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Light Quanta · Paper 1, §5 Heuristic Foundation

Independent Configurations and the Gas Analogy

How counting independent configurations produces an entropy law depending on volume as n ln(V/V₀), matching Wien-regime radiation and establishing the heuristic light-quantum concept.

Light Quanta · §5 Statistical Microstate Counting

Independent configurations and Boltzmann entropy

How counting independent configurations produces an entropy depending on volume as n ln V, and why locking the positions together gives V rather than V^n.

Predict Mode · Microstate Reasoning

With 10 independent points, what is the chance that all sit in the left half (f = 1/2)?

Total volume V₀ (Full box)Subvolume V = 0.500 V₀ (50%)
Subvolume VV₀ − V
Mode: Independent PointsW = fⁿ = (0.50)^4 = 6.2500e-2

Binomial distribution: Points inside subvolume P(k)

k = 0 .. 4
0
1
2
3
4

Green bar at k = 4 represents all points inside: W = P(4) = f^4.

Microstate enumeration: Total microstates: 16. Favorable: 1. Exact ratio: 1 / 16.

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Calculated Microstate & Entropy Outputs

Physical QuantitySymbolic FormCalculated ValuePhysical Meaning
Relative State ProbabilityW = (V/V₀)ⁿ = fⁿ6.250000e-2Probability that all n independent points are found in V
Natural Logarithm ln Wn ln f-2.772589Proportional to the entropy difference ΔS / k_B
Dimensionless Entropy Change ΔS/k_Bn ln(V/V₀)-2.772589Matches Wien-regime radiation entropy S - S₀ = (E / hν) k_B ln(V/V₀)
Base-10 Logarithm log₁₀ Wn log₁₀ f-1.204120Order of magnitude (e.g. 10^-18 for n = 60)

Paper Assumptions (§5 as printed)

  • No favored part of the space or direction in volume V_0.
  • Negligible interactions among the n movable points.
  • Other movable points may also be present without altering the independent distribution.
  • No assumption is needed about the laws of motion of the points.

What this model leaves out (not modeled)

  • Interactions between points.
  • Gas dynamics or time evolution.
  • Radiation itself (this is the gas and dilute-solution analogy, not a model of light).
  • Correlations other than the fully locked case.

The Independence Argument in Einstein 1905 §5

In §5 of the 1905 light-quanta paper, Einstein applies Boltzmann's principle S − S₀ = (R/N) lg W to an ideal gas of n movable points in volume V₀. If the points move independently with no favored position or direction, the statistical probability that all n points are found in a subvolume V is simply:

W = (V / V₀)ⁿ  ⟹  S − S₀ = (R / N) n ln(V / V₀)

Comparing this gas entropy with the monochromatic radiation entropy found in §4, S − S₀ = (E / hν) k_B ln(V / V₀), leads directly to the conclusion: monochromatic radiation behaves energetically as if it consists of E / (hν) independent energy quanta of magnitude hν.

Read the original German source text and translation for §5 →