Companion Notes Maxwell (1860)

Illustrations of the Dynamical Theory of Gases

A reader's companion to the 1860 paper by James Clerk Maxwell

Compiled by T. M. Jones 2026

Abstract

In 1860 James Clerk Maxwell published a dense and beautiful paper on the kinetic theory of gases. The version most people encounter today is a static PDF, difficult to search, annotate, or navigate. This project reconstructs the paper as a living document in vanilla HTML: every symbol annotated, every equation numbered, glossary included.

The first paper felt like Cirque du Soleil with me on the trapeze. Survived. Boltzmann Section II (1872) is hard to parse. The second with Einstein (1905) was calmer in German after the OCR restoration, more like putting on a collared shirt. The third Maxwell (1860) arrived as an artifact in a day. Three papers, three methods, one curator getting comfortable using multiple AIs, learning when to push, when to bail, and when to bump the machine.

I · 1872 Artifact Here ↗

This restoration is the third in a series built with multiple AI models. The sequence reflects the order of reconstruction:

I · 1872 Boltzmann — H-theorem companion ↗ II · 1905 Einstein — Brownian motion companion ↗

III · 1860 Maxwell — kinetic theory this paper

Methods

This paper demonstrates the evolution of a methodology across three reconstructions. The Maxwell run was fast by design. By this point the pattern was clear: identify a favorable candidate text, know which model is strong at which task, deploy accordingly, read everything, and intervene when the machine drifts. The ringleader leads. The models build the artifact. The curator writes the companion.

What this paper does

Maxwell's 1860 paper is the founding document of the kinetic theory of gases as a rigorous mathematical discipline. It is not long. It is not comfortable yet historic. It arrives at one of the most consequential distributions in all of physics, the Maxwell speed distribution: through an argument so compressed that readers have been re-deriving it ever since to convince themselves it actually works.

The paper divides into two parts. Part I treats perfectly elastic spheres and derives the equilibrium velocity distribution (Propositions I through V). Part II uses that distribution to predict transport properties: viscosity, diffusion, heat conduction, and arrives at the scandalous result that viscosity is independent of density. Maxwell thought this was probably wrong. Experiment eventually proved him right.

These notes follow Part I closely, annotating the logical structure, unpacking the notation Maxwell took for granted, and flagging the moments where the argument is doing more work than it appears to be doing.

Atoms are just a thought - unprovable

1 — Collisions are elastic and spherically symmetric

Hard sphere collisions conserve kinetic energy and momentum. The line of centres at impact determines everything. Velocities along that line are exchanged; velocities perpendicular to it are untouched. This geometry is the engine of Props. I through III.

2 — The distribution is Gaussian by logical necessity, not by assumption

Maxwell does not assume the bell curve. He derives it from a single constraint: that the distribution in each velocity component must be independent of the others, and that the joint distribution can only depend on speed, not direction. The Gaussian is the unique solution to that functional equation.

3 — The equilibrium state is not imposed — it is proved

Maxwell shows that collisions drive the system toward the distribution, not merely that the distribution is consistent with equilibrium. This is a precursor to Boltzmann's H-theorem, published twelve years later, which made the irreversibility argument precise.

4 — The paper carries two embarrassments Maxwell flags himself

First: viscosity independent of pressure, which he thinks is probably an artifact. Second: the equipartition theorem predicts specific heats that disagree with experiment. He says so plainly. Both problems were eventually resolved — one quickly by experiment confirming him, the other requiring quantum mechanics a half-century later.

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The Argument — Proposition IV Maxwell (1860)

The logical spine: Proposition IV

Proposition IV is the centre of the paper. Everything before it builds the collision machinery. Everything after it applies the result. The argument runs as follows, reconstructed step by step.

1 Set up coordinates. Let be the three components of a particle's velocity. By symmetry of the container, each component has the same unknown distribution function .

2 Independence. Because the three axes are at right angles, knowing the x-velocity tells you nothing about y or z. The joint probability density for all three is therefore the product \(f(x)\,f(y)\,f(z)\).

3 Isotropy constraint. Because the container is isotropic, the joint distribution cannot depend on direction — only on the scalar . So the product \(f(x)\,f(y)\,f(z)\) must equal some function \(\varphi(x^2 + y^2 + z^2)\).

4 Solve the functional equation. The only continuous function satisfying \(f(x)\,f(y)\,f(z) = \varphi(x^2+y^2+z^2)\) for all \(x, y, z\) is the exponential: \(f(x) = C\,e^{Ax^2}\).

5 Sign of A. If , particle count increases with speed and the integral over all velocities diverges. So \(A = -1/\alpha^2\) for some .

6 Normalize. Integrating \(N\,C\,e^{-x^2/\alpha^2}\) from \(-\infty\) to \(+\infty\) must give \(N\). The Gaussian integral yields \(C = 1/(\alpha\sqrt{\pi})\).

7 Convert to speed distribution. Converting from Cartesian velocity components to spherical speed introduces the \(v^2\) factor from the shell volume \(4\pi v^2\,dv\), giving Equation I.

Why this is not circular

The independence of components (step 2) might seem assumed rather than proved. Maxwell is relying on the isotropy of the container and the rotational symmetry argument: in a gas in equilibrium with no preferred direction, there is no mechanism to correlate x and y velocities. The isotropy constraint (step 3) then does the rest.

The functional equation unpacked

The pivot of the whole argument is Equation IX in Maxwell's numbering:

\[ f(x)\,f(y)\,f(z) = \varphi(x^2 + y^2 + z^2) \]

(IX)

The isotropy constraint — product of three 1D functions depends only on radial distance

To see why the Gaussian is the only solution: differentiate both sides with respect to \(x\), holding \(y\) and \(z\) fixed. You get \(f'(x)\,f(y)\,f(z) = 2x\,\varphi'(r^2)\). Divide by the original equation: \(f'(x)/f(x) = 2x\,\varphi'(r^2)/\varphi(r^2)\). The right side must be a function of \(x\) alone, so \(\varphi'(r^2)/\varphi(r^2)\) must be constant. This forces \(\varphi\) to be exponential in \(r^2\), and therefore \(f\) to be exponential in \(x^2\). That is: Gaussian, and nothing else.

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Key Equations — I through XIII Maxwell (1860)

The equations, annotated

Maxwell numbers his results I through XIII in the 1860 paper. What follows is each equation reproduced with the notation decoded, grouped by function.

Collision geometry — Props. I and II

The impact parameter probability (Eq. III)

For an incoming sphere approaching a target of combined radius \(s\)s: sum of the two sphere radii. Any line of approach within distance s of the target centre results in a collision., the impact parameter \(r\)r: perpendicular distance from the target centre to the incoming trajectory. Zero means head-on; s means grazing. is uniformly distributed across the collision cross-section, giving:

\[ \frac{2r\,dr}{s^2} \]

(III)

Probability that the impact parameter lies between r and r + dr

This is just the fraction of the collision disk's area occupied by an annular ring of radius \(r\) and width \(dr\). Area of ring: \(2\pi r\,dr\). Area of disk: \(\pi s^2\). Ratio: \(2r\,dr/s^2\). The \(\pi\) cancels. The uniformity assumption (all positions in the disk equally likely) is the physical content.

Deflection angle probability (Eq. IV)

The geometry of a hard sphere collision links impact parameter \(r\) to deflection angle \(\phi\)φ: angle between incoming direction and direction after impact. From 0 (no deflection) to π (head-on reversal). via \(r = s\sin(\phi/2)\). Substituting into Eq. III:

\[ \tfrac{1}{2}\sin\phi\,d\phi \]

(IV)

Probability of deflection angle lying between φ and φ + dφ

The critical corollary, via Eq. VI: the probability of the rebound direction falling in any solid angle element \(\omega\)ω: any small patch of area on the unit sphere, in steradians. The total sphere is 4π steradians. on the unit sphere is \(\omega/4\pi\) — uniform over all directions. After collision, every direction of rebound is equally probable. This is what makes the subsequent statistical argument tractable.

The Maxwell distribution — Prop. IV

The 1D component distribution (Eq. XIII)

\[ f(x) = \frac{1}{\alpha\sqrt{\pi}}\,e^{-x^2/\alpha^2} \]

(XIII)

Probability density for one velocity component — the normalized Gaussian

This is a Gaussian with mean zero and standard deviation \(\alpha/\sqrt{2}\)α/√2: the standard deviation. Maxwell parameterizes by α rather than σ. The two are related by σ = α/√2. Modern texts use σ² = kT/m.. Mean zero follows from isotropy: there is no preferred direction in a gas at rest, so \(\langle x \rangle = 0\). The width parameter \(\alpha\)α: Maxwell's width parameter. Connects to temperature via α² = 2kT/m, though Maxwell does not make this identification explicit in 1860. That linkage was clarified in subsequent work. is left as a free parameter — Maxwell does not yet identify it with temperature.

The 3D speed distribution (Eq. I)

\[ N\,\frac{4}{\alpha^3\sqrt{\pi}}\,v^2\,e^{-v^2/\alpha^2}\,dv \]

(I)

Number of particles with speed between v and v + dv — the Maxwell distribution

The \(v^2\) factor is the surface area of a shell of radius \(v\) in velocity space, divided by \(4\pi\) and folded into the prefactor. This is the distribution that gives the Maxwell speed distribution its characteristic right-skewed shape: rising from zero, peaking at the most probable speed \(v_p = \alpha\), then falling off exponentially at high speeds.

Three speeds to know

From this distribution Maxwell derives three characteristic speeds. The most probable speed is \(v_p = \alpha\). The mean speed is \(\bar{v} = 2\alpha/\sqrt{\pi}\) (his Result 3). The root-mean-square speed is \(v_{rms} = \alpha\sqrt{3/2}\), since \(\langle v^2 \rangle = \frac{3}{2}\alpha^2\) (his Result 4, Eq. II). These satisfy \(v_p < \bar{v} < v_{rms}\), as Maxwell notes: the mean square speed exceeds the square of the mean speed, "as it ought to be."

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Notation · Context · What came next Maxwell (1860)

Maxwell's notation vs. modern usage

Reading Maxwell directly requires adjusting to notation that has since been standardized differently. The table below gives the most important translations.

Maxwell writes	Modern form	What it means
\(\alpha\)	\(\sqrt{2kT/m}\)	Width of the distribution. Maxwell leaves this as a free parameter; later work identifies it with temperature \(T\), Boltzmann's constant \(k\), and particle mass \(m\).
\(f(x)\,dx\)	\(P(v_x)\,dv_x\)	Probability density for one velocity component. Maxwell writes it as number of particles; modern convention is probability (divide by N).
\(\varphi(r^2)\)	—	The auxiliary radial function in the functional equation (Eq. IX). Has no standard modern name — it appears in the derivation and is eliminated immediately.
\(v\)	\(\|\mathbf{v}\|\) or \(c\)	Speed (scalar magnitude of velocity). Some older texts use \(c\) for speed to distinguish from velocity vector \(\mathbf{v}\).
\(s\)	\(\sigma\) or \(d\)	Collision diameter (sum of radii). Modern kinetic theory uses \(\sigma\) or \(d\).
vis viva	kinetic energy	Maxwell uses the old term. Vis viva for a particle is \(mv^2\); kinetic energy is \(\frac{1}{2}mv^2\). The factor of two is conventional, not physical.

Where the paper sits in history

Before Maxwell (1860)

Bernoulli, Joule, Krönig, and Clausius had all proposed kinetic models of gases. Clausius had derived the mean free path. But none had tackled the distribution of velocities in a statistically rigorous way. The velocity was typically assumed uniform across all particles. A simplification Maxwell demolishes by showing the equilibrium distribution is necessarily a spread.

The immediate scandal: viscosity

Part II of the paper predicts that the viscosity of a gas is independent of its pressure. Maxwell writes that this result appears to be "not consistent with the ordinary notions respecting the viscosity of gases." He went home and measured it himself. It was true. This experimental confirmation by a theorist building apparatus in his kitchen is one of the more dramatic moments in the history of physics.

What Maxwell did not solve

The paper has two known difficulties, which Maxwell states plainly. The equipartition theorem, derived for elastic spheres, predicts that every degree of freedom of a molecule shares equally in the kinetic energy. For a diatomic molecule this gives a specific heat ratio \(\gamma = C_p/C_v = 1.4\), but for a sphere with three rotational degrees it gives \(1.33\). Experiment says \(1.4\) for air. Maxwell calls this "a real difficulty." The resolution required quantum mechanics: rotational degrees of freedom of diatomic molecules are not fully excited at room temperature.

The link to Boltzmann (1872)

Maxwell proves that collisions produce the equilibrium distribution but does not give a rigorous proof that any initial distribution must evolve toward it. That required Boltzmann's H-theorem twelve years later: Boltzmann constructed a functional \(H = \int f \log f\,dv\) and proved \(dH/dt \leq 0\), with equality only at the Maxwell distribution. The Maxwell distribution is the unique equilibrium state; Boltzmann showed why every other state is transient.

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Symbol	Name	What it actually is
The Distribution — Core Parameters
\(\alpha\)	Width parameter	Maxwell's single free parameter. Controls the spread of the Gaussian. Wider \(\alpha\) = faster particles. Connects to thermodynamics via \(\alpha^2 = 2kT/m\), though Maxwell does not make that identification in 1860. He leaves it for experiment to determine.
\(N\)	Total particles	The whole number of spheres. Every count of particles in an interval is expressed as \(N \times\) (a fraction). Think Avogadro-scale. Maxwell normalizes everything to \(N\).
\(f(x)\)	Velocity pdf	The probability density for one Cartesian component of velocity. Determined to be Gaussian by the functional equation argument. Same functional form applies to \(y\) and \(z\) — that isotropy is the key physical input of Prop. IV.
\(\varphi(r^2)\)	Radial function	Auxiliary function in Eq. IX. Whatever the 3D distribution looks like, isotropy forces it to depend only on \(r^2 = x^2+y^2+z^2\). This constraint, combined with independence, forces the Gaussian. \(\varphi\) itself drops out once \(f\) is pinned down.
\(C\)	Normalization constant	The prefactor making the distribution integrate to 1. Fixed by requiring \(\int_{-\infty}^{+\infty} NC\,e^{-x^2/\alpha^2}\,dx = N\). The Gaussian integral gives \(C = 1/(\alpha\sqrt{\pi})\). This is Eq. XII.
\(A\)	Exponent constant	The constant multiplying \(x^2\) in the exponent of the raw solution \(f(x) = Ce^{Ax^2}\). Must be negative: positive \(A\) gives a distribution that grows with speed, making the total count infinite. Maxwell sets \(A = -1/\alpha^2\).
Velocity Components & Speed
\(x, y, z\)	Velocity components	Cartesian projections of a particle's velocity. Maxwell treats them as statistically independent: the three axes are at right angles, so knowing \(x\) tells you nothing about \(y\) or \(z\). This independence is the physical pivot of Prop. IV.
\(v\)	Speed	Scalar magnitude: \(v = \sqrt{x^2+y^2+z^2}\). Direction stripped away. Converting from Cartesian components to speed introduces the spherical shell factor \(4\pi v^2\,dv\), which produces the \(v^2\) in the Maxwell distribution (Eq. I).
\(v_p\)	Most probable speed	The speed at the peak of the distribution. From Eq. I: \(v_p = \alpha\). Not stated explicitly by Maxwell but follows immediately from setting the derivative of the distribution to zero.
\(\bar{v}\)	Mean speed	Maxwell's Result (3): \(\bar{v} = 2\alpha/\sqrt{\pi} \approx 1.128\,\alpha\). Larger than \(v_p\) because the distribution is right-skewed: the high-speed tail pulls the mean above the peak.
\(v_{rms}\)	RMS speed	Root of Maxwell's Result (4): \(v_{rms} = \alpha\sqrt{3/2} \approx 1.225\,\alpha\). Largest of the three characteristic speeds. Proportional to \(\sqrt{T/m}\), connecting kinetic energy to temperature.
Collision Geometry — Props. I–III
\(s\)	Collision diameter	Sum of the radii of two colliding spheres. A collision occurs if and only if the line of approach passes within distance \(s\) of the target centre. The collision cross-section is then a disk of area \(\pi s^2\).
\(r\)	Impact parameter	Perpendicular distance from the target centre to the incoming line of motion. Ranges from 0 (head-on) to \(s\) (grazing). Uniformly distributed across the collision disk — that uniformity is the content of Eq. III.
\(\phi\)	Deflection angle	Angle between incoming direction and direction after impact. Related to impact parameter by \(r = s\sin(\phi/2)\). The key result of Prop. II is that all deflection angles are equally probable — which means all rebound directions are equally probable (Eq. VI).
\(\omega\)	Solid angle	Area element on the unit sphere, in steradians. Total sphere: \(4\pi\) sr. The probability of rebound into \(\omega\) is \(\omega/4\pi\) — uniform. This uniformity is what makes the collision mechanics compatible with the isotropic equilibrium distribution.
\(G\)	Centre of mass	Centre of gravity of the two colliding spheres. Its velocity is conserved through the collision (momentum conservation). Maxwell uses \(G\) as the pivot for the geometric decomposition in Prop. III: decompose into motion of \(G\) plus relative motion, handle each, recompose.
Named Results — Quick Reference
Eq. I	Maxwell distribution	\(N\,\frac{4}{\alpha^3\sqrt{\pi}}\,v^2\,e^{-v^2/\alpha^2}\,dv\) — the 3D speed distribution. The \(v^2\) factor is the spherical shell volume element. Right-skewed, rising from zero, peaking at \(v_p = \alpha\), decaying exponentially. The whole point of the paper.
Eq. IX	Functional equation	\(f(x)\,f(y)\,f(z) = \varphi(x^2+y^2+z^2)\) — the isotropy constraint. Product of three independent 1D functions depends only on radial distance. Unique continuous solution: Gaussian. The logical core of Prop. IV.
Eq. XIII	Normalized 1D Gaussian	\(f(x) = \frac{1}{\alpha\sqrt{\pi}}\,e^{-x^2/\alpha^2}\) — the velocity component distribution. Symmetric about zero, normalized to integrate to 1. The building block from which the 3D distribution is assembled.
Result (3)	Mean speed	\(\bar{v} = 2\alpha/\sqrt{\pi}\). The arithmetic mean of the speed distribution. Obtained by computing \(\int_0^\infty v \cdot [Eq.\,I]\,dv / N\). Requires the integral \(\int_0^\infty u^3 e^{-u^2}\,du = 1/2\).
Result (4)	Mean square speed	\(\langle v^2 \rangle = \frac{3}{2}\alpha^2\). Mean kinetic energy per unit mass is \(\frac{1}{2}\langle v^2 \rangle = \frac{3}{4}\alpha^2\). This exceeds \(\bar{v}^2 = 4\alpha^2/\pi \approx 1.27\,\alpha^2\) — Maxwell notes this with satisfaction. Connects to temperature once \(\alpha^2 = 2kT/m\) is identified.