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Ensemble entropy and potential identities

Physical scope

Finite canonical, grand and isothermal-isobaric preparations with finite entropy and energy moments and differentiable normalizers; identities use the explicitly chosen reference measures and fixed microscopic models.

Statement

Adopt the canonical, grand and isothermal-isobaric preparations of Canonical thermal preparation, Grand canonical particle-exchange preparation and Isothermal-isobaric joint preparation. Assume their positive finite normalizers, finite first moments and integrable $ρ \log ρ$ , and assume the local parameter envelopes of Dominated exponential normalization and covariance for each normalizer's sufficient statistics. If a Hamiltonian may be $+ \infty$ , restrict its reference to the measurable set where it is finite; the Gibbs density is zero off that set, and the restricted reference remains sigma-finite. Thus every statistic passed to the exponential theorem is finite real-valued. In each preparation, write $U : = ⟨ H ⟩$ for the mean Hamiltonian; on the grand disjoint union let $N$ be the sector index and write $\overset{ˉ}{N} : = ⟨ N ⟩$ . Define the reference-relative entropy by $S = - k_{B} \int ρ \log ρ d m$ using the exact joint reference of each preparation. Then the finite canonical free energy, grand potential and isothermal-isobaric potential satisfy $F = - k_{B} T \log Z = U - T S, d F = - S d T$ at fixed canonical model and reference; $Ω = - k_{B} T \log Ξ = U - T S - μ \overset{ˉ}{N}, d Ω = - S d T - \overset{ˉ}{N} d μ;$ and, with $B = H + P_{e x t} V$ , $G_{i s o P} = - k_{B} T \log Δ = ⟨ H ⟩ + P_{e x t} ⟨ V ⟩ - T S, d G_{i s o P} = - S d T + ⟨ V ⟩ d P_{e x t} .$ For every normalized competing density $p$ with the same reference and the finite energy and entropy conditions of Reference-relative entropy and Gibbs variational inequality, one also has $⟨ H ⟩_{p} - T S [p] \geq F$ , with equality exactly at the canonical density almost everywhere. The differentials hold at fixed remaining model data; integer $N$ is not a differentiable variable. The identities do not assert thermodynamic-limit equivalence or derive any preparation from ignorance.

Facts & Assumptions

Given: The three adopted normalized exponential laws, integrable entropy and energy terms, fixed declared references, and local parameter envelopes.

[A1]

Each relevant density is normalized and has finite first energy/number/ volume moments and $\int ∣ ρ \log ρ ∣ d m < \infty$ .

[A2]

The Hamiltonian and the chosen reference are independent of $T$ ; grand and pressure derivatives are taken with the other displayed controls fixed.

[A3]

Any competing canonical density in the variational clause has finite energy and absolutely integrable $p \log p$ against the same reference.

[F1]

The ensemble probability laws and reference conventions are the three adopted preparations (Canonical thermal preparation, Grand canonical particle-exchange preparation, Isothermal-isobaric joint preparation).

[F2]

Normalizer derivatives equal means under their local exponential envelopes (Dominated exponential normalization and covariance).

[F3]

The entropy integral is interpreted against the selected reference measure (Statistical states, densities, observables and expectations); for a Gibbs density the relative-entropy identity and equality case hold (Reference-relative entropy and Gibbs variational inequality).

[F4]

Temperature, energy and Boltzmann-constant units satisfy $β = (k_{B} T)^{- 1}$ (SI quantities and unit conventions).

Proof

Given: [A1]–[A3] and the preparation, differentiation, entropy and unit interfaces [F1]–[F4].

1.1A1A2A3F1F2F3F4algebra

For a canonical density $ρ = Z^{- 1} e^{- β H}$ , $\log ρ = - β H - \log Z$ . Integrating against $ρ$ gives $S = k_{B} (\log Z + β U)$ , so $- k_{B} T \log Z = U - T S$ because $k_{B} T β = 1$ . For any competitor $p$ , [F3] gives $D (p ∥ ρ) = β (⟨ H ⟩_{p} - T S [p] - F) \geq 0$ , with equality exactly when $p = ρ$ almost everywhere. At fixed model, $U = - \partial_{β} \log Z$ by [F2]. Differentiating $F = - β^{- 1} \log Z$ and using $d β / d T = - 1 / (k_{B} T^{2})$ gives $\partial_{T} F = - S$ , proving the canonical differential.

1.2A1A2F1F2F3F4algebra

On the grand disjoint union, $\log ρ = - β (H - μ N) - \log Ξ$ . Integration gives $S = k_{B} (\log Ξ + β (U - μ \overset{ˉ}{N}))$ , hence $Ω = - k_{B} T \log Ξ = U - T S - μ \overset{ˉ}{N}$ . Applying [F2] to the sufficient statistics $H$ and $N$ gives $\partial_{μ} \log Ξ = β \overset{ˉ}{N}$ and $- \partial_{β} \log Ξ ∣_{μ} = U - μ \overset{ˉ}{N}$ . Thus $\partial_{μ} Ω = - \overset{ˉ}{N}$ and $\partial_{T} Ω = - S$ , proving the grand differential at fixed model.

2.1A1A2F1F2F3F4algebra∎

For the joint isothermal-isobaric density, $\log ρ = - β (H + P_{e x t} V) - \log Δ$ . Integration gives $S = k_{B} (\log Δ + β ⟨ H + P_{e x t} V ⟩)$ and therefore the displayed $G_{i s o P}$ . Applying [F2] to $B = H + P_{e x t} V$ gives $- \partial_{β} \log Δ ∣_{P} = ⟨ B ⟩$ , while $\partial_{P} \log Δ = - β ⟨ V ⟩$ ; hence $\partial_{T} G_{i s o P} = - S$ and $\partial_{P} G_{i s o P} = ⟨ V ⟩$ . This proves the stated differential and completes the conditional identities.

Ensemble entropy and potential identities

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources