Ergodic
Energy contours, enclosed phase-space volume, and finite-width energy shells for a harmonic oscillator in canonical coordinates (q,p), followed by rational and irrational angle trajectories on a two-oscillator fixed-action torus. The guide distinguishes ergodicity on that torus from ergodicity on the full energy surface. Incorporates the former Momentum Shell entry. See ergodic theory and statistical mechanics.
The interior counts states below E; the annulus counts a finite energy interval; exact-energy dynamics stays on the contour. Opposite edges of each angle plot are identified. An irrational frequency ratio produces a dense trajectory over infinite time, while the drawing shows only a finite segment. Uniform-angle ergodicity applies to the fixed-action torus: uncoupled oscillators conserve their individual actions, so this does not establish microcanonical ergodicity on the full energy surface.
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Code
ergodic.typ (112 lines)
#import "@preview/cetz:0.5.2": canvas, draw
#import draw: circle, content, line, rect
#set page(width: 660pt, height: auto, margin: 20pt, fill: none)
#set text(font: "Avenir Next", size: 10.5pt, fill: rgb("#19324f"))
#set par(leading: 0.55em)
#let panel = block.with(
width: 100%,
inset: 12pt,
radius: 7pt,
fill: rgb("#cdd3da"),
breakable: false,
)
// A one-degree-of-freedom oscillator uses canonical coordinates (q,p), not two positions.
#let oscillator_energy(q, p, mass: 1, stiffness: 1) = p * p / (2 * mass) + stiffness * q * q / 2
#let shell_energy = 1
#let shell_width = .25
#let shell_axes(energy, mass: 1, stiffness: 1) = (
calc.sqrt(2 * energy / stiffness),
calc.sqrt(2 * mass * energy),
)
#let angle_point(time, ratio) = (calc.rem(time, 1), calc.rem(time * ratio, 1))
#let orbit(ratio, turns: 1) = canvas(length: 1pt, {
rect((0, 0), (210, 125), fill: rgb("#cdd3da"), stroke: rgb("#78828c") + .6pt)
let previous = angle_point(0, ratio)
for step in range(1, turns * 120 + 1) {
let point = angle_point(step / 120, ratio)
// Break at identified edges: never draw a spurious chord across a wrap.
if calc.abs(point.at(0) - previous.at(0)) < .1 and calc.abs(point.at(1) - previous.at(1)) < .1 {
line(
(210 * previous.at(0), 125 * previous.at(1)),
(210 * point.at(0), 125 * point.at(1)),
stroke: rgb("#0b5fa5") + .65pt,
)
}
previous = point
}
content((105, -15), $theta_1: 0 arrow.r 2pi$)
content((-12, 62), rotate(-90deg, reflow: true)[$theta_2: 0 arrow.r 2pi$])
})
#panel[
#text(size: 14pt, weight: "bold")[1 One oscillator: volume, contour, shell]
#v(9pt)
#grid(
columns: (280pt, 1fr),
gutter: 14pt,
align: horizon,
align(center, canvas(length: 1pt, {
let inner = shell_axes(shell_energy)
let outer = shell_axes(shell_energy + shell_width)
let scale_q = 70
let scale_p = 45
circle(
(0, 0),
radius: (scale_q * outer.at(0), scale_p * outer.at(1)),
fill: rgb("#c6d8d2"),
stroke: rgb("#286744") + 1pt,
)
circle(
(0, 0),
radius: (scale_q * inner.at(0), scale_p * inner.at(1)),
fill: rgb("#cdd3da"),
stroke: rgb("#0b5fa5") + 1.5pt,
)
line((-125, 0), (125, 0), stroke: rgb("#19324f") + .7pt, mark: (end: "stealth"))
line((0, -83), (0, 83), stroke: rgb("#19324f") + .7pt, mark: (end: "stealth"))
content((130, 0), $q$)
content((0, 90), $p$)
content((52, 20), $H < E$)
content((45, -84), text(fill: rgb("#0b5fa5"))[$H=E$])
line((45, -75), (55, -53), stroke: rgb("#0b5fa5") + .7pt)
content((-60, 90), text(fill: rgb("#286744"))[$E < H < E+Delta E$])
})),
[
$H(q,p) = p^2/(2m) + k q^2/2$
#v(6pt)
$q_(max)=sqrt(2E/k), quad p_(max)=sqrt(2 m E)$
#v(6pt)
Interior: $H < E$\ Blue contour: $H=E$\ Green shell: $E < H < E+Delta E$
#v(6pt)
Axes scaled independently; shell exaggerated.
],
)
]
#v(12pt)
#grid(
columns: (1fr, 1fr),
gutter: 12pt,
panel[
#text(size: 14pt, weight: "bold")[2 Rational: closed orbit]
#v(8pt)
#align(center, orbit(2))
#v(8pt)
#align(center)[$omega_2 \/ omega_1=2$ · Opposite edges identified.]
],
panel[
#text(size: 14pt, weight: "bold")[3 Irrational: dense orbit]
#v(8pt)
#align(center, orbit(calc.sqrt(2), turns: 24))
#v(8pt)
#align(center)[$omega_2 \/ omega_1=sqrt(2)$ · Finite segment shown.]
],
)
#v(12pt)
#block(
width: 100%,
inset: 10pt,
radius: 5pt,
fill: rgb("#c6d8d2"),
breakable: false,
)[Irrational flow is ergodic on its *fixed-action torus* with uniform angle measure, not generally on the full energy surface.]