forked from bai/curriculum-project-hub
sjfhsjfh
c73a2c903f
Real-content fixture for the end-to-end pipeline. TH-141 (39 parts: 22 segment, 15 lemma, 2 example) migrated from the prototype's typst `#let parts` manifest to ADR-0008: declarative manifest.toml (project+info+ordered parts+targets) + per-element element.toml (kind + scalars; examples carry `source`). Content .typ files copied byte-identical (no math corruption); per-element main.typ + meta.toml dropped (wiring is now generated). Part order matches source exactly; 5 lemmas have no proof.typ (optional); no cross-file imports / paralearn refs / figs. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
14 lines
654 B
Typst
14 lines
654 B
Typst
设下半空间体相分子的数密度为 $n_v = 1 \/ d^3$。固定一个表面分子作为参考分子，对其与下半空间所有体相分子的相互作用求和。由对称性，相对参考分子距离 $r$ 在下半空间内的体积元为 $dif V = 2 pi r^2 dif r$。下限取 $r = d$，上限取 $infinity$：
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$ Delta U = 2 pi n_v integral_d^infinity u(r) thin r^2 dif r . $
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代入 $u(r)$，用 $integral r^2 dot r^(-n) dif r = r^(3 - n) \/ (3 - n)$ 化简，得
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$ Delta U tilde.op epsilon . $
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把 $Delta U$ 乘以单位面积分子数 $n_s tilde.op 1 \/ d^2$，由 @骨架公式 即得
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$ sigma tilde.op epsilon / d^2 , $
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即 @LJ标度。
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