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

Track A4: Gaussian mass energies and stability

Track A4: Gaussian mass energies and stability · lines 886–909

If $K_{ii} = k_{0} > 0$ and

i max j \neq = i \sum \abs K_{ij} \leq ε k_{0}, ε < 1,

then

\frac{( 1 - ε ) k _{0}}{2} \norm s_{2}^{2} \leq E_{G} (s) \leq \frac{( 1 + ε ) k _{0}}{2} \norm s_{2}^{2} .

This follows from $2 \abs s_{i} s_{j} \leq s_{i}^{2} + s_{j}^{2}$ . It gives linear scaling for full patterns in that regime.

If all $f_{i} = f_{*}$ , aligned energy is quadratic: $n^{2} G \norm f_{*}_{C}^{2} /2$ . If each domain contains $d$ identical profiles and every domain profile has the same Coulomb norm, independent domain signs give mean $n d G \norm f_{*}_{C}^{2} /2$ . Neither regime is an asserted molecular geometry.

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$i max j \neq = i \sum \abs K_{ij} \leq ε k_{0}, ε < 1,$
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$\frac{( 1 - ε ) k _{0}}{2} \norm s_{2}^{2} \leq E_{G} (s) \leq \frac{( 1 + ε ) k _{0}}{2} \norm s_{2}^{2} .$

Track A4: Gaussian mass energies and stability

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