indistinguishables

Indistinguishables are entities that cannot be distinguished by any measurement or intrinsic property. In physics a collection of identical particles is indistinguishable because swapping two particles does not produce a new physical state. In mathematics and combinatorics, objects are indistinguishable when relabelings do not produce a new configuration; for example, multisets or partitions count identical items without labeling.

In quantum mechanics, indistinguishability is fundamental. Particles such as electrons, photons, or atoms of the same species have wavefunctions that must be symmetric (bosons) or antisymmetric (fermions) under exchange. This leads to Bose-Einstein or Fermi-Dirac statistics and distinct physical consequences like the Pauli exclusion principle for fermions.

For statistical mechanics, indistinguishability affects the counting of microstates. The proper treatment avoids the Gibbs paradox and yields a factor 1/N! in the partition function, ensuring extensive entropy and correct thermodynamic limits.

In chemistry and condensed matter, indistinguishable particles require methods that respect symmetry, such as Slater determinants for fermions. In combinatorics and mathematics, indistinguishability reduces the number of distinct configurations; counting often uses equivalence relations, Burnside's lemma, or Pólya enumeration to count distinct arrangements.