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A single Fourier density mode under SSEP hydrodynamics

Example

Fix $d \geq 1$ , $k \in Z^{d}$ , $a \in [0, 1]$ , and $b \in R$ with $∣ b ∣ \leq \min (a, 1 - a)$ . For initial density $ρ_{0} (x) = a + b \cos (2 π k \cdot x)$ , the continuum heat profile is

$ρ_{t} (x) = a + b \exp (- 4 π^{2} ∣ k ∣^{2} t) \cos (2 π k \cdot x), t \geq 0.$

For any sequence of SSEP initial laws whose empirical measures converge in probability against every smooth periodic test to this $ρ_{0} (x) d x$ , the empirical-field conclusion of the diffusive heat-limit theorem holds uniformly for $0 \leq t \leq T$ for each finite $T$ . This is a conditional model prediction, not a measured diffusion profile.

Example references

Fourier heat flow on the continuum torus

Diffusive heat limit for symmetric exclusion

Facts & Assumptions

Given: The Axiom of Choice; $d \geq 1$ , $k \in Z^{d}$ , $a \in [0, 1]$ , $∣ b ∣ \leq \min (a, 1 - a)$ ; and an SSEP initial-law sequence satisfying the theorem's empirical-profile hypothesis.

[A1]

Assume the Axiom of Choice, meaning every family of nonempty sets has a choice function, as required by the heat-limit theorem (The Axiom of Choice).

[F1]

Fourier heat flow on the continuum torus: the weak heat solution is unique and its Fourier modes evolve with multiplier $e^{- 4 π^{2} ∣ k ∣^{2} t}$ .

[F2]

Diffusive heat limit for symmetric exclusion: under the stated initial empirical-profile convergence, SSEP empirical fields converge uniformly in time against each smooth periodic test to the heat profile.

Verification

1.1givenalgebra

The constant function has Laplacian zero. Direct differentiation of $\cos (2 π k \cdot x)$ gives $Δ \cos (2 π k \cdot x) = - 4 π^{2} ∣ k ∣^{2} \cos (2 π k \cdot x)$ . Therefore the displayed $ρ_{t}$ solves $\partial_{t} ρ_{t} = Δ ρ_{t}$ and has the stated initial value. The bound on $b$ gives $0 \leq ρ_{0} \leq 1$ .

2.1A1F1F2step 1.1∎

By uniqueness in [F1], this explicit solution is the heat profile used in [F2]. Applying [F2] to the assumed SSEP sequence gives the uniform-in-time convergence statement. The formula is therefore a conditional model prediction and contains no empirical observation.

A single Fourier density mode under SSEP hydrodynamics

Example

Example references

Facts & Assumptions

Verification

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