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Diffusion-scale walk and recurrence/coarse-entropy counterexamples

Physical scope

A finite IID symmetric walk under diffusive scaling, an explicitly defined periodic rotor, and a four-state measure-preserving permutation. Results are conditional model calculations; no collision-to-diffusion limit, path-space convergence, experimental observation, or universal entropy law is asserted.

Statement

Assume the Axiom of Choice. The following are separate calculations for three explicitly defined models.

Diffusive-scale walk. Let $D > 0$ have units m $^{2}$ s $^{- 1}$ , fix $t \geq 0$ seconds, and let $(δ t_{j})_{j \geq 1}$ be any positive sequence tending to zero, with $ℓ_{j} = \sqrt{2 D δ t_{j}}$ metres and $n_{j} = ⌊ t / δ t_{j} ⌋$ . For each $j$ , on the finite product space $Ω_{j} = {- 1, 1}^{n_{j}}$ with uniform probability, let $Ξ_{i}$ be the coordinate increments and $S_{j} = ℓ_{j} \sum_{i = 1}^{n_{j}} Ξ_{i} .$ For the dimensionless lattice index $r \in Z$ , define $u_{n} (r) : = 2^{- n} # {(x_{1}, \dots, x_{n}) \in {- 1, 1}^{n} : \sum_{i = 1}^{n} x_{i} = r},$ with the empty sum equal to zero. Then $E S_{j} = 0$ , $Var (S_{j}) = n_{j} ℓ_{j}^{2}$ , and the lattice probabilities satisfy $u_{n + 1} (r) = \frac{1}{2} u_{n} (r - 1) + \frac{1}{2} u_{n} (r + 1)$ with $u_{0} (r) = 1_{{r = 0}}$ . For every wave number $k \in R$ in m $^{- 1}$ , $φ_{S_{j}} (k) = \cos (ℓ_{j} k)^{n_{j}} .$ The laws of $S_{j}$ converge weakly on $R$ to $N (0, 2 D t)$ , the normal probability law with mean zero and variance $2 D t$ (in m $^{2}$ ). The conclusion is fixed-time marginal convergence only.

Periodic rotor. Let $T : = R / (2 π Z)$ with its quotient topology, and let the phase space $X = T \times R$ have its metrizable product topology. Let $q \in T$ be a dimensionless angle, $p \in R$ have units kg m $^{2}$ s $^{- 1}$ , and $I > 0$ have units kg m $^{2}$ . Define the model energy $H (q, p) = p^{2} / (2 I)$ and, for time $s \in R$ measured in seconds, the flow $Φ_{s} (q, p) = (q + p s / I m o d 2 π, p) .$ For each $p_{0} \neq 0$ , the orbit is periodic with period $T = 2 π I / ∣ p_{0} ∣$ seconds. Let $ν$ be normalized arc length on $T$ , define the continuous inclusion $ι_{p_{0}} (q) = (q, p_{0})$ , and for each Borel $E \subseteq X$ set $μ_{0} (E) = ν (ι_{p_{0}}^{- 1} (E))$ . Thus $μ_{0}$ is the uniform angle probability on the fixed- $p_{0}$ orbit. For this invariant probability, mixing means $μ_{0} (B \cap Φ_{s}^{- 1} C) \to μ_{0} (B) μ_{0} (C)$ for all Borel $B, C \subseteq X$ as $s \to \infty$ . The rotor is not mixing: for the half-circle arc $A = {([q], p_{0}) : q \in [0, π)}$ , where $[q] = q + 2 π Z \in T$ , the mixing correlation fails along times $s = m T$ with positive integers $m$ . For any $q_{0} = [θ_{0}] \in T$ , the point-supported initial law $δ_{(q_{0}, p_{0})}$ (unit mass at $(q_{0}, p_{0})$ ) has no weak limit as $s \to \infty$ ; the bounded continuous witness $g ([θ], p) = \cos (θ - θ_{0})$ is well-defined because cosine is $2 π$ -periodic.

Finite coarse-entropy model. Let $Ω = {0, 1, 2, 3}$ with uniform reference probability and permutation $T (0) = 1, T (1) = 2, T (2) = 3, T (3) = 0$ . Start from $π_{0} = (1 / 2, 1 / 2, 0, 0)$ , and evolve by pushforward under $T$ . For the coarse partition ${{0, 1}, {2, 3}}$ , let $(r_{1}, r_{2})$ be any probability vector with $r_{i} \geq 0$ and $r_{1} + r_{2} = 1$ , and define its coarse Shannon entropy by $H_{c} (r_{1}, r_{2}) = - \sum_{i : r_{i} > 0} r_{i} \log r_{i}$ in nats. Define fine entropy analogously for a probability vector on the four states, using $0 \log 0 = 0$ . The coarse entropies at times $0, 1, 2$ are $0, \log 2, 0$ , while the fine entropy is $\log 2$ at all three times. Thus deterministic invertibility and preservation of the uniform reference measure do not imply monotone coarse entropy.

The smooth-flow result Fine entropy under a specified incompressible periodic flow has different hypotheses: it concerns positive $C^{1}$ densities on a flat torus transported by supplied jointly $C^{2}$ volume-preserving diffeomorphisms and preserves their dimensionless fine differential entropy. It supplies no monotonicity conclusion for coarse entropy in a finite permutation. These models report no observations and prove no diffusion limit from deterministic collisions.

Facts & Assumptions

Given: The Axiom of Choice; the finite product walk laws; the stated rotor flow and dimensions; the four-state permutation and initial probability vector; and the definitions of the entropies in the Statement.

[F1]

Declared example preparations and thermodynamic interpretation: product preparations are separately specified probability laws in consuming model statements, not consequences of flow and not assertions that a laboratory realizes them.

[F2]

Finite probability spaces, outcome weights, events, and event probabilities: a finite probability space is a finite set with nonnegative outcome weights summing to one.

[F3]

Real random variables on finite probability spaces and their finite distributions: a finite real random variable is a function from the finite outcome set to $R$ .

[F4]

Expectation of a real random variable on a finite probability space: expectation is the finite weighted sum.

[F5]

Characteristic function of a real random variable: $φ_{X} (k) = E [e^{i k X}]$ ; for a Borel law $μ$ , $φ_{μ} (k) = \int e^{i k x} μ (d x)$ .

[F6]

Standard normal and normal laws: $N (m, σ^{2})$ is the law of $m + σ Z$ for a standard normal $Z$ , including $N (m, 0) = δ_{m}$ .

[F7]

Characteristic function of a normal law: assuming AC, $N (m, σ^{2})$ has characteristic function $e^{i m k - σ^{2} k^{2} / 2}$ for every real $k$ , including $σ = 0$ .

[F8]

Characteristic function criterion for weak convergence: assuming AC, Borel probability laws on $R$ converge weakly to a specified Borel probability law if their characteristic functions converge pointwise at every real frequency.

[F9]

The Axiom of Choice: AC states that every family of nonempty sets has a choice function; the weak-convergence and normal-law suppliers explicitly require this full axiom.

[F10]

The derivatives of sine and cosine are cosine and minus sine: $\sin^{'} = \cos$ , $\cos^{'} = - \sin$ , $\sin 0 = 0$ , and $\cos 0 = 1$ .

[F11]

The natural logarithm as the inverse of the exponential function: $\log$ is the inverse of $\exp$ on $(0, \infty)$ .

[F12]

The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: $\log^{'} (x) = 1 / x$ for $x > 0$ .

[F13]

Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ : products and quotients of differentiable functions obey the usual derivative rules where the denominator is nonzero.

[F14]

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ : derivatives of differentiable compositions are compositions of derivatives.

[F15]

Peano's form: the normalized Taylor remainder tends to zero: if $f$ is twice differentiable on an open neighbourhood of $0$ , then $f (u) = f (0) + f^{'} (0) u + \frac{1}{2} f^{''} (0) u^{2} + o (u^{2})$ .

[F16]

The one-dimensional torus and its normalized Haar integral: under countable choice, on $R / Z$ the probability measure is the pushforward of Lebesgue measure on $[0, 1)$ and is translation-invariant; rescaling the angle by $2 π$ gives normalized arc length on $R / (2 π Z)$ .

[F17]

Fine entropy under a specified incompressible periodic flow: its current same-run draft statement says that for a specified jointly $C^{2}$ diffeomorphic flow on a flat torus with identity initial map, $C^{1}$ divergence-free velocity, and positive bounded $C^{1}$ probability density, the dimensionless fine differential entropy of the transported density is finite and conserved; the item's proof is complete conditional on its stated mathematical suppliers, but independent review remains pending.

[F18]

Weak convergence of borel probability measures: weak convergence of Borel probability measures on a metric space means convergence of integrals against every bounded continuous real-valued function.

[F19]

Variance of a finite real random variable: on a finite probability space, $Var (Y) = E [(Y - E [Y])^{2}]$ .

[A1]

For each walk index $j$ , the coordinates on ${- 1, 1}^{n_{j}}$ are assigned the explicit uniform product weights $2^{- n_{j}}$ ; the different $j$ may use different finite spaces.

[A2]

The rotor law is the explicit flow in the Statement; no flow-existence theorem, ergodic hypothesis, or coupling to another system is assumed.

[A3]

The four-state probability evolves by the stated deterministic pushforward, with no stochastic reset.

Proof

Given: The definitions and model assumptions in the Statement and Facts & Assumptions; assume full AC for the weak-convergence and normal-law suppliers.

1.1givenA1F1F2F3F4F19algebra

Under the product-preparation convention [F1], for a fixed $j$ the uniform weight on ${- 1, 1}^{n_{j}}$ is a probability law, including the singleton case $n_{j} = 0$ . Each increment has mean zero and square one by [F2]–[F4]. Independence follows from the product weights, so the cross terms in $E (\sum_{i} Ξ_{i})^{2}$ vanish; hence $E S_{j} = 0$ and, by [F19], $Var (S_{j}) = n_{j} ℓ_{j}^{2}$ . Conditioning on the last increment gives $u_{n + 1} (r) = \frac{1}{2} u_{n} (r - 1) + \frac{1}{2} u_{n} (r + 1)$ , with the stated initial mass.

1.2givenA2F9F16algebra

For the rotor, write $R_{s} (q) = q + p_{0} s / I$ modulo $2 π$ . At $T = 2 π I / ∣ p_{0} ∣$ , the angle increment is $2 π$ when $p_{0} > 0$ and $- 2 π$ when $p_{0} < 0$ , so $R_{T}$ is the identity. The stated full AC includes the countable-choice assumption of [F16]. For every Borel $E \subseteq X$ , $ι_{p_{0}}^{- 1} (Φ_{s}^{- 1} E) = R_{s}^{- 1} (ι_{p_{0}}^{- 1} E)$ ; translation invariance in [F16] therefore gives $μ_{0} (Φ_{s}^{- 1} E) = μ_{0} (E)$ , and in particular $μ_{0}$ is invariant under the rotor flow.

1.3givenA3F11algebra

Under the four-state pushforward, the first three probability vectors are $π_{0} = (1 / 2, 1 / 2, 0, 0), π_{1} = (0, 1 / 2, 1 / 2, 0), π_{2} = (0, 0, 1 / 2, 1 / 2) .$ The map $T$ is a permutation, so it maps the uniform reference weights $1 / 4$ to themselves; it also leaves the multiset of the evolving fine probabilities unchanged, and each fine entropy is $- \frac{1}{2} \log \frac{1}{2} - \frac{1}{2} \log \frac{1}{2} = \log 2$ . The coarse cell probabilities for ${{0, 1}, {2, 3}}$ are $(1, 0)$ , $(1 / 2, 1 / 2)$ , and $(0, 1)$ , giving coarse entropies $0, \log 2, 0$ under the zero-term convention. This proves the stated decrease.

2.1givenA1F5F6F7F10F11F12F13F14F15step 1.1algebra

By independence and [F5], the characteristic function is $φ_{S_{j}} (k) = \prod_{i = 1}^{n_{j}} E (e^{i k ℓ_{j} Ξ_{i}}) = \cos (ℓ_{j} k)^{n_{j}} .$ Fix $k$ . Continuity of cosine and $\cos 0 = 1$ make $\cos u > 0$ for all sufficiently small $∣ u ∣$ . For such $u$ , put $f (u) = \log \cos u$ . By the chain rule, $f^{'} (u) = - \sin u / \cos u$ and $f^{'} (0) = 0$ . From [F10], $\sin u / u \to \sin^{'} (0) = 1$ and $\cos u \to 1$ , so the difference quotient $f^{'} (u) / u = - \sin u / (u \cos u)$ tends to $- 1$ ; hence $f^{''} (0) = - 1$ . The same derivative rules make $f$ twice differentiable on a neighbourhood of zero, so [F15] gives $\log \cos u = - u^{2} / 2 + o (u^{2})$ . For $t > 0$ , $n_{j} δ t_{j} \to t$ because $0 \leq t - n_{j} δ t_{j} < δ t_{j}$ , and therefore $n_{j} \log \cos (ℓ_{j} k) = - \frac{n_{j} ℓ_{j}^{2} k^{2}}{2} + n_{j} o (ℓ_{j}^{2}) ⟶ - D t k^{2} .$ Indeed $n_{j} ℓ_{j}^{2} = 2 D n_{j} δ t_{j} \to 2 D t$ , so $n_{j} ℓ_{j}^{2}$ is bounded, while $o (ℓ_{j}^{2}) / ℓ_{j}^{2} \to 0$ ; their product proves the remainder tends to zero. For $t = 0$ , $n_{j} = 0$ and every law is $δ_{0}$ . Thus in all cases $φ_{S_{j}} (k) \to e^{- D t k^{2}} = φ_{N (0, 2 D t)} (k)$ for every real $k$ , using [F6]–[F7] for the target law.

2.2givenF16F18step 1.2algebra

The measure definition gives $μ_{0} (A) = ν ([0, π)) = 1 / 2$ . At each $s = m T$ , $Φ_{s}$ is the identity on the support of $μ_{0}$ , so $μ_{0} (A \cap Φ_{s}^{- 1} A) = 1 / 2 \neq 1 / 4 = μ_{0} (A)^{2}$ , contradicting the mixing condition stated in the Statement. For $q_{0} = [θ_{0}]$ , the function $g ([θ], p) = \cos (θ - θ_{0})$ is well-defined and continuous on the quotient because cosine is continuous and $2 π$ -periodic, and it is bounded by $1$ . Its values are $1$ at times $m T$ and $- 1$ at times $m T + T / 2$ , so its expectation has no limit. By [F18], weak convergence would force convergence of this expectation, so the evolving point laws have no weak limit.

2.3givenF17step 1.3

The mathematical result [F17] assumes a smooth diffeomorphic flow on a flat torus, divergence-free velocity, and a positive bounded $C^{1}$ density. The atomic four-state law in step 1.3 is not such a density or flow; its coarse-entropy decrease neither invokes nor contradicts that fine differential-entropy result. It establishes no monotonicity law for other coarse-graining procedures.

3.1givenF7F8F9step 1.1step 2.1

The right-hand side of step 2.1 is the characteristic function of the specified Borel probability $N (0, 2 D t)$ . Apply the reverse implication of [F8] to the Borel laws of $S_{j}$ to obtain weak convergence. The only non-finite-choice strength used here is the full AC explicitly carried by [F7]–[F9]; the finite product construction is in step 1.1 and its pointwise characteristic calculation is in step 2.1.

4.1givenA1A2A3step 1.1step 2.1step 3.1step 1.2step 1.3∎

The walk limit in steps 1.1–2.1 concerns only fixed-time marginal laws under the stated independent-increment scaling; no path-space limit, deterministic collision limit, experiment, observed frequency, or empirical inference is part of the hypotheses or conclusion.

Remarks

The full Axiom of Choice includes the countable-choice assumption of the normalized torus-measure definition [F16] and is also required by the imported normal-law and weak-convergence interfaces [F7]–[F9]. The recurrence, finite permutation, and finite product calculations themselves are choice-free. A finite histogram is not an underlying probability law, and no histogram is supplied as one here.

Diffusion-scale walk and recurrence/coarse-entropy counterexamples

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources