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Concave entropy, constrained maxima, and local stability

Physical scope

C2 single-phase entropy charts and explicitly convex affine slices of permitted exchange coordinates; the radial-curvature conclusion applies only to the separately designated common-constitutive-law bulk cone with its adopted entropy transport and common reference conventions.

Statement

Let $C$ be an open convex region of dimensionless numerical coordinates for a compound of independent separated systems, and let $S_{t o t} : C \to R$ be the adopted $C^{2}$ concave total entropy on a smooth single-phase chart. Let the feasible set be the affine slice $F = C \cap {x : A x = b}$ for a fixed real linear map $A$ and vector $b$ . For the following conclusions, assume the mathematical input that for differentiable concave $f$ on an open convex $C$ , an interior point of $C \cap {x : A x = b}$ whose derivative vanishes on $\ker A$ is a global feasible maximizer; strict concavity gives at most one maximizer; for $C^{2}$ concave $f$ , the Hessian quadratic form is negative semidefinite. Under this input, if $x_{*} \in F$ is interior relative to this slice and $D S_{t o t} (x_{*})$ vanishes on $\ker A$ , then $x_{*}$ is a global entropy maximum on $F$ . If $S_{t o t}$ is strictly concave on $F$ , that maximum is unique. The Hessian conclusion does not by itself give strict stability.

The mathematical input is supplied by Concave supporting bound, affine maxima, and Hessian sign on the stated region and admissible affine segments.

There is a further conditional scaling conclusion. On a designated open single-phase bulk cone $C_{c} \subseteq D_{c}$ with the common constitutive law adopted by Common constitutive entropy law for a designated bulk family, the $C^{2}$ entropy Hessian has zero radial direction: $D^{2} S_{c} (x) [h, x] = 0$ for every $x \in C_{c}$ and every vector $h$ . This branch adopts coordinate-preserving entropy transport between the distinct scaled-system tag and one fixed material-law chart, with common species, units and reference conventions. The null-direction conclusion is not asserted for the general tagged family from Concave macroscopic fundamental entropy and Additive separated systems and tagged bulk scaling alone.

Facts & Assumptions

Given: A declared affine feasible slice $F$ of an open convex chart, a $C^{2}$ concave entropy sum, and an interior point stationary along every feasible exchange direction. For the final conditional clause only, adopt the common-constitutive-law bulk postulate on its designated open single-phase cone $C_{c}$ . This is a conditional model deduction; it makes no empirical claim.

[A1]

Concave macroscopic fundamental entropy: Entropy is finite and concave on each specified convex region, additive for the admitted independent separated systems, and $C^{2}$ only on designated open single-phase charts. Its tagged bulk clause is $S_{c^{(λ)}} (δ_{λ} x) = λ S_{c} (x)$ .

[F1]

Isolated compound and feasible equilibrium set: The feasible set is the model-declared subset fixed by totals, allowed exchange directions, and retained partitions; it need not be convex or attain a maximum unless further conditions or a postulate provide them.

[F2]

Additive separated systems and tagged bulk scaling: The scaling map sends coordinates of tag $c$ to a possibly distinct tag $c^{(λ)}$ ; the statement expressly does not identify $c^{(λ)}$ with $c$ .

[A3]

Common constitutive entropy law for a designated bulk family: For the radial clause only, the designated open single-phase cone $C_{c}$ admits every positive scaling, and the coordinate-preserving transport $j_{c, λ} (y) = y$ from its scaled-system chart preserves entropy: $S_{c^{(λ)}} (y) = S_{c} (j_{c, λ} (y))$ . This is additional adopted physical constitutive information with common species, units and energy/entropy reference conventions; the distinct system tags are retained.

[L1]

Concave supporting bound, affine maxima, and Hessian sign: differentiability and concavity along each admissible segment give $f (y) \leq f (x) + D f (x) (y - x)$ . On an affine feasible set, stationarity on $\ker A$ gives a global maximum when every segment from the stationary point to a feasible point is admitted and satisfies concavity. Strict concavity on a convex feasible slice gives uniqueness. At a $C^{2}$ point, local midpoint concavity implies a negative-semidefinite Hessian. The supplier assumes no convexity outside the explicitly used region and no physical attainment postulate.

[L2]

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ : The total derivative of a composition is the composition of total derivatives; it applies to the scaling curve in step 1.3.

Proof

1.1givenA1F1L1algebra

The physical entropy sum is concave on the specified chart $C$ and additive for the admitted separated systems by [A1]. The given affine-slice hypothesis and [F1] identify $F = C \cap {x : A x = b}$ ; since $C$ is convex, each segment between feasible points stays in $C$ and in the affine slice. Thus for every $y \in F$ , the segment from $x_{*}$ to $y$ is an admissible entropy segment, and the given stationarity on $\ker A$ meets [L1]. Its supporting bound gives $S_{t o t} (y) \leq S_{t o t} (x_{*})$ , so $x_{*}$ is a global maximum. If $S_{t o t} ∣_{F}$ is strictly concave, [L1] gives uniqueness because $F$ is convex. These are claims on the stated chart and feasible slice, not an assertion that all of $D_{c}$ is convex.

1.2A1L1givenalgebra

At every $x \in C$ and every vector $v$ , openness permits both $x \pm t v \in C$ for sufficiently small $∣ t ∣$ ; concavity gives $S_{t o t} (x + t v) + S_{t o t} (x - t v) \leq 2 S_{t o t} (x)$ . The adopted $C^{2}$ chart hypothesis therefore meets [L1]'s local Hessian clause and gives $D^{2} S_{t o t} (x) [v, v] \leq 0$ . Semidefiniteness does not assert strict negative curvature in every nonzero direction, strict concavity, or uniqueness without their further hypotheses.

1.3givenA1A3F2L2

For $x \in C_{c}$ and $λ > 0$ , [A3] gives $S_{c} (λ x) = S_{c^{(λ)}} (δ_{λ} x)$ by coordinate-preserving entropy transport, and [A1] gives $S_{c^{(λ)}} (δ_{λ} x) = λ S_{c} (x)$ . Thus the same function on $C_{c}$ is degree-one homogeneous even though the physical system tags remain distinct under [F2]. Define $g (λ) = S_{c} (λ x)$ near $1$ ; its derivative exists on this smooth open chart, and differentiating $g (λ) = λ S_{c} (x)$ at $1$ using [L2] gives $D S_{c} (x) [x] = S_{c} (x)$ for every $x \in C_{c}$ .

2.1step 1.3A1L1algebra∎

Differentiate the identity in step 1.3 in an arbitrary direction $h$ . For $t \neq 0$ , the difference quotient of its left side is $\frac{D S_{c} (x + t h) [x + t h] - D S_{c} (x) [x]}{t} = \frac{(D S_{c} (x + t h) - D S_{c} (x)) [x]}{t} + D S_{c} (x + t h) [h] .$ Since $S_{c}$ is $C^{2}$ , $D S_{c}$ is differentiable by [A1] and [L1]; the limit is $D^{2} S_{c} (x) [h, x] + D S_{c} (x) [h]$ . The derivative of the right side of the Euler identity is $D S_{c} (x) [h]$ , so subtraction gives $D^{2} S_{c} (x) [h, x] = 0$ for every $h$ . In particular, the radial direction has zero Hessian quadratic form and the Hessian is not negative definite on the full cone.

Remarks

The actual entropy postulate uses the distinct tag $c^{(λ)}$ and does not identify it with $c$ . The additional common-constitutive-law branch [A3] explicitly supplies the transport needed for the M4 radial conclusion; the general tagged postulates alone do not supply it. The exact concavity prerequisite [L1] supplies the affine-maximum and Hessian-sign clauses on the admitted chart. A finite-dimensional countermodel is recorded on the companion page.

Concave entropy, constrained maxima, and local stability

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources