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Independent finite entropy-exchange equilibrium

Statement

Let $r \geq 1$ be finite. For each cell $i = 1, \dots, r$ , let $U_{i} \subseteq R^{2}$ be open and convex, with coordinates $(E_{i}, n_{i})$ , and let $S_{i} \in C^{1} (U_{i}, R)$ be concave. The coordinates may be assigned SI dimensions $[E_{i}] = J$ , $[n_{i}] = 1$ , and $[S_{i}] = J K^{- 1}$ ; the particle-number coordinate is a continuous thermodynamic variable. Fix constants $N_{i} > 0$ (dimensionless), $Φ_{i} \in R$ (volts), and a common charge $q \in R$ (coulombs). Define $E (x) : = \sum_{i = 1}^{r} (N_{i} E_{i} + q Φ_{i} n_{i}), N (x) : = \sum_{i = 1}^{r} n_{i}, S_{t o t} (x) : = \sum_{i = 1}^{r} S_{i} (E_{i}, n_{i}) .$ For a supplied feasible interior point $x^{*} \in U : = \prod_{i} U_{i}$ , let $A : R^{2 r} \to R^{2}$ be the linear map whose rows are the coefficients of $(E, N)$ . Call $x^{*}$ stationary under independent exchange variations when $D S_{t o t} (x^{*}) [h] = 0$ for every $h \in \ker A$ . Assume $\partial_{E_{i}} S_{i} (x_{i}^{*}) > 0$ and define $\frac{1}{T_{i}} : = \partial_{E_{i}} S_{i} (x_{i}^{*}), \frac{μ_{i}}{T_{i}} : = - \partial_{n_{i}} S_{i} (x_{i}^{*}) .$ Then $x^{*}$ is stationary if and only if there are constants $T_{K} > 0$ and $μ_{K} \in R$ such that, for every cell, $N_{i} T_{i} = T_{K}, N_{i} μ_{i} + q Φ_{i} = μ_{K} .$ If each $S_{i}$ is concave, every such stationary point is a global maximizer of $S_{t o t}$ on the feasible set in $U$ . If each $S_{i}$ is strictly concave, that maximizer is unique whenever it exists. No existence or boundary attainment of a stationary point is asserted. These are finite-dimensional conditional deductions; the coefficients $N_{i}, Φ_{i}$ are inputs, not predictions about a physical system.

Facts & Assumptions

Given: The finite product domain, cell entropies, positive coefficients $N_{i}$ , fixed $q, Φ_{i}$ , the two affine constraints, and a feasible interior point $x^{*}$ .

[A1]

$r \geq 1$ , every $U_{i} \subseteq R^{2}$ is open and convex, $S_{i} \in C^{1} (U_{i}, R)$ is concave, $N_{i} > 0$ , and $q, Φ_{i}$ are fixed real constants.

[A2]

The point $x^{*} \in U$ satisfies $A x^{*} = (E_{0}, N_{0})$ for fixed values $(E_{0}, N_{0})$ , and $\partial_{E_{i}} S_{i} (x_{i}^{*}) > 0$ for every cell $i$ .

[A3]

For the forward implication and the concavity conclusion only, assume $x^{*}$ is stationary under independent exchange variations, meaning $D S_{t o t} (x^{*}) [h] = 0$ for every $h \in \ker A$ .

[A4]

For the converse implication only, assume there are constants $T_{K} > 0$ and $μ_{K} \in R$ satisfying the two redshift equations in the Statement.

Proof

1.1givenA1algebra

Order coordinates as $(E_{1}, n_{1}, \dots, E_{r}, n_{r})$ . The first row of $A$ has $E_{i}$ coefficient $N_{i} > 0$ , while the second row has every $E_{i}$ coefficient zero and every $n_{i}$ coefficient one. Hence the two rows are linearly independent and $rank A = 2$ . Therefore $\dim (\ker A)^{⊥} = 2$ . The range of $A^{T}$ is contained in $(\ker A)^{⊥}$ because $A^{T} y \cdot h = y \cdot A h = 0$ whenever $h \in \ker A$ ; $A^{T}$ is injective since $A$ has rank two. Its range also has dimension two, so $range (A^{T}) = (\ker A)^{⊥}$ .

2.1givenA2A3step 1.1algebra

Under the forward hypothesis [A3], $\nabla S_{t o t} (x^{*})$ is orthogonal to $\ker A$ , hence lies in $range (A^{T})$ by step 1.1. Thus there are unique multipliers $λ, ν \in R$ with, for each $i$ , $\partial_{E_{i}} S_{i} (x_{i}^{*}) = λ N_{i}, \partial_{n_{i}} S_{i} (x_{i}^{*}) = λ q Φ_{i} + ν .$ Since $\partial_{E_{i}} S_{i} (x_{i}^{*}) > 0$ and $N_{i} > 0$ , $λ > 0$ .

3.1givenA1A2A3A4step 2.1algebra

Under [A3], set $T_{K} : = 1 / λ > 0$ and $μ_{K} : = - ν / λ$ using the multipliers of step 2.1. The first multiplier equation gives $1 / T_{i} = λ N_{i}$ , so $N_{i} T_{i} = T_{K}$ . The second gives $- μ_{i} / T_{i} = λ q Φ_{i} + ν$ . Multiplying by $- N_{i} T_{i}$ and using $N_{i} T_{i} = 1 / λ$ yields $N_{i} μ_{i} + q Φ_{i} = - ν / λ = μ_{K}$ . Conversely, assume [A4] and set $λ : = 1 / T_{K}$ and $ν : = - μ_{K} / T_{K}$ . Then $\partial_{E_{i}} S_{i} (x_{i}^{*}) = 1 / T_{i} = N_{i} / T_{K} = λ N_{i}$ , and $\partial_{n_{i}} S_{i} (x_{i}^{*}) = - μ_{i} / T_{i} = - N_{i} μ_{i} / T_{K} = (q Φ_{i} - μ_{K}) / T_{K} = λ q Φ_{i} + ν$ . Thus $\nabla S_{t o t} (x^{*}) = A^{T} (λ, ν)$ . For every $h \in \ker A$ , $D S_{t o t} (x^{*}) [h] = (λ, ν) \cdot A h = 0$ ; feasibility is [A2], so $x^{*}$ is stationary. This proves the reverse implication without using the forward stationarity hypothesis.

4.1givenA1A2A3step 2.1algebra∎

For the concavity conclusion assume [A3]. For any other feasible $y \in U$ , $A (y - x^{*}) = 0$ . By step 2.1, $\nabla S_{t o t} (x^{*}) = A^{T} (λ, ν)$ , hence $D S_{t o t} (x^{*}) [y - x^{*}] = 0$ . Put $φ (t) : = S_{t o t} (x^{*} + t (y - x^{*}))$ for $0 \leq t \leq 1$ . For $0 < s < t \leq 1$ , concavity gives $φ (s) \geq (1 - s / t) φ (0) + (s / t) φ (t)$ , so $(φ (s) - φ (0)) / s \geq (φ (t) - φ (0)) / t$ . Letting $s ↓ 0$ and using differentiability shows $φ (t) - φ (0) \leq t φ^{'} (0) = 0$ , so $S_{t o t} (y) \leq S_{t o t} (x^{*})$ . If every $S_{i}$ is strictly concave and $y \neq x^{*}$ , at least one cell coordinate differs; its entropy is strictly concave along the segment, while every other cell entropy is concave, so their sum $φ$ is strictly concave. If $y$ were also a maximizer, strict concavity would give $φ (t) > (1 - t) φ (0) + t φ (1) = φ (0)$ for $0 < t < 1$ , contradicting the tangent bound. Thus no distinct feasible $y$ is also a maximizer. This proves the conditional maximum and uniqueness claims without asserting that any stationary point exists.

Remarks

The proof uses only the finite affine constraint map and differentiable concave functions. It does not import a statistical sampling rule, a Gibbs state, an equation of state, or a gravitational field equation. In the neutral case $q = 0$ , the particle relation reduces to $N_{i} μ_{i} = μ_{K}$ without division by $q$ .

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.