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Fixed lapse two gas normalization

Physical scope

Grand-canonical classical ideal gas on a prescribed flat finite box with constant dimensionless lapse two and a stated Killing-generator normalization; the local temperature parameter is T_K/2 and is not a reported measurement.

Example

Let $B = (0, L)^{3}$ with $L > 0$ , $I_{x^{0}} \subset R$ open, $h_{i j} = δ_{i j}$ , and constant dimensionless lapse $N = 2$ on the prescribed static metric $g = - 4 (d x^{0})^{2} + δ_{i j} d x^{i} d x^{j}$ , with $x^{0} = c t$ . Choose positive particle mass $m > 0$ in kilograms, $c > 0$ in metres per second, action reference $h_{*} > 0$ in joule-seconds, inverse Killing-energy parameter $β_{K} > 0$ in inverse joules, and dimensionless fugacity $z > 0$ . Select the grand-canonical branch, in which particle number is summed rather than fixed. Since the metric coefficients are independent of $x^{0}$ , choose the Killing generator $ξ = \partial_{x^{0}}$ to define $T_{K} = (k_{B} β_{K})^{- 1}$ . This fixes a normalization convention for the generator; $T_{K}$ is not identified with the proper temperature at the wall.

Facts & Assumptions

Given: The spatial box is bounded with positive finite volume, the static metric is prescribed, its lapse is the constant $N = 2$ , the Killing generator has the stated normalization, $m, c, h_{*}, β_{K}, z > 0$ , and the grand-canonical branch is selected.

[F1]

On a prescribed bounded-lapse static background, the selected grand-canonical classical gas has $0 < Z_{1} < \infty$ , $Ξ = e^{z Z_{1}} \in (0, \infty)$ , and $T_{l o c} = T_{K} / N$ . (Constructed static gas redshift and hydrostatics)

Verification

1.1givenF1

The constant lapse satisfies $0 < N_{0} \leq N \leq N_{1} < \infty$ with $N_{0} = N_{1} = 2$ , the flat spatial box has positive finite volume, the classical parameters are positive, and the grand-canonical branch is selected; thus [F1] applies.

2.1F1step 1.1algebra∎

Apply [F1] to the selected branch: $Z_{1}$ is finite and positive and $Ξ = e^{z Z_{1}}$ is finite and positive. Substituting $N = 2$ gives $T_{l o c} = T_{K} / 2$ . These are normalization and parameter identities in the selected model and report no observed temperature.

Depends on

Constructed static gas redshift and hydrostatics

Fixed lapse two gas normalization

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources