Binomial entropy of independent two-level particles
= Binomial entropy of independent two-level particles
{title2=$S=k_B\ln\binom{N}{N_+}$}
For independently labelled particles with nondegenerate levels, choosing which $N_+$ particles occupy the upper level gives a binomial <multiplicity>. With $x=(N_+-N_-)/N$, its large-number <entropy> is $Nk_B[\ln2-(1+x)\ln(1+x)/2-(1-x)\ln(1-x)/2]$. This counting assumes distinguishable or localized degrees of freedom, not indistinguishable particles sharing one pair of quantum orbitals.