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Fine entropy under a specified incompressible periodic flow

Statement

Use $\log$ for the natural logarithm of The natural logarithm as the inverse of the exponential function. Let $T^{n} : = R^{n} / Λ$ be a flat translation torus, where $Λ = Z ℓ_{1} + \dots + Z ℓ_{n}$ for a linearly independent basis $ℓ_{1}, \dots, ℓ_{n}$ , with the smooth structure and Euclidean metric descended from $R^{n}$ . Let $μ = c d a$ be its fixed translation-invariant volume measure in dimensionless translation coordinates, with $c > 0$ . Let $I \subseteq R$ be an open interval containing $0$ . Suppose $χ : I \times T^{n} \to T^{n}$ is jointly $C^{2}$ , each $χ_{t}$ is a diffeomorphism, and $χ_{0} = id$ . Let $π : T T^{n} \to T^{n}$ be the tangent-bundle projection and let $b : I \times T^{n} \to T T^{n}$ have $C^{1}$ chart coefficients with $π (b (t, a)) = a$ and $\partial_{t} χ (t, a) = b (t, χ (t, a))$ . In translation charts write $b = \sum_{i} b^{i} \partial_{x^{i}}$ and define $div b = \sum_{i} \partial_{x^{i}} b^{i}$ . Assume $div b = 0$ in these charts. Let $f_{0} \in C^{1} (T^{n})$ satisfy $0 < m \leq f_{0} \leq M < \infty$ and $\int f_{0} d μ = 1$ , and define $f_{t} : = f_{0} \circ χ_{t}^{- 1}$ . Then $f_{t}$ is a probability density for every $t \in I$ , the integral $H (f_{t}) : = - \int_{T^{n}} f_{t} \log f_{t} d μ$ is finite, and $H (f_{t}) = H (f_{0})$ . Here $H$ is the dimensionless fine entropy of this density; multiplication by a physical constant such as $k_{B}$ is a separate interpretation.

Facts & Assumptions

Given: The flat torus, fixed measure $μ = c d a$ , jointly $C^{2}$ diffeomorphisms $χ_{t}$ , divergence-free velocity, and positive $C^{1}$ probability density $f_{0}$ bounded above and below as in the Statement.

[F1]

Jacobian identity for a specified smooth diffeomorphic flow: on the flat translation torus, for a jointly $C^{2}$ diffeomorphic family $χ_{t}$ with $χ_{0} = id$ and a $C^{1}$ time-dependent vector field $b$ satisfying $\partial_{t} χ (t, a) = b (t, χ (t, a))$ , the local-chart Jacobian satisfies $\partial_{t} J = J div b$ , is positive, and equals $1$ when the chart divergence $\sum_{i} \partial_{x^{i}} b^{i}$ vanishes.

[F2]

Change of variables for an injective $C^{1}$ map on a compact Jordan set: an injective $C^{1}$ map with invertible derivative on an open neighborhood of a compact Jordan set obeys the Riemann change-of-variables formula for bounded integrable functions.

[F3]

The natural logarithm as the inverse of the exponential function defines $\log$ on $(0, \infty)$ as the inverse of the real exponential; Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm makes it continuous there. In particular it is bounded and continuous on every compact interval $[m, M] \subset (0, \infty)$ .

Proof

1.1givenF1

Apply [F1] to the given family. Its hypotheses are exactly the supplied $C^{2}$ diffeomorphisms, identity initial map and $C^{1}$ divergence-free velocity, so every local translation-chart Jacobian is $J_{t} = 1$ .

2.1givenF1F2step 1.1

For fixed $t$ , around each point choose a source chart and a smaller coordinate neighborhood whose closure maps under $χ_{t}$ into a target chart. Continuity and compactness give finitely many closed coordinate boxes $Q_{j}$ with interiors covering the torus and with each $χ_{t} (Q_{j})$ inside a target chart. Put $E_{1} = Q_{1}$ and $E_{j} = Q_{j} ∖ ⋃_{i < j} int (Q_{i})$ ; each $E_{j}$ is compact, the pieces cover the torus, and any overlap lies on one of the box faces. Each $E_{j}$ is Jordan because its boundary lies in a finite union of coordinate faces, which are compact $C^{1}$ hypersurface patches. A $C^{1}$ graph patch has bounded slope on a compact subpatch, so its mesh- $h$ box cover has total $n$ -volume $O (h)$ and hence content zero. On compact chart neighborhoods $χ_{t}$ has bounded derivative, so it maps these zero-content boundaries to zero-content boundaries. Each compact piece and its image therefore have Jordan boundaries and lie in source and target charts; [F2] applies there. Summing over the pieces, overlaps do not affect the integrals, translation chart changes preserve $d a$ , and the constant $c$ is the same on both sides. Thus for every continuous $φ$ on the torus, $\int φ \circ χ_{t} d μ = \int φ J_{t} d μ = \int φ d μ$ .

3.1givenstep 2.1

The function $f_{t} = f_{0} \circ χ_{t}^{- 1}$ is positive and continuous. The substitution of step 2.1 with $φ = f_{t}$ gives $\int f_{t} d μ = \int f_{0} d μ = 1$ , so $f_{t}$ is normalized.

4.1givenF3step 2.1algebra∎

The bounds $m \leq f_{0} \leq M$ also hold for $f_{t}$ ; by [F3], $\log$ is continuous on $[m, M]$ , so $f_{t} \log f_{t}$ is bounded and continuous on the compact torus and its integral is finite. Applying step 2.1 to $φ = f_{t} \log f_{t}$ gives $\int f_{t} \log f_{t} d μ = \int (f_{t} \log f_{t}) \circ χ_{t} d μ = \int f_{0} \log f_{0} d μ$ , since $f_{t} \circ χ_{t} = f_{0}$ . Negating proves $H (f_{t}) = H (f_{0})$ .

Remarks

This is a statement about supplied smooth mathematical flow data. It does not assert existence of a flow for a vector field, Hamiltonian structure, relaxation, mixing, ergodicity, monotonicity of coarse entropy, or a physical entropy interpretation.

Fine entropy under a specified incompressible periodic flow

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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Sources