distinguishables

Distinguishables are objects or particles that can be individually identified by some property, label, or location. The concept is used across physics, mathematics, and information theory to decide when two configurations should be regarded as the same or different.

In classical physics, swapping two identical particles with fixed labels or trajectories yields a different microstate, so the particles are treated as distinguishable. This assumption underpins Boltzmann counting and plays a role in discussions of the Gibbs paradox, where entropy changes depend on whether particles are treated as distinguishable.

In quantum mechanics, identical particles are fundamentally indistinguishable in the sense that physical states are invariant under permutation; the correct description uses symmetric (bosons) or antisymmetric (fermions) states. However, in many experiments, particles occupy separate spatial modes or carry distinct quantum numbers, providing effective distinguishability for practical measurements.

In combinatorics and information theory, the distinction between distinguishable and indistinguishable objects affects counting and probability. Labeled (distinguishable) objects yield different counts than unlabeled (indistinguishable) ones, with implications for partition problems, occupancy problems, and entropy calculations.

Operationally, whether a system is treated as distinguishable depends on the measurement context and the resolution of the description. Distinguishability thus informs modeling choices, entropy considerations, and the interpretation of quantum statistics.