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Grand-canonical number fluctuation identity

In the grand canonical ensemble, particle-number fluctuations equal the μ-derivative of the mean: Var(N) = (1/β) ∂⟨N⟩/∂μ = (1/β²) ∂² ln Ξ/∂μ².
Grand-canonical number fluctuation identity

Statement

In the with grand partition function

Ξ(β,μ)=Tr exp⁡ ⁣(−β(H−μN)), \Xi(\beta,\mu)=\mathrm{Tr}\,\exp\!\big(-\beta(H-\mu N)\big),

the particle-number variance satisfies

Var(N)  =  1β ∂∂μ⟨N⟩  =  1β2 ∂2∂μ2ln⁡Ξ(β,μ). \mathrm{Var}(N)\;=\;\frac{1}{\beta}\,\frac{\partial}{\partial \mu}\langle N\rangle \;=\;\frac{1}{\beta^2}\,\frac{\partial^2}{\partial \mu^2}\ln \Xi(\beta,\mu).

Key hypotheses

  • The grand canonical state exists for the given (β,μ)(\beta,\mu) and is differentiable in μ\mu.
  • Differentiation under the trace/integral is justified.

Conclusions

  • Number susceptibility (response of ⟨N⟩\langle N\rangle to μ\mu) equals a fluctuation: ∂μ⟨N⟩=β Var(N)\partial_\mu\langle N\rangle = \beta\,\mathrm{Var}(N).
  • This is the particle-number analog of canonical fluctuation identities (compare ).

Proof idea / significance

From ∂μln⁡Ξ=β⟨N⟩\partial_\mu\ln\Xi=\beta\langle N\rangle (see ), differentiate once more in μ\mu: ∂μ2ln⁡Ξ=β ∂μ⟨N⟩\partial_\mu^2\ln\Xi = \beta\,\partial_\mu\langle N\rangle. A direct differentiation of the expectation also gives ∂μ⟨N⟩=β Var(N)\partial_\mu\langle N\rangle=\beta\,\mathrm{Var}(N) by the same covariance algebra as in the canonical case. This connects compressibility-like responses to equilibrium density fluctuations.