# What are distinguishable and indistinguishable particles in statistical mechanics?

What are distinguishable and indistinguishable particles in statistical mechanics? While learning different distributions in statistical mechanics I came across this doubt; Maxwell-Boltzmann distribution is used for solving distinguishable particle and Fermi-Dirac, Bose-Einstein for indistinguishable particles. What is the significance of these two terms in these distributions?

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Besides this problem, there are quite a few features of classical Statistical Mechanics which are reminiscent of QM, although they were developed one or two decades before QM. It would be very interesting to clarify once and for all how much actual QM is contained in these "classical" assumptions of Statistical Mechanics. Examples of these problematic classical features include all equations where $\hbar$ actually appears (e.g. the Sackur-Tetrode Eq., the integration measure in phase space ${\frac{1}{N!}$$\frac{dp\ dq}{2 \pi \hbar}$$}^N$) as well as the third principle of Thermodynamics. –  Lupercus Sep 16 '12 at 12:30

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 weighting factor.

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Since there is no way in which the molecules can be labeled, the particles are indistinguishable. On the other hand, if the assembly is a crystal, the molecules can be labeled in accord with the positions they occupy in the crystal lattice and can be considered distinguishable.

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what you meant by molecules can't be labeled? Is it gaseous molecules –  Eka Dec 11 '13 at 6:11