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Grand-canonical particle number identity

In the grand canonical ensemble, the mean particle number is the μ-derivative of log Ξ, equivalently the −μ-derivative of the grand potential.
Grand-canonical particle number identity

Statement

In the at inverse temperature β\beta and chemical potential μ\mu, let the grand partition function be

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

(or the analogous classical phase-space integral). Then the mean particle number satisfies

⟨N⟩  =  1β ∂∂μln⁡Ξ(β,μ). \langle N\rangle \;=\; \frac{1}{\beta}\,\frac{\partial}{\partial \mu}\ln \Xi(\beta,\mu).

Equivalently, with the defined by Ω(β,μ)=−(1/β)ln⁡Ξ(β,μ)\Omega(\beta,\mu)=-(1/\beta)\ln \Xi(\beta,\mu),

⟨N⟩  =  − ∂Ω∂μ∣β. \langle N\rangle \;=\; -\,\frac{\partial \Omega}{\partial \mu}\Big|_{\beta}.

Key hypotheses

  • The is well-defined for the given (β,μ)(\beta,\mu).
  • Differentiation with respect to μ\mu can be interchanged with the trace/integral.

Conclusions

  • μ\mu is conjugate to NN: changing changes ⟨N⟩\langle N\rangle via a derivative of ln⁡Ξ\ln\Xi.
  • Thermodynamic form: ⟨N⟩\langle N\rangle is the negative μ\mu-derivative of Ω\Omega.

Proof idea / significance

Differentiate Ξ(β,μ)\Xi(\beta,\mu) with respect to μ\mu: ∂μΞ=β Tr ⁣(N e−β(H−μN))\partial_\mu \Xi = \beta\,\mathrm{Tr}\!\left(N\,e^{-\beta(H-\mu N)}\right). Divide by Ξ\Xi to convert to an expectation, giving ∂μln⁡Ξ=β⟨N⟩\partial_\mu \ln\Xi=\beta\langle N\rangle. This is the basic “conjugate variable” identity underpinning particle-number control by μ\mu.