PX284 - I3 - grand canonical ensemble

PX284 - B4 - ensembles

microcanonical ensemble

isolated, so fixed energy

\begin{matrix} P_{i} = \frac{1}{Ω} \\ S = k_{B} \ln Ω \\ Ω = \exp (β T S) \end{matrix}

canonical ensemble

exchanges energy with a reservoir, so fixes $T$

\begin{matrix} P_{i} = \frac{\exp (- β E_{i})}{Z} \\ F = - k_{B} T \ln Z \\ Z = \exp (- β F) \end{matrix}

grand canonical ensemble

exchanges energy and particles with a reservoir, so fixes $T$ and $μ$
considering a system with an energy state, $E_{i}$ , and particle state, $N_{i}$ , connected with a very large reservoir with an energy, $U - E_{i}$ , and $N - N_{i}$ particles
assuming $U ≫ E_{i}$ and $N ≫ N_{i}$ , the entropy of the reservoir is given by :

\begin{aligned} S (U - E_{i}, N - N_{i}) & \approx S (U, N) - E_{i} {(\frac{\partial S}{\partial U})}_{N, U} - N_{i} {(\frac{\partial S}{\partial N})}_{N, U} \\ = S (U, N) - \frac{1}{T} (E_{i} - μ N_{i}) \end{aligned}

as for seen the canonical ensemble:

P_{i} (E_{i}, N_{i}) \propto Ω (U - E_{i}, N - N_{i})

where, $Ω$ is the number of microstates of the bath

but: $S = k_{B} \ln Ω ⟹ Ω = \exp (β T S)$

P_{i} \propto \exp (β T S)

⟹ P_{i} \propto \exp (- β (E_{i} - μ N_{i}))

after normalizing:

P_{i} = \exp \frac{- β (E_{i} - μ N_{i})}{Z}

where, $Z = \sum_{i} \exp (- β (E_{i} - μ N_{i}))$

this is the gibbs distribution, and $Z$ is the grand partition function
note: if $μ = 0 : P_{i} = \exp (- β E_{i}) / Z$ , ie. the boltzmann distribution
a grand potential can also be defined:

Φ_{G} = - k_{B} T \ln Z

such that:

Z = \exp (- β Φ_{G})