Manuscript Oai227-A@1 · Oai227-A.tex · section · 7f45f607

The two autocorrelations and the averaging convention

Stage 0: freeze the classical model and the external claim › The two autocorrelations and the averaging convention · lines 324–364

Oai227-A@1#eq:physical-correlation, Oai227-A@1#eq:scaled

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Let $f_{i} (σ) = σ_{i}$ . For physical waiting time $w \geq 0$ and lag $ℓ \geq 0$ , define

C_{n, J}^{stat} (ℓ) C_{n, J}^{q} (w, ℓ) = \frac{1}{n} i = 1 \sum^{n} π [f_{i} P_{ℓ}^{J} f_{i}], = \frac{1}{n} i = 1 \sum^{n} E_{U_{n}}^{J} [σ_{i} (w) σ_{i} (w + ℓ)] .

These are means of individual-spin autocorrelations, not autocorrelations of the total magnetization.

Put

v_{n} = n^{1/3}, T_{n} = n^{2/3},

and define

A_{n, J} (t) = v_{n} C_{n, J}^{stat} (T_{n} t), B_{n, J} (s, t) = v_{n} C_{n, J}^{q} (T_{n} s, T_{n} t) .

Initial spins, clocks, and heat-bath resampling are averaged in these functions. Disorder is not averaged. Consequently $A_{n, J}$ and $B_{n, J}$ are random functions only when $J$ is sampled.

Displayed formulas in this passage

Oai227-A@1#eq:physical-correlation · line 328
$C_{n, J}^{stat} (ℓ) C_{n, J}^{q} (w, ℓ) = \frac{1}{n} i = 1 \sum^{n} π [f_{i} P_{ℓ}^{J} f_{i}], = \frac{1}{n} i = 1 \sum^{n} E_{U_{n}}^{J} [σ_{i} (w) σ_{i} (w + ℓ)] .$
Oai227-A@1#eq:scaled · line 349
$A_{n, J} (t) = v_{n} C_{n, J}^{stat} (T_{n} t), B_{n, J} (s, t) = v_{n} C_{n, J}^{q} (T_{n} s, T_{n} t) .$
unlabeled · line 345
$v_{n} = n^{1/3}, T_{n} = n^{2/3},$