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Gibbs pressure and entropy with physical reference units

Physical scope

Bounded finite-alphabet translation-invariant finite-range lattice models under the stated DLR and SI reference conventions; conditional model deductions only.

Statement

Assume the adopted interacting Gibbs-equilibrium postulate for a bounded finite-range translation-invariant lattice model in dimension $d \geq 1$ , Countable Choice, lattice spacing $a > 0$ metres, and temperature $T > 0$ kelvin. For any stationary DLR equilibrium state with dimensionless pressure per site $p$ , Shannon entropy density $h_{S h}$ in nats per site, and mean energy $u$ in joules per site, the conditional model prediction is $P = \frac{k_{B} T}{a^{d}} p, s = \frac{k_{B}}{a^{d}} h_{S h}, f = \frac{u}{a^{d}} - T s = - \frac{k_{B} T}{a^{d}} p .$ The generalized pressure density $P$ and free-energy density $f$ have units ${J m}^{- d}$ ; for $d = 3$ , $P$ is in pascals ( $P a = {J m}^{- 3}$ ). The reference spin entropy density $s$ has units ${J K}^{- 1} m^{- d}$ . This is a consequence of the bounded-lattice DLR variational identity; it makes no claim that the model describes a material.

Facts & Assumptions

Given: The adopted equilibrium model, a stationary DLR state for a bounded finite-range lattice interaction, SI parameters as in the local definition, and Countable Choice.

[F1]

The equilibrium meaning is adopted as a postulate for a chosen Hamiltonian and reference; it does not imply empirical accuracy or equilibration (Interacting classical Gibbs equilibrium adoption).

[F2]

The dimensionless pressure and entropy/energy densities satisfy $p = h_{S h} - β u$ for stationary DLR states under Countable Choice (DLR states attain the bounded lattice pressure variational principle).

[F3]

$β = (k_{B} T)^{- 1}$ , $P = k_{B} T p / a^{d}$ , and $s = k_{B} h_{S h} / a^{d}$ in the declared SI convention (Classical Gibbs quantities and SI reference conventions).

[A1]

Countable Choice is the measure-extension assumption carried by [F2] (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

1.1A1F2algebra

Under Countable Choice [A1], [F2] gives $p = h_{S h} - β u$ . Multiplication by $- k_{B} T / a^{d}$ gives $- k_{B} T p / a^{d} = - k_{B} T h_{S h} / a^{d} + (k_{B} T β) u / a^{d}$ .

2.1F1F3step 1.1algebra∎

By [F3], $k_{B} T β = 1$ , $s = k_{B} h_{S h} / a^{d}$ , and $P = k_{B} T p / a^{d}$ . Substitution in step 1.1 gives $f = u / a^{d} - T s = - P$ . This exact conclusion is conditional on [F1] and the declared reference convention, not an empirical claim.

Gibbs pressure and entropy with physical reference units

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

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