Physics theoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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Thermal fluctuations in the specified linear harmonic bath model

Physical scope

Conditional predictions of the specified one-dimensional linear Ornstein-Uhlenbeck model after stationary Gaussian preparation; in the postulate’s common fixed Galilean rest chart with constant trap-and-bath center-of-mass velocity throughout the modeled interval; not empirical claims and not general laws for arbitrary baths.

Statement

Assume the Axiom of Choice The Axiom of Choice: every family of nonempty sets has a choice function. It is inherited from the Brownian and Itô-integral suppliers used for the stationary Gaussian model.

Under Linear Ornstein-Uhlenbeck bath model, prepare $X_{0} \sim N (0, V)$ independently of future Brownian increments, where $V = k_{B} T_{b} / k$ . Then the conditional linear model has stationary covariance $C (τ) = V e^{- μ k ∣ τ ∣}$ . With angular-frequency convention $S (ω) = (2 π)^{- 1} \int_{R} C (τ) e^{i ω τ} d τ$ , its spectrum is $S (ω) = μ k_{B} T_{b} / (π [(μ k)^{2} + ω^{2}])$ . For every averaging time $T > 0$ , $Var (T^{- 1} \int_{0}^{T} X_{t} d t) \leq 2 k_{B} T_{b} / (μ k^{2} T)$ . These are model predictions, not reported observations.

Facts & Assumptions

Given: The Axiom of Choice and the postulated harmonic potential, fixed temperature, positive mobility/stiffness, Einstein relation, and stationary Gaussian initial law.

[A1]

Linear Ornstein-Uhlenbeck bath model: use its supplied mechanical center-of-mass data and one constant rest boost throughout the modeled time interval; $γ = μ k > 0$ , $σ^{2} = 2 μ k_{B} T_{b}$ , and the stochastic differential is interpreted by the supplied integral equation.

[F1]

Ornstein–Uhlenbeck equation, transition law, and fluctuations: actual current carrier status is draft; its inspected statement assumes the Brownian filtration and initial-data conditions of Ornstein–Uhlenbeck integral equation, and for $V = σ^{2} / (2 γ)$ gives stationary covariance $V e^{- γ ∣ τ ∣}$ , spectrum $V γ / [π (γ^{2} + ω^{2})]$ , and time-average variance at most $2 V / (γ T)$ .

Proof

Given: The Axiom of Choice and [A1], [F1], $T_{b}, k, μ > 0$ , and $X_{0} \sim N (0, k_{B} T_{b} / k)$ independent of future increments.

1.1A1givenalgebra

The physical postulate gives $V = σ^{2} / (2 γ) = (2 μ k_{B} T_{b}) / (2 μ k) = k_{B} T_{b} / k$ . This has units of square metres because $k_{B} T_{b}$ is joules and $k$ is joules per square metre.

2.1F1step 1.1

The inspected mathematical supplier's stationary clause applies with this $V$ and $γ = μ k$ , so $C (τ) = V e^{- μ k ∣ τ ∣}$ and $S (ω) = V μ k / [π ((μ k)^{2} + ω^{2})] = μ k_{B} T_{b} / [π ((μ k)^{2} + ω^{2})]$ .

3.1A1F1step 1.1step 2.1∎

The same supplier's time-average bound becomes $2 V / (γ T) = 2 k_{B} T_{b} / (μ k^{2} T)$ . All three formulas are conditional deductions of the adopted model; no apparatus, observed value, or measured uncertainty is asserted.

Depends on

Dependency tree · 2 levels

Sources

U. Seifert, Stochastic thermodynamics, fluctuation theorems, and molecular machines