Boltzmann's entropy formula

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: S = k B ln ⁡ W {displaystyle S=k_{mathrm {B} }ln W}     (1) W = N ! ∏ i N i ! {displaystyle W={frac {N!}{prod _{i}N_{i}!}}}     (2) S = − k B ∑ p i ln ⁡ p i {displaystyle S=-k_{mathrm {B} }sum p_{i}ln p_{i}}     (3) S B = − N k B ∑ i p i ln ⁡ p i {displaystyle S_{mathrm {B} }=-Nk_{mathrm {B} }sum _{i}p_{i}ln p_{i}}     (4) 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 as simply k) and 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. 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 W 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 and momenta 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 N identical particles, of which Ni are in the i-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. W was historically misinterpreted as literally meaning the number of microstates, and that is what it usually means today. W can be counted using the formula for permutations where i ranges over all possible molecular conditions and '!' denotes factorial. The 'correction' in the denominator is due to the fact that identical particles in the same condition are indistinguishable. W is sometimes called the 'thermodynamic probability' since it is an integer greater than one, while mathematical probabilities are always numbers between zero and one.