Manuscript Psy02-B@2 · Psy02-B-v2.tex · theorem · cfd06e2e

Theorem: Exact filtered variance

Track A3: concentration and temporal fluctuations › Bath covariances and filters · lines 684–710 · proof included or linked

Psy02-B@2#thm:filters

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The symmetrized covariance is

C_{B} (τ) = j \sum g_{j}^{2} coth (β ℏ ν_{j} /2) cos ν_{j} τ,

and

Var (F_{k}) = j \sum g_{j}^{2} coth (β ℏ ν_{j} /2) \abs k (ν_{j})^{2} .

For $S_{B} = \int e^{i ν τ} C_{B} (τ) d τ$ ,

S_{B} = π j \sum g_{j}^{2} coth (β ℏ ν_{j} /2) (δ_{ν_{j}} + δ_{- ν_{j}}), Var (F_{k}) = \frac{1}{2 π} \int \abs k^{2} S_{B} (d ν) .

Proof. Thermal anomalous moments vanish and different modes are independent. Expand the anticommutator, then integrate against $k (t) k (s)$ . Fourier transformation of a cosine gives the two spectral atoms.

Displayed formulas in this passage

unlabeled · line 687
$C_{B} (τ) = j \sum g_{j}^{2} coth (β ℏ ν_{j} /2) cos ν_{j} τ,$
unlabeled · line 692
$Var (F_{k}) = j \sum g_{j}^{2} coth (β ℏ ν_{j} /2) \abs k (ν_{j})^{2} .$
unlabeled · line 698
$S_{B} = π j \sum g_{j}^{2} coth (β ℏ ν_{j} /2) (δ_{ν_{j}} + δ_{- ν_{j}}), Var (F_{k}) = \frac{1}{2 π} \int \abs k^{2} S_{B} (d ν) .$

Theorem: Exact filtered variance

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