Matching the radiation and gas entropy coefficients

When faint light of one colour is squeezed into half the room, its entropy falls by the same rule as a gas of independent particles does. Matching the two rules says how many particles the light would have to contain, and so how much energy each one carries, an amount set by the colour of the light.

Section 4 found that radiation of energy E in a narrow band at frequency ν, where Wien's law holds, changes its entropy with volume as S − S₀ = (E/βν) ln(v/v₀). Section 5 used Boltzmann's principle, S − S₀ = (R/N) ln W: for n independent moving points the probability that all of them are in a part v of the volume v₀ is W = (v/v₀)ⁿ, so S − S₀ = (R/N) n ln(v/v₀). Section 6 writes the radiation result in the same form, S − S₀ = (R/N) ln[(v/v₀)^{(N/R)(E/βν)}], and reads off the probability that all the radiation energy is in v. The exponent plays the part of n: the energy behaves as if it consisted of n = NE/(Rβν) independent quanta, each of size Rβν/N, which is hν. At the defaults, 9.06 × 10⁻⁹ J at 600 THz, that is 2.28 × 10¹⁰ quanta of 2.48 eV. Halving the volume lowers the radiation's entropy by 2.18 × 10⁻¹³ J/K, as it would for a gas of that many points; the instrument's gas of 10 points loses 9.57 × 10⁻²³ J/K. Einstein then compared the mean quantum of a whole Wien spectrum, 3(R/N)T, with a molecule's mean kinetic energy, (3/2)(R/N)T: at 3000 K, 0.776 eV against 0.388 eV, a factor of 2.

Boltzmann's principle ties entropy to probability: S − S₀ = (R/N) ln W, where R/N, the gas constant divided by the number of molecules in a mole, is 1.381 × 10⁻²³ J/K. Take n independent points moving about a box of volume v₀. The chance that one of them is, at a given moment, in a part v of the box is v/v₀. Because they move independently, the chance that all n are there together is the product, (v/v₀)ⁿ. With 10 points and half the box, that is (1/2)¹⁰ = 1/1024. The entropy difference is then (R/N) ln[(v/v₀)ⁿ] = (R/N) n ln(v/v₀) = 1.381 × 10⁻²³ × 10 × ln 0.5 = −9.57 × 10⁻²³ J/K. Now the radiation. Section 4 gives S − S₀ = (E/βν) ln(v/v₀). Pull R/N out in front, E/βν = (R/N) × (N/R)(E/βν), and use the rule that a number times a logarithm is the logarithm of a power: S − S₀ = (R/N) ln[(v/v₀)^{(N/R)(E/βν)}]. Set this beside the gas. The two have the same form if n = (N/R)(E/βν). Divide the energy by that count to find the energy of one part: E/n = Rβν/N. With today's constants Rβ/N is Planck's h, 6.626 × 10⁻³⁴ J s, so each part carries hν = 6.626 × 10⁻³⁴ × 6.00 × 10¹⁴ = 3.976 × 10⁻¹⁹ J, or 2.48 eV. The defaults hold E = 9.056 × 10⁻⁹ J, so n = 9.056 × 10⁻⁹/(3.976 × 10⁻¹⁹) = 2.28 × 10¹⁰. The radiation's coefficient E/βν is n times R/N, 2.28 × 10¹⁰ × 1.381 × 10⁻²³ = 3.145 × 10⁻¹³ J/K, and halving the volume changes the entropy by 3.145 × 10⁻¹³ × ln 0.5 = −2.18 × 10⁻¹³ J/K. The probability that all that light is in one half at once is 1/2 raised to the power 2.28 × 10¹⁰, too small to write out. Last, the mean quantum. Over a whole Wien spectrum at temperature T, the energy per unit frequency is αν³e^−βν/T, and the number of quanta per unit frequency is that divided by Rβν/N; the ratio of the two integrals is 3(R/N)T. At 3000 K that is 3 × 1.381 × 10⁻²³ × 3000 = 1.243 × 10⁻¹⁹ J, or 0.776 eV, twice a gas molecule's (3/2)(R/N)T = 0.388 eV. A 600 THz quantum, at 2.48 eV, is 3.20 times the mean.

/** Effective independent particle count E / (h*nu) or E / (B*nu). */
export function effectiveIndependentCount(
  E: number,
  nu: CyclicFrequency | number,
  setOrB?: ConstantSet | number,
): Readonly<{
  status: "value";
  quantityId: "effectiveIndependentCount";
  unit: "";
  count: number;
  quantumEnergy: number;
  quantumEnergyEv: number;
  linearRepresentable: true;
}> {
  let elementEnergy: number;
  if (typeof setOrB === "number") {
    elementEnergy = setOrB * nu;
  } else if (setOrB && "entries" in setOrB) {
    const { h } = quantumConstantsSI(setOrB, constantReaders);
    elementEnergy = h * nu;
  } else {
    // Modern SI
    const h = 6.62607015e-34;
    elementEnergy = h * nu;
  }

  const count = E / elementEnergy;
  const quantumEnergyEv = elementEnergy / JOULES_PER_EV;

  return {
    status: "value",
    quantityId: "effectiveIndependentCount",
    unit: "",
    count,
    quantumEnergy: elementEnergy,
    quantumEnergyEv,
    linearRepresentable: true,
  };
}

/**
 * Mean quantum energy over a full Wien spectrum: <epsilon> = 3 * k_B * T (paper 1 §6).
 * Distinct from meanResonatorEnergy (k_B * T) and quantumEnergy (h * nu).
 */
export function meanQuantumEnergyWien(
  T: number,
  set: ConstantSet,
  options: { epsilonW?: number } = {},
): MeanQuantumEnergyWienResult {
  const { kB, h } = quantumConstantsSI(set, constantReaders);

  const meanWienJoules = 3 * kB * T;
  const meanWienEv = meanWienJoules / JOULES_PER_EV;

  const resonatorJoules = kB * T;
  const resonatorEv = resonatorJoules / JOULES_PER_EV;

  const moleculeKineticEv = (1.5 * resonatorJoules) / JOULES_PER_EV;
  const ratioToMoleculeKinetic = meanWienEv / moleculeKineticEv; // Exactly 2

  // Ratio h * nu / (3 * k_B * T) at 600 THz
  const nu600THz = 6.0e14;
  const quantumEnergy600THz = h * nu600THz;
  const ratioAt600THz = quantumEnergy600THz / meanWienJoules;

  const epsilonW = options.epsilonW ?? 0.01;
  const wienBoundaryX = Math.log(1 / epsilonW);

  // Shares below boundary x0
  const energyShareBelowBoundary = gamma4(wienBoundaryX) / 6;
  const countShareBelowBoundary = gamma3(wienBoundaryX) / 2;

  return {
    status: "value",
    meanQuantumEnergyWien: meanWienJoules,
    meanQuantumEnergyWienEv: meanWienEv,
    meanResonatorEnergy: resonatorJoules,
    meanResonatorEnergyEv: resonatorEv,
    moleculeKineticEnergyEv: moleculeKineticEv,
    ratioToMoleculeKinetic,
    ratioAt600THz,
    integrationRange: "all-positive-frequencies",
    wienAdmittedBoundaryX: wienBoundaryX,
    energyShareBelowBoundary,
    countShareBelowBoundary,
    modelStatus: "stipulated-model-extrapolated-beyond-admitted-regime",
  };
}

// Paper 1, §6: matching the entropy coefficients
// Radiation entropy:     S - S_0 = (E / (beta * nu)) * ln(V / V_0)
// Boltzmann gas entropy: S - S_0 = (R / N) * n * ln(V / V_0)
//
// Equating the exponents in W = (V / V_0)^n:
// n_eff = (N / R) * (E / (beta * nu)) = E / (h * nu)
// Energy per quantum: epsilon = E / n_eff = (R * beta * nu) / N = h * nu
//
// Mean quantum energy over a Wien spectrum:
// <epsilon> = 3 * (R / N) * T = 3 * k_B * T, twice a molecule's 1.5 * k_B * T