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

Bath covariances and filters

Track A3: concentration and temporal fluctuations › Bath covariances and filters · lines 753–777

This regression calculation uses the declared Markov process. It is not substituted into the memory-bearing thermal bath.

Let

ζ (d ν) = j \sum g_{j}^{2} coth (β ℏ ν_{j} /2) δ_{ν_{j}} (d ν) .

For two admissible measures,

t \leq T sup \abs χ_{ζ} (t) - χ_{ζ^{'}} (t) \leq 4 \int min {T^{2} /2, 2/ ν^{2}} \abs ζ - ζ^{'} (d ν),

\abs Var_{ζ} (F_{k}) - Var_{ζ^{'}} (F_{k}) \leq \int \abs k (ν)^{2} \abs ζ - ζ^{'} (d ν) .

These follow by integrating the exact formulas and using $1 - cos x \leq min {x^{2} /2, 2}$ . They are bath-specific estimates, not renamed scalar input bounds.

Displayed formulas in this passage

unlabeled · line 759
$ζ (d ν) = j \sum g_{j}^{2} coth (β ℏ ν_{j} /2) δ_{ν_{j}} (d ν) .$
unlabeled · line 764
$t \leq T sup \abs χ_{ζ} (t) - χ_{ζ^{'}} (t) \leq 4 \int min {T^{2} /2, 2/ ν^{2}} \abs ζ - ζ^{'} (d ν),$
unlabeled · line 769
$\abs Var_{ζ} (F_{k}) - Var_{ζ^{'}} (F_{k}) \leq \int \abs k (ν)^{2} \abs ζ - ζ^{'} (d ν) .$

Bath covariances and filters

Displayed formulas in this passage

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