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This foundational formula connects the macroscopic 热力学 quantity of entropy (S) with the number of possible microscopic arrangements, or microstates (W), corresponding to the system’s macroscopic state. The equation, [latex]S = k_B \ln W[/latex], reveals that entropy is a measure of statistical disorder or randomness. The constant [latex]k_B[/latex] is the Boltzmann constant, linking energy at the particle level with temperature.
Boltzmann’s entropy formula provides a statistical definition for the thermodynamic concept of entropy, which was previously defined by Rudolf Clausius in terms of heat transfer ([latex]dS = \frac{\delta Q}{T}[/latex]). Boltzmann’s breakthrough was to link this macroscopic quantity to the statistical properties of the system’s constituent particles. A ‘macrostate’ is defined by macroscopic variables like pressure, volume, and temperature. A ‘microstate’ is a specific configuration of the positions and momenta of all individual particles. The key insight is that a single macrostate can be realized by an enormous number of different microstates. The quantity W, sometimes called the statistical weight or thermodynamic probability, is this number.
The formula implies that the equilibrium state of an isolated system, which is the state of maximum entropy according to the Second Law of Thermodynamics, is simply the most probable macrostate—the one with the largest number of corresponding microstates (largest W). The logarithmic relationship is crucial because it ensures that entropy is an extensive property. If you combine two independent systems, their total entropy is the sum of their individual entropies ([latex]S_{tot} = S_1 + S_2[/latex]), while the total number of microstates is the product ([latex]W_{tot} = W_1 W_2[/latex]). The logarithm turns this product into a sum: [latex]k_B \ln(W_1 W_2) = k_B \ln W_1 + k_B \ln W_2[/latex]. This formula is famously engraved on Boltzmann’s tombstone in Vienna.
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