Editing Indistinguishable particles (section)

== Statistical properties ==

=== 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.

=== Statistical properties of bosons and fermions ===

There are important differences between the statistical behavior of bosons and fermions, which are described by [[Bose–Einstein statistics]] and [[Fermi–Dirac statistics]] respectively. Roughly speaking, bosons have a tendency to clump into the same quantum state, which underlies phenomena such as the [[laser]], [[Bose–Einstein condensate|Bose–Einstein condensation]], and [[superfluid]]ity. Fermions, on the other hand, are forbidden from sharing quantum states, giving rise to systems such as the [[Fermi gas]]. This is known as the Pauli Exclusion Principle, and is responsible for much of chemistry, since the electrons in an atom (fermions) successively fill the many states within [[Electron shell|shells]] rather than all lying in the same lowest energy state.

The differences between the statistical behavior of fermions, bosons, and distinguishable particles can be illustrated using a system of two particles. The particles are designated A and B. Each particle can exist in two possible states, labelled <math>|0\rangle</math> and <math>|1\rangle</math>, which have the same energy.

The composite system can evolve in time, interacting with a noisy environment. Because the <math>|0\rangle</math> and <math>|1\rangle</math> states are energetically equivalent, neither state is favored, so this process has the effect of randomizing the states. (This is discussed in the article on [[quantum entanglement]].) After some time, the composite system will have an equal probability of occupying each of the states available to it. The particle states are then measured.

If A and B are distinguishable particles, then the composite system has four distinct states: <math>|0\rangle|0\rangle</math>, <math>|1\rangle|1\rangle</math>, <math>|0\rangle|1\rangle</math>, and <math>|1\rangle|0\rangle</math>. The probability of obtaining two particles in the <math>|0\rangle</math> state is 0.25; the probability of obtaining two particles in the <math>|1\rangle</math> state is 0.25; and the probability of obtaining one particle in the <math>|0\rangle</math> state and the other in the <math>|1\rangle</math> state is 0.5.

If A and B are identical bosons, then the composite system has only three distinct states: <math>|0\rangle|0\rangle</math>, <math>|1\rangle|1\rangle</math>, and <math>\frac{1}{\sqrt{2}}(|0\rangle|1\rangle + |1\rangle|0\rangle)</math>. When the experiment is performed, the probability of obtaining two particles in the <math>|0\rangle</math> state is now 0.33; the probability of obtaining two particles in the <math>|1\rangle</math> state is 0.33; and the probability of obtaining one particle in the <math>|0\rangle</math> state and the other in the <math>|1\rangle</math> state is 0.33. Note that the probability of finding particles in the same state is relatively larger than in the distinguishable case. This demonstrates the tendency of bosons to "clump".

If A and B are identical fermions, there is only one state available to the composite system: the totally antisymmetric state <math>\frac{1}{\sqrt{2}}(|0\rangle|1\rangle - |1\rangle|0\rangle)</math>. When the experiment is performed, one particle is always in the <math>|0\rangle</math> state and the other is in the <math>|1\rangle</math> state.

The results are summarized in Table 1:

{| class="wikitable" style="margin:auto"
|+ Table 1: Statistics of two particles
|-
! Particles !! Both 0 !! Both 1 !! One 0 and one 1
|-
| Distinguishable|| align=center |0.25|| align=center |0.25|| align=center |0.5
|-
| Bosons|| align=center |0.33|| align=center |0.33|| align=center |0.33
|-
| Fermions|| align=center |0|| align=center |0|| align=center |1
|}

As can be seen, even a system of two particles exhibits different statistical behaviors between distinguishable particles, bosons, and fermions. In the articles on [[Fermi–Dirac statistics]] and [[Bose–Einstein statistics]], these principles are extended to large number of particles, with qualitatively similar results.