Kinetic theory & equipartition

Pressure from molecular collisions, Maxwell–Boltzmann, equipartition, mean free path.

The macroscopic state variables — pressure, temperature, density — are statistical summaries of the molecular world. Kinetic theory is the bridge that recovers each of them from the underlying motion: $p$ from the rate of momentum delivery, $T$ from the mean molecular kinetic energy, $\gamma$ from the active molecular degrees of freedom, the Boltzmann factor from the canonical ensemble.

Pressure from molecular collisions

Imagine a gas of $N$ identical point particles of mass $m$ in a cubic box of volume $V$ . Each particle has a velocity $\mathbf{v}$ drawn from some isotropic distribution. The pressure on a wall is the time-averaged rate at which the wall receives momentum per unit area — exactly the impulse-momentum theorem from the mechanics chapter applied per collision.

▶ p = (1/3) n m ⟨v²⟩

Consider the wall at $x = L$ (area $A$ , with the box volume $V = AL$ ). A molecule hitting this wall with $x$ -component $v_x > 0$ rebounds elastically with $x$ -component $-v_x$ , delivering an impulse $2 m v_x$ to the wall.

The number of molecules with $x$ -velocity in $[v_x, v_x + dv_x]$ that strike the wall per unit time is $\tfrac12 n A |v_x|\, f(v_x)\, dv_x$ (only the half moving toward the wall reach it). The pressure — force per unit area — is the total momentum delivery per unit time per unit area,

p \;=\; \int_{-\infty}^\infty 2 m v_x \cdot \tfrac12 n |v_x|\, f(v_x)\, dv_x \;=\; n m \langle v_x^2\rangle.

For an isotropic distribution $\langle v_x^2\rangle = \langle v_y^2\rangle = \langle v_z^2\rangle = \tfrac13 \langle v^2\rangle$ , so

p \;=\; \tfrac13 n m \langle v^2 \rangle.

Rewriting in terms of the mean kinetic energy $\langle\varepsilon\rangle = \tfrac12 m \langle v^2\rangle$ ,

p V \;=\; \tfrac23 N \langle \varepsilon \rangle.

Comparing with the ideal-gas law $pV = N k_B T$ identifies the mean translational kinetic energy as $\langle \varepsilon \rangle = \tfrac32 k_B T$ .

The factor of $1/3$ comes from the three Cartesian components of velocity sharing the mean kinetic energy equally; this isotropy is the kinetic statement of equilibrium.

Number of particles N = 50 Characteristic speed v₀ = 2.00 ≈ √(k_BT/m)

Each particle hitting the right wall delivers an impulse 2m|v_x| (elastic bounce). Summing these over time and dividing by wall length and elapsed time gives the measured pressure. The kinetic theory predicts p = nm⟨v_x²⟩ — the two numbers agree to within statistical noise.

The simulation accumulates the impulse delivered to the right wall and computes the pressure. The result tracks the theoretical $nm\langle v_x^2\rangle$ to within statistical noise. Speed up the particles (raise the “temperature” slider) and the impulse rate rises; double the density (raise $N$ ) and the pressure doubles.

The Maxwell–Boltzmann distribution

The form of $f(v_x)$ is fixed not by mechanics but by statistical mechanics. The argument: in equilibrium the joint distribution of $(v_x, v_y, v_z)$ must be (i) factorisable across components (by independence), (ii) isotropic (depend on $|\mathbf{v}|$ only), and (iii) consistent with $\langle\varepsilon\rangle = \tfrac32 k_B T$ . The only functional form satisfying all three is Gaussian:

f(v_x) \;=\; \left(\frac{m}{2\pi k_B T}\right)^{1/2} e^{-m v_x^2/(2 k_B T)}.

Going from the Cartesian-component density to the speed density requires the spherical shell factor $4\pi v^2\, dv$ :

f(v) \;=\; 4\pi \left(\frac{m}{2\pi k_B T}\right)^{3/2} v^2\, e^{-m v^2 / (2 k_B T)}.

This is the Maxwell–Boltzmann speed distribution. It has three useful characteristic speeds, each the natural one for a different question:

Most-probable speed $v_p = \sqrt{2 k_B T/m}$ — the peak of the distribution, the typical speed of a random molecule.
Mean speed $\langle v\rangle = \sqrt{8 k_B T/(\pi m)}$ — the right speed for collision rate arguments (rate ∝ ⟨v⟩).
Root-mean-square speed $v_\text{rms} = \sqrt{3 k_B T/m}$ — the right speed for pressure (which involves $\langle v^2\rangle$ ).

They sit in the fixed ratio $\sqrt{2} : \sqrt{8/\pi} : \sqrt{3} \approx 1.41 : 1.60 : 1.73$ . Note that the speed of sound is less than all three: $c = \sqrt{\gamma R T / M} = \sqrt{\gamma/3}\, v_\text{rms} \approx 0.68\, v_\text{rms}$ for $\gamma = 7/5$ .

v_p (most probable)410 m/s

⟨v⟩ (mean)463 m/s

v_rms502 m/s

c (speed of sound)343 m/s

Gas: Temperature: 293 K (20 °C)

The most-probable, mean, and RMS speeds always stand in the ratio √2 : √(8/π) : √3 ≈ 1.41 : 1.60 : 1.73. The speed of sound is the smallest of the four: it equals √(γ/3) ≈ 0.68 of v_rms for diatomic gas. Drop the temperature and the whole distribution contracts toward zero like √T.

Slide temperature; the distribution shifts and broadens as $\sqrt{T}$ . Switch gases; lighter molecules (H₂, He) have far higher thermal speeds at the same temperature. The speed of sound is marked for comparison.

Equipartition

In classical statistical mechanics, every quadratic degree of freedom in the Hamiltonian contributes $\tfrac12 k_B T$ to the mean energy. The argument is elementary:

▶ ½α⟨q²⟩ = ½k_BT, the equipartition integral

For a coordinate $q$ entering the Hamiltonian as $\tfrac12 \alpha q^2$ , the Boltzmann-weighted mean is

\langle \tfrac12 \alpha q^2\rangle \;=\; \frac{\int_{-\infty}^\infty \tfrac12 \alpha q^2\, e^{-\beta \alpha q^2/2}\, dq}{\int_{-\infty}^\infty e^{-\beta \alpha q^2/2}\, dq}.

The denominator is the Gaussian integral $\sqrt{2\pi/(\beta\alpha)}$ . The numerator is computed by recognising it as $-d/d\beta$ of the denominator:

\int q^2\, e^{-\beta \alpha q^2/2}\, dq \;=\; -\frac{2}{\alpha}\frac{d}{d\beta}\!\sqrt{\frac{2\pi}{\beta\alpha}} \;=\; \frac{1}{\beta}\sqrt{\frac{2\pi}{\beta\alpha}}.

The ratio is $1/(2\beta) = \tfrac12 k_B T$ , independent of $\alpha$ .

The independence from $\alpha$ is the heart of equipartition: a stiffer coordinate (large $\alpha$ ) is narrower in its Boltzmann distribution (smaller $\langle q^2\rangle$ ), but the mean potential energy $\tfrac12 \alpha \langle q^2\rangle$ is unchanged. This is what makes equipartition such a clean accounting tool.

spring α1.00

temperature k_BT1.00

σ_q = √(k_BT/α)1.000

∫ ½αq² p(q) dq0.5000

½ k_BT0.5000

Spring constant α = 1.00 Temperature k_BT = 1.00

Vary α: the parabola widens or narrows, and the Gaussian narrows or widens in compensation. The integral of energy × density is *unchanged* at fixed T; it always equals ½k_BT. This is equipartition — each quadratic degree of freedom contributes exactly ½k_BT regardless of stiffness.

Vary $\alpha$ at fixed $T$ : the parabola widens or narrows, the Gaussian narrows or widens in compensation, and the integrand area is invariant — it always equals $\tfrac12 k_B T$ .

Each translational direction is one quadratic degree of freedom; each rotational axis with nonzero moment of inertia is another; each vibrational mode contributes two (one kinetic, one potential). For an ideal gas:

Molecule	Active DOF at room T	$U / N$	$c_v / R$	$\gamma$
Monatomic (He, Ar)	3 (trans.)	$\tfrac32 k_B T$	$3/2$	$5/3 \approx 1.67$
Diatomic (N₂, O₂, air)	5 (3 trans. + 2 rot.)	$\tfrac52 k_B T$	$5/2$	$7/5 = 1.4$
Triatomic (CO₂, H₂O)	6+	$\tfrac{6+}{2} k_B T$	$6/2 = 3$	$4/3 \approx 1.33$

The vibrational degrees of freedom of N₂ and O₂ are frozen out at room temperature: the vibrational quantum is much larger than $k_B T \approx 25\,\text{meV}$ , so vibrations contribute negligibly to the heat capacity. This is the first hint that classical equipartition is incomplete; it is corrected by quantum statistics for stiff oscillators.

Mean free path

A molecule with diameter $d$ moving through a gas of number density $n$ sweeps out a collision cross-section $\sigma = \pi d^2$ per unit length. The mean distance between collisions is therefore

\ell \;=\; \frac{1}{\sqrt{2}\, n \pi d^2}.

(The factor of $\sqrt{2}$ accounts for the motion of the targets.) The collision rate is $\langle v\rangle / \ell$ , which for air at STP is $\sim 7\times 10^9$ per second — on any timescale of acoustic interest, billions of collisions occur and the continuum picture is rock-solid.

n (m⁻³)1.00e+25

d (m)3.16e-10

λ = 1/(√2 n π d²)2.25e-7 m

L (system)1.00e-2 m

Kn = λ/L2.25e-5

log₁₀ n = 25.0 (n = 1.00e+25 m⁻³) log₁₀ d = -9.5 (d = 0.32 nm) log₁₀ L = -2.0 (L = 1.00e-2 m)

Air at STP has n ≈ 2.5×10²⁵ m⁻³ and d ≈ 0.3 nm, giving λ ≈ 70 nm — much smaller than any acoustic system, so the continuum picture holds. Lower n (e.g. upper-atmosphere) or shrink L (e.g. MEMS) and Kn rises; once Kn ≳ 1 the gas is rarefied and the Navier-Stokes equation fails.

Slide density and molecular diameter; read off the mean free path and the Knudsen number $\mathrm{Kn} = \ell/L$ for a chosen system size. Continuum hydrodynamics requires $\mathrm{Kn} \ll 1$ ; the upper-atmosphere and MEMS-scale flows where $\mathrm{Kn}$ approaches unity require the more general kinetic theory beyond Navier–Stokes.

The Boltzmann factor and thermal activation

Beyond equilibrium velocity distributions, the same exponential — the Boltzmann factor $e^{-E/k_B T}$ — is the universal weight for any process requiring an energy $E$ . In a two-state system with energy gap $\Delta E$ , the equilibrium population ratio is

\frac{n_\text{excited}}{n_\text{ground}} \;=\; e^{-\Delta E / k_B T}.

ΔE1.00

k_BT1.00

ΔE / k_BT1.00

e^(−ΔE/k_BT)3.68e-1

Energy gap ΔE = 1.00 Temperature k_BT = 1.00

The Boltzmann factor governs every thermally activated process. When ΔE/k_BT = 1 the excited state holds e⁻¹ ≈ 37% of the ground-state population; at ΔE/k_BT = 10 the ratio is 4×10⁻⁵ — effectively zero. The dimensionless ratio ΔE/k_BT, not ΔE alone, decides which states matter.

Slide $\Delta E$ and $T$ . When $\Delta E \sim k_B T$ the two states populate comparably; when $\Delta E \gg k_B T$ the excited state is essentially empty. The dimensionless ratio $\Delta E/k_B T$ — not $\Delta E$ alone — decides which physics matters at a given temperature.

This activation factor appears throughout the bookshelf — in the nucleation barrier of the free-energy chapter, in the open-probability of a mechanically gated channel (Hearing Ch 4.6), and in the chemical-reaction rate $k = A e^{-E_a/k_BT}$ of Arrhenius kinetics. The exponential is what makes thermal physics so sensitive to temperature: a 50 mV shift in a 130 mV gating signal at body temperature (k_BT ≈ 27 meV) changes the open probability by a factor of $e^{50/27} \approx 6$ .

Brownian motion as visible kinetic theory

A particle large enough to see but small enough to be jostled by molecular collisions undergoes a random walk: each collision delivers a small, random impulse, and the particle’s position diffuses as $\langle x^2\rangle = 2 D t$ in one dimension. The diffusion constant connects to the friction coefficient (Stokes drag, viscosity & diffusion chapter) by the Einstein relation $D = k_B T / \gamma_\text{drag}$ . The random-walk mathematics is in Math Foundations Ch 10.3; the physics is in the next chapter on viscosity and diffusion.

⏳ The history — From Bernoulli's bouncing balls to Boltzmann's H-theorem

Daniel Bernoulli in Hydrodynamica (1738) gave the first kinetic derivation of pressure: he modelled a gas as a swarm of point particles bouncing off the walls of a container and recovered $p \propto v^2$ from rate-of-momentum arguments alone. The result was ahead of its time; chemistry was still pre-Daltonian and the idea of atoms was philosophically suspect.

The modern kinetic theory dates to the 1850s and 1860s. Rudolf Clausius (1857) gave the first rigorous derivation of $p = \tfrac13 n m \langle v^2\rangle$ and introduced the mean free path. James Clerk Maxwell (1860) wrote down the equilibrium velocity distribution. Ludwig Boltzmann (1872) gave a dynamical derivation using his H-theorem, showing that an arbitrary initial distribution evolves toward the Maxwell form under collisions.

The molecular reality of gases was disputed for another generation. Einstein’s 1905 quantitative theory of Brownian motion and Perrin’s confirming measurements (1908) put the matter to rest: atoms are real, kinetic theory is exact in its classical limit, and the macroscopic gas laws are the statistical consequence.

For the cross-book applications — kinetic-theory route to the speed of sound, Brownian-motion hearing floor, Boltzmann-factor activation in nucleation and channel gating — see the key examples sub-page.