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Linear barotropic equation of state and its enthalpy coordinates

Example

Fix a dimensionless constant $w$ with $0 < w < 1$ , take $J = (0, \infty)$ , choose a reference energy density $ϵ_{*} > 0$ , and set

$p (ϵ) = w ϵ .$

Choose $N_{*} > 0$ with units $m^{- 3}$ as in Strict charged barotropic tensor system. Then $ϵ + p = (1 + w) ϵ > 0$ and $p^{'} = w \in (0, 1)$ on $J$ , so this is an instance of the strict EOS. The EOS-defined number-density and enthalpy coordinates are

$N (ϵ) = N_{*} {(\frac{ϵ}{ϵ_{*}})}^{1 / (1 + w)}, r (ϵ) = \frac{\sqrt{w}}{1 + w} \log (\frac{ϵ}{ϵ_{*}}) .$

Here the ratios and $r$ are dimensionless. The map $r : J \to R$ is strictly increasing and onto; in these variables $N = N_{*} e^{r / \sqrt{w}}$ . These are mathematical EOS variables, not a microscopic derivation of particle number.

Facts & Assumptions

Given: $0 < w < 1$ , $ϵ > 0$ , $ϵ_{*} > 0$ , $N_{*} > 0$ , and the EOS-defined functions of Strict charged barotropic tensor system and Strict charged-fluid symmetric reduction.

[F1]

Strict charged barotropic tensor system: $N (ϵ) = N_{*} \exp (\int_{ϵ_{*}}^{ϵ} (z + p (z))^{- 1} d z)$ , with $N_{*} > 0$ .

[F2]

Strict charged-fluid symmetric reduction: $r (ϵ) = \int_{ϵ_{*}}^{ϵ} \sqrt{p^{'} (z)} / (z + p (z)) d z$ and $r^{'} = \sqrt{p^{'}} / (ϵ + p) > 0$ for the strict EOS.

Proof

technique · direct integral evaluation and differentiation

1.1givenalgebra

The derivative of $p (ϵ) = w ϵ$ is $p^{'} = w$ . Since $w > 0$ and $ϵ > 0$ , $p^{'} = w \in (0, 1)$ and $ϵ + p = (1 + w) ϵ > 0$ on $J$ . Thus the strict EOS hypotheses hold.

1.2F1givenalgebra

For every $z > 0$ , $(z + p (z))^{- 1} = ((1 + w) z)^{- 1}$ . Substituting this in [F1] and integrating gives $N = N_{*} \exp ((1 + w)^{- 1} \log (ϵ / ϵ_{*})) = N_{*} (ϵ / ϵ_{*})^{1 / (1 + w)}$ . The logarithm is dimensionless because both arguments are energy densities.

2.1F2givenstep 1.2algebra∎

Since $\sqrt{p^{'} (z)} = \sqrt{w}$ , [F2] gives $r = (\sqrt{w} / (1 + w)) \log (ϵ / ϵ_{*})$ . Differentiating yields $r^{'} = \sqrt{w} / ((1 + w) ϵ) > 0$ , so $r$ is strictly increasing. As $ϵ ↓ 0$ its logarithm tends to $- \infty$ , and as $ϵ \to \infty$ it tends to $+ \infty$ ; the positive prefactor preserves these limits, so its range is all of $R$ . Solving the displayed formula gives $ϵ / ϵ_{*} = \exp ((1 + w) r / \sqrt{w})$ ; inserting this in step 1.2 yields $N = N_{*} e^{r / \sqrt{w}}$ .

Linear barotropic equation of state and its enthalpy coordinates

Example

Facts & Assumptions

Proof

Depends on

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