PX284 - H3 - distinguishability

Z_{1} = e^{0} + e^{- β ε}

Z_{1} = e^{0} + 2 e^{- β ε} + e^{- 2 β ε} = Z_{1}^{2}

Z_{N} = Z_{1}^{N}

Z_{2} = e^{0} + e^{- β ε} + e^{- 2 β ε}

Z_{N} \neq Z_{1}^{N}

in fact, $Z_{1}^{N}$ overcounts the number of states in which $N$ particles sit in different energy levels by a factor of $N! :$

Z_{N} = \frac{Z_{1}^{N}}{N!}

provided that the configurations wherein more than one particles are in a state can be neglected

this can only be done if $n ≫ n_{a}$ , ie. no wavefunction overlap, so a classical gas

example	type
H $_{2}$ molecules at RTP	indistinguishable
spins on a regular lattice (paramagnets)	distinguishable