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}

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

© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.