It is widely believed that the concept of distinguishability of entities in an ensemble is a classical attribute. The origin of the belief may be traced in Boltzmann's argument that when two identical atoms interchange their places in phase space, a new dynamical state is created, because we can follow the movement of each atom continuously during the process of interchange and hence the two states can be distinguished. In quantum mechanics, continuous tracking is impossible and hence Boltzmann's counting process is wrong. This leads to the concept of indistinguishability of atoms. But atomic ensembles are not the only ones we encounter in statistical mechanics. An alternative example is the ensemble of non-interacting harmonic oscillators of identical frequency, as we find in the Hamiltonian version of electromagnetic radiation field and of the elastic waves in a solid in the harmonic approximation. In working out the statistical mechanics of these ensembles, we have a dilemma. Do we consider them as distinguishable or indistinguishable entities? The question is significant because in this case the entities are not real objects, they are mathematical constructs. Here, appeal to mechanics does not help. The only way to decide the question is to work out both tile distinguishability and indistinguishability statistics and see which one is consistent with thermodynamics. In tile atomic systems we know that distinguishability leads to Gibbs' paradox. The correct statistics is either Bose-Einstein (B-E) or Fermi-Dirac (F-D).
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