Particle number expectation value
Set
| context | w … grand canonical weight |
| definiendum | ⟨ˆN⟩(β,μ):=∑∞N=0wN(β,μ)⋅N |
Discussion
The notation “⟨ˆN⟩” is chosen for the function because we can also introduce the sequence of observables ˆN defined to give us the particle number of each canonical ensemble, i.e. ˆNN=N, and then the above coincides with the proper grand canonical expectation value of ˆN. Notice that this ˆN is sometimes denoted by N, which can get a little confusing.
Theorems
- Given the grand potential Ω(β,μ), we find
| ⟨ˆN⟩=−∂∂μΩ |
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- For the deviation of the particle number, we find
| 1β∂∂μ⟨ˆN⟩=⟨ˆN2⟩−⟨ˆN⟩2 |
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- The entry grand canonical partition function shortly discusses free bosons and fermions.
Context
