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Constructing observables from log partition functions

Ensemble averages and fluctuations are obtained by differentiating the log of the partition function with respect to conjugate parameters.
Constructing observables from log partition functions

A central reason partition functions are useful is that derivatives of their logarithms generate equilibrium observables and their fluctuations. This is the concrete bridge between microscopic definitions (microstates and energies) and macroscopic thermodynamic response functions.

Throughout, expectations are taken in the relevant ensemble.

Canonical ensemble: derivatives of log⁡ZN\log Z_N

Let ZN(β,V)Z_N(\beta,V) be the (constructed in ). Then:

Energy and energy fluctuations

The mean energy is obtained from

⟨H⟩  =  −∂∂β log⁡ZN(β,V). \langle H \rangle \;=\; -\frac{\partial}{\partial \beta}\,\log Z_N(\beta,V).

The energy variance is the second derivative

Var(H)  =  ∂2∂β2 log⁡ZN(β,V), \mathrm{Var}(H) \;=\; \frac{\partial^2}{\partial \beta^2}\,\log Z_N(\beta,V),

which is the basic instance of the general principle that second derivatives of log⁡Z\log Z give .

This links directly to heat capacity: at fixed (N,V)(N,V),

CV  =  (∂⟨H⟩∂T)N,V  =  kBβ2 Var(H), C_V \;=\; \left(\frac{\partial \langle H\rangle}{\partial T}\right)_{N,V} \;=\; k_B \beta^2\, \mathrm{Var}(H),

connecting the fluctuation formula to and the .

Conjugate observables via parameter derivatives

Suppose the Hamiltonian depends on an external parameter λ\lambda (e.g. a field strength), H=HλH=H_\lambda. Then

∂∂λ log⁡ZN(β,V;λ)  =  −β ⟨∂Hλ∂λ⟩. \frac{\partial}{\partial \lambda}\,\log Z_N(\beta,V;\lambda) \;=\; -\beta\,\Bigl\langle \frac{\partial H_\lambda}{\partial \lambda}\Bigr\rangle.

Thus the observable conjugate to λ\lambda is obtained as a derivative of log⁡Z\log Z.

Grand canonical ensemble: derivatives of log⁡Ξ\log \Xi

Let Ξ(β,μ,V)\Xi(\beta,\mu,V) be the (constructed in ). Then:

Mean particle number and its fluctuations

The mean particle number is

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

The particle-number variance is

Var(N)  =  ∂2∂(βμ)2 log⁡Ξ(β,μ,V), \mathrm{Var}(N) \;=\; \frac{\partial^2}{\partial (\beta\mu)^2}\,\log \Xi(\beta,\mu,V),

and more generally mixed derivatives give .

Energy–number covariance

Because Ξ\Xi depends on both β\beta and βμ\beta\mu, mixed derivatives encode correlations between energy and particle number. A standard example is

∂2∂β ∂(βμ)log⁡Ξ  =  −Cov(H,N), \frac{\partial^2}{\partial \beta\,\partial (\beta\mu)} \log \Xi \;=\; -\mathrm{Cov}(H,N),

where Cov\mathrm{Cov} is the of the joint fluctuations in the grand canonical ensemble.

Log partition functions as cumulant generators

A useful viewpoint is that log⁡Z\log Z and log⁡Ξ\log \Xi are cumulant generating objects for the observables conjugate to the parameters. This is formalized in constructions like and .

Physical interpretation

  • First derivatives of log⁡Z\log Z or log⁡Ξ\log \Xi give equilibrium “equations of state” (mean values of conserved or conjugate quantities).
  • Second derivatives quantify equilibrium fluctuations and linear response (susceptibilities), linking to and related response coefficients.
  • Taking the logarithm is essential: it converts products of partition functions (from composing subsystems) into sums, making derivatives match extensive thermodynamic behavior (see ).