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No-sinks cost offsets give a weak global entropy

Statement

Assume the Axiom of Choice. Let $F$ be the decorated cost of Decorated finite-chain costs with catalysts, and assume the no-sinks condition

$F (Γ, Γ^{'}) < + \infty ⟺ F (Γ^{'}, Γ) < + \infty$

for every type pair. Let $V_{F}$ be the real difference space of Cost-zero type offsets and their difference space. Then there is an additive extensive offset $B : G \to R$ such that

$- F (Γ^{'}, Γ) \leq B (Γ) - B (Γ^{'}) \leq F (Γ, Γ^{'}) .$

If every admitted cross-type transition $x ≺ y$ from $Γ$ to $Γ^{'}$ satisfies the calibrated cost inequality of Catalyst-chain cost characterizes cross-type transitions, then $σ_{Γ} (x) + B (Γ)$ is nondecreasing across every such transition. Only this weak inequality is asserted; no-gap equalities and strict extensions are not concluded.

Facts & Assumptions

Given: The no-sinks condition, A1–A6 for cross-type transitions, the finite-chain costs, and the Axiom of Choice.

[A1]

The Axiom of Choice: adopt AC for the Hahn–Banach dominated-extension and basis-extension steps in the vector-space supplier.

[F1]

Cost-zero type offsets and their difference space and The cost-zero offset construction is a real vector space: $V_{F}$ is a real vector space of formal type differences, with composition additive and positive scaling homogeneous.

[F2]

Algebraic properties of decorated-chain costs: $F (Γ, Γ) = 0$ , $F$ is translation-invariant, positively homogeneous and subadditive, finite reverse witnesses bound forward costs, and costs have no negative cycles.

[F3]

Infinite-valued no-sinks Hahn–Banach sandwich and conditional offsets: under AC, a sublinear $p : V \to R \cup {+ \infty}$ with $p (0) = 0$ and symmetric finite domain admits a linear $L$ with $- p (- v) \leq L (v) \leq p (v)$ .

[F4]

Catalyst-chain cost characterizes cross-type transitions: every admitted transition $x ≺ y$ satisfies $σ_{Γ^{'}} (y) - σ_{Γ} (x) \geq F (Γ, Γ^{'})$ .

Proof

Given: [A1], [F1]–[F4], and the no-sinks condition.

1.1A1F2given

If $F (Γ, Γ^{'}) < + \infty$ , no-sinks gives a finite reverse witness of some real cost $c$ . Concatenating it with any forward witness makes a closed chain, so [F2] gives forward cost at least $- c$ . Thus $F (Γ, Γ^{'}) > - \infty$ whenever it is finite; all cost values are in $R \cup {+ \infty}$ .

2.1F1F2step 1.1algebra

For a pair class $v = [Γ, Γ^{'}] \in V_{F}$ , define $p (v) : = F (Γ, Γ^{'})$ . Suppose $(Γ, Γ^{'}) \equiv (Λ, Λ^{'})$ via $C$ , so $P = Γ \times Λ^{'} \times C \approx_{0} Q = Λ \times Γ^{'} \times C$ . Put $D = Γ^{'} \times Λ^{'} \times C$ . Since $P \approx_{0} Q$ , the triangle inequality gives $F (P, D) \leq F (Q, D)$ and $F (Q, D) \leq F (P, D)$ ; the costs have no $- \infty$ by step 1.1, so these inequalities imply equality. Catalyst invariance from [F2] gives $F (P, D) = F (Γ, Γ^{'})$ and $F (Q, D) = F (Λ, Λ^{'})$ . Therefore $p$ is independent of the pair representative.

2.2F1F2givenstep 1.1

The identity $F (Γ, Γ) = 0$ gives $p (0) = 0$ . Homogeneity of $F$ gives $p (t v) = t p (v)$ for $t > 0$ , and its product subadditivity gives $p (v + w) \leq p (v) + p (w)$ . The no-sinks condition says $p (v) < + \infty$ exactly when $p (- v) < + \infty$ . By step 1.1, $p$ has no $- \infty$ value.

3.1A1F1F3step 2.2algebra

Apply [F3] to $V_{F}$ and $p$ . It gives a real-linear $L : V_{F} \to R$ with $- p (- v) \leq L (v) \leq p (v)$ for all $v$ . Put $B (Γ) = L ([Γ, 1])$ , where $1$ is the empty unit type. Since $[Γ \times Γ^{'}, 1] = [Γ, 1] + [Γ^{'}, 1]$ , $B$ is additive. Since $[t Γ, 1] = t [Γ, 1]$ (the unit type satisfies $t 1 = 1$ ), $B (t Γ) = t B (Γ)$ for $t > 0$ . Also $[Γ, Γ^{'}] = [Γ, 1] - [Γ^{'}, 1]$ , so applying the sandwich to this pair gives the displayed two-sided cost bound.

4.1F4step 3.1algebra∎

For an admitted transition, [F4] gives $σ_{Γ^{'}} (y) - σ_{Γ} (x) \geq F (Γ, Γ^{'})$ , while step 3.1 gives $B (Γ) - B (Γ^{'}) \leq F (Γ, Γ^{'})$ . Subtracting the latter from the former yields $[σ_{Γ^{'}} (y) + B (Γ^{'})] - [σ_{Γ} (x) + B (Γ)] \geq 0.$ Thus the offset entropy is nondecreasing across every admitted transition.

Remarks

The no-sinks condition is an explicit hypothesis about cross-type reachability. AC is used by Infinite-valued no-sinks Hahn–Banach sandwich and conditional offsets and is carried here. The result does not assert a no-gap condition, a unique offset, or an exact entropy principle for all transitions.

Depends on

Dependency tree · 2 levels

Sources

Lieb and Yngvason, The Physics and Mathematics of the Second Law of Thermodynamics, §VI.B, pp. 76–80