assume that the energy levels form a smooth continuum, which is done in the derivations of the equipartition theorem below. Historically, the failures of the classical equipartition theorem to explain specific heats and blackbody radiation were critical in showing the need for a new theory of matter and radiation, namely, quantum mechanics and quantum field theory.

To illustrate this breakdown, consider the average energy in a single (quantum) harmonic oscillator. Its energy levels are given by E_n = nħω, where ħ = h/2π is Planck's constant , ω 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

$P(E_{n})={\frac {e^{-\beta n\hbar \omega }}{Z}}$

where β=1/k_BT and the denominator Z is the partition function, here a geometric series

$Z=\sum _{n=0}^{\infty }e^{-\beta n\hbar \omega }={\frac {1}{1-e^{-\beta \hbar \omega }}}$

Its average energy is given by

$\langle H\rangle =\sum _{n=0}^{\infty }E_{n}P(E_{n})={\frac {1}{Z}}\sum _{n=0}^{\infty }n\hbar \omega e^{-\beta n\hbar \omega }={\frac {-1}{Z}}{\frac {\partial Z}{\partial \beta }}=-{\frac {\partial \log Z}{\partial \beta }}$

Substitution of the formula for Z gives the final result

$\langle H\rangle =\hbar \omega {\frac {e^{-\beta \hbar \omega }}{1-e^{-\beta \hbar \omega }}}$

At high temperatures, when the thermal energy k_BT is much greater than the spacing ħω between energy levels, the exponential argument βħω is much less than one and the average energy becomes k_BT, in agreement with the equipartition theorem. However, at low temperatures, when βħω>>1, the average energy goes to zero — the higher excited energy levels are "frozen out". For 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. For example, this reasoning was used by Albert Einstein to resolve the ultraviolet catastrophe of blackbody 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-ω modes goes to zero as ω 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.

Symmetry in a quantum system introduces another, more subtle correction to the equipartition theorem. Excited rotational states are impossible for systems with a continuous symmetry, such as the rotation of a diatomic gas about the axis connecting the centers of the atoms. Since such states do not exist, they are effectively frozen out; hence, the system has fewer effective degrees of freedom. Thus, the diatomic molecule has a molar specific heat of (5/2)R, instead of (7/2)R; the vibrational degree of freedom is frozen out at room temperature, and one rotational degree of freedom is frozen out because of symmetry. At even lower temperatures, the two remaining rotational modes are frozen out, giving a molar specific heat of (3/2)R, corresponding to the three degrees of translational freedom.

Derivations

Derivations of the equipartition theorem involve averages over phase space, which is the set of generalized coordinates q and their conjugate momenta p needed to completely specify the state of the system. The symbol dΓ represents an infinitesimal volume of phase space

$d\Gamma =\prod _{i}dq_{i}dp_{i}$

We use the symbol Γ to represent the volume of the phase space where the energy H lies between two limits, E and E+ΔE

$\Gamma (E,\Delta E)=\int _{H\in \left[E,E+\Delta E\right]}d\Gamma$

It is generally assumed that ΔE is very small, ΔE<<E. Similarly, Σ represents the total volume of phase space where the energy is less than E.

$\Sigma (E)=\int _{H<E}d\Gamma$

Since ΔE is very small, the following integrations are equivalent

$\int _{H\in \left[E,E+\Delta E\right]}d\Gamma \ldots =\Delta E{\frac {\partial }{\partial E}}\int _{H<E}d\Gamma \ldots$

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

$\Gamma =\Delta E\ {\frac {\partial \Sigma }{\partial E}}=\Delta E\ \rho (E)$

where ρ(E) is the density of states. By the usual definitions of statistical mechanics, the entropy S equals k_B log Σ(E), and the temperature T is defined by

${\frac {1}{T}}={\frac {\partial S}{\partial E}}=k_{b}{\frac {\partial \log \Sigma }{\partial E}}=k_{b}{\frac {1}{\Sigma }}\left({\frac {\partial \Sigma }{\partial E}}\right)$

Derivations of the equipartition theorem can be found in many statistical mechanics textbooks, both for the microcanonical ensemble and for the canonical ensemble.

Derivation for the canonical ensemble

In the canonical ensemble, the system is in thermal equilibrium with an infinite heat bath at temperature T (in Kelvin), and the probability of each state in phase space is given by its Boltzmann factor times a normalization factor ${\mathcal {N}}$. The probabilities should sum to one

${\mathcal {N}}\int d\Gamma e^{-\beta H(p,q)}=1$

where b = 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

${\mathcal {N}}\int d\Gamma _{k}\left[e^{-\beta H(p,q)}x_{k}\right]_{x_{k}=a}^{x_{k}=b}+{\mathcal {N}}\int d\Gamma e^{-\beta H(p,q)}x_{k}\beta {\frac {\partial H}{\partial x_{k}}}=1$

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

${\mathcal {N}}\int d\Gamma e^{-\beta H(p,q)}x_{k}{\frac {\partial H}{\partial x_{k}}}=\langle x_{k}{\frac {\partial H}{\partial x_{k}}}\rangle ={\frac {1}{\beta }}=k_{B}T$

Here, the averaging symbolized by $\langle \ldots \rangle$ is the ensemble average taken over the canonical ensemble.

Derivation for the microcanonical ensemble

In the microcanonical ensemble, the system is effectively isolated from the rest of the world. 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

$\langle x_{m}{\frac {\partial H}{\partial x_{n}}}\rangle =\left({\frac {1}{\Gamma }}\right)\ \int _{H\in \left[E,E+\Delta E\right]}d\Gamma \ x_{m}{\frac {\partial H}{\partial x_{n}}}$

$=\left({\frac {\Delta E}{\Gamma }}\right)\ {\frac {\partial }{\partial E}}\int _{H<E}d\Gamma \ x_{m}{\frac {\partial H}{\partial x_{n}}}=\left({\frac {1}{\rho }}\right)\ {\frac {\partial }{\partial E}}\int _{H<E}d\Gamma \ x_{m}{\frac {\partial \left(H-E\right)}{\partial x_{n}}}$

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

$\int _{H<E}d\Gamma \ x_{m}{\frac {\partial \left(H-E\right)}{\partial x_{n}}}=\int _{H<E}d\Gamma \ {\frac {\partial }{\partial x_{n}}}\left[x_{m}\left(H-E\right)\right]-\int _{H<E}d\Gamma \ \delta _{mn}\left(H-E\right)$

The first term on the right-hand side is zero, since it can be re-cast an integral of (H - E) on the hypersurface where H = E.

$\int _{H<E}d\Gamma \ x_{m}{\frac {\partial \left(H-E\right)}{\partial x_{n}}}=\delta _{mn}\int _{H<E}d\Gamma \left(E-H\right)$

Substitution of this result in the above equation

$\langle x_{m}{\frac {\partial H}{\partial x_{n}}}\rangle =\delta _{mn}\left({\frac {1}{\rho }}\right)\ {\frac {\partial }{\partial E}}\int _{H<E}d\Gamma \left(E-H\right)=\delta _{mn}\left({\frac {1}{\rho }}\right)\ \int _{H<E}d\Gamma =\delta _{mn}\left({\frac {1}{\rho }}\right)\ \Sigma$

yields the equipartition theorem

$\langle x_{m}{\frac {\partial H}{\partial x_{n}}}\rangle =\delta _{mn}\left({\frac {\partial E}{\partial \Sigma }}\right)\ \Sigma =\delta _{mn}{\frac {\partial E}{\partial \log \Sigma }}=\delta _{mn}k_{B}T$

See also

• Degrees of freedom (physics and chemistry)
• Virial theorem
• Kinetic theory
• Heat capacity
• Statistical mechanics
• Quantum statistical mechanics
• Ultraviolet catastrophe

Further reading

• Pathria, RK (1972). Statistical Mechanics. Pergamon Press. pp. pp. 43–48, 73–74. ISBN 0-08-016747-0. {{cite book}}: |pages= has extra text (help)

• Huang, K (1987). Statistical Mechanics (2nd ed. ed.). John Wiley and Sons. pp. pp. 136–138. ISBN 0-471-81518-7. {{cite book}}: |edition= has extra text (help); |pages= has extra text (help)

• Pauli, W (1973). Pauli Lectures on Physics: Volume 4. Statistical Mechanics. MIT Press. pp. 27–40. ISBN 0-262-16049-8.

• Khinchin, AI (1949). Mathematical Foundations of Statistical Mechanics (G. Gamow, translator). New York: Dover Publications. pp. pp. 93–98. ISBN 0-486-63896-0. {{cite book}}: |pages= has extra text (help)

• Mohling, F (1982). Statistical Mechanics: Methods and Applications. John Wiley and Sons. pp. pp. 137–139, 270–273, 280, 285–292. ISBN 0-470-27340-2. {{cite book}}: |pages= has extra text (help)

• Landau, LD (1980). Statistical Physics, Part 1 (3rd ed. ed.). Pergamon Press. pp. pp. 129–132. ISBN 0-08-023039-3. {{cite book}}: |edition= has extra text (help); |pages= has extra text (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)

• Mandl, F (1971). Statistical Physics. John Wiley and Sons. pp. 213–219. ISBN 0-471-56658-6.

• Tolman, RC (1938). The Principles of Statistical Mechanics. New York: Dover Publications. pp. pp. 93–98. ISBN 0-486-63896-0. {{cite book}}: |pages= has extra text (help)

• Tolman, RC (1927). Statistical Mechanics, with Applications to Physics and Chemistry. Chemical Catalog Company. pp. pp. 72–81. {{cite book}}: |pages= has extra text (help)

External links

• Amazingly cool applet demonstrating equipartition in real time for a mixture of monatomic and diatomic gases