  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.
