Particle Statistics

#import "@preview/cetz:0.5.3": canvas, draw
#import "@preview/cetz-plot:0.1.4": plot
#import draw: content
#let label-size = 12pt
#let paragraph-size = 14pt
#let heading-size = 16pt

#let card_body(title, body, caption) = block(
  width: 100%,
  inset: 12pt,
  radius: 8pt,
  fill: rgb("#cdd3da"),
  breakable: false,
)[
  #text(size: heading-size, weight: "bold", title)
  #v(8pt)
  // Measure unconstrained artwork before scaling, including content wider than its card.
  #layout(size => {
    let artwork = text(size: label-size, body)
    std.scale(size.width / measure(artwork).width * 100%, reflow: true, artwork)
  })
  #v(7pt)
  #text(size: paragraph-size, caption)
]

#let card-grid(columns: 2, ..cards) = layout(size => {
  let rows = cards
    .pos()
    .chunks(columns)
    .map(row => {
      let ratios = row.map(card => {
        let bounds = measure(text(size: label-size, card.at(1)))
        bounds.width / bounds.height
      })
      let available = size.width - 12pt * (row.len() - 1) - 24pt * row.len()
      grid(
        columns: ratios.map(ratio => 24pt + available * ratio / ratios.sum()),
        gutter: 12pt,
        ..row.map(args => card_body(..args)),
      )
    })
  stack(dir: ttb, spacing: 12pt, ..rows)
})

#let takeaway(body) = block(
  width: 100%,
  inset: 12pt,
  radius: 6pt,
  fill: rgb("#c6d8d2"),
  breakable: false,
  text(size: paragraph-size, body),
)

#set page(width: 780pt, height: auto, margin: 22pt, fill: none)
#set text(font: "Avenir Next", size: paragraph-size, fill: rgb("#19324f"))
#set par(leading: 0.55em)

#let (blue, orange, green, teal) = (rgb("#0b5fa5"), rgb("#b04f08"), rgb("#12793f"), rgb("#008580"))
#let legend-fill = draw.set-style(legend: (fill: rgb("#cdd3da")))
// Compact legend with a thin border.
#let compact-legend = (item: (spacing: 0.15), padding: 0.15, stroke: 0.5pt)
#let axis-mark = (end: "stealth", fill: black)

// Mean occupation per single-particle state; zero-point energy is not occupation.
#let bose-occupation(x) = {
  assert(x > 0, message: "Bose occupation requires epsilon > mu")
  1 / (calc.exp(x) - 1)
}
#let fermi-dirac(x) = 1 / (calc.exp(x) + 1)
#let boltzmann(x) = 1 / calc.exp(x)

// === 1  Compare occupation laws ===
#let figure-0 = canvas(length: 1.45cm, {
  legend-fill
  draw.set-style(axes: (
    x: (mark: axis-mark, label: (anchor: "south-east", offset: -0.2)),
    y: (mark: axis-mark, label: (anchor: "north-west", offset: -0.2)),
  ))

  plot.plot(
    size: (8, 5),
    x-label: $beta (epsilon - mu)$,
    y-label: $macron(n)$,
    x-min: -7,
    x-max: 7,
    y-min: 0,
    y-max: 1.8,
    x-tick-step: 2,
    y-tick-step: 0.5,
    axis-style: "school-book",
    x-grid: true,
    y-grid: true,
    legend: "inner-north-east",
    legend-style: compact-legend,
    {
      // Start the Bose curve above zero, where its occupation diverges.
      for (label, distribution, color, minimum, samples) in (
        ("Bose–Einstein", bose-occupation, blue, 0.1, 200),
        ("Boltzmann", boltzmann, orange, -1, 100),
        ("Fermi–Dirac", fermi-dirac, green, -7, 100),
      ) {
        plot.add(
          style: (stroke: color + 1.5pt),
          domain: (minimum, 7),
          samples: samples,
          label: label,
          distribution,
        )
      }
    },
  )
})

// === 2  Fermions: a smeared step ===
#let figure-1 = canvas(length: 1.04cm, {
  legend-fill
  draw.set-style(axes: (
    x: (mark: axis-mark, label: (anchor: "north-east")),
    y: (mark: axis-mark, label: (anchor: "south-west")),
  ))
  // Chemical potential, half-width of the thermal smearing bracket around it, energy cutoff
  let (chem-pot, smear, x-max) = (1, 0.2, 2.3)

  plot.plot(
    size: (8, 7),
    x-label: [$epsilon$ / $mu$],
    y-label: $n(epsilon)$,
    x-min: 0,
    x-max: x-max,
    y-min: 0,
    y-max: 1.2,
    x-tick-step: .5,
    y-tick-step: .5,
    y-ticks: (0.5, 1),
    axis-style: "left",
    legend: "inner-south-west",
    legend-style: compact-legend,
    {
      for (beta, color) in ((5, blue), (25, orange)) {
        plot.add(
          style: (stroke: color + 1.5pt),
          domain: (0, x-max),
          samples: 150,
          x => fermi-dirac(beta * (x - chem-pot)),
          label: $k_"B" T = mu \/ #beta$,
        )
      }

      // T = 0 step function
      let points = ((0, 1), (chem-pot, 1), (chem-pot, 0), (x-max, 0))
      plot.add(style: (stroke: green + 1.5pt), points, label: $T = 0$)

      // Bracket the thermal smearing width around the chemical potential
      for x in (chem-pot - smear, chem-pot + smear) {
        plot.add-vline(x, style: (stroke: (dash: "dashed", thickness: 0.5pt)))
      }
      plot.add-hline(1.1, min: chem-pot - smear, max: chem-pot + smear, style: (
        stroke: (thickness: 0.5pt),
        mark: (symbol: "stealth", stroke: 0.5pt, fill: black, scale: .1),
      ))
    },
  )

  content((3.5, 6.5), $prop 1 \/ beta$, anchor: "south")
})

// === 3  Bosons: shared states ===
#let figure-2 = canvas(length: 1.45cm, {
  legend-fill
  plot.plot(
    size: (8, 6),
    x-min: 0,
    x-max: 4,
    y-min: 0,
    y-max: 4,
    x-label: none,
    y-label: $macron(n)$,
    x-tick-step: 1,
    y-tick-step: 1,
    axis-style: "left",
    legend: "inner-north-east",
    legend-style: compact-legend,
    {
      for (temp, color) in ((0.5, orange), (1, teal), (2, blue)) {
        plot.add(
          style: (stroke: color + 1.5pt),
          domain: (0.03, 4),
          samples: 240,
          x => bose-occupation(x / temp),
          label: [$T \/ T_0 = #temp$],
        )
      }
    },
  )
  draw.content((4, -.8), $(epsilon-mu) \/ (k_"B" T_0)$)
})

// === 4  The same variable in three formulas ===
#let figure-3 = box(width: 230pt, height: 330pt, align(center + horizon, text(
  size: paragraph-size,
  (
    (blue, [Bose–Einstein], $macron(n) = 1/(e^x-1)$),
    (green, [Fermi–Dirac], $macron(n) = 1/(e^x+1)$),
    (orange, [Boltzmann limit], $macron(n) approx e^(-x)$),
  )
    .map(((color, name, formula)) => [#text(fill: color, strong(name))\ #formula])
    .join([\ #v(13pt)]),
)))

Count the mean occupation of a single-particle state. Quantum statistics changes how particles share states; temperature controls how sharply energies are selected.
#v(14pt)
#card-grid(
  (
    [1  Compare occupation laws],
    figure-0,
    [At $x=beta(epsilon-mu) >> 1$, all three approach $exp(-x)$. Bosons require $x>0$: their occupation diverges as $epsilon$ approaches $mu$ from above.],
  ),
  (
    [2  Fermions: a smeared step],
    figure-1,
    [Each state has occupation between zero and one. Heating rounds the step around the chemical potential $mu$ over an energy range of order $k_"B" T$.],
  ),
  (
    [3  Bosons: shared states],
    figure-2,
    [Bosons can accumulate in the same state. At fixed positive $epsilon-mu$, increasing temperature increases its mean occupation; there is no Pauli ceiling of one. $T_0$ is a reference temperature; $mu$ is held fixed.],
  ),
  (
    [4  The same variable in three formulas],
    figure-3,
    [Here $epsilon$ is the state’s energy, $mu$ the chemical potential, and $beta=1 \/ (k_"B" T)$. Spin degeneracy counts separate states; it does not change the occupation limit per state.],
  ),
)
#v(12pt)
#takeaway[*Occupation is not a speed probability density.* The separate Maxwell–Boltzmann speed plot includes the number of states at each speed and a normalization factor. The complex-plane Bose surface serves a different, contour-integration purpose.]