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Fluctuation formulas from log Z

How derivatives of the log partition function generate variances, covariances, and response coefficients in equilibrium ensembles.
Fluctuation formulas from log Z

In equilibrium statistical mechanics, the logarithm of the partition function is the central generator of both mean values and fluctuations. Starting from the in the , one can systematically obtain fluctuation and response formulas by differentiating log⁡Z\log Z with respect to the parameters that couple to observables.

Setup: parameters conjugate to observables

Let a family of Hamiltonians depend smoothly on parameters λ=(λ1,…,λm)\lambda=(\lambda_1,\dots,\lambda_m), and define the canonical partition function

Z(β,λ)  =  ∫Γexp⁡ ⁣(−β H(x;λ)) dμ(x), Z(\beta,\lambda) \;=\; \int_{\Gamma} \exp\!\big(-\beta\,H(x;\lambda)\big)\, \mathrm{d}\mu(x),

where β\beta is the and dμ\mathrm{d}\mu is the underlying measure on (see for the classical case). The associated is

⟨A⟩β,λ  =  1Z(β,λ)∫ΓA(x) e−βH(x;λ) dμ(x). \langle A\rangle_{\beta,\lambda} \;=\; \frac{1}{Z(\beta,\lambda)}\int_{\Gamma} A(x)\,e^{-\beta H(x;\lambda)}\,\mathrm{d}\mu(x).

A particularly important (and common) parametrization is a linear coupling to observables AiA_i:

H(x;λ)  =  H0(x)  −  ∑i=1mλi Ai(x). H(x;\lambda) \;=\; H_0(x) \;-\; \sum_{i=1}^m \lambda_i\,A_i(x).

In that case, each λi\lambda_i is a “field” conjugate to AiA_i.

First derivatives: means

Differentiate log⁡Z\log Z under the integral sign (when justified; in practice this is ensured by standard dominated-convergence conditions). One obtains the general identity

∂∂λilog⁡Z(β,λ)  =  −β ⟨∂H∂λi⟩β,λ. \frac{\partial}{\partial \lambda_i}\log Z(\beta,\lambda) \;=\; -\beta\,\Big\langle \frac{\partial H}{\partial \lambda_i}\Big\rangle_{\beta,\lambda}.

For linear couplings ∂λiH=−Ai\partial_{\lambda_i}H=-A_i, this becomes

∂∂λilog⁡Z(β,λ)  =  β ⟨Ai⟩β,λ. \frac{\partial}{\partial \lambda_i}\log Z(\beta,\lambda) \;=\; \beta\,\langle A_i\rangle_{\beta,\lambda}.

A special case is differentiation with respect to β\beta:

∂∂βlog⁡Z(β,λ)  =  −⟨H⟩β,λ. \frac{\partial}{\partial \beta}\log Z(\beta,\lambda) \;=\; -\langle H\rangle_{\beta,\lambda}.

Thus log⁡Z\log Z encodes the mean energy directly.

These identities are the core of .

Second derivatives: variances and covariances

Second derivatives produce fluctuations. For linear couplings, the mixed second derivative is

∂2∂λi ∂λjlog⁡Z(β,λ)  =  β2 Covβ,λ(Ai,Aj), \frac{\partial^2}{\partial \lambda_i\,\partial \lambda_j}\log Z(\beta,\lambda) \;=\; \beta^2\,\mathrm{Cov}_{\beta,\lambda}(A_i,A_j),

where Cov(Ai,Aj)=⟨AiAj⟩−⟨Ai⟩⟨Aj⟩\mathrm{Cov}(A_i,A_j)=\langle A_iA_j\rangle-\langle A_i\rangle\langle A_j\rangle (see ).

In particular,

∂2∂λi2log⁡Z(β,λ)  =  β2 Varβ,λ(Ai), \frac{\partial^2}{\partial \lambda_i^2}\log Z(\beta,\lambda) \;=\; \beta^2\,\mathrm{Var}_{\beta,\lambda}(A_i),

recovering the as a derivative of log⁡Z\log Z.

For energy fluctuations (no λ\lambda needed),

∂2∂β2log⁡Z(β)  =  Varβ(H), \frac{\partial^2}{\partial \beta^2}\log Z(\beta) \;=\; \mathrm{Var}_\beta(H),

since ∂βlog⁡Z=−⟨H⟩\partial_\beta \log Z=-\langle H\rangle and ∂β⟨H⟩=−Var(H)\partial_\beta \langle H\rangle=-\mathrm{Var}(H).

These are the canonical “fluctuation formulas” (see also ).

Physical interpretation: response = fluctuations

If λ\lambda is a physical field, then ∂λ⟨A⟩\partial_{\lambda}\langle A\rangle is a linear response coefficient (a susceptibility). From the identities above,

∂∂λj⟨Ai⟩β,λ  =  β Covβ,λ(Ai,Aj), \frac{\partial}{\partial \lambda_j}\langle A_i\rangle_{\beta,\lambda} \;=\; \beta\,\mathrm{Cov}_{\beta,\lambda}(A_i,A_j),

so response is controlled by equilibrium fluctuations. In common cases this is exactly the equilibrium form of a fluctuation–dissipation relation (compare with ).

Higher derivatives: cumulants

More generally, all higher derivatives of log⁡Z\log Z generate higher-order connected moments (cumulants). This perspective is made explicit in and in .

Finally, the thermodynamic potentials inherit these derivative structures via F=−(1/β)log⁡ZF=-(1/\beta)\log Z (see and ): convexity/concavity properties of log⁡Z\log Z translate into stability conditions and positivity of variances.