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Isobaric volume and enthalpy fluctuations

Physical scope

Finite isothermal-isobaric preparation with fixed declared joint reference, positive temperature and pressure, and uniform local envelopes for volume and enthalpy moments.

Statement

For the joint preparation of Isothermal-isobaric joint preparation, write $H = H_{N}$ for its Hamiltonian and set $B = H + P_{e x t} V$ . Suppose the local exponential envelopes hold for $(B, V)$ on the measurable finite-energy set $S_{H} : = {H < \infty}$ , with the joint reference restricted to $S_{H}$ . The Gibbs weight is zero off $S_{H}$ ; there $B$ and $V$ are finite real-valued statistics and the restricted reference remains sigma-finite. At fixed $T > 0$ and $P_{e x t} > 0$ , $\partial_{P_{e x t}} \log Δ = - β ⟨ V ⟩, \partial_{P_{e x t}} ⟨ V ⟩ = - β Var (V),$ $- \partial_{β} \log Δ ∣_{P_{e x t}} = ⟨ B ⟩, C_{P} : = \partial_{T} ⟨ B ⟩ ∣_{P_{e x t}} = \frac{Var (B)}{k_{B} T^{2}} \geq 0.$ When $⟨ V ⟩ > 0$ , the model isothermal compressibility $κ_{T} : = - ⟨ V ⟩^{- 1} \partial_{P_{e x t}} ⟨ V ⟩$ equals $β Var (V) / ⟨ V ⟩$ . These are finite preparation responses for the specified $d V / V_{0}$ reference; they need not equal macroscopic response functions absent an additional limit argument.

Facts & Assumptions

Given: The adopted joint volume-phase law, positive finite normalizer, fixed reference data and uniform local exponential-moment envelopes.

[A1]

The pressure prior is $d V / V_{0}$ and the normalizer is positive finite by the adopted Isothermal-isobaric joint preparation. The local exponential envelopes on $S_{H}$ dominating the required powers of $B$ and $V$ are the additional hypotheses in the Statement.

[A2]

$H$ , the shape reference and $V_{0}$ are temperature independent, and $β = (k_{B} T)^{- 1}$ with $k_{B} > 0$ (SI quantities and unit conventions).

[F1]

Under the common envelope, derivatives of log normalizers are means and covariances of their sufficient statistics (Dominated exponential normalization and covariance).

Proof

Given: [A1]–[A2] and the joint preparation and exponential-derivative interfaces [F1].

1.1A1F1algebra

In $Δ = \int e^{- β H - β P_{e x t} V} d ν$ , differentiate with respect to $P_{e x t}$ using [F1], with statistic $V$ and coefficient $- β P_{e x t}$ . This gives $\partial_{P} \log Δ = - β ⟨ V ⟩$ . Differentiating that expectation once more gives $\partial_{P} ⟨ V ⟩ = - β Var (V)$ . The definition of $κ_{T}$ then gives its displayed formula.

2.1A2F1step 1.1algebra∎

At fixed pressure, use sufficient statistic $B = H + P_{e x t} V$ with parameter $- β$ . [F1] yields $- \partial_{β} \log Δ = ⟨ B ⟩$ and $\partial_{β} ⟨ B ⟩ = - Var (B)$ . Since $d β / d T = - 1 / (k_{B} T^{2})$ , the chain rule gives $C_{P} = Var (B) / (k_{B} T^{2})$ . Both variances are nonnegative integrals of squares.

Isobaric volume and enthalpy fluctuations

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources