Boltzmann's entropy formula

In statistical mechanics, Boltzmann's entropy formula (also known as the Boltzmann–Planck equation, not to be confused with the more general Boltzmann equation, which is a partial differential equation) is a probability equation relating the entropy ${\displaystyle S}$, also written as ${\displaystyle S_{\mathrm {B} }}$, of an ideal gas to the multiplicity (commonly denoted as ${\displaystyle \Omega }$ or ${\displaystyle W}$), the number of real microstates corresponding to the gas's macrostate:

${\displaystyle S=k_{\mathrm {B} }\ln W}$

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where ${\displaystyle k_{\mathrm {B} }}$ is the Boltzmann constant (also written as simply ${\displaystyle k}$) and equal to 1.380649 × 10⁻²³ J/K, and ${\displaystyle \ln }$ is the natural logarithm function (or log base e, as in the image above).

In short, the Boltzmann formula shows the relationship between entropy and the number of ways the atoms or molecules of a certain kind of thermodynamic system can be arranged. What is important to note is that W is not all possible states of the system, but ways the system can be arranged and still have the same properties from perspective of external observer. So for example when system contains 5 particles of gas and given amount of energy distributed between them for example [1,1,2,3,4]. Energy distribution can be realized as [1,2,1,3,4] where index represent a particle, but the distribution can also be realized as [2,1,1,3,4] after swapping first two and so forth. W is measure of all possible way the distribution can be realized. When W is small for given distribution that distribution has small entropy, when W is large for given distribution it has a large entropy.