Categories: Physics, Quantum mechanics, Statistics.

Bose-Einstein statistics

Bose-Einstein statistics describe how bosons, which do not obey the Pauli exclusion principle, will distribute themselves across the available states in a system at equilibrium.

Consider a single-particle state \(s\), which can contain any number of bosons. Since the occupation number \(N_s\) is variable, we turn to the grand canonical ensemble, whose grand partition function \(\mathcal{Z_s}\) is as follows, where \(\varepsilon_s\) is the energy per particle, and \(\mu\) is the chemical potential:

\[\begin{aligned} \mathcal{Z}_s = \sum_{N_s = 0}^\infty \Big( \exp\!(- \beta (\varepsilon_s - \mu)) \Big)^{N_s} = \frac{1}{1 - \exp\!(- \beta (\varepsilon_s - \mu))} \end{aligned}\]

The corresponding thermodynamic potential is the Landau potential \(\Omega\), given by:

\[\begin{aligned} \Omega_s = - k T \ln{\mathcal{Z_s}} = k T \ln\!\Big( 1 - \exp\!(- \beta (\varepsilon_s - \mu)) \Big) \end{aligned}\]

The average number of particles \(\expval{N_s}\) is found by taking a derivative of \(\Omega\):

\[\begin{aligned} \expval{N_s} = - \pdv{\Omega_s}{\mu} = k T \pdv{\ln{\mathcal{Z_s}}}{\mu} = \frac{\exp\!(- \beta (\varepsilon_s - \mu))}{1 - \exp\!(- \beta (\varepsilon_s - \mu))} \end{aligned}\]

By multitplying both the numerator and the denominator by \(\exp\!(\beta(\epsilon_s \!-\! \mu))\), we arrive at the standard form of the Bose-Einstein distribution \(f_B\):

\[\begin{aligned} \boxed{ \expval{N_s} = f_B(\varepsilon_s) = \frac{1}{\exp\!(\beta (\varepsilon_s - \mu)) - 1} } \end{aligned}\]

This tells the expected occupation number \(\expval{N_s}\) of state \(s\), given a temperature \(T\) and chemical potential \(\mu\). The corresponding variance \(\sigma_s^2\) of \(N_s\) is found to be:

\[\begin{aligned} \boxed{ \sigma_s^2 = k T \pdv{\expval{N_s}}{\mu} = \expval{N_s} \big(1 + \expval{N_s}\big) } \end{aligned}\]

References

  1. H. Gould, J. Tobochnik, Statistical and thermal physics, 2nd edition, Princeton.

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