Editing Indistinguishable particles (section)

=== Statistical effects of indistinguishability ===

The indistinguishability of particles has a profound effect on their statistical properties. To illustrate this, consider a system of ''N'' distinguishable, non-interacting particles. Once again, let ''n''<sub>''j''</sub> denote the state (i.e. quantum numbers) of particle ''j''. If the particles have the same physical properties, the ''n''<sub>''j''</sub>s run over the same range of values. Let ''ε''(''n'') denote the [[energy]] of a particle in state ''n''. As the particles do not interact, the total energy of the system is the sum of the single-particle energies. The [[partition function (statistical mechanics)|partition function]] of the system is
: <math> Z = \sum_{n_1, n_2, \ldots, n_N} \exp\left\{ -\frac{1}{kT} \left[ \varepsilon(n_1) + \varepsilon(n_2) + \cdots + \varepsilon(n_N) \right] \right\} </math>
where ''k'' is the [[Boltzmann constant]] and ''T'' is the [[temperature]]. This expression can be [[factorization|factored]] to obtain
: <math> Z = \xi^N </math>
where
: <math> \xi = \sum_n \exp\left[ - \frac{\varepsilon(n)}{kT} \right].</math>

If the particles are identical, this equation is incorrect. Consider a state of the system, described by the single particle states [''n''<sub>1</sub>, ..., ''n''<sub>''N''</sub>]. In the equation for ''Z'', every possible permutation of the ''n''s occurs once in the sum, even though each of these permutations is describing the same multi-particle state. Thus, the number of states has been over-counted.

If the possibility of overlapping states is neglected, which is valid if the temperature is high, then the number of times each state is counted is approximately ''N''<nowiki>!</nowiki>. The correct partition function is
: <math> Z = \frac{\xi^N}{N!}.</math>

Note that this "high temperature" approximation does not distinguish between fermions and bosons.

The discrepancy in the partition functions of distinguishable and indistinguishable particles was known as far back as the 19th century, before the advent of quantum mechanics. It leads to a difficulty known as the [[Gibbs paradox]]. [[Willard Gibbs|Gibbs]] showed that in the equation ''Z'' = ''ξ''<sup>''N''</sup>, the [[entropy (thermodynamics)|entropy]] of a classical [[ideal gas]] is
: <math>S = N k \ln \left(V\right) + N f(T)</math>
where ''V'' is the [[volume]] of the gas and ''f'' is some function of ''T'' alone. The problem with this result is that ''S'' is not [[Extensive variable|extensive]] – if ''N'' and ''V'' are doubled, ''S'' does not double accordingly. Such a system does not obey the postulates of [[thermodynamics]].

Gibbs also showed that using ''Z'' = ''ξ''<sup>''N''</sup>/''N''! alters the result to
: <math>S = N k \ln \left(\frac{V}{N}\right) + N f(T)</math>
which is perfectly extensive.