Gibbs paradox and a possible mechanism of like-charge attraction in colloids

Based on a reconsideration of the Gibbs paradox, we show that a residual, non-extensive term in entropy turns up upon mixing identical particles, whether they are indistinguishable or not. The positive contribution from this residual entropy leads to a decrease in free energy, and we suggest that this entropic mechanism may serve as a source of like-charge attractions between a pair of colloidal particles or other macroions. For a system of two colloidal particles along with their neutralizing counterions, such decrease in free energy is of a few thermal energies and therefore crucial to the effective interaction between the particles.