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When can particles be distinguished?
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Protons are said to be ''identical particles'' - which has a precise
formal definition that translates to the informal meaning that they
don't have an identity in themselves but get them only through their
environment.
So one can identify a proton by ''the proton passing at time t near
position x'', or ''the proton just detected by the Geiger counter''
but it makes no sense to say that ''this proton is the same as that
one'' unless you know (or assume) that it followed a predictable
(approximate) trajectory and you know (or assume) in addition that no
other proton could have taken its place.
The same holds for all elementary particles, but also for composite
particles if they are small enough. Particles begin to become
distinguishable when they either are confined to a lattice (such as
atoms in a crystal; then they are distinguishable by their position),
or when they have so much internal structure (e.g., macromolecules)
that the structure of any two is distinct enough to make them
experimentally distinguishable by measuring their different internal
structure.
On the deepest level, particles are indistinguishable if and only if
they have the same quantum numbers (mass, spin, and charges).
However, in statistical mechanics one ofte studies effective theories
where there are additional means of distinguishing particles.
Two important examples:
1. In modeling molecular fluids, two atoms on the same molecule are
distinguishable if and only if there is no molecular symmetry
interchanging the two atoms, and two atoms in different molecules are
distinguishable if and only if there is no congruent matching of the
two molecules such that the two atoms correspond to each other.
2. In modeling the solid state, one typically assumes that the atoms
are confined to lattice sites, and that each site is occupied at most
once.. In this case, the position in the lattice is a distinguishable
label, which makes all atoms distinguishable.
The computational relevance of the distinction is that permutations of
(in)distinguishable particles (don't) count towards the weighing
factor.