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Finite bath can stabilize negative heat capacity exchange

Example

Take a negative-capacity subsystem at an interior energy $E_{*}$ with Gibbs-volume temperature $T_{*} > 0$ and capacity $C_{G} (E_{*}) < 0$ , as in Finite-bath entropy curvature and specified exchange stability. Set $C_{b} = ∣ C_{G} (E_{*}) ∣ / 2$ . For any energy offset $E_{0} \in R$ in joules, reference temperature $T_{r e f} > 0$ in kelvin, and additive entropy constant $S_{0}$ in $J K^{- 1}$ , take $S_{b} (E_{b}) = C_{b} \log (\frac{E_{b} - E_{0}}{C_{b} T_{r e f}}) + S_{0}, E_{b} > E_{0},$ and choose $E_{b} = E_{0} + C_{b} T_{*}$ . The argument of the logarithm is dimensionless.

Facts & Assumptions

Given: A negative-capacity finite subsystem at common temperature $T_{*}$ and the selected finite-bath exchange model.

[F1]

For the specified finite-bath entropy and exchange law, $0 < C_{b} < ∣ C_{G} ∣$ gives strict negative combined-entropy curvature and local exponential attraction under $\dot{x} = L S_{t o t}^{'} (x)$ . (Finite-bath entropy curvature and specified exchange stability)

Verification

1.1givenalgebra

Differentiating the displayed bath entropy gives $S_{b}^{'} (E_{b}) = C_{b} / (E_{b} - E_{0})$ , so $T_{b} = 1 / S_{b}^{'} (E_{b}) = (E_{b} - E_{0}) / C_{b}$ and $d E_{b} / d T_{b} = C_{b}$ . The chosen $E_{b}$ makes $T_{b} = T_{*}$ .

2.1F1step 1.1algebra

Since $C_{b} = ∣ C_{G} ∣ / 2$ , the reciprocal-capacity sum is $1 / C_{G} + 1 / C_{b} = - 1 / ∣ C_{G} ∣ + 2 / ∣ C_{G} ∣ = 1 / ∣ C_{G} ∣ > 0$ . At the matched-temperature critical point, [F1] gives $S_{t o t}^{''} = - \frac{1}{T_{*}^{2}} (\frac{1}{C_{G}} + \frac{1}{C_{b}}) = - \frac{1}{T_{*}^{2} ∣ C_{G} ∣} < 0.$ Thus the combined entropy has a strict local maximum in this exchange direction.

3.1F1step 2.1∎

The selected effective law $\dot{x} = L S_{t o t}^{'} (x)$ has nonnegative entropy production and exponentially attracts this local maximum by [F1]. This is a constructed mathematical reservoir model, not empirical data or a generic gravitational stability conclusion.

Depends on

Finite-bath entropy curvature and specified exchange stability

Finite bath can stabilize negative heat capacity exchange

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources