Boltzmann entropy
In statistical mechanics, Boltzmann's equation is a probability equation relating the entropy S of an ideal gas to the quantity W, the number of real microstates corresponding to the gas' macrostate:
where kB is the Boltzmann constant (also written with k), which is equal to 1.38065 × 10−23 J/K.
In short, the Boltzmann formula shows the relationship between entropy and the number of ways the atoms or molecules of a thermodynamic system can be arranged. In 1934, Swiss physical chemist Werner Kuhn successfully derived a thermal equation of state for rubber molecules using Boltzmann's formula, which has since come to be known as the entropy model of rubber.
The equation was originally formulated by Ludwig Boltzmann between 1872 and 1875, but later put into its current form by Max Planck in about 1900. To quote Planck, "the logarithmic connection between entropy and probability was first stated by L. Boltzmann in his kinetic theory of gases".
The value of was originally intended to be proportional to the Wahrscheinlichkeit (the German word for probability) of a macroscopic state for some probability distribution of possible microstates—the collection of (unobservable) "ways" the (observable) thermodynamic state of a system can be realized by assigning different positions(x) and momenta(p) to the various molecules. Interpreted in this way, Boltzmann's formula is the most general formula for the thermodynamic entropy. However, Boltzmann's paradigm was an ideal gas of identical particles, of which are in the -th microscopic condition (range) of position and momentum. For this case, the probability of each microstate of the system is equal, so it was equivalent for Boltzmann to calculate the number of microstates associated with a macrostate. was historically misinterpreted as literally meaning the number of microstates, and that is what it usually means today. can be counted using the formula for permutations
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