Manuscript En01-C@1 · En01-C.tex · section · 00a2161c

White-noise convention and transfer

Physical support-displacement noise › White-noise convention and transfer · lines 799–831

Let $u (t)$ be a centred classical support displacement. For a quadratic potential,

\frac{1}{2} κ (t) (x - u (t))^{2} = \frac{1}{2} κ (t) x^{2} - κ (t) u (t) x + \frac{1}{2} κ (t) u (t)^{2} .

The last term is scalar and contributes no commutator. For an exactly quadratic translated potential this expansion is exact. For a general potential, its neglected terms require a local approximation bound.

We use the two-sided angular-frequency convention

C_{u} (τ) = E [u (t + τ) u (t)], S_{u} (ω) = \int_{R} e^{iω τ} C_{u} (τ) d τ,

C_{u} (τ) = \frac{1}{2 π} \int_{R} e^{- iω τ} S_{u} (ω) d ω .

The white convention is

C_{u} (τ) = S_{u} δ (τ),

so the two-sided spectrum is the constant $S_{u}$ . One-sided spectra require their corresponding conversion; no one-sided factor is inserted silently.

Displayed formulas in this passage

unlabeled · line 803
$\frac{1}{2} κ (t) (x - u (t))^{2} = \frac{1}{2} κ (t) x^{2} - κ (t) u (t) x + \frac{1}{2} κ (t) u (t)^{2} .$
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$C_{u} (τ) = E [u (t + τ) u (t)], S_{u} (ω) = \int_{R} e^{iω τ} C_{u} (τ) d τ,$
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$C_{u} (τ) = \frac{1}{2 π} \int_{R} e^{- iω τ} S_{u} (ω) d ω .$
unlabeled · line 825
$C_{u} (τ) = S_{u} δ (τ),$

White-noise convention and transfer

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