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The stationary harmonic bath has a Lorentzian position spectrum

Physical scope

A calculated stationary spectrum and averaging bound in the adopted scalar harmonic Ornstein-Uhlenbeck model; in the specified common fixed Galilean rest chart with constant trap-and-bath center-of-mass velocity throughout the modeled interval; no measured spectrum is asserted.

Example

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.

Use mobility $μ > 0$ , stiffness $k > 0$ , and fixed bath temperature $T_{b} > 0$ . Under the adopted linear bath model, the stationary position spectrum has angular-frequency convention $S (ω) = (2 π)^{- 1} \int_{R} C (τ) e^{i ω τ} d τ$ .

Facts & Assumptions

Given: The Axiom of Choice and the stationary Gaussian preparation of the specified linear model.

[F1]

Linear Ornstein-Uhlenbeck bath model: its supplied center-of-mass data and constant velocity select one common rest chart throughout the modeled interval; $γ = μ k$ and $σ^{2} = 2 μ k_{B} T_{b}$ .

[F2]

Thermal fluctuations in the specified linear harmonic bath model: $C (τ) = (k_{B} T_{b} / k) e^{- μ k ∣ τ ∣}$ and the exact model spectrum is known.

Verification

Given: The Axiom of Choice and $γ = μ k > 0$ and $V = k_{B} T_{b} / k$ .

1.1F1F2algebra

Substituting $V = k_{B} T_{b} / k$ and $γ = μ k$ into the exact spectrum supplied by [F2] gives $S (ω) = μ k_{B} T_{b} / (π [(μ k)^{2} + ω^{2}])$ .

2.1F2step 1.1algebra∎

The same supplier bound becomes $Var (T^{- 1} \int_{0}^{T} X_{t} d t) \leq 2 k_{B} T_{b} / (μ k^{2} T)$ , which tends to zero as $T \to \infty$ . Both formulas are conditional model predictions, not measurements.

Depends on

Dependency tree · 2 levels

Sources

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