Grand canonical ensemble

The grand canonical ensemble represents an otherwise closed system in thermal and chemical equilibrium with a heat bath and particle reservoir. As such, its natural variables are

$𝜇$ — Chemical potential
$𝑉$ — System volume
$𝑇$ — Temperature of the heat bath (and thus the system)

and microstates of all energies $𝐸 𝑖$ and particle numbers $𝑁 𝑖$ are accessible to the system, with probabilities at equilibrium given by the distribution

𝑃 𝑖 = e x p (- 𝐸 𝑖 - 𝜇 𝑁 𝑖 𝑘 𝐵 𝑇) Z Z = \sum 𝑖 e x p (- 𝐸 𝑖 - 𝜇 𝑁 𝑖 𝑘 𝐵 𝑇) = e x p (- Φ 𝑘 𝐵 𝑇)

where $Z$ is the grand canonical partition function, related to the Landau free energy $Φ = - 𝑘 𝐵 𝑇 l n Z$ . The determination of $Z$ depends on the statistics used.

Maxwell-Boltzmann statistics
Modified Boltzmann approximation
Fermi-Dirac statistics
Bose-Einstein statistics

Derivation

derivation

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Grand canonical ensemble

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