A bin full of blue balls or a bin full of blue, red, and yellow balls?
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I have seen this sort of questions quite often. Some asked whether a shuffled deck of cards would have a higher entropy than a sorted one. These questions taken literally do not have good answers. The entropy is dominated by the thermodynamic properties of those balls (cards), not their colors (order). It's like asking whether two charges would attract or repel given their masses, while the electrostatic force would dominate the gravitational pull.
But if the "balls" are not understood as classical macroscopic objects, but as microscopic particles and colors are to distinguish their types, then this question could be linked to the Gibbs paradox and becomes an interesting topic to discuss. A bulk of molecules of the same type, according to Gibbs, has a smaller entropy than a bulk of distinguishable molecules by $k_B\ln N!$, with $N$ being the number of molecules. This is because if the molecules are indistinguishable, swapping any two of them would not yield a different microstate. So the ensemble of distinguishable particles would overcount the number of microstates $\Omega$ by a factor of $N!$. Then recall that entropy
$$S=k_B\ln\Omega,$$
where $k_B$ is Boltzmann's constant. Gibbs' counting factor $\ln N!$ is a constant for closed systems with a fixed number of particles. But it resolves the paradox of why mixing two types of gasses yields a mixing entropy but "mixing" the same type of gas does not.
Zhuoran He
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