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Finite-bath entropy curvature and specified exchange stability

Statement

Let $I_{1}, I_{2} \subseteq R$ be open energy intervals (energies in joules), and let $S_{i} : I_{i} \to R$ be $C^{2}$ entropies measured in joules per kelvin. Fix a total energy $E_{t o t}$ in joules and define the open exchange domain $I : = {x \in I_{1} : E_{t o t} - x \in I_{2}}$ . For $x \in I$ , set $E_{1} = x$ , $E_{2} = E_{t o t} - x$ , and $T_{i} (E_{i}) : = 1 / S_{i}^{'} (E_{i})$ wherever $S_{i}^{'} (E_{i}) > 0$ . At a point $x \in I$ with $T_{1} (E_{1}) = T_{2} (E_{2}) = T > 0$ and $T_{i}^{'} (E_{i}) \neq 0$ , define the finite nonzero heat capacities $C_{i} (E_{i}) : = 1 / T_{i}^{'} (E_{i}) = d E_{i} / d T_{i}$ . For $S_{t o t} (x) = S_{1} (x) + S_{2} (E_{t o t} - x),$ one has $S_{t o t}^{''} (x) = - \frac{1}{T^{2}} (\frac{1}{C_{1}} + \frac{1}{C_{2}}) .$ In particular, if $C_{1} < 0 < C_{2}$ , the critical point is a strict local entropy maximum in this energy-exchange direction whenever $0 < C_{2} < ∣ C_{1} ∣$ .

This condition is realized by taking $S_{1} = S_{G}$ from Negative Gibbs-volume capacity in a finite regulated shell at an interior branch energy $E_{*}$ (for example, $r_{0} / R < δ < 1 / 16$ ), where $T_{*} = T_{G} (E_{*}) > 0$ and $C_{1} = C_{G} (E_{*}) < 0$ . The explicit phase-volume formula makes $S_{1}$ $C^{2}$ on an open energy interval around $E_{*}$ . Choose a finite bath entropy $S_{b} (E_{b}) = C_{b} \log (\frac{E_{b} - E_{0}}{C_{b} T_{r e f}}) + S_{0}, E_{b} > E_{0},$ with $C_{b} > 0$ in $J K^{- 1}$ , $E_{0}$ in $J$ , $T_{r e f} > 0$ in $K$ , and $S_{0}$ in $J K^{- 1}$ . It has $T_{b} = (E_{b} - E_{0}) / C_{b}$ and heat capacity $C_{b}$ ; choose $E_{b} = E_{0} + C_{b} T_{*}$ to match the subsystem temperature $T_{*}$ and choose $C_{b} < ∣ C_{1} ∣$ .

Separately adopt, on the open domain $I$ and with time $t$ in seconds, the effective scalar exchange equation $\dot{x} = L S_{t o t}^{'} (x), L > 0 J K s^{- 1} .$ For each strict local maximum $x_{*} \in I$ with $S_{t o t}^{''} (x_{*}) < 0$ , there are $a, r > 0$ such that $I_{0} = [x_{*} - r, x_{*} + r] \subset I$ , $S_{t o t}^{''} \leq - a$ on $I_{0}$ , and every solution with $x (0) \in I_{0}$ exists for all $t \geq 0$ , remains in $I_{0}$ , and satisfies $∣ x (t) - x_{*} ∣ \leq e^{- L a t} ∣ x (0) - x_{*} ∣$ . Thus the equilibrium solution $x (t) \equiv x_{*}$ attracts those initial states exponentially. Along every solution, $d S_{t o t} / d t = L (S_{t o t}^{'})^{2} \geq 0$ . This is stability of the specified one-dimensional exchange law, not of a general relativistic star or a causal transport equation.

Facts & Assumptions

Given: The entropy functions, their positive temperatures and finite nonzero heat capacities, a common-temperature critical point, and the separately adopted scalar exchange equation.

[F1]

At an interior regulated Kepler branch energy $E_{*}$ , the Gibbs-volume response formulas supply $T_{*} = T_{G} (E_{*}) > 0$ and finite $C_{1} = C_{G} (E_{*}) < 0$ . (Negative Gibbs-volume capacity in a finite regulated shell)

[F2]

A continuous vector field locally Lipschitz in its state variable has a unique local solution through every point of its open ODE domain. (Picard-Lindelöf local existence and uniqueness for first-order systems)

[F3]

A Picard-Lindelof solution whose graph approaches a compact subset of its open ODE domain at a finite endpoint extends past that endpoint. (A solution whose graph approaches a compact interior region at a finite endpoint extends past that endpoint)

[F4]

For positive $μ, A, r_{0}, h_{*}$ and $R > 16 r_{0}$ , the branch $A / R < e < A / (16 r_{0})$ has $δ = e r_{0} / A$ and $Ω (- e) = K e^{- 3 / 2} J (δ)$ , where $K = (4 π) (4 π / 3) (2 μ)^{3 / 2} A^{3} / h_{*}^{3} > 0$ , $J (δ) = \int_{δ}^{1} s^{1 / 2} (1 - s)^{3 / 2} d s$ , and $S_{G} = k_{B} \log Ω$ . (Regulated relative Kepler phase volume and shell law)

Proof

1.1givenalgebra

Since $S_{i}^{'} = 1 / T_{i}$ , differentiation with respect to energy gives $S_{i}^{''} = - T_{i}^{- 2} (d T_{i} / d E_{i}) = - 1 / (T_{i}^{2} C_{i})$ .

2.1givenstep 1.1algebra

The first derivative of $S_{t o t}$ is $S_{1}^{'} (x) - S_{2}^{'} (E_{t o t} - x)$ , so it vanishes exactly at common temperature. The second derivative is the sum $S_{1}^{''} (x) + S_{2}^{''} (E_{t o t} - x)$ , which by step 1.1 is the stated expression. If $C_{1} < 0 < C_{2}$ , then $S_{t o t}^{''} < 0$ exactly when $1 / C_{2} > 1 / ∣ C_{1} ∣$ , equivalently $C_{2} < ∣ C_{1} ∣$ ; continuity and the second-derivative test give a strict local maximum.

3.1F1F4step 2.1algebra

On $0 < δ < 1$ , the function $f (δ) = δ^{1 / 2} (1 - δ)^{3 / 2}$ has continuous derivative. The identity $J (δ + h) - J (δ) = - \int_{δ}^{δ + h} f (s) d s$ and continuity of $f$ give $J^{'} = - f$ , so $J^{''} = - f^{'}$ is continuous; also $J > 0$ . Thus [F4] makes $Ω$ positive and $C^{2}$ , and $S_{G} = k_{B} \log Ω$ is $C^{2}$ on an open interval about the interior energy $E_{*}$ . For the displayed bath entropy, $S_{b}^{'} = C_{b} / (E_{b} - E_{0})$ , hence $T_{b} = (E_{b} - E_{0}) / C_{b}$ and $d E_{b} / d T_{b} = C_{b}$ . Choose $0 < C_{b} < ∣ C_{G} (E_{*}) ∣$ , $E_{b}^{*} = E_{0} + C_{b} T_{*}$ , and $E_{t o t} = E_{*} + E_{b}^{*}$ . Both entropies are $C^{2}$ near these energies, their positive temperatures agree, and step 2.1 gives strictly negative total entropy curvature.

4.1F2F3givenstep 3.1algebra∎

At the stipulated maximum $x_{*}$ with $S_{t o t}^{''} (x_{*}) < 0$ , continuity gives a compact interval $I_{0} = [x_{*} - r, x_{*} + r] \subset I$ and $a > 0$ with $S_{t o t}^{''} \leq - a$ there. Integrating from $x_{*}$ , where $S_{t o t}^{'} = 0$ , gives $(x - x_{*}) S_{t o t}^{'} (x) \leq - a (x - x_{*})^{2}$ . The vector field $L S_{t o t}^{'}$ on $R \times I$ is $C^{1}$ and hence locally Lipschitz, so [F2] gives a unique local solution. At the two endpoints of $I_{0}$ the field points strictly inward, preventing a first exit, including for endpoint initial data. The chain rule gives $d S_{t o t} / d t = L (S_{t o t}^{'})^{2} \geq 0$ along every solution and, for those in $I_{0}$ , $\frac{d}{d t} ∣ x - x_{*} ∣^{2} \leq - 2 L a ∣ x - x_{*} ∣^{2} .$ Multiplying by $e^{2 L a t}$ and differentiating yields $∣ x (t) - x_{*} ∣ \leq e^{- L a t} ∣ x (0) - x_{*} ∣$ . If its maximal forward endpoint $b$ were finite, its graph for $0 \leq t < b$ would lie in the compact set $[0, b] \times I_{0} \subset R \times I$ , so [F3] would extend the solution, a contradiction. The local basin therefore exists for all forward times and converges exponentially.

Finite-bath entropy curvature and specified exchange stability

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

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