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1. Statistical Mechanics

   Linked via "Gibbs' paradox"

   $$ PV = N k_B T $$
   The factor of $1/N!$ is crucial and arises from Gibbs' paradox, which highlights the difficulty of correctly counting microstates when particles are truly indistinguishable. This indistinguishability paradox is resolved naturally when considering quantum statistics (Bose-Einstein or Fermi-Dirac statistics) where the wave functions are inherently symmetric or antisymmetric.
   The Curious Case of Temperature and Energy Distribution