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Liouville transport and fine-grained entropy invariance

Statement

Let $D \subseteq Γ_{N}$ be an open invariant domain for a $C^{2}$ Hamiltonian $H$ , and let its actual Hamiltonian flow $Φ_{t} : D \to D$ be defined as a $C^{1}$ diffeomorphism for each time $t$ considered, with $Φ_{- t} = Φ_{t}^{- 1}$ . Let $m_{N} = c_{N} Λ_{N}$ be the Liouville reference measure of Classical configuration, cotangent phase and reference measures. If $P_{0} = ρ_{0} m_{N}$ is a probability state, define its transported density by $ρ_{t} (z) = ρ_{0} (Φ_{- t} z) .$ Then $ρ_{t}$ is normalized and, for every measurable $f$ for which the integrals exist, $\int_{D} f (z) ρ_{t} (z) d m_{N} (z) = \int_{D} f (Φ_{t} z) ρ_{0} (z) d m_{N} (z) .$ If additionally $\int_{D} ∣ ρ_{0} \log ρ_{0} ∣ d m_{N} < \infty$ , with $0 \log 0 : = 0$ , then the fine-grained reference-relative entropy $S (ρ) : = - k \int_{D} ρ \log ρ d m_{N} (k > 0)$ is constant under this transport. This is an identity for the transported density; it asserts no relaxation, equilibration or entropy increase.

Facts & Assumptions

Given: The invariant domain, actual diffeomorphic Hamiltonian flow, normalized initial density and the stated entropy integrability.

[A1]

$H \in C^{2} (D, R)$ and for each considered $t$ , the actual flow $Φ_{t}$ is a $C^{1}$ diffeomorphism of $D$ onto itself with inverse $Φ_{- t}$ .

[A2]

$ρ_{0} \geq 0$ , $\int_{D} ρ_{0} d m_{N} = 1$ , and $\int_{D} ∣ ρ_{0} \log ρ_{0} ∣ d m_{N} < \infty$ for the entropy assertion.

[F1]

The phase reference has Liouville volume density (Classical configuration, cotangent phase and reference measures).

[F2]

In a canonical chart, $X_{H} = (H_{p}, - H_{q})$ is $C^{1}$ . For its $C^{1}$ flow, differentiating the flow integral equation in the initial point gives ${\dot{J}}_{t} = D X_{H} (Φ_{t} z) J_{t}$ with $J_{0} = I$ . Differentiating the determinant gives $\frac{d}{d t} \det J_{t} = (div X_{H}) (Φ_{t} z) \det J_{t} = 0$ : the continuous mixed second partials of $H$ agree. Thus each canonical local flow has Jacobian one at the stated $C^{2}$ regularity. The smooth case is also Liouville volume preservation.

[F3]

A measure-preserving map preserves integrals of nonnegative measurable and integrable functions (Integral invariance under measure-preserving maps).

[F4]

The nonnegative change-of-variables theorem holds with Countable Choice (A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F5]

Densities, expectations and integrability are interpreted relative to the declared measure (Statistical states, densities, observables and expectations).

Proof

Given: [A1]–[A2] and the Liouville and measure-transport interfaces [F1]–[F5].

1.1A1A2F1F2F3F4

By [F1]–[F2], canonical local flow maps have Jacobian one and preserve the constant coordinate reference density. Composition retains this property along each flow segment. Apply [F4] to obtain preservation of measure on each source/target canonical chart pair. A countable atlas partitions $D$ into countably many disjoint Borel chart pieces; summing their measure identities proves global $m_{N}$ -preservation. Apply [F3] to the nonnegative function $ρ_{0}$ and the inverse flow to obtain $\int_{D} ρ_{t} d m_{N} = \int_{D} ρ_{0} (Φ_{- t} z) d m_{N} (z) = \int_{D} ρ_{0} (z) d m_{N} (z) = 1.$ Thus $ρ_{t}$ is a probability density.

2.1A1F3F4step 1.1algebra

For nonnegative measurable $f$ , apply [F3] to $f ρ_{t}$ under $Φ_{t}$ ; then $ρ_{t} (Φ_{t} z) = ρ_{0} (z)$ , giving the displayed transport identity. For integrable signed $f$ , apply the same argument separately to its positive and negative parts, whose integrals are finite by the stated existence condition.

3.1A2F3F5step 1.1∎

The same measure-preserving substitution applied to the nonnegative function $ρ_{0} \log^{+} ρ_{0}$ and to $ρ_{0} \log^{-} ρ_{0}$ separately shows that both parts of $ρ_{t} \log ρ_{t}$ have the same integrals as those of $ρ_{0} \log ρ_{0}$ . They are finite in sum by [A2], and therefore $\int_{D} ρ_{t} \log ρ_{t} d m_{N} = \int_{D} ρ_{0} \log ρ_{0} d m_{N} .$ Multiplication by $- k$ proves $S (ρ_{t}) = S (ρ_{0})$ .

Liouville transport and fine-grained entropy invariance

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources