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A noncompact phase domain with no attained mixture maximum

Physical scope

A constructed continuous entropy extension that is positive and concave on a convex ambient domain but restricted to a noncompact generator set omitting its maximizing barycenter; it makes no material claim.

Statement refuted

Let $d \geq 1$ , let $K \subseteq C \subseteq R^{d}$ with $K$ noncompact and $C$ convex, and let $P$ be a nonempty family of Borel probability measures on $K$ sharing a prescribed barycenter. The statement refuted is that $\sup_{P \in P} \int_{K} s d P$ is always attained whenever $s = \tilde{s} ∣_{K}$ for a bounded, positive, continuous concave function $\tilde{s} : C \to R$ . Here convexity of $C$ means $(1 - t) x + t y \in C$ for $x, y \in C$ and $t \in [0, 1]$ , and concavity means

\tilde{s} ((1 - t) x + t y) \geq (1 - t) \tilde{s} (x) + t \tilde{s} (y) (x, y \in C, t \in [0, 1]) .

Example

Let $K = (- 1, 0) \cup (0, 1)$ , which is noncompact: the open cover $U_{n} = (- 1, - 1 / n) \cup (1 / n, 1)$ for $n \geq 2$ has no finite subcover. Let $C = [- 1, 1]$ and use dimensionless coordinate $x$ . For constants $s_{0} > a > 0$ in $J m o l^{- 1} K^{- 1}$ , define the ambient entropy extension and its restriction by

\tilde{s} (x) = s_{0} - a x^{2} (x \in C), s = \tilde{s} ∣_{K} .

For $x, y \in C$ and $t \in [0, 1]$ ,

\tilde{s} ((1 - t) x + t y) = (1 - t) \tilde{s} (x) + t \tilde{s} (y) + a t (1 - t) (x - y)^{2} \geq (1 - t) \tilde{s} (x) + t \tilde{s} (y),

so $\tilde{s}$ is concave on the convex ambient domain. It is continuous, bounded, and positive since $s_{0} - a > 0$ . The admissible generator set $K$ excludes the maximizing point $x = 0$ . Declare all Borel probability measures on $K$ with fixed barycenter $\overset{ˉ}{x} = 0$ feasible, with entropy per mole $\int_{K} s d P$ . For $y \in K$ , define the point mass $δ_{y} (A) = 1$ if $y \in A$ and $0$ otherwise for every Borel set $A \subseteq K$ . It is countably additive because at most one set in a disjoint family can contain $y$ , and $δ_{y} (K) = 1$ .

Facts & Assumptions

Given: The noncompact Borel domain $K$ , the restricted entropy $s$ , and all Borel probability measures on $K$ with mean zero as the feasible family.

[F1]

The Borel sigma-algebra of a topological space, Probability measures and probability spaces, and Measure space and integral: Feasible mixtures are countably additive Borel probability measures of total mass one, and their integrals use the stated measure convention.

Counterexample

1.1givenF1algebra

Since $0 < s_{0} - a \leq s (x) < s_{0}$ on $K$ , every feasible $s$ is nonnegative and bounded. For $n \geq 2$ , put $A_{n} : = {x \in K : ∣ x ∣ \geq 1 / n}$ . Their union is $K$ ; since $P (K) = 1$ , countable additivity implies $P (A_{n}) > 0$ for some $n$ . On $A_{n}$ , $s \leq s_{0} - a / n^{2}$ , and on its complement $s \leq s_{0}$ . For any nonnegative simple function $φ \leq s$ , the measure integral definition [F1] therefore gives

\int_{K} φ d P \leq (s_{0} - a / n^{2}) P (A_{n}) + s_{0} P (K ∖ A_{n}) = s_{0} - a P (A_{n}) / n^{2} < s_{0} .

Indeed, refine the finitely many level sets of $φ$ by $A_{n}$ and its complement; each simple-function coefficient is bounded by the corresponding pointwise bound on that piece, so the simple-integral formula [F1] gives the inequality. The strict upper bound is independent of $φ$ , hence its supremum also satisfies $\int_{K} s d P < s_{0}$ .

2.1givenF1step 1.1algebra∎

For each $0 < ϵ < 1$ , the measure $P_{ϵ} = \frac{1}{2} δ_{- ϵ} + \frac{1}{2} δ_{ϵ}$ is a Borel probability measure on $K$ , has mean zero, and satisfies $\int_{K} s d P_{ϵ} = s_{0} - a ϵ^{2}$ . Letting $ϵ ↓ 0$ shows that the supremum is $s_{0}$ . Step 1.1 shows that no feasible measure attains it. Thus the false statement fails even with a bounded positive continuous concave extension, because the admissible generator set excludes its maximizing barycenter.

A noncompact phase domain with no attained mixture maximum

Statement refuted

Example

Facts & Assumptions

Counterexample

Depends on

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