In October 1877, 33-year-old Austrian physicist Ludwig Boltzmann (1844–1906) published a monumental paper in Sitzungsberichte der Wiener Akademie (Vol. 76) titled “Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung…”

Boltzmann introduced discrete mathematical energy elements $\epsilon$ into statistical mechanics and derived the world-famous statistical entropy formula:

$$S = k_{\text{B}} \log W$$

Discrete Energy States & Statistical Probability

To calculate the probability of energy distributions across $N$ gas molecules, Boltzmann partitioned total energy $E$ into discrete energy packages $\epsilon, 2\epsilon, 3\epsilon, \dots$:

1. Permutation of Microstates ($W$)

Boltzmann defined thermodynamic probability $W$ (Wahrscheinlichkeit) as the total number of distinct microstate permutations corresponding to a given macroscopic thermodynamic state:

$$W = \frac{N!}{n_0! \, n_1! \, n_2! \dots}$$

2. The Logarithmic Entropy Equation

Using Stirling’s approximation ($\ln N! \approx N \ln N - N$), Boltzmann proved that thermodynamic entropy $S$ is directly proportional to the logarithm of microstate multiplicity:

$$S = k_{\text{B}} \ln W$$

Where $k_{\text{B}} = 1.380649 \times 10^{-23}\,\text{J/K}$ is Boltzmann’s Constant (named by Max Planck).

The Catalyst for Quantum Mechanics

Boltzmann introduced discrete energy steps $\epsilon$ as a mathematical trick, planning to take the limit $\epsilon \rightarrow 0$.

In December 1900, Max Planck adopted Boltzmann’s 1877 discrete energy state combinatorics $S = k_{\text{B}} \ln W$, but discovered that $\epsilon$ cannot be zero—it must be finite and proportional to frequency:

$$\epsilon = h\nu$$

Starting the Quantum Revolution! Boltzmann’s formula $S = k \log W$ is famously engraved on his gravestone in the Zentralfriedhof in Vienna.

Key Takeaways

  • Year: 1877
  • Key Figure: Ludwig Boltzmann (Austrian Physicist)
  • Core Discovery: Formulated Statistical Entropy ($S = k_{\text{B}} \log W$) and introduced discrete energy states into physics.
  • Microscopic Meaning: Proved entropy is the measure of microscopic atomic microstate multiplicity.
  • Quantum Relevance: Directly enabled Max Planck’s 1900 quantum energy hypothesis ($\epsilon = h\nu$).