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Aren't classical, identical particles, always indistinguishable?

Consider monitoring the trajectory of one particle. After it collides with an identical particle how would one continue to keep track of which particle is which? However, textbooks on statistical mechanics routinely discuss classical, identical (but distinguishable) particles in the context of the Gibbs Paradox (https://en.wikipedia.org/wiki/Gibbs_paradox) and other situations.

Following query does not address the issue: Distinguishing identical particles

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In classical mechanics, you can actually "watch" the particles. You can track the position of each particle before and after the collision. You can label one particle as "particle 1" and the other as "particle 2", and those labels will stay consistent as you watch the system do what it does.

In QM that is not the case. We only observe our system through measurements, but before those measurements we cannot "watch" the particles do what they are doing. Therefore, there is no way to give the particles unique labels. This is why we impose certain symmetries under particle exchange for the state vectors of systems of indistinguishable particles.

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