= Boltzmann's entropy formula
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{title2=$S=k_B\log\Omega$}
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= Boltzmann entropy
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Why does a <gas> spread through an available container? One part of the answer is counting: a nearly uniform <macrostate> can be realized by vastly more <microstates> than a <macrostate> with every particle crowded into one corner. <Boltzmann's entropy formula> turns that multiplicity into <entropy>. The useful insight is that a large <entropy> measures how many microscopic realizations fit a specified macroscopic description, rather than how untidy a picture looks.
For a specified <macrostate> $M$ containing $\Omega(M)$ accessible, equally likely <microstates>, write
$$
S(M)=k_B\ln\Omega(M),\qquad p_i=\frac1{\Omega(M)}\quad(i\text{ compatible with }M).
$$
Here $k_B$ is the <Boltzmann constant>, and $\ln$ is the <natural logarithm>, also written $\log$ in this article's title. The count $\Omega(M)$ is dimensionless; <entropy> has units of <energy> divided by <temperature>. Always specify the <macrostate>, the constraints and the model's microscopic resolution before counting. A <microcanonical ensemble> provides a familiar equilibrium setting for the equal-<probability> assumption. https://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node57.html[MIT, §7.4: equiprobable microstates and S = k ln Ω]{ref}
The <natural logarithm> compresses an enormous count into a manageable number and makes products of counts into sums of <entropies>; see <additivity of Boltzmann entropy>. With this counting convention, one allowed <microstate> has zero <Boltzmann entropy>. This zero refers to the chosen model and resolution; it does not by itself assign an absolute <entropy> to a real material.
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