Classical Statistics and Equipartition

A macroscopic system is assembled from a number of atoms of order $10^{22}$. Classically, its future is fixed by the equations of motion for every constituent given the state of motion at one instant, but solving $10^{22}$ coupled equations is not a program anyone can run. The measurable properties of bulk matter can nonetheless be predicted, because conservation of energy and momentum applied to a large ensemble fixes the probable behavior of the system, and the probable behavior is what a measurement reports.¹ This is statistical mechanics: the observable properties of a system are averages over a probability distribution of microscopic states, and the distribution, not the trajectories, is the object of study.

The central task is to find how a fixed amount of energy distributes itself among the particles of a system in thermal equilibrium. The particles exchange energy through collisions, so any one particle's energy fluctuates above and below the mean over time. Classical statistical mechanics asserts that the values the energy takes on follow one specific probability law, the Boltzmann distribution, and that from it the properties of the whole system follow.

The Boltzmann distribution

Classically the system is a large ensemble of identical but distinguishable particles: all alike, yet in principle trackable through a collision, like billiard balls with numbers painted on them. Boltzmann derived the law giving the probable number of particles occupying each available energy state in such an ensemble at equilibrium.

The Boltzmann constant is the gas constant per molecule,

so that at room temperature $kT\approx 0.026\mathrm{eV}$, a number worth memorizing: it is the energy scale that separates easily excited from frozen out. A state costing much more than $kT$ above the ground state is almost never occupied; a state within $kT$ is populated freely.

A single state is rarely the object of interest. The number of distinct states at energy $E$ — the statistical weight or degeneracy $g(E)$ — multiplies the occupation probability, so the number of particles with energy $E$ is

When the spectrum is dense, $E$ is treated as continuous and $g(E)$ becomes the density of states, defined so that $g(E)\mathbf{d}E$ counts the states with energy between $E$ and $E+\mathbf{d}E$. The exponential decay of $f_B$ with energy is the single most important feature: it is drawn below as the falling population of a ladder of levels.

Two examples show the distribution at work.

The law of atmospheres. For an ideal gas in a uniform gravitational field, a molecule of mass $m$ at height $z$ has energy $E=p^2/2m+mgz$. Integrating the Boltzmann factor over the momentum components leaves the height dependence alone,

so the density falls off exponentially with altitude. For air ($M=28.6$) at $T=300\mathrm{K}$, the ratio of densities at $1\mathrm{km}$ and at ground is $e^{-mg(1000)/kT}=e^{-0.113}=0.893$, giving $\rho (1000)=1.154kg/{\mathrm{m}}^3$ from a ground value of $1.292kg/{\mathrm{m}}^3$.³

Populations of atomic levels. The ratio of populations of two levels is the ratio of Boltzmann factors weighted by degeneracies,

For the first excited state of hydrogen, $E_2-E_1=10.2\mathrm{eV}$ with $g_2/g_1=4$. At room temperature $kT\approx 0.026\mathrm{eV}$ and

which is why a jar of hydrogen at room temperature does not glow with the Balmer series. At the Sun's surface, $T=5800\mathrm{K}$ and $kT\approx 0.50\mathrm{eV}$, giving $n_2/n_1\approx 4e^{-20.4}\approx 10^{-8}$: about $10^{15}$ atoms per mole sit in the first excited state at any instant, enough to produce the observed absorption lines.

Temperature and entropy

Temperature acquires a precise microscopic meaning through the number of microstates. A macrostate (fixed total energy, volume, particle number) is realized by many microstates — detailed assignments of energy to individual particles. The number of microstates $W$ consistent with a macrostate is its multiplicity, and Boltzmann's relation defines the entropy as its logarithm.

Absolute temperature is then the rate at which entropy responds to added energy at fixed volume,

A low-entropy macrostate concentrates the energy in a few particles; the high-entropy macrostate spreads it out. There are vastly more ways to spread energy than to concentrate it, so an isolated system drifts toward the spread arrangement, and this drift is the second law.

The Maxwell speed distribution

The Boltzmann distribution applied to the kinetic energy of gas molecules reproduces Maxwell's velocity and speed distributions, derived a few years before Boltzmann's general law. Assuming the three velocity components are independent and the distribution depends only on speed, the normalized velocity distribution is a product of three Gaussians,

Each component is symmetric about zero, so $⟨v_x⟩=0$: on average a gas goes nowhere. The distribution of speeds $v=|\mathbf{v}|$ is obtained by multiplying the density in velocity space by the volume $4\pi v^2\mathbf{d}v$ of the spherical shell of radius $v$,

The $v^2$ factor pulls $n(v)$ to zero at the origin and the exponential pulls it to zero at large $v$, so the distribution peaks at an intermediate speed and is asymmetric, with a long high-speed tail.

Three characteristic speeds summarize the curve. The most probable speed$v_m$ is at the peak, the average speed $⟨v⟩$ is the mean, and the rms speed $v_{\text{rms}}$ is the square root of the mean square, tied to the average kinetic energy. They are computed from the distribution by integration and ordered $v_m<⟨v⟩<v_{\text{rms}}$.

All three scale as $\sqrt{T/m}$: heavier molecules move slower at the same temperature, and any gas speeds up as $\sqrt{T}$. For nitrogen ($m=4.68\times 10^{-26}\mathrm{kg}$) at $300\mathrm{K}$,

about eight percent below $v_{\text{rms}}$.

The distribution of kinetic energy

Changing variables from speed to kinetic energy $E=\frac{1}{2}m{v}^2$ in the Maxwell distribution gives the distribution of molecular kinetic energy,

where the $E^{1/2}$ is the density of states carried over from the $v^2$ shell factor. Multiplying by $E$ and integrating gives the average translational kinetic energy,

independent of the molecular mass. This is the microscopic content of temperature: absolute temperature is a direct measure of the mean translational kinetic energy per molecule.

That $⟨E⟩$ does not depend on mass has a consequence for planetary atmospheres. A gas escapes a planet over $\sim 10^8$ years if its average speed reaches about one-sixth of the escape speed. At $T=300\mathrm{K}$ every molecule has $⟨E⟩=\frac{3}{2}kT=6.21\times 10^{-21}\mathrm{J}$; for hydrogen ($m=3.34\times 10^{-27}\mathrm{kg}$) this gives $v\approx 1.93km/s$, above one-sixth of Earth's $11.2km/s$ escape speed. Hydrogen therefore leaks away, and its absence from the atmosphere puts a lower bound on the age of Earth of $10^8$ years.⁵

Equipartition of energy

The result $⟨E⟩=\frac{3}{2}kT$ splits into $\frac{1}{2}kT$ for each of the three translational velocity components. This is one instance of a general theorem.

The theorem follows because every quadratic term in the energy is a variable integrated against the same Gaussian Boltzmann factor, and each such integral returns $\frac{1}{2}kT$. A monatomic gas molecule has three translational degrees of freedom and $⟨E⟩=\frac{3}{2}kT$. A one-dimensional harmonic oscillator has two — the coordinate $x$ (potential energy $\frac{1}{2}\kappa x^2$) and the velocity $v_x$ (kinetic energy $\frac{1}{2}m{v}_x^2$) — and averages $kT$.

Heat capacities of gases

The molar heat capacity at constant volume, $C_V=(\partial U/\partial T{)}_V$ with $U$ the internal energy of a mole, is the equipartition theorem's most direct experimental test. A rigid diatomic molecule modeled as a dumbbell can translate along three axes and rotate about the two axes perpendicular to the bond, for five degrees of freedom:

Rotation about the bond axis is neglected. The average energy is $\frac{5}{2}kT$ per molecule, so $U=\frac{5}{2}RT$ per mole and $C_V=\frac{5}{2}R$. That $C_V\approx \frac{5}{2}R$ for nitrogen and oxygen led Clausius to infer that these gases are diatomic rotors.

A nonrigid molecule vibrates along the bond, adding a kinetic and a potential term, two more degrees of freedom, predicting $C_V=\frac{7}{2}R$. Yet the measured values for most diatomic gases match $\frac{5}{2}R$ with no vibrational contribution, and equipartition offers no reason why some degrees of freedom should be inactive.

The failure sharpens with temperature. The heat capacity of ${\mathrm{H}}_2$ depends on $T$, contradicting equipartition, which predicts a constant. Below about $60\mathrm{K}$ hydrogen behaves as a monatomic gas with $C_V=\frac{3}{2}R$; between $250$ and $700\mathrm{K}$ it is a rigid rotor with $\frac{5}{2}R$; only near dissociation does it approach $\frac{7}{2}R$. Degrees of freedom switch on one at a time as $T$ rises, in a staircase that classical mechanics cannot explain.

Heat capacities of solids

For solids, equipartition gives the Dulong-Petit law. Modeling each atom as a three-dimensional oscillator bound by springs, the vibrational energy has six quadratic terms — three kinetic, three potential,

so $⟨E⟩=6\times \frac{1}{2}kT=3kT$ and $U=3RT$ per mole.

At high temperature every solid obeys it. But below a material-dependent critical temperature $C_V$ drops toward zero as $T\to 0$, the critical temperature being lower for soft solids such as lead and higher for hard ones such as diamond. Equipartition, temperature-independent by construction, has no account of this fall.

A second failure is quantitative. The classical free-electron picture of a metal treats roughly one conduction electron per atom as a gas, which by equipartition should add $\frac{3}{2}R$ to the heat capacity. Metals show no such extra contribution: their heat capacities scarcely exceed those of insulators. Both failures — the low-temperature collapse of $C_V$ and the missing electron contribution — trace to the same source. Classical mechanics is the wrong mechanics for atoms. The energy of an oscillator or a rotor is quantized, degrees of freedom whose quantum spacing exceeds $kT$ are frozen out, and the electrons obey an exclusion principle that keeps almost all of them from sharing thermal energy. The repair requires the quantum distributions, and the fermion gas explains the metals. The search for an understanding of specific heats was, historically, one of the roads into the quantum theory itself.