Categories: Physics, Quantum mechanics, Statistics.

Bose-Einstein distribution

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

Consider a single-particle state ss, which can contain any number of bosons. Since the occupation number NN is variable, we use the grand canonical ensemble, whose grand partition function Z\mathcal{Z} is as shown below, where ε\varepsilon is the energy per particle, and μ\mu is the chemical potential. We evaluate the sum in Z\mathcal{Z} as a geometric series:

Z=∑N=0∞(e−β(ε−μ))N=11−e−β(ε−μ)\begin{aligned} \mathcal{Z} = \sum_{N = 0}^\infty \Big( e^{-\beta (\varepsilon - \mu)} \Big)^{N} = \frac{1}{1 - e^{-\beta (\varepsilon - \mu)}} \end{aligned}

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

Ω=−kTln⁡Z=kTln⁡ ⁣(1−e−β(ε−μ))\begin{aligned} \Omega = - k T \ln{\mathcal{Z}} = k T \ln\!\big( 1 - e^{-\beta (\varepsilon - \mu)} \big) \end{aligned}

The average number of particles ⟨N⟩\expval{N} in ss is then found by taking a derivative of Ω\Omega:

⟨N⟩=−∂Ω∂μ=kT∂ln⁡Z∂μ=e−β(ε−μ)1−e−β(ε−μ)\begin{aligned} \expval{N} = - \pdv{\Omega}{\mu} = k T \pdv{\ln{\mathcal{Z}}}{\mu} = \frac{e^{-\beta (\varepsilon - \mu)}}{1 - e^{-\beta (\varepsilon - \mu)}} \end{aligned}

By multiplying both the numerator and the denominator by eβ(ε ⁣− ⁣μ)e^{\beta(\varepsilon \!-\! \mu)}, we arrive at the standard form of the Bose-Einstein distribution fBf_B:

⟨N⟩=fB(ε)=1eβ(ε−μ)−1\begin{aligned} \boxed{ \expval{N} = f_B(\varepsilon) = \frac{1}{e^{\beta (\varepsilon - \mu)} - 1} } \end{aligned}

This gives the expected occupation number ⟨N⟩\expval{N} of state ss with energy ε\varepsilon, given a temperature TT and chemical potential μ\mu.

References

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