Um estudo de escala da dinâmica de estados homogêneos

❯◆■❱❊❘❙■❉❆❉❊ ❉❊ ❇❘❆❙❮▲■❆

■◆❙❚■❚❯❚❖ ❉❊ ❋❮❙■❈❆

P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ❋❮❙■❈❆

❯♠ ❡st✉❞♦ ❞❡ ❡s❝❛❧❛ ❞❛ ❞✐♥â♠✐❝❛ ❞❡ ❡st❛❞♦s ❤♦♠♦❣ê♥❡♦s

❞♦ s✐st❡♠❛ ❣r❛✈✐t❛❝✐♦♥❛❧ ✉♥✐❞✐♠❡♥s✐♦♥❛❧

▲②❞✐❛♥❡ ❋❡rr❡✐r❛ ❞❡ ❙♦✉③❛

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ▼❛r❝♦ ❆✳ ❆♠❛t♦

❇r❛sí❧✐❛

❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦✱ ♣r✐♠❡✐r❛♠❡♥t❡✱ ❛ ❉❡✉s ♣♦r ✐❧✉♠✐♥❛r ♠❡✉s ♣❛ss♦s ❡ ♠❡ ❞❛r ❢♦rç❛ ♣❛r❛ s✉♣❡r❛r ♦s ♠♦♠❡♥t♦s ❞❡ ❞✐✜❝✉❧❞❛❞❡s q✉❡ ✈✐✈✐ ♥❡st❡s ❞♦✐s ❛♥♦s✳

❆♦ ♣r♦❢❡ss♦r ❆♠❛t♦✱ ♣❡❧❛ ❝♦♥✜❛♥ç❛ ❡ ♣❡❧♦ ❛♣♦✐♦ q✉❡ ♦ s❡♥❤♦r ♠❡ ❞❡✉✱ s❡♠♣r❡ ♠❡ ♠♦t✐✈❛♥❞♦✱ ❞❛♥❞♦ ❜♦♥s ❝♦♥s❡❧❤♦s ❡ ♠❡ ❛❥✉❞❛♥❞♦ ❝♦♠ ♠✐♥❤❛s ❞ú✈✐❞❛s ✐♥✜♥✐t❛s✳

❆♦ ♠❡✉ ♣❛✐ ◆é❧✐♦✱ ♣❡❧♦ ❝❛r✐♥❤♦ ❡ ❛♠♦r ✐♥❝♦♥❞✐❝✐♦♥❛❧✳ ❊✉ só ❝♦♥s❡❣✉✐ ❝❤❡❣❛r ❛té ❛q✉✐ ♣♦rq✉❡ ✈♦❝ê ❡st❛✈❛ ❛♦ ♠❡✉ ❧❛❞♦ ♦ t❡♠♣♦ t♦❞♦✳ ❆ ♠✐♥❤❛ ♠ã❡ ❘✐t❛✱ q✉❡ s❡♠♣r❡ ❢♦✐ ✉♠ ❡①❡♠♣❧♦ ❞❡ ♠✉❧❤❡r ❡ ♠ã❡✳ ❊ ♠❡s♠♦ ♥ã♦ ❡st❛♥❞♦ ♠❛✐s ♣r❡s❡♥t❡ ❡♠ ✈✐❞❛✱ ❢♦✐ ✉♠❛ ❢♦♥t❡ ❞❡ ✐♥s♣✐r❛çã♦ ♣❛r❛ ♠✐♠✳

❆s ♠✐♥❤❛s ✐r♠ãs ▲✉r②❛♥❡ ❡ ❨❛♥❛r❛✱ q✉❡ sã♦ ♠✐♥❤❛s ♠❡❧❤♦r❡s ❛♠✐❣❛s ❡ ❝♦♠♣❛♥❤❡✐r❛s ❞❡ ✈✐❛❣❡♥s✱ ♣❡❧♦ ❛♣♦✐♦ ❡ ❛♠✐③❛❞❡✳

❆♦ ❘♦❞r✐❣♦✱ ♦ ♠❡✉ ❛♠✐❣♦ ❡ ♥❛♠♦r❛❞♦✱ ♣♦r ❡st❛r ❛♦ ♠❡✉ ❧❛❞♦ ♠❡ ♠♦t✐✈❛♥❞♦✱ ✐♥❝❡♥✲ t✐✈❛♥❞♦ ❡ t♦r❝❡♥❞♦ ♣♦r ♠✐♠✳ ❖❜r✐❣❛❞❛ ♣♦r t✉❞♦✱ ✈♦❝ê ❢♦✐ ♦ ♠❡❧❤♦r ♣r❡s❡♥t❡ q✉❡ ❣❛♥❤❡✐ ♥❡st❡ ♠❡str❛❞♦✳

❆♦s ❛♠✐❣♦s ❞❛ ❯♥❇ ✭❆rt❤✉r✱ ❆❦✐r❛✱■❣♦r✱ ❏♦sé✱ ◆❡②♠❛r✱ ▼♦✐sés✱ ❏✉❝é❧✐❛✱✳✳✳✮ ❡ ❛♦s ❛♠✐✲ ❣♦s ❞❛ ✈✐❞❛ ✭❆♠♣❛r♦✱ ❆✈❛♥✐✱ ❆♥❞ré✱ ❇r✉♥❛✱ ❏ú❧✐❛✱ ❚❛t✐❛♥❡✱ ▲❡tí❝✐❛✱✳✳✳✮ ♣♦r ❢❛③❡r❡♠ ♣❛rt❡ ❞❛ ♠✐♥❤❛ ✈✐❞❛✳

❆♦s ❢❛♠✐❧✐❛r❡s✱ t✐♦ ❈❧é③✐♦ ❡ ❈❧❡♦♥✐❝❡ ❡ ♣r✐♠♦s ✭❘❛✐❧♠❛✱ ❘✉❜s✱ ❏✉❝é❧✐❛✱ ❏♦❝✐r❛✱ ❏♦s✐❧â♥✐❛ ❡ ❏♦s✐❡♥❡✮ ✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦ ❘❛♥✐✈❛r ❡ ❛ ❉❛②❛♥❡✱ ♣♦r ❡st❛r❡♠ s❡♠♣r❡ ♣r❡s❡♥t❡✱ ✈♦❝ês ❢❛③❡♠ ❛ ❞✐❢❡r❡♥ç❛ ♥❛ ♠✐♥❤❛ ✈✐❞❛✳

❘❡s✉♠♦

❊st❛❞♦s q✉❛s❡ ❡st❛❝✐♦♥ár✐♦s ❞❡ s✐st❡♠❛s ❝♦♠ ✐♥t❡r❛çõ❡s ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡ tê♠ s✐❞♦ ❡st✉✲ ❞❛❞♦s ♥♦s ú❧t✐♠♦s q✉✐♥③❡ ❛♥♦s✳ ❯♠ ♣♦t❡♥❝✐❛❧ ❞❡ ✐♥t❡r❛çõ❡s ❡♠ ♣❛r❡s é ❞✐t♦ s❡r ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡ s❡ ❞❡❝❛✐ ❡♠ ❧♦♥❣❛s ❞✐stâ♥❝✐❛s ❝♦♠♦ r−α _{❝♦♠ α ≤ d ❡ d é ❛ ❞✐♠❡♥sã♦ ❡s♣❛❝✐❛❧✳}

❊q✉❛çõ❡s ❝✐♥ét✐❝❛s ♣❛r❛ s✐st❡♠❛s ✐♥t❡r❛❣❡♥t❡s ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡ ✉s✉❛❧♠❡♥t❡ ♣♦❞❡♠ s❡r ♦❜t✐❞❛s ❞❛ ❤✐❡r❛rq✉✐❛ ❇❇❑●❨ ❬✶✼❪ ❧❡✈❛♥❞♦ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ❛ ❝♦♥tr✐❜✉✐çã♦ ❞❛ ❢✉♥çã♦ ❞❡ ❝♦rr❡❧❛çã♦ ❞❡ ❞♦✐s ❝♦r♣♦s✱ ❛ q✉❛❧ é ❞❛ ♦r❞❡♠ 1/N ❬✸❪ ❡ r❡s✉❧t❛ ❡♠ ✉♠❛ ❡s❝❛❧❛ ❞❡ t❡♠♣♦

❞❡ r❡❧❛①❛çã♦ ❝♦❧✐s✐♦♥❛❧ ♣r♦♣♦r❝✐♦♥❛❧ ❛ ◆✳ ❆ ❡q✉❛çã♦ ❞❡ ❇❛❧❡s❝✉✲▲❡♥❛r❞ ♣❛r❛ ✉♠ s✐st❡♠❛ ❤♦♠♦❣ê♥❡♦ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ ❛♥✉❧❛✲s❡ ❞❡✈✐❞♦ à ❢✉♥çã♦ ❞❡❧t❛ ❞❡ ❉✐r❛❝✱ ♣♦rt❛♥t♦ t❡r♠♦s ❞❡ ❛❧t❛ ♦r❞❡♠ ❞❡✈❡♠ s❡r ♠❛♥t✐❞♦s q✉❛♥❞♦ ❛ ❤✐❡r❛rq✉✐❛ ❢♦r tr✉♥❝❛❞❛✱ ❛ss✐♠✱ ❧❡✈❛♥❞♦ ❛ ✉♠ ❡s❝❛❧♦♥❛♠❡♥t♦ ❞✐❢❡r❡♥t❡ ❞❛ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞♦ ❡st❛❞♦ ❤♦♠♦❣ê♥❡♦✳ ❉❡✈❡ s❡r ♥❛t✉r❛❧ ❡s✲ ♣❡r❛r q✉❡ ♥♦ ♣r❡s❡♥t❡ ❝❛s♦ ❛s ❝♦rr❡çõ❡s ❝♦❧✐s✐♦♥❛✐s ♣r❡❞♦♠✐♥❛♥t❡s ♣❛r❛ ❛ ❡q✉❛çã♦ ❝✐♥ét✐❝❛ ✈ê♠ ❞❡ t❡r♠♦s ❞❡ ❛❧t❛ ♦r❞❡♠ ♣r♦♣♦r❝✐♦♥❛✐s ❛ 1/N2_{✱ s✉❣❡r✐♥❞♦ ✉♠❛ ❡s❝❛❧❛ ❞❡ r❡❧❛①❛çã♦}

♣r♦♣♦r❝✐♦♥❛❧ ❛ N2_{✳ ❊♠ ✉♠ tr❛❜❛❧❤♦ ♣ré✈✐♦ ❬✾❪ ♠♦str❛✲s❡ q✉❡ ♣❛r❛ ♦s ♠♦❞❡❧♦s ❍▼❋ ❡}

❆❜str❛❝t

◗✉❛s✐✲❙t❛t✐♦♥❛r② ❙t❛t❡s ♦❢ ❧♦♥❣✲r❛♥❣❡ ✐♥t❡r❛❝t✐♥❣ s②st❡♠s ❤❛✈❡ ❜❡❡♥ st✉❞✐❡❞ ❛t ❧❡♥❣t❤ ♦✈❡r t❤❡ ❧❛st ✜❢t❡❡♥ ②❡❛rs✳ ❆ ♣❛✐r ✐♥t❡r❛❝t✐♦♥ ♣♦t❡♥t✐❛❧ ✐s s❛✐❞ t♦ ❜❡ ❧♦♥❣ r❛♥❣❡❞ ✐❢ ✐t ❞❡❝❛②s ❛t ❧♦♥❣ ❞✐st❛♥❝❡s ❛s r−α _{✇✐t❤ α ≤d ✇❤❡r❡ d ✐s t❤❡ s♣❛t✐❛❧ ❞✐♠❡♥s✐♦♥✳ ❑✐♥❡t✐❝ ❡q✉❛t✐♦♥s ❢♦r}

❧♦♥❣✲r❛♥❣❡ ✐♥t❡r❛❝t✐♥❣ s②st❡♠s ✉s✉❛❧❧② ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ ❇❇●❑❨ ❤✐❡r❛r❝❤② ❬✶✼❪ ❜② t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t ❝♦♥tr✐❜✉t✐♦♥s ❢r♦♠ t❤❡ t✇♦✲❜♦❞② ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s✱ ✇❤✐❝❤ ❛r❡ ♦❢ ♦r❞❡r 1/N ❬✸❪ t❤❛t r❡s✉❧t ✐♥ ❛ t✐♠❡ s❝❛❧❡ ♦❢ ❝♦❧❧✐s✐♦♥❛❧ r❡❧❛①❛t✐♦♥ ♣r♦♣♦rt✐♦♥❛❧ t♦ N✳

❚❤❡ ❇❛❧❡s❝✉✲▲❡♥❛r❞ ❡q✉❛t✐♦♥ ❢♦r ❛ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❤♦♠♦❣❡♥❡♦✉s s②st❡♠ ✈❛♥✐s❤❡s ✐❞❡♥✲ t✐❝❛❧❧② ❞✉❡ t♦ t❤❡ ❉✐r❛❝ ❞❡❧t❛ ❢✉♥❝t✐♦♥✳ ❚❤❡r❡❢♦r❡ ❤✐❣❤❡r ♦r❞❡r t❡r♠s ♠✉st ❜❡ ❦❡♣t ✇❤❡♥ tr✉♥❝❛t✐♥❣ t❤❡ ❤✐❡r❛r❝❤②✱ ❧❡❛❞✐♥❣ t♦ ❛ ❞✐✛❡r❡♥t s❝❛❧✐♥❣ ♦❢ t❤❡ t✐♠❡ ❡✈♦❧✉t✐♦♥ ♦❢ ❛ ❤♦♠♦✲ ❣❡♥❡♦✉s st❛t❡✳ ■t ✇♦✉❧❞ ❜❡ ♥❛t✉r❛❧ t♦ ❡①♣❡❝t t❤❛t ✐♥ t❤❡ ♣r❡s❡♥t ❝❛s❡ t❤❡ ♣r❡❞♦♠✐♥❛♥t ❝♦❧❧✐s✐♦♥❛❧ ❝♦rr❡❝t✐♦♥s t♦ t❤❡ ❦✐♥❡t✐❝ ❡q✉❛t✐♦♥ ❝♦♠❡ ❢r♦♠ ❤✐❣❤❡r ♦r❞❡r t❡r♠s ♣r♦♣♦rt✐♦♥❛❧ t♦ 1/N2_{✱ t❤✐s ✐♠♣❧✐❡s ❛ r❡❧❛①❛t✐♦♥ s❝❛❧✐♥❣ ♣r♦♣♦rt✐♦♥❛❧ t♦N}2_{✳ ■♥ ❛ ♣r❡✈✐♦✉s r❡♣♦rt ❬✾❪ ✐t}

❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ✷

■ ❍✐❡r❛rq✉✐❛ ❇❇❑●❨ ✹

✶✳✶ ❊q✉❛çã♦ ❞❡ ▲❛❣r❛♥❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ ❊q✉❛çã♦ ❞❡ ❍❛♠✐❧t♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✸ ❚r❛♥s❢♦r♠❛çõ❡s ❈❛♥ô♥✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✹ ❊q✉❛çã♦ ❞❡ ▲✐♦✉✈✐❧❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✺ ❍✐❡r❛rq✉✐❛ ❇❇❑●❨ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✻ ❆♥á❧✐s❡ ❞❡ Pr✐❣♦❣✐♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

■■ ❊q✉❛çõ❡s ❝✐♥ét✐❝❛s ✸✻

✷✳✶ ❋✉♥çã♦ ❈♦rr❡❧❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✷ ❊q✉❛çã♦ ❞❡ ❱❧❛s♦✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✸ ❊q✉❛çã♦ ❞❡ ▲❛♥❞❛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✷✳✹ ❊q✉❛çã♦ ❞❡ ❇♦❧t③♠❛♥♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✷✳✺ ❊q✉❛çã♦ ❞❡ ▲❡♥❛r❞✲❇❛❧❡s❝✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻

■■■ ❘❡s✉❧t❛❞♦s ✼✵

✸✳✶ ❍▼❋ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✸✳✷ ❊q✉❛çõ❡s ❝✐♥ét✐❝❛s ♣❛r❛ ♦ ❍▼❋ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✸✳✸ ❙✐st❡♠❛ ❣r❛✈✐t❛❝✐♦♥❛❧ ✶❉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹ ✸✳✹ ❊q✉❛çõ❡s ❝✐♥ét✐❝❛s ♣❛r❛ ♦ s✐st❡♠❛ ❣r❛✈✐t❛❝✐♦♥❛❧ ✶❉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✽ ✸✳✺ ❈♦♥❝❧✉sã♦ ❡ ♣❡rs♣❡❝t✐✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✷

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✵✳✵✳✶ ❊✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞❛ ♠❛❣♥❡t✐③❛çã♦ ♣❛r❛ ♦ ❍▼❋✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✹✳✶▼♦✈✐♠❡♥t♦ ❞♦ ✈♦❧✉♠❡ ♥♦ ❡s♣❛ç♦ ❜✐❞✐♠❡♥s✐♦♥❛❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵

✶✳✺✳✷❚❡r♠♦s ❞♦ s♦♠❛tór✐♦ ✶✳✺✳✷✶✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹

✶✳✺✳✸❱ért✐❝❡s ❳ ❡ ❨✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺

✶✳✺✳✹❉✐❛❣r❛♠❛ ❞❡ ❡✈♦❧✉çã♦ ❞❡f1(x1)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺

✶✳✺✳✺❉✐❛❣r❛♠❛ ❞❡ ❡✈♦❧✉çã♦ ❞❡f2(x1, x2)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

✶✳✺✳✻❉✐❛❣r❛♠❛ ❞❡ ❡✈♦❧✉çã♦ ❞❡f3(x1, x2, x3)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

✶✳✻✳✼❋✉♥çã♦ ❞✐str✐❜✉✐çã♦ r❡❞✉③✐❞❛fs(N)♥♦ ❧✐♠✐t❡ t❡r♠♦❞✐♥â♠✐❝♦✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

✶✳✻✳✽❱❡t♦r❡s ✐x✱ ✐y ❡ ✐z✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

✶✳✻✳✾❉✐❛❣r❛♠❛ ❞❡ρ3

k′_ak′_mk′_ak′

b(α, m, a, b|..., t)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹

✶✳✻✳✶✵P♦ssí✈❡✐s t✐♣♦s ❞❡ ❞✐❛❣r❛♠❛s ❞❡ ✉♠❛ ✐♥t❡r❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹

✶✳✻✳✶✶P♦ssí✈❡✐s t✐♣♦s ❞❡ ❞✐❛❣r❛♠❛s ♣❛r❛ ✶◦ _{✐♥t❡r❛çã♦✳} ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹

✶✳✻✳✶✷P♦ssí✈❡✐s ❞✐❛❣r❛♠❛s ♣❛r❛ ❛ ✷◦ ✐♥t❡r❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

✶✳✻✳✶✸❉✐❛❣r❛♠❛s ♣❛r❛ρ2

1(t)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

✶✳✻✳✶✹❉✐❛❣r❛♠❛s ♣❛r❛ρ2

❦l(t)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

✷✳✶✳✶ ❉✐❛❣r❛♠❛ ❞❛ f(x1)❝♦♠ ❛ ❝♦rr❡❧❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✷✳✶✳✷ ❉✐❛❣r❛♠❛ ❞❛ f2(x1, x2)❝♦♠ ❛ ❝♦rr❡❧❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽

✷✳✶✳✸ ❉✐❛❣r❛♠❛s ❝❛♥❝❡❧❛❞♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾

✷✳✶✳✹ ❉✐❛❣r❛♠❛s ❞❛ ❡✈♦❧✉çã♦ ❞❡g2(x1, x2)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾

✷✳✷✳✺ ❉✐❛❣r❛♠❛s ❞❛ ❡✈♦❧✉çã♦ ❞❡f(x1)❞❡ ♦r❞❡♠λ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷

✷✳✷✳✻ P♦t❡♥❝✐❛❧ ♣❛r❛ ❞✐❢❡r❡♥t❡s ♣✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸

✷✳✸✳✼ ❉✐❛❣r❛♠❛ ❞❡ ❡✈♦❧✉çã♦ ❞❡ g2(x1, x2, t)♣❛r❛ ♦r❞❡♠λ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹

✷✳✸✳✽ ❚r❛❥❡tór✐❛ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❝♦♠ ✈❡❧♦❝✐❞❛❞❡ ❱✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻

✷✳✸✳✾ ❋✐❣✉r❛ ❡①tr❛í❞❛ ❞❡ ❬✸❪✳ ❙✐st❡♠❛ ❞❡ r❡❢❡rê♥❝✐❛ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ ●✭❣✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵

✷✳✹✳✶✵ ❚r❛❥❡tór✐❛ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❝♦❧✐❞✐♥❞♦ ❝♦♠ ♦ ❝❡♥tr♦ ❞❡ ♠❛ss❛ ❖✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷

✷✳✹✳✶✶ ❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❡ ❬✸❪✳ ❊sq✉❡♠❛ ♣❛r❛ ❝♦♥t❛❣❡♠ ❞❡ ♣❛rtí❝✉❧❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹

✷✳✺✳✶✷ ❊sq✉❡♠❛ ❞♦ ♣❧❛s♠❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼

✷✳✺✳✶✸❉✐❛❣r❛♠❛ ❞❛ fα_(x

1, t)♣❛r❛ ♣❧❛s♠❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵

✷✳✺✳✶✹❉✐❛❣r❛♠❛ ❞❛ g2αβ(x1, x2, t)♣❛r❛ ♣❧❛s♠❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵

✸✳✶✳✶ P❛rtí❝✉❧❛s ♥♦ ❍▼❋✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✸✳✷✳✷ ❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❡ ❬✶✵❪✳ ❚❡♠♣♦ ❞❡ r❡❧❛①❛çã♦ ❡♠ ❢✉♥çã♦ ❞♦ ♥ú♠❡r♦ ❞❡ ♣❛rtí❝✉❧❛s ◆✱

♣❛r❛ ◆ ❣r❛♥❞❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸

▲✐st❛ ❞❡ ❚❛❜❡❧❛s

▲✐st❛ ❞❡ ❙í♠❜♦❧♦s

f

s(

x1, ..., x

s) ❢✉♥çã♦ ❞✐str✐❜✉✐çã♦ r❡❞✉③✐❞❛ ❞❡ s ♣❛rtí❝✉❧❛s

ρ

_kj,kn,...

(

v

j

,

v

n

, ...

|

...

; 0)

❝♦♠♣♦♥❡♥t❡ ❞❡ ❋♦✉r✐❡r ❝✉❥♦s t❡r♠♦s ❛ ❡sq✉❡r❞❛ ❞❛ ❜❛rr❛ tê♠

✈❡t♦r ❞❡ ♦♥❞❛ ♥ã♦✲♥✉❧♦✱ ❛s ❞❡♠❛✐s ✈❡❧♦❝✐❞❛❞❡s sã♦ ❡s❝r✐t❛s ❛ ❞✐r❡✐t❛ ❞❛ ❜❛rr❛ ♣♦✐s ♥ã♦ sã♦ r❡❧❡✈❛♥t❡s

ρ

0_{k}

(

|

;

t

)

♦ ♣r✐♠❡✐r♦ t❡r♠♦ ❞❛ ❡q✉❛çã♦ ✶✳✻✳✺✹✱ ♣❛r❛ ♥❂✵✱ q✉❡ r❡♣r❡s❡♥t❛ ❛ ❝♦♠♣♦♥❡♥t❡ ❞❡ ❋♦✉r✐❡r ❞❛ ❢✉♥çã♦ ❞✐str✐❜✉✐çã♦ ♣❛r❛ ◆ ♣❛rtí❝✉❧❛s ♥♦ ✐♥st❛♥t❡ t✱ t❛❧ q✉❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❛s ♣❛rtí❝✉❧❛s ❝♦♠ ✈❡t♦r ❞❡ ♦♥❞❛ ♥ã♦✲♥✉❧♦ ❡stã♦ ❛ ❡sq✉❡r❞❛ ❞❛ ❜❛rr❛ ❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❛s ❞❡♠❛✐s ♣❛rtí❝✉❧❛s ❡stã♦ ❛ ❞✐r❡✐t❛

n

i(

r

)

❞❡♥s✐❞❛❞❡ ❞❡ í♦♥s ♥❛ ♣♦s✐çã♦ r

n

e(

r

)

❞❡♥s✐❞❛❞❡ ❞❡ ❡❧étr♦♥s ♥❛ ♣♦s✐çã♦r

˜

g

₂αβ ❝♦♠♣♦♥❡♥t❡ ❞❡ ❋♦✉r✐❡r ❞❛ ❝♦rr❡❧❛çã♦ ❡♥tr❡ ❞✉❛s ♣❛rtí❝✉❧❛s✱ t❛❧ q✉❡✱α ❡ β sã♦

❛s ❡s♣é❝✐❡s ❞❛s ♣❛rtí❝✉❧❛s ✭❡❧étr♦♥s ♦✉ í♦♥s✮

g

_ij t❡r♠♦ ❥ ❞❛ ❡①♣❛♥sã♦ ❞❛ ❝♦rr❡❧❛çã♦ ❣ ♣❛r❛ ✐ ♣❛rtí❝✉❧❛s

˜

g

_ij ❝♦♠♣♦♥❡♥t❡ ❞❡ ❋♦✉r✐❡r ❞❡ g_ij

e

−κ|x−x′|

é ✉♠❛ ❢✉♥çã♦ ❞❡ tr✐❛❣❡♠✱ t❛❧ q✉❡κ →0

❈❛♣ít✉❧♦

■♥tr♦❞✉çã♦

◆❡st❡ tr❛❜❛❧❤♦ ✐r❡♠♦s ❡st✉❞❛r ❛s ❡q✉❛çõ❡s ❝✐♥ét✐❝❛s ❝♦♠ ✐♥t❡r❛çõ❡s ❞❡ ❧♦♥❣♦ ❛❧✲ ❝❛♥❝❡✳

❚❛✐s s✐st❡♠❛s sã♦ ❝❛r❛❝t❡r✐③❛❞♦s ♣♦r ♣♦t❡♥❝✐❛✐s ❞❡❝❛✐♥❞♦ ❝♦♠ r−α_{✱ ♣❛r❛ α < d ❝♦♠ ❞}

s❡♥❞♦ ❛ ❞✐♠❡♥sã♦ ❞♦ ❡s♣❛ç♦✳

P❛r❛ ❞❡✜♥✐r s✐st❡♠❛s ❝♦♠ ✐♥t❡r❛çã♦ ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡✱ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ♦ ♣♦t❡♥❝✐❛❧ ♣❛r❛ ✉♠❛ ❞❛❞❛ ♣❛rtí❝✉❧❛ q s✐t✉❛❞❛ ♥♦ ❝❡♥tr♦ ❞❛ ❡s❢❡r❛ ❞❡ r❛✐♦ ❘ ❡ ✈♦❧✉♠❡ ❱✳ ❱❛♠♦s ♦♠✐t✐r ❛ ✐♥t❡r❛çã♦ ❞❛ ♠❛tér✐❛ s✐t✉❛❞❛ ❛ ✉♠❛ ❞✐stâ♥❝✐❛ ε≪R✳

❙❡ ❛s ♣❛rtí❝✉❧❛s ✐♥t❡r❛❣❡♠ ✈✐❛ ✉♠ ♣♦t❡♥❝✐❛❧ ♣r♦♣♦r❝✐♦♥❛❧ ❛ 1

rα✱ ❝♦♠ r s❡♥❞♦ ❛ ❞✐stâ♥❝✐❛

❞❛ ♦r✐❣❡♠ ♦♥❞❡ ❡stá ❛ ♣❛rtí❝✉❧❛ ❛té ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r ♥♦ ❡s♣❛ç♦✱ ❡♥tã♦

U =

Z R

ε

4πr2drq V

1

rα = 4πρ

Z R

ε

r2−αdr∝ r3−α

R ε ,

❝♦♠ ρ ❛ ❞❡♥s✐❞❛❞❡ ❞❛ ♣❛rtí❝✉❧❛✳

◗✉❛♥❞♦ ✐♥❝r❡♠❡♥t❛♠♦s ♦ r❛✐♦ ❘✱ ❛ ❝♦♥tr✐❜✉✐çã♦ ❞❡✈✐❞♦ ❛ s✉♣❡r❢í❝✐❡ ❞❛ ❡s❢❡r❛ R3−α_✱

♣♦❞❡ s❡r ♥❡❣❧✐❣❡♥❝✐❛❞❛ q✉❛♥❞♦ α > 3✱ ♠❛s ❞✐✈❡r❣❡ s❡ α < 3✳ ❊❢❡✐t♦s ❞❡ s✉♣❡r❢í❝✐❡

sã♦ ✐♠♣♦rt❛♥t❡s ❡ ♣♦rt❛♥t♦ ❛ ❛❞✐t✐✈✐❞❛❞❡ ♥ã♦ é ❝✉♠♣r✐❞❛✳ ❯♠ s✐st❡♠❛ é ♥ã♦ ❛❞✐t✐✈♦ s❡

E 6=E1+E2+...+En✱ ❛ s♦♠❛ ❞❛s ❡♥❡r❣✐❛s ❞❛s ♣❛rt❡s é ❞✐❢❡r❡♥t❡ ❞❛ ❡♥❡r❣✐❛ t♦t❛❧✳ ❙❡ ❛

❡♥❡r❣✐❛ ❞❛ ✐♥t❡r❢❛❝❡ ✭s✉♣❡r✜❝✐❛❧✮ ♥ã♦ ♣♦❞❡ s❡r ♥❡❣❧✐❣❡♥❝✐❛❞❛ ❡♥tã♦ ♦ s✐st❡♠❛ é ♥ã♦ ❛❞✐t✐✈♦✳ ❙❡ ❣❡♥❡r❛❧✐③❛♠♦s ♣❛r❛ s✐st❡♠❛s ❡♠ ❞ ❞✐♠❡♥sõ❡s✱ ❢❛❝✐❧♠❡♥t❡ ♠♦str❛♠♦s q✉❡ ❛ ❡♥❡r❣✐❛ ✈❛✐ s❡r ♥ã♦ ❛❞✐t✐✈❛ s❡ ❛ ❡♥❡r❣✐❛ ♣♦t❡♥❝✐❛❧ V(r) ❝♦♠♣♦rt❛✲s❡ ❝♦♠♦

V(r)∼ 1

rα,

❝♦♠ α < d ❛ ❧♦♥❣❛s ❞✐stâ♥❝✐❛s✳

❆❧é♠ ❞❛ ♥ã♦ ❛❞✐t✐✈✐❞❛❞❡ ❡st❡s s✐st❡♠❛s ❛♣r❡s❡♥t❛♠ ❛s s❡❣✉✐♥t❡s ❝❛r❛❝t❡ríst✐❝❛s✿

• ■♥❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ ❡♥s❡♠❜❧❡✿ ❡♠ s✐st❡♠❛s ❛✉t♦✲❣r❛✈✐t❛♥t❡s ♣♦♥t♦s tr✐❝rít✐❝♦s ♥♦ ❡♥✲ s❡♠❜❧❡ ❝❛♥ô♥✐❝♦ ❡ ♠✐❝r♦❝❛♥ô♥✐❝♦ ♥ã♦ ❝♦✐♥❝✐❞❡♠❀

• ❝❛❧♦r ❡s♣❡❝í✜❝♦ ♥❡❣❛t✐✈♦✿ ◆♦ ❡♥s❡♠❜❧❡ ❝❛♥ô♥✐❝♦ ♦ ✈❛❧♦r ♠é❞✐♦ ❞❛ ❡♥❡r❣✐❛ é

hEi=−∂lnZ

∂β =

P

i

Eie−βEi

Z ,

❝♦♠ ❩ ❛ ❢✉♥çã♦ ❞❡ ♣❛rt✐çã♦ ❡ ♦ ❝❛❧♦r ❡s♣❡❝í✜❝♦ é

cv =

∂hEi

∂T ∝

(E− hEi)2 >0,

❈❆P❮❚❯▲❖ ✳ ■◆❚❘❖❉❯➬➹❖ ✸

♦✉ s❡❥❛✱ ♦ ❝❛❧♦r ❡s♣❡❝í✜❝♦ ❝❛♥ô♥✐❝♦ é s❡♠♣r❡ ♣♦s✐t✐✈♦✳ ❊st❡ ♥ã♦ é ♦ ❝❛s♦ ♣❛r❛ s✐st❡♠❛s ❝♦♠ ✐♥t❡r❛çõ❡s ❛✉t♦ ❣r❛✈✐t❛♥t❡s✳ ❯s❛♥❞♦ ♦ t❡♦r❡♠❛ ❞♦ ✈✐r✐❛❧ ♣❛r❛ t❛✐s ♣❛rtí❝✉❧❛s

2hEci+hEpoti= 0

E =hEci+hEpoti=− hEci

❈♦♠♦ ❛ ❡♥❡r❣✐❛ ❝✐♥ét✐❝❛ Ec é ♣♦r ❞❡✜♥✐çã♦ ♣r♦♣♦r❝✐♦♥❛❧ à t❡♠♣❡r❛t✉r❛✱ t❡♠♦s q✉❡

cv =

∂hEi

∂T ∝

∂E ∂hEci

<0.

❙✐st❡♠❛s ❝♦♠ ♣♦t❡♥❝✐❛❧ ❣r❛✈✐t❛❝✐♦♥❛❧ ♦✉ ❝♦✉❧♦♠❜✐❛♥♦ sã♦ ❡①❡♠♣❧♦s ❞❡ s✐st❡♠❛s ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡ ❬✽❪✳

❆ ❞✐♥â♠✐❝❛ ❞❡ss❡s s✐st❡♠❛s é ❞❡s❝r✐t❛ ♣♦r três ❡stá❣✐♦s✱ ❛ ♣❛rt✐r ❞❡ ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s ❡s♣❡❝í✜❝❛s✿

✶✳ ❘❡❧❛①❛çã♦ ✈✐♦❧❡♥t❛✿ ❡stá❣✐♦ ✐♥✐❝✐❛❧✱ ♥❡st❡ ❡stá❣✐♦ ❛s ♣❛rtí❝✉❧❛s ♠✉❞❛♠ s✉❛ ❡♥❡r❣✐❛ s✐❣♥✐✜❝❛t✐✈❛♠❡♥t❡ ❡♠ ✉♠ ❝✉rt♦ ❡s♣❛ç♦ ❞❡ t❡♠♣♦ ❬✶✶❪✳

✷✳ ❊st❛❞♦s q✉❛s❡ ❡st❛❝✐♦♥ár✐♦s ✭◗❙❙✮✿ ❡st❡s ❡st❛❞♦s ❡stã♦ ❢♦r❛ ❞♦ ❡q✉✐❧í❜r✐♦ ❡ ♣♦ss✉❡♠ ✉♠ t❡♠♣♦ ❞❡ ✈✐❞❛ q✉❡ ❛✉♠❡♥t❛ ❝♦♠ ♦ ♥ú♠❡r♦ ❞❡ ♣❛rtí❝✉❧❛s✳ ❊st❡s ❡st❛❞♦s ♥ã♦ sã♦ ❞❡s❝r✐t♦s ♣❡❧❛ ❡st❛tíst✐❝❛ ❞❡ ❇♦❧t③♠❛♥♥❬✻❪✳

✸✳ ❊st❛❞♦ ❞❡ ❡q✉✐❧í❜r✐♦✳

❆ ❋✐❣ ✵✳✵✳✶ ❞❡s❝r❡✈❡ ❡ss❡s três ❡stá❣✐♦s ♣❛r❛ ❛ ♠❛❣♥❡t✐③❛çã♦ ❡♠ ❢✉♥çã♦ ❞♦ t❡♠♣♦ ♣❛r❛ ♦ ❍▼❋✭❍❛♠✐❧t♦♥✐❛♥ ▼❡❛♥ ❋✐❡❧❞✮ ♣❛r❛ ❞✐❢❡r❡♥t❡s ♥ú♠❡r♦s ❞❡ ♣❛rtí❝✉❧❛s✱ ♦ ♥ú♠❡r♦ ❞❡ ♣❛rtí❝✉❧❛s ❛✉♠❡♥t❛ ❞❛ ❡sq✉❡r❞❛ ♣❛r❛ ❛ ❞✐r❡✐t❛✳

❋✐❣✉r❛ ✵✳✵✳✶✿ ❊✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞❛ ♠❛❣♥❡t✐③❛çã♦ ♣❛r❛ ♦ ❍▼❋✳

◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ ❞❡st❡ tr❛❜❛❧❤♦ ✈❛♠♦s ❢❛③❡r ✉♠❛ ❜r❡✈❡ r❡✈✐sã♦ ❞❛ ♠❡❝â♥✐❝❛ ❝❧ás✲ s✐❝❛✱ ❝❤❡❣❛r ♥❛ ❡q✉❛çã♦ ❞❡ ▲✐♦✉✈✐❧❧❡✱ ❡st✉❞❛r ✉♠ ♠ét♦❞♦ ❞❡s❡♥✈♦❧✈✐❞♦ ♣♦r Pr✐❣♦❣✐♥❡ ♣❛r❛ r❡s♦❧✈❡r ❡st❛ ❡q✉❛çã♦ ✉s❛♥❞♦ sér✐❡ ❞❡ ❋♦✉r✐❡r ❡ ❡♥❝♦♥tr❛r ❛ ❤✐❡r❛rq✉✐❛ ❇❇❑●❨✳

❈❛♣ít✉❧♦ ■

❍✐❡r❛rq✉✐❛ ❇❇❑●❨

◆❡st❡ ❝❛♣ít✉❧♦ r❡✈✐s❛r❡♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❞❛ ♠❡❝â♥✐❝❛ ❝❧áss✐❝❛ ❡ ✉s❛r❡♠♦s ♦s ❝♦♥❝❡✐t♦s ❛❞q✉✐r✐❞♦s ♣❛r❛ ♦❜t❡r ❛ ❡q✉❛çã♦ ❞❡ ▲✐♦✉✈✐❧❧❡✱ q✉❡ é ❛ ❡q✉❛çã♦ ❜❛s❡ ♣❛r❛ ♦❜t❡r♠♦s ❛ ❤✐❡r❛rq✉✐❛ ❇❇❑●❨✳

✶✳✶ ❊q✉❛çã♦ ❞❡ ▲❛❣r❛♥❣❡

❙❛❜❡♠♦s ❞❛ ✷◦ _{❧❡✐ ❞❡ ◆❡✇t♦♥ q✉❡✿}

F=dp

dt =p.˙

P❛r❛ ✉♠ s✐st❡♠❛ ❞❡ ♣❛rtí❝✉❧❛s✱ t❡♠♦s q✉❡✿

Fj =

X

i

Fij +F(je) =p˙j =mj¨rj,

♦✉ s❡❥❛✱ ❛ ❢♦rç❛ r❡s✉❧t❛♥t❡ ♥❛ ♣❛rtí❝✉❧❛ ❥ é ❞❡✈✐❞♦ à ❢♦rç❛ ❡①t❡r♥❛F(_je) ❡ ❛s ❢♦rç❛s ✐♥t❡r♥❛s Fij q✉❡ ♦✉tr❛s ♣❛rtí❝✉❧❛s ❡①❡r❝❡♠ ♥❛ ♣❛rtí❝✉❧❛ ❥✳ ▼❛s ♥❡st❛s ❡q✉❛çõ❡s ♥ã♦ ♣♦❞❡♠♦s

❝♦♥t❛❜✐❧✐③❛r ♦s ✈í♥❝✉❧♦s✱ q✉❡ sã♦ r❡str✐çõ❡s ❣❡♦♠étr✐❝❛s ♦✉ ❝✐♥❡♠át✐❝❛ ❛♦ ♠♦✈✐♠❡♥t♦✳ ◗✉❛♥❞♦ ❡st❡ ✈í♥❝✉❧♦ é ❤♦❧ô♥♦♠♦ ♣♦❞❡♠♦s ❡s❝r❡✈ê✲❧♦ ❡♠ ❢✉♥çã♦ ❞❛s ❝♦♦r❞❡♥❛❞❛s✱ t❛❧ q✉❡✿

f(r1,r2, ...,rN, t) = 0.

P❛r❛ ✉♠ s✐st❡♠❛ ❞❡ ◆ ♣❛rtí❝✉❧❛s✱ t❡♠♦s ✸◆ ❝♦♦r❞❡♥❛❞❛s ✐♥❞❡♣❡♥❞❡♥t❡s✳ ❙❡ ❡①✐st❡♠ ❦ ❡q✉❛çõ❡s ♣❛r❛ ♦s ✈í♥❝✉❧♦s✱ ❡♥tã♦ ❡①✐st❡♠ ✭✸◆✲❦✮ ❝♦♦r❞❡♥❛❞❛s ✐♥❞❡♣❡♥❞❡♥t❡s(q1, q2, ..., q3N−k)✱

q✉❡ sã♦ ❛s ❝♦♦r❞❡♥❛❞❛s ❣❡♥❡r❛❧✐③❛❞❛s ❞❡st❡ s✐st❡♠❛✳ ▲♦❣♦✱

r1 = r1(q1, q2, ..., q3N−k, t)

. . .

rN = rN(q1, q2, ..., q3N−k, t),

✭✶✳✶✳✶✮

♦♥❞❡ ♦s ✈í♥❝✉❧♦s ❡stã♦ ✐♥❝❧✉s♦s ✐♠♣❧✐❝✐t❛♠❡♥t❡✳

❱❛♠♦s ❞❡✜♥✐r ✉♠ ❞❡s❧♦❝❛♠❡♥t♦ ✈✐rt✉❛❧ δri ❝♦♠♦ ✉♠ ❞❡s❧♦❝❛♠❡♥t♦ ✐♥✜♥✐t❡s✐♠❛❧✱ q✉❡

♥ã♦ ✈✐♦❧❛ ♦s ✈í♥❝✉❧♦s ❡ ♦❝♦rr❡ ❡♠ ✉♠ ✐♥st❛♥t❡ t ✜①♦✳ ❏á ♦ ❞❡s❧♦❝❛♠❡♥t♦ r❡❛❧dr♦❝♦rr❡ ♥♦

✐♥st❛♥t❡ ❞t✳ ❈♦♥s✐❞❡r❛♥❞♦ ❛ s✉♣❡r❢í❝✐❡✱ q✉❡ r❡str✐♥❣❡ ♦ ♠♦✈✐♠❡♥t♦ ❞❛ ♣❛rtí❝✉❧❛✱ ❧✐s❛✱ t❡♠♦s q✉❡ ❛ ❢♦rç❛ ❞❡ ❝♦♥t❛t♦ ❡♥tr❡ ❛ s✉♣❡r❢í❝✐❡ ❡ ❛ ♣❛rtí❝✉❧❛ ♥ã♦ t❡♠ ❝♦♠♣♦♥❡♥t❡ t❛♥❣❡♥❝✐❛❧✳ ▲♦❣♦✱

X

i

fi.δri = 0 ✭tr❛❜❛❧❤♦ ✈✐rt✉❛❧ t♦t❛❧ ❞❛s ❢♦rç❛s ❞❡ ✈í♥❝✉❧♦✮

✶✳✶✳ ❊◗❯❆➬➹❖ ❉❊ ▲❆●❘❆◆●❊ ✺

♦♥❞❡ fi é ❛ ❢♦rç❛ ❞❡ ✈í♥❝✉❧♦✳ ❆ ❢♦rç❛ ❞❡ ✈í♥❝✉❧♦ ♣r❡s❡r✈❛ ❛s r❡str✐çõ❡s ❣❡♦♠étr✐❝❛s ♦✉

❝✐♥ét✐❝❛s✳ ❈♦♠♦ ✈✐♠♦s

Fi =p˙i −→ Fi−p˙i = 0

▲♦❣♦✿

(Fi−p˙i)·δri = 0

X

i

(Fi−p˙i)·δri = 0

s❛❜❡♥❞♦ q✉❡ ♣♦❞❡♠♦s ❡s❝r❡✈❡r Fi =F(ia)+fi✱ ♦♥❞❡F(ia) é ❛ ❢♦rç❛ ❛♣❧✐❝❛❞❛ ♥❛ ♣❛rtí❝✉❧❛ ✐✱

t❡♠♦s q✉❡✿

X

i

F(_ia)−p˙_i·δr_i+X

i

f_i·δr_i = 0

X

i

F(_ia)−p˙_i·δr_i = 0 ✭✶✳✶✳✷✮ q✉❡ é ♦ Pr✐♥❝í♣✐♦ ❞❡ ❉✬❛❧❡♠❜❡r❣✳

◆❛ ❡q✉❛çã♦ ✶✳✶✳✶✱ t❡♠♦s ri = ri(q1, q2, ..., qn, t)✱ ♦♥❞❡ ♥ é ♦ ♥ú♠❡r♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s

✐♥❞❡♣❡♥❞❡♥t❡s✳ ❊♥tã♦✿

vi = ˙ri =

X

k

∂ri

∂qk

˙

qk+

∂ri

∂t. ✭✶✳✶✳✸✮

❈♦♠♦ ♦ ❞❡s❧♦❝❛♠❡♥t♦ ✈✐rt✉❛❧ ♦❝♦rr❡ ♥♦ ♠❡s♠♦ ✐♥st❛♥t❡ ❞❡ t❡♠♣♦✱ ❡♥tã♦✿

δri =

X

j

∂ri

∂qj

δqj.

❊♥tã♦ ♦ tr❛❜❛❧❤♦ ✈✐rt✉❛❧ ❞❡Fi é✿

X

i

Fi·δri =

X

i,j

Fi·

∂ri

∂qj

δqj

= X

j

Qj δqj, ✭✶✳✶✳✹✮

❝♦♠

Qj ≡

X

i

F_i· ∂ri

∂qj

❛ ❢♦rç❛ ❣❡♥❡r❛❧✐③❛❞❛✳ ❚❡♠♦s q✉❡ ✿

X

i

˙

pi·δri =

X

i

mi¨ri·δri

= X

i,j

mi¨ri·

∂ri

∂qj

δqj

X

i

mi¨ri.

∂r_i ∂qj =X i d dt mi˙ri.

∂r_i ∂qj

−mi˙ri

d dt

∂r_i ∂qj ❡ d dt ∂ri ∂qj

= ∂˙ri

∂qj

=X

k

∂2_r

i

∂qj∂qk

˙

qk+

∂2_r

i

∂qj∂t

= ∂vi

✶✳✶✳ ❊◗❯❆➬➹❖ ❉❊ ▲❆●❘❆◆●❊ ✻

❉❛ ❡q✉❛çã♦ ✶✳✶✳✸✱ t❡♠♦s q✉❡✿

∂vi

∂q˙j

= ∂ri

∂qj

.

▲♦❣♦✿

X

i

mi¨ri.

∂ri ∂qj =X i d dt mivi.

∂vi

∂q˙j

−mivi.

∂vi ∂qj . P♦rt❛♥t♦✿ X i ˙

pi.δri =

X j ( d dt " ∂ ∂q˙i

X

i

mivi2

2 !# − ∂ ∂qj X i

mivi2

2

!)

δqj ✭✶✳✶✳✺✮

❙✉❜st✐t✉✐♥❞♦ ✶✳✶✳✹ ❡ ✶✳✶✳✺ ❡♠ ✶✳✶✳✷ ❡ ❝♦♥s✐❞❡r❛♥❞♦P

i miv2

i

2 ❝♦♠♦ ❛ ❡♥❡r❣✐❛ ❝✐♥ét✐❝❛ ❞♦

s✐st❡♠❛ ✭❚✮✱ t❡♠♦s✿

X j d dt ∂T ∂q˙i

−∂T

∂qi

−Qj

δqj = 0.

❖s ❞❡s❧♦❝❛♠❡♥t♦s ✈✐rt✉❛✐s δqj sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s✱ ❧♦❣♦ ♦s t❡r♠♦s q✉❡ ♠✉❧t✐♣❧✐❝❛♠ δqj

sã♦ ♥✉❧♦s✿

d dt

∂T ∂q˙i

− ∂T

∂qi

−Qj = 0, ✭✶✳✶✳✻✮

t❡♥❞♦ ♥ ❡q✉❛çõ❡s ❞❡st❛ ❢♦r♠❛✳

P❛r❛ ❢♦rç❛s q✉❡ ❞❡r✐✈❛♠ ❞♦ ♣♦t❡♥❝✐❛❧V(r1,r2, ...,r_N, t)✿ Fi =−∇V(r1,r2, ...,rN, t),

♦✉ s❡❥❛✱ ♥ã♦ r❡str✐♥❣✐♠♦s ❛ s✐st❡♠❛s ❝♦♥s❡r✈❛t✐✈♦s✱ ♣♦✐s ❱ ❞❡♣❡♥❞❡ ❡①♣❧✐❝✐t❛♠❡♥t❡ ❞♦ t❡♠♣♦✳ ❊♥tã♦✿

Qj =

X

i

Fi.

∂ri

∂qj

=−X

i

∇V.∂ri ∂qj

=−∂V

∂qj

.

❙✉❜st✐t✉✐♥❞♦ ❡st❛ ❡q✉❛çã♦ ❡♠ ✶✳✶✳✻✱t❡♠♦s✿

d dt

∂T ∂q˙i

−∂(T −V)

∂qi

= 0.

❈♦♠♦ ❱ ♥ã♦ ❞❡♣❡♥❞❡ ❞❛s ✈❡❧♦❝✐❞❛❞❡s ❣❡♥❡r❛❧✐③❛❞❛s✱

d dt

∂L ∂q˙i

− ∂L

∂qi

= 0, ✭✶✳✶✳✼✮

♦♥❞❡ L=T −V✱ é ❛ ❧❛❣r❛♥❣✐❛♥❛✳ ❊st❛ é ❛ ❡q✉❛çã♦ ❞❡ ▲❛❣r❛♥❣❡✳

❆ ❧❛❣r❛♥❣✐❛♥❛ é ✉♠❛ ❢✉♥çã♦ ❞❡ q✱q˙❡ t✳ P♦❞❡♠♦s ❞❡✜♥✐r ♦ ♠♦♠❡♥t♦ ❝♦♥❥✉❣❛❞♦ ❛ ♣❛rt✐r ❞❛ ❧❛❣r❛♥❣✐❛♥❛✱ ♣❛r❛ t❛♥t♦ s❡❥❛ L(q,q, t˙ )✱ ❡♥tã♦✿

∂L ∂q˙i

= ∂T

∂q˙i

−∂V

∂q˙i

= ∂T

∂q˙i

= ∂

∂q˙i

X

j

miq˙i2

2

!

=miq˙i =pi

P♦rt❛♥t♦ ♦ ♠♦♠❡♥t♦ ❝♦♥❥✉❣❛❞♦ à ❝♦♦r❞❡♥❛❞❛qi é ❬✶✹❪✿

pi =

∂L ∂q˙i

✶✳✷✳ ❊◗❯❆➬➹❖ ❉❊ ❍❆▼■▲❚❖◆ ✼

✶✳✷ ❊q✉❛çã♦ ❞❡ ❍❛♠✐❧t♦♥

❆ ❡q✉❛çã♦ ❞❡ ▲❛❣r❛♥❣❡ ❣❡r❛ ♥ ❡q✉❛çõ❡s ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ q✉❡ r❡q✉❡r ✷♥ ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s ♣❛r❛ r❡s♦❧✈ê✲❧❛✳

❏á ♥♦ ❢♦r♠❛❧✐s♠♦ ❤❛♠✐❧t♦♥✐❛♥♦ t❡♠♦s ✷♥ ❡q✉❛çõ❡s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✱ t❛❧ q✉❡ ❛ ❞✐✲ ♥â♠✐❝❛ é ❞❡s❝r✐t❛ ♣♦r q1, q2, ...qn, p1, p2, ..., pn✱ ♦♥❞❡ pi é ♦ ♠♦♠❡♥t♦ ❝♦♥❥✉❣❛❞♦ ❞❛❞♦ ♣♦r

✶✳✶✳✽✳ ❆s ✈❛r✐á✈❡✐s ✭q✱♣✮ sã♦ ❝♦♥❤❡❝✐❞❛s ❝♦♠♦ ✈❛r✐á✈❡✐s ❝❛♥ô♥✐❝❛s✳ ❆ ♠✉❞❛♥ç❛ ❞❛s ✈❛r✐á✲ ✈❡✐s (q,q˙) ♣❛r❛ ✭q✱♣✮ é r❡❛❧✐③❛❞❛ ❛tr❛✈és ❞❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❡❣❡♥❞r❡✱ ❝♦♥❢♦r♠❡ ❞❡s❝r✐t♦ ❛ s❡❣✉✐r✳

❆ ❞✐❢❡r❡♥❝✐❛❧ ❞❡L(q,q, t˙ ) é ❡s❝r✐t❛ ❝♦♠♦✿

dL=

n

X

i=1

∂L ∂qi

dqi+

∂L ∂q˙i

dq˙i

+ ∂L

∂tdt

❉❛ ❡q✉❛çã♦ ❞❡ ▲❛❣r❛♥❣❡ ✶✳✶✳✼✱ t❡♠♦s q✉❡✿ ˙

pi =

∂L ∂qi

.

❉❛í✱

dL=

n

X

i=1

( ˙pidqi +pidq˙i) +

∂L

∂tdt. ✭✶✳✷✳✾✮

❆ ❤❛♠✐❧t♦♥✐❛♥❛ ❍✭q✱♣✱t✮ é ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❡❣❡♥❞r❡ ❞❛ ❧❛❣r❛♥❣✐❛♥❛✱ ❞❡✜♥✐❞❛ ♣♦r✱

H(q, p, t) =

n

X

i

˙

qipi−L(q,q, t˙ ).

❉✐❢❡r❡♥❝✐❛♥❞♦ ❡st❛ ❡q✉❛çã♦ ❡ ✉s❛♥❞♦ ✶✳✷✳✾✱ t❡♠♦s

dH(q, p, t) =

n

X

i

( ˙qi dpi+pi dq˙i)−dL(q,q, t˙ )

=

n

X

i

( ˙qi dpi+pi dq˙i)− n

X

i=1

( ˙pidqi+pidq˙i)−

∂L ∂tdt

=

n

X

i

( ˙qi dpi−p˙i dqi)−

∂L ∂tdt.

P♦r ♦✉tr♦ ❧❛❞♦✿

dH(q, p, t) = X

i

∂H ∂qi

dqi+

∂H ∂pi

dpi

+ ∂H

∂t dt.

❈♦♠♣❛r❛♥❞♦ ❛s ❞✉❛s ❡q✉❛çõ❡s ❛❝✐♠❛✱ t❡♠♦s q✉❡✿ ˙

pi = −

∂H ∂qi

˙

qi =

∂H ∂pi

−∂L

∂t =

∂H

∂t ✭✶✳✷✳✶✵✮

♣❛r❛ ✐❂✶✱✳✳✳✱♥✳ ❊st❛s sã♦ ❛s ❡q✉❛çõ❡s ❞❡ ❍❛♠✐❧t♦♥✳

✶✳✸✳ ❚❘❆◆❙❋❖❘▼❆➬Õ❊❙ ❈❆◆Ô◆■❈❆❙ ✽

❝❤❛♠❛❞♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡✳

P♦❞❡♠♦s ❛♣❧✐❝❛r ❛s ❡q✉❛çõ❡s ❞❡ ❍❛♠✐❧t♦♥ ♣❛r❛ ♦❜t❡r ❛ ❡q✉❛çã♦ ❞❡ ♠♦✈✐♠❡♥t♦ ♣❛r❛ ✈❛r✐á✈❡✐s ❞✐♥â♠✐❝❛s✳ ❙❡❥❛ u(q, p, t)✱ ❡♥tã♦✿

du

dt =

X

i

∂u ∂qi

˙

qi+

∂u ∂pi

˙

pi

+∂u

∂t

= X

i

∂u ∂qi

∂H ∂pi

− ∂u

∂pi

∂H ∂qi

+∂u

∂t,

❞❡✜♥✐♥❞♦ ♦ ♣❛rê♥t❡s❡s ❞❡ P♦✐ss♦♥ s❡♥❞♦ [A, B] =P

i

∂A ∂qi

∂B ∂pi −

∂A ∂pi

∂B ∂qi

✳ ❚❡♠♦s✿

du

dt = [u, H] + ∂u

∂t. ✭✶✳✷✳✶✶✮

✶✳✸ ❚r❛♥s❢♦r♠❛çõ❡s ❈❛♥ô♥✐❝❛s

❯♠ s✐st❡♠❛ ♣♦❞❡ s❡r ❞❡s❝r✐t♦ ♣♦r ✉♠❛ ❣❛♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❣❡♥❡r❛❧✐③❛❞❛s✳ ❖✉ s❡❥❛✱ ♣♦❞❡♠♦s ❡s❝♦❧❤❡r ❛s ✈❛r✐á✈❡✐s q✉❡ s✐♠♣❧✐✜q✉❡♠ ❛s ❡q✉❛çõ❡s ❞❡ ❍❛♠✐❧t♦♥✳

❯♠❛ tr❛♥s❢♦r♠❛çã♦ ❝❛♥ô♥✐❝❛ é ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧ q✉❡ ♣r❡s❡r✈❛ ❛s ❡q✉❛çõ❡s ❞❡ ❍❛♠✐❧t♦♥✳

■♥✐❝✐❛❧♠❡♥t❡ ✈❛♠♦s ✐♥tr♦❞✉③✐r ♦ ❝♦♥❝❡✐t♦ ❞♦ ♣r✐♥❝í♣✐♦ ❞❡ ❍❛♠✐❧t♦♥ ♣❛r❛ ❡♥❝♦♥tr❛r ❛s tr❛♥s❢♦r♠❛çõ❡s ❝❛♥ô♥✐❝❛s✳ ❖ ♣r✐♥❝í♣✐♦ ❞❡ ❍❛♠✐❧t♦♥ ❞✐③ q✉❡✱ ❞❛❞♦ ✉♠ s✐st❡♠❛ ❞❡s❝r✐t♦ ♣❡❧❛ ❧❛❣r❛♥❣✐❛♥❛ L(q,q, t˙ ) ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ ❛çã♦✱

S =

Z t2

t1

L(q,q, t˙ )dt

t❛❧ q✉❡ ❡st❛ é ♠í♥✐♠❛ ♠❛♥t❡♥❞♦ ✜①♦ ♦s ♣♦♥t♦s ✐♥✐❝✐❛❧ ❡ ✜♥❛❧ ❞❛ tr❛❥❡tór✐❛✳ ❙❛❜❡♥❞♦ ✐st♦✱ ❡st❛♠♦s ✐♥t❡r❡ss❛❞♦s ❡♠ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ✐♥✈❡rsí✈❡❧ t❛❧ q✉❡✿

Qi = Qi(q, p, t)

Pi = Pi(q, p, t)

❡ ❛ ♥♦✈❛ ❤❛♠✐❧t♦♥✐❛♥❛ ❑✭◗✱P✱t✮ q✉❡ ♣r❡s❡r✈❛ ❛ ❢♦r♠❛ ❤❛♠✐❧t♦♥✐❛♥❛✱ ˙

Qi = ∂K_∂Pi P˙i =−_∂Qi∂K.

❉♦ ♣r✐♥❝í♣✐♦ ❞❡ ❍❛♠✐❧t♦♥ t❡♠♦s q✉❡✿

δS = δ

Z t2

t1

L(q,q, t˙ )dt = 0

= δ Z t2

t1 (

X

i

piq˙i−H(q, p, t)

)

dt= 0 ✭✶✳✸✳✶✷✮

P♦❞❡♠♦s r❡❡s❝r❡✈❡r ❡st❛ ❡q✉❛çã♦ ♣❛r❛ ❛s ♥♦✈❛s ❝♦♦r❞❡♥❛❞❛s✱

δ Z t2

t1 (

X

i

PiQ˙i−K(Q, P, t)

)

dt = 0. ✭✶✳✸✳✶✸✮

P❛r❛ q✉❡ ✶✳✸✳✶✷ ❡ ✶✳✸✳✶✸ s❡❥❛♠ s❛t✐s❢❡✐t❛s s✐♠✉❧tâ♥❡❛♠❡♥t❡✿

X

i

piq˙i−H(q, p, t) =

X

i

PiQ˙i−K(Q, P, t) +

dφ(q, p, t)

✶✳✸✳ ❚❘❆◆❙❋❖❘▼❆➬Õ❊❙ ❈❆◆Ô◆■❈❆❙ ✾

♦♥❞❡✱

δ Z t2

t1 dφ

dtdt =δφ(q(t2), p(t2), t2)−δφ(q(t1), p(t1), t1) = 0

♣♦✐s δ(q(t2)) =δ(q(t1)) = 0 =δ(p(t2)) =δ(p(t1))✳

❆ ❡①✐stê♥❝✐❛ ❞❡K(Q, P, t) ❡ φ(q, p, t) q✉❡ s❛t✐s❢❛③ ✶✳✸✳✶✹ ❣❛r❛♥t❡ q✉❡ ❛ tr❛♥s❢♦r♠❛çã♦ é ❝❛♥ô♥✐❝❛✳ ❉❡ ✶✳✸✳✶✹✱ tê♠ s❡ q✉❡✿

X

i

pidqi−

X

i

PidQi−(H−K)dt=dφ.

❖✉ s❡❥❛φ é ✉♠❛ ❢✉♥çã♦ ❞❡ q✱ ◗ ❡ t✱ ❧♦❣♦✿

F1(q, Q, t) = φ(q, p(q, Q, t), t),

t❛❧ q✉❡

pi =

∂F1 ∂qi

Pi = −

∂F1 ∂Qi

K(Q, P, t) = H(q, p, t) + ∂F1

∂t .

❆ ❢✉♥çã♦F1(q, Q, t) é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❢✉♥çã♦ ❣❡r❛❞♦r❛ ❞❛ tr❛♥s❢♦r♠❛çã♦✳ ❆ tr❛♥s❢♦r✲ ♠❛çã♦ (q, p)−→ (Q, P)é ❝❛♥ô♠✐❝❛ ♣♦r ❝♦♥str✉çã♦✳

P♦❞❡♠♦s t❛♠❜é♠ ❝♦♥str✉✐r ✉♠❛ ❢✉♥çã♦ ❣❡r❛❞♦r❛ q✉❡ ❞❡♣❡♥❞❛ ❞❡ ✭q✱P✮ ❛♦ ✐♥✈és ❞❡ ✭q✱◗✮✱ ✈❛♠♦s ❞❡✜♥✐✲❧❛ s❡♥❞♦✿

F2(q, P, t) = X

i

PiQi+F1

= X

i

PiQi(q, P, t) +φ(q, p(q, P, t), t).

❉✐❢❡r❡♥❝✐❛♥❞♦ ❡st❛ ❡q✉❛çã♦✱ t❡♠♦s✿

dF2 = X

i

(PidQi+QidPi) +dφ

= X

i

(pidqi+QidPi)−(H−K)dt.

▲♦❣♦✿

pi =

∂F2 ∂qi

Qi =

∂F2 ∂Pi

K = H(q, p, t) + ∂F2

∂t .

✶✳✹✳ ❊◗❯❆➬➹❖ ❉❊ ▲■❖❯❱■▲▲❊ ✶✵

✶✳✹ ❊q✉❛çã♦ ❞❡ ▲✐♦✉✈✐❧❧❡

❈♦♥s✐❞❡r❡ ✉♠❛ r❡❣✐ã♦ ❘0 ♥♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡✱ ❝♦♠♦ ♥❛ ✜❣✉r❛ ✶✳✹✳✶✳ ❆s ❝♦♦r❞❡♥❛❞❛s x= (x1, x2,..., x2n)r❡♣r❡s❡♥t❛♠ ❛s ✈❛r✐á✈❡✐s(q,p)✱ ❝♦♥s✐❞❡r❛♠♦s ✉♠ ✈❡t♦rF= (F1, F2, ..., F2n)

t❛❧ q✉❡ Fi r❡♣r❡s❡♥t❛ ♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❡ ✶✳✷✳✶✵✱ ♦✉ s❡❥❛✱ x˙ =F(x,t)✳ ❆ r❡❣✐ã♦ ❘0 r❡♣r❡s❡♥t❛

❛ ❝♦♥✜❣✉r❛çã♦ ✐♥✐❝✐❛❧ ❞♦ s✐st❡♠❛✱ ♦✉ s❡❥❛✱ ✈❛♠♦s ❝♦♥s✐❞❡r❛r t=✵✳ ◆✉♠ ✐♥st❛♥t❡ t ♣♦st❡✲ r✐♦r ✉♠ ♣♦♥t♦ x s❡ ♠♦✈❡ ♣❛r❛ X(x,t) ❡♠ ❘t✳ ◆♦ ❡s♣❛ç♦ ❜✐❞✐♠❡♥s✐♦♥❛❧ ✭q✱♣✮✭♣♦❞❡♠♦s

❣❡♥❡r❛❧✐③❛r ♣❛r❛ q✉❛❧q✉❡r ❞✐♠❡♥sã♦✮ ♦ ✈♦❧✉♠❡ é

V (t) =

Z

R_t

dX1dX2 =

Z

R₀

Jdx1dx2,

t❛❧ q✉❡ ❏ é ♦ ❥❛❝♦❜✐❛♥♦ ❞❛ tr❛♥s❢♦r♠❛çã♦ X(x,t)✿

J =

∂X1

∂x1

∂X1

∂x2

∂X2

∂x1

∂X2

∂x2 .

❋✐❣✉r❛ ✶✳✹✳✶✿ ▼♦✈✐♠❡♥t♦ ❞♦ ✈♦❧✉♠❡ ♥♦ ❡s♣❛ç♦ ❜✐❞✐♠❡♥s✐♦♥❛❧✳

P❛r❛ t ♣❡q✉❡♥♦✱ ♣♦❞❡♠♦s ❛♣r♦①✐♠❛rX ♣♦r✱ X = X(x,0) +tdX

dt (x,0) +O t 2

= x+tF(x,0) +O t2

✭✶✳✹✳✶✺✮ ❝♦♥s✐❞❡r❛♥❞♦ q✉❡ x˙ =F(x,t)✳

❉❡ ✶✳✹✳✶✺✱ t❡♠♦s q✉❡✿

X1 =x1+tF1(x,0) +O(t2₎ X2 =x2+tF2(x,0) +O(t2_).

❙✉❜st✐t✉✐♥❞♦ ❛ r❡❧❛çã♦ ❛❝✐♠❛ ♥♦ ❥❛❝♦❜✐❛♥♦ ♦❜t❡♠♦s✿

J = ∂X1

∂x1 ∂X2

∂x2 − ∂X1

∂x2 ∂X2

∂x1

=

1 +t∂F1

∂x1 1 +t

∂F2 ∂x2

−t2∂F1 ∂x2

∂F2 ∂x1

= 1 +t

∂F1 ∂x1 +

∂F2 ∂x2

t=0

+O t2

✶✳✹✳ ❊◗❯❆➬➹❖ ❉❊ ▲■❖❯❱■▲▲❊ ✶✶

❊♥tã♦ ♦ ✈♦❧✉♠❡ ❞❡ ❘t é

V (t) =

Z

R₀

(1 +t∇.F(x,0))dx1dx2+O t2 .

▲♦❣♦✱

dV (t)

dt

t=0

= lim

t→0

V (t)−V (0)

t

=

Z

R₀

∇.F(x,0)dx1dx2.

P♦❞❡♠♦s ❛♣❧✐❝❛r ❡st❛ ❡q✉❛çã♦ ♣❛r❛ q✉❛❧q✉❡r t✱ t❛❧ q✉❡✿

dV (t)

dt =

Z

R_t

∇ ·F(x,t)dx1dx2.

P❛r❛ ✉♠ s✐st❡♠❛ q✉❡ ♦❜❡❞❡❝❡ ❛s ❡q✉❛çõ❡s ❞❡ ❍❛♠✐❧t♦♥✱

∇.F = ∂F1

∂x1

+∂F2

∂x2

= ∂

∂q

∂H ∂p

+ ∂

∂p

−∂H

∂q

= 0.

▲♦❣♦✱

dV (t)

dt = 0.

❊♥tã♦ ♦ ♠♦✈✐♠❡♥t♦ ❞❡ s✐st❡♠❛s ❤❛♠✐❧t♦♥✐❛♥♦s ♣r❡s❡r✈❛ ♦ ✈♦❧✉♠❡ ♥♦ ❡s♣❛ç♦ ✭q✱♣✮✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ t❡♦r❡♠❛ ❞❡ ▲✐♦✉✈✐❧❧❡❬✶✸❪✳

❆ ♣❛rt✐r ❞♦ t❡♦r❡♠❛ ❞❡ ▲✐♦✉✈✐❧❧❡ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ❛ ❡q✉❛çã♦ ❞❡ ▲✐♦✉✈✐❧❧❡✳ P❛r❛ t❛♥t♦✱ s❡❥❛ ✉♠ s✐st❡♠❛ r❡♣r❡s❡♥t❛❞♦ ♣♦r ✉♠ ♣♦♥t♦ ♥♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡✳ ❱❛♠♦s ❞❡✜♥✐r ♦ ❡♥s❡♠❜❧❡ ❝♦♠♦ s❡♥❞♦ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ s✐st❡♠❛s ♥♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡✳ ❆ ❞❡♥s✐❞❛❞❡ ❞❡ ♣♦♥t♦s ♥♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡ ❉✭q✱♣✱t✮ é ❞❡✜♥✐❞❛ ❝♦♠♦

D(q, p, t) = dN

dV .

♦♥❞❡ ❞◆ é ♦ ♥ú♠❡r♦ ❞❡ ♣♦♥t♦s ✭s✐st❡♠❛s✮ ♥♦ ✈♦❧✉♠❡ ❞❱✳

❈♦♠♦ ♣♦❞❡♠♦s ♦❜s❡r✈❛r ♥❛ ✜❣✉r❛ ✶✳✹✳✶ ❛ ❡✈♦❧✉çã♦ ❞❡ ❘0 ♥♦ t❡♠♣♦ ❢❛③ ❝♦♠ q✉❡ ❛

❢♦r♠❛ ❞❡st❡ ✈♦❧✉♠❡ ♠✉❞❡ ♥♦ t❡♠♣♦✱ ♠❛s ♦ ✈♦❧✉♠❡ ♣❡r♠❛♥❡❝❡ ❝♦♥st❛♥t❡✳ ❖ ♥ú♠❡r♦ ❞❡ s✐st❡♠❛s ❞❡♥tr♦ ❞❡ ❘0 t❛♠❜é♠ ♣❡r♠❛♥❡❝❡ ❝♦♥st❛♥t❡✱ ♣♦✐s s❡ ❛❧❣✉♠ ♣♦♥t♦ ❛tr❛✈❡ss❛r ❛

❜♦r❞❛✱ ❡♠ ❛❧❣✉♠ ✐♥st❛♥t❡✱ s✉❛ tr❛❥❡tór✐❛ ❝r✉③❛ ❛ tr❛❥❡tór✐❛ ❞❡ ✉♠ ♣♦♥t♦ q✉❡ ❡stá ♥❛ s✉♣❡r❢í❝✐❡✳ ◆❡st❡ ❝❛s♦ ❡❧❡s t❡r✐❛♠ ❛s ♠❡s♠❛s ❝♦♦r❞❡♥❛❞❛s ♥♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡ ❞❡s❞❡ q✉❡ ❛ ❞✐♥â♠✐❝❛ ❞♦ s✐st❡♠❛ é ❞❡t❡r♠✐♥❛❞❛ ✉♥✐❝❛♠❡♥t❡ ♣❡❧❛ ❧♦❝❛❧✐③❛çã♦ ♥♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡✳ ❊ss❡s ❞♦✐s ♣♦♥t♦s ✈✐❛❥❛r✐❛♠ ❥✉♥t♦s✱ ❡♥tã♦ ✉♠ s✐st❡♠❛ ♥✉♥❝❛ ♣♦❞❡ s❛✐r ❞♦ ✈♦❧✉♠❡✱ ❢❛③❡♥❞♦ ❝♦♠ q✉❡ ♦ ♥ú♠❡r♦ ❞❡ s✐st❡♠❛s t❛♠❜é♠ ♣❡r♠❛♥❡ç❛ ✜①♦✳ ▲♦❣♦D(q, p, t)é ❝♦♥st❛♥t❡ ♥♦ t❡♠♣♦✱ ♦✉ s❡❥❛✱

dD(q, p, t)

dt = 0,

♦✉ ❝♦♠♦ ✈✐♠♦s ❡♠ ✶✳✷✳✶✶✿

∂D

∂t + [D, H] = 0 ✭✶✳✹✳✶✻✮

✶✳✺✳ ❍■❊❘❆❘◗❯■❆ ❇❇❑●❨ ✶✷

✶✳✺ ❍✐❡r❛rq✉✐❛ ❇❇❑●❨

P❛r❛ ✉♠ s✐st❡♠❛ ❞❡ ◆ ♣❛rtí❝✉❧❛s ✈❛♠♦s ❝♦♥s✐❞❡r❛r xi= (qi,pi)✱ ♦♥❞❡ qi é ❛ ❝♦✲

♦r❞❡♥❛❞❛ ❣❡♥❡r❛❧✐③❛❞❛ ❡ pi ♦ ♠♦♠❡♥t♦ ❝♦♥❥✉❣❛❞♦ ❝❛♥ô♥✐❝♦✳ ❙❡ ♦ ❡st❛❞♦ x1, x2, ..., xN

❡stá ♦❝✉♣❛❞♦✱ ❡♥tã♦ ❛ ♣❛rtí❝✉❧❛ ✶ ❡stá ♥♦ ❡st❛❞♦ x1✱ ❛ ♣❛rtí❝✉❧❛ ✷ ❡stá ❡♠ x2 ❡ ❛ss✐♠

♣♦r ❞✐❛♥t❡✳ ❱❛♠♦s ❞❡✜♥✐r F(x1, x2, ..., xN)✭❛ ❞❡♣❡♥❞ê♥❝✐❛ ♥♦ t❡♠♣♦ s❡rá ♦❝✉❧t❛❞❛✮ ❝♦♠♦

s❡♥❞♦ ❛ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ◆ ♣❛rtí❝✉❧❛s✳ ❆ ❢✉♥çã♦ F(x1, x2, ..., xN) é

s✐♠étr✐❝❛ s♦❜ ♠✉❞❛♥ç❛ ❞❡ ❛r❣✉♠❡♥t♦s✱

F (x1, x2,..., xj, ..., xk, ..., xN) =F (x1, x2,..., xk, ..., xj, ..., xN).

❱❛♠♦s ❞❡✜♥✐r f1(x1)✱ ❛ ❢✉♥çã♦ ❞✐str✐❜✉✐çã♦ r❡❞✉③✐❞❛ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛✱ t❛❧ q✉❡✿

f1(x1) = N Z

dx2...dxNF(x1, x2, ..., xN).

❊ ❛ ❢✉♥çã♦ ❞✐str✐❜✉✐çã♦ r❡❞✉③✐❞❛ ❞❡ s ♣❛rtí❝✉❧❛sfs(x1, x2, ..., xs)✱ ♣❛r❛ s≤◆

fs(x1, ..., xs) =

N! (N −s)!

Z

dxs+1...dxNF (x1, x2, ..., xN),

fs(x1, ..., xs)r❡♣r❡s❡♥t❛ ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❡♥❝♦♥tr❛r s ♣❛rtí❝✉❧❛s ❡♠x1, ..., xs✳

❆ ❝♦♥❞✐çã♦ ❞❡ ♥♦r♠❛❧✐③❛çã♦ ❞❡fs(x1, ..., xs) é

Z

dx1...dxsfs(x1, x2, ..., xs) =

N!

(N −s)!. ✭✶✳✺✳✶✼✮ ❈♦♥s✐❞❡r❡ ✉♠ s✐st❡♠❛ ❞❡ ◆ ♣❛rtí❝✉❧❛s ✐❞ê♥t✐❝❛s✱ t❛❧ q✉❡ ❛ ❤❛♠✐❧t♦♥✐❛♥❛ é

H(x1, x2, ..., xN) = N

X

i=1 p2

i

2m + X

i≤

N

X

k

V (qi−qk).

P♦❞❡♠♦s r❡❡s❝r❡✈❡r ❛ ❡q✉❛çã♦ ✶✳✹✳✶✻ ❡♠ t❡r♠♦s ❞❡ ♦♣❡r❛❞♦r❡s ❧✐♦✉✈✐❧✐❛♥♦s✱ ❞❡✜♥✐❞♦s ❛ s❡❣✉✐r✱ t❛❧ q✉❡

∂tF (q, p, t) = [H, F(q, p, t)]

= LF(q, p, t) ♦♥❞❡

LF ≡ X

j ∂H ∂qj ∂F ∂pj − ∂H ∂pj ∂F ∂qj = X j ( X i≤ N X k

∂V (qi−qk)

∂qj

∂F ∂pj

− pj

m ∂F ∂qj ) = X i≤ N X k

∂V (qi−qk)

∂qi

∂F ∂pi

+∂V (qi−qk)

∂qk ∂F ∂pk −X j pj m ∂F ∂qj = X i≤ N X k

∂V (qi−qk)

∂qi

∂F ∂pi

− ∂V (qi−qk)

∂qi ∂F ∂pk −X j pj m ∂F ∂qj

= −X

j vj ∂F ∂qj +X i≤ N X k

∂V (qi−qk)

✶✳✺✳ ❍■❊❘❆❘◗❯■❆ ❇❇❑●❨ ✶✸

❢❛③❡♥❞♦

vj = pj_m, ∇j ≡ _∂qj∂ , ∂j ≡ _∂pi∂ , ∂jn ≡ _∂pj∂ −_∂pn∂ ,

♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ▲✱ ❝♦♠♦✿

L=L0+L′ =X

j

L0_j +X

j≤

N

X

k

L′_jk

♦♥❞❡

L0_j = −vj∇j,

L′_jk = (∇jVjk)∂jk.

❖✉ s❡❥❛✱

∂tF = N

X

j=1

L0_jF +X

j≤

N

X

n=1

L′_jnF. ✭✶✳✺✳✶✽✮

Z

dx1...dxNF (x1, x2, ..., xN) = 1.

P❛r❛ ♦s ♣ró①✐♠♦s ❝á❧❝✉❧♦s ✈❛♠♦s ❝♦♥s✐❞❡r❛r

R

∂qi∂F_∂qi = 0 e

R

∂pi_∂p∂F_i = 0, ✭✶✳✺✳✶✾✮

♣♦✐s ❡st❛s sã♦ ✐♥t❡❣r❛✐s ❞❡ s✉♣❡r❢í❝✐❡✳

■♥t❡❣r❛♥❞♦ ❛ ❡q✉❛çã♦ ✶✳✺✳✶✽ s♦❜ ❛s ♣❛rtí❝✉❧❛s s✰✶✱✳✳✳✱◆ ❡ ♠✉❧t✐♣❧✐❝❛♥❞♦ ♣♦r N! (N−s)!✱

t❡♠♦s✿

∂tfs =

N! (N −s)!

Z

dxs+1...dxN

( _N X

j=1

L0_jF +X

j≤

N

X

n=1 L′_jnF

)

. ✭✶✳✺✳✷✵✮

❈♦♥s✐❞❡r❛♥❞♦ ♦ ♣r✐♠❡✐r♦ t❡r♠♦ ❞♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❛ ❡q✉❛çã♦ ❛❝✐♠❛✱

N! (N −s)!

N

X

j=1 Z

dxs+1...dxNvj

∂F ∂qj

= N! (N −s)!

s

X

j=1 Z

dxs+1...dxNvj

∂F ∂qj

+ N! (N −s)!

N

X

j=s+1 Z

dxs+1...dxNvj

∂F ∂qj

= N! (N −s)!

s

X

j=1 Z

dxs+1...dxNvj

∂F ∂qj = s X j=1

L0_jfs(x1, x2, ..., xs)

❖ s❡❣✉♥❞♦ t❡r♠♦ é ❞✐✈✐❞✐❞♦ ❡♠ três ❝❛s♦s✿

N! (N −s)!

Z

dxs+1...dxN N X j=1≤ N X n=1

L′_jnF = N! (N −s)!

Z

dxs+1...dxN s X j=1≤ s X n=1 L′_jnF

+ N! (N −s)!

Z

dxs+1...dxN s

X

j=1

N

X

n=s+1 L′_jnF

+ N! (N −s)!

Z

dxs+1...dxN N

X

j=s+1≤

N

X

n=s+1 L′_jnF

✶✳✺✳ ❍■❊❘❆❘◗❯■❆ ❇❇❑●❨ ✶✹

❊st❡s três ❝❛s♦s ❡stã♦ r❡♣r❡s❡♥t❛❞♦s ♥❛ ✜❣✉r❛ ✶✳✺✳✷✿

❋✐❣✉r❛ ✶✳✺✳✷✿ ❚❡r♠♦s ❞♦ s♦♠❛tór✐♦ ✶✳✺✳✷✶✳

✶✳ ❥ ❡ ♥ ❡stã♦ ❞❡♥tr♦ ❞♦ ❣r✉♣♦ ✭✶✱✳✳✳✱s✮✿

N! (N −s)!

Z

dxs+1...dxN s

X

j=1≤

s

X

n=1

L′_jnF =

s

X

j=1≤

s

X

n=1

L′_jn N!

(N −s)!

Z

dxs+1...dxNF

=

s

X

j=1≤

s

X

n=1

L′_jnfs(x1, x2, ..., xs).

✷✳ ❥ ♣❡rt❡♥❝❡ ❛♦ ❣r✉♣♦ ✭✶✱✳✳✳✱s✮ ❡ ♥ ❛♦ ❣r✉♣♦ ✭s✰✶✱✳✳✳✱◆✮✳ ❯s❛♥❞♦ ♦ ❢❛t♦ ❞❡ ❋ s❡r ✉♠❛ ❢✉♥çã♦ s✐♠étr✐❝❛ s♦❜ ♠✉❞❛♥ç❛ ❞❡ ♣❛rtí❝✉❧❛s✿

N! (N −s)!

Z

dxs+1...dxN s

X

j=1

N

X

n=s+1 L′_jnF

= N! (N −s)!

Z

dxs+1...dxN s

X

j=1

N

X

n=s+1

∇jVjn

∂F ∂pj

− ∂F

∂pn

= N! (N −s)!

Z

dxs+1...dxN s

X

j=1

(N −s)∇jVj(s+1)

∂F ∂pj

− ∂F

∂p(s+1)

=

s

X

j=1 Z

dxs+1∇jVj(s+1)∂j(s+1)

N! (N −s−1)!

Z

dxs+2...dxNF

=

s

X

j=1 Z

dxs+1L

′

j(s+1)fs+1(x1, x2, ..., xs+1).

✸✳ ❥ ❡ ♥ ❡stã♦ ❞❡♥tr♦ ❞♦ ❣r✉♣♦ ✭s✰✶✱✳✳✳✱◆✮✿

N! (N −s)!

Z

dxs+1...dxN N

X

j=s+1≤

N

X

n=s+1

L′_jnF = 0

✶✳✺✳ ❍■❊❘❆❘◗❯■❆ ❇❇❑●❨ ✶✺

P♦rt❛♥t♦ s✉❜st✐t✉✐♥❞♦ ❡st❡s r❡s✉❧t❛❞♦s ❡♠ ✶✳✺✳✷✵✱t❡♠♦s✿

∂tfs(x1, x2, ..., xs) = s

X

j=1

L0_jfs(x1, x2, ..., xs) + s

X

j≤

s

X

n=1

L′_jnfs(x1, x2, ..., xs)

+

s

X

j=1 Z

dxs+1L

′

j(s+1)fs+1(x1, x2, ..., xs+1) ✭✶✳✺✳✷✷✮

♣❛r❛ 1≤s≤N✳

❆ ❡q✉❛çã♦ ❛❝♦♣❧❛❞❛ ❛❝✐♠❛ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❡q✉❛çã♦ ❇❇❑●❨✱ ❡st❛ ❡q✉❛çã♦ t❡♠ ❡str✉t✉r❛ ❤✐❡r❛rq✉✐❝❛✱ ♦✉ s❡❥❛✱ ♣❛r❛ ❞❡t❡r♠✐♥❛r ❛ ❢✉♥çã♦ ❞✐str✐❜✉✐çã♦ ❞❡ s ♣❛rtí❝✉❧❛s é ♥❡❝❡ssár✐♦ ❝♦♥❤❡❝❡r ❛ ❢✉♥çã♦ ❞✐str✐❜✉✐çã♦ ❞❡ s✰✶ ♣❛rtí❝✉❧❛s✳

❆ ❡q✉❛çã♦ ✶✳✺✳✷✷ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ❛tr❛✈és ❞❡ ❞✐❛❣r❛♠❛s✳ ❱❛♠♦s ❝♦♥s✐❞❡r❛r

fs(x1, x2, ..., xs) ✉♠❛ ❧✐♥❤❛ ❤♦r✐③♦♥t❛❧✱ ♦ ♦♣❡r❛❞♦r L0j ♥ã♦ é r❡♣r❡s❡♥t❛❞♦ ❣r❛✜❝❛♠❡♥t❡✱

♣♦✐s ❡st❡ ♥ã♦ r❡♣r❡s❡♥t❛ ♥❡♥❤✉♠❛ ✐♥t❡r❛çã♦✱ ♥ã♦ ♠✉❞❛ ♦ ❡st❛❞♦ ❞❛s ♣❛rtí❝✉❧❛✳ ❏á ♦ ♦♣❡r❛❞♦r L′_jn✱ ❝♦♠♦ r❡♣r❡s❡♥t❛ ❛s ✐♥t❡r❛çõ❡s ❡♥tr❡ ♣❛rtí❝✉❧❛s ❡ ❡st❛s ✐♥t❡r❛çõ❡s ❛❢❡t❛♠ ♦

❡st❛❞♦ ❞♦ s✐st❡♠❛✱ ✈❛♠♦s r❡♣r❡s❡♥tá✲❧♦ ❝♦♠♦ ✉♠ ✈ért✐❝❡ q✉❡ ✉♥❡ ❛s ❧✐♥❤❛s ❞❡ ❥ ❡ ♥ ❞♦ ❧❛❞♦ ❞✐r❡✐t♦✳

◆❛ ❡q✉❛çã♦ ✶✳✺✳✷✷ ❤á ❞♦✐s t❡r♠♦s q✉❡ ❡♥✈♦❧✈❡ ♦ ♦♣❡r❛❞♦r L′_jn✳ ❯♠ t❡r♠♦ r❡♣r❡s❡♥t❛

❛♣❡♥❛s ❛ ✐♥t❡r❛çã♦ ❡♥tr❡ ✉♠ ❣r✉♣♦ ❞❡ ♣❛rtí❝✉❧❛s✱ ♥❡st❡ ❝❛s♦ ✈❛♠♦s r❡♣r❡s❡♥t❛r ❝♦♠♦ ❞✉❛s ❧✐♥❤❛s ✐♥❞♦ ♣❛r❛ ❛ ❡sq✉❡r❞❛✱ ❡st❡ t✐♣♦ ❞❡ ❞✐❛❣r❛♠❛ ✈❛✐ s❡r ❝❤❛♠❛❞♦ ❞❡ ✈ért✐❝❡ ❳ ✭✈❡r ✶✳✺✳✸✮✳

❖ s❡❣✉♥❞♦ t❡r♠♦ ❝♦♠ L′_j(s+1) ❡♥✈♦❧✈❡ ❛ ✐♥t❡r❛çã♦ ❡♥tr❡ ❛s ♣❛rtí❝✉❧❛s✱ ❥ ❡ s✰✶✱ ❡ ❛ ✐♥✲

t❡❣r❛çã♦ ❡♠ s✰✶✱ q✉❡ r❡♣r❡s❡♥t❛ ❛ ♠✉❞❛♥ç❛ ❞❡ ❡st❛❞♦ ❞❡st❛ ♣❛rtí❝✉❧❛ ♣❛r❛ ♦ ❡st❛❞♦ ❞❛ ♣❛rtí❝✉❧❛ ❥✳ ❯♠❛ s✐♠♣❧❡s ❧✐♥❤❛ ❞♦ ❧❛❞♦ ❡sq✉❡r❞♦ r❡♣r❡s❡♥t❛ ❡st❛ ✐♥t❡❣r❛çã♦✱ ❛ ❡st❡ t✐♣♦ ❞❡ ❞✐❛❣r❛♠❛ ✈❛♠♦s ❝❤❛♠❛r ❞❡ ✈ért✐❝❡ ❨ ✭✈❡r ✶✳✺✳✸✮✳

❋✐❣✉r❛ ✶✳✺✳✸✿ ❱ért✐❝❡s ❳ ❡ ❨✳

P❛r❛ s❂✶

∂tf1(x1) = L01f1(x1) + Z

dx2L′₁₂f2(x1, x2).

❋✐❣✉r❛ ✶✳✺✳✹✿ ❉✐❛❣r❛♠❛ ❞❡ ❡✈♦❧✉çã♦ ❞❡f1(x1)✳

P❛r❛ s❂✷

∂tf2(x1, x2) = L01f2(x1, x2) +L02f2(x1, x2) +L

′

12f2(x1, x2)

+

Z dx3L

′

13f3(x1, x2, x3) + Z

dx3L

′

✶✳✻✳ ❆◆➪▲■❙❊ ❉❊ P❘■●❖●■◆❊ ✶✻

❋✐❣✉r❛ ✶✳✺✳✺✿ ❉✐❛❣r❛♠❛ ❞❡ ❡✈♦❧✉çã♦ ❞❡f2(x1, x2)✳

P❛r❛ s❂✸✱

∂tf3(x1, x2, x3) = L01f3(x1, x2, x3) +L02f3(x1, x2, x3) +L03f3(x1, x2, x3)

+L′₁₂f3(x1, x2, x3) +L′₁₃f3(x1, x2, x3) +L′₂₃f3(x1, x2, x3) +

Z

dx4L′₁₄+L′₂₄+L′₃₄f4(x1, x2, x3)

❋✐❣✉r❛ ✶✳✺✳✻✿ ❉✐❛❣r❛♠❛ ❞❡ ❡✈♦❧✉çã♦ ❞❡f3(x1, x2, x3)✳

❊ ❛ss✐♠ ♣♦r ❞✐❛♥t❡ ❬✸❪✳

✶✳✻ ❆♥á❧✐s❡ ❞❡ Pr✐❣♦❣✐♥❡

✶✳✻✳ ❆◆➪▲■❙❊ ❉❊ P❘■●❖●■◆❊ ✶✼

❙✐st❡♠❛s ♠❛❝r♦s❝ó♣✐♦s sã♦ ❝❛r❛❝t❡r✐③❛❞♦s ❡♠ ❤♦♠♦❣ê♥❡♦s ❡ ✐♥♦♠♦❣ê♥❡♦s✳ ❙❡ ♦s s✐st❡✲ ♠❛s sã♦ ❤♦♠♦❣ê♥❡♦s ❡♥tã♦ ❛ ❢✉♥çã♦ ❞✐str✐❜✉✐çã♦ ❞❡ ◆ ♣❛rtí❝✉❧❛s ❋✭x1✱x2✱✳✳✳✱xN✮ é ✐♥✈❛r✐✲

❛♥t❡ s♦❜ tr❛♥s❧❛çã♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s ✭ql✮→✭ql✰❜✮✱ ♦✉ s❡❥❛✱

F (q1+a,q2+a, ...,qN +a,v1,v2, ...,vN, t) =F (q1,q2, ...,qN,v1,v2, ...,vN, t).

✭✶✳✻✳✷✸✮ ❊st❛ ✐❣✉❛❧❞❛❞❡ ✐♠♣❧✐❝❛ q✉❡ ❛ ❢✉♥çã♦ ❞✐str✐❜✉✐çã♦ r❡❞✉③✐❞❛ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ é✿

f1(q1,v1, t) = N Z

dq2...dqNdv2...dvNF (q1,q2, ...,qN,v1,v2, ...,vN, t)

= N Z

dq2...dqNdv2...dvNF (q1+a,q2+a, ...,qN +a,v1,v2, ...,vN, t)

= f1(q1+a,v1, t).

P♦rt❛♥t♦

f1(q1,v1, t) = cϕ1(v1, t). ✭✶✳✻✳✷✹✮

❊ ❛ ❢✉♥çã♦ ❞✐str✐❜✉✐çã♦ r❡❞✉③✐❞❛ ❞❡ ❞✉❛s ♣❛rtí❝✉❧❛s ❞❡♣❡♥❞❡ ❞❛ ❞✐stâ♥❝✐❛ r❡❧❛t✐✈❛ ❞❛s ❞✉❛s ♣❛rtí❝✉❧❛s✿

f2(q1,q2,v1,v2, t) = N(N −1) Z

dq3...dqNdv3...dvNF (q1,q2, ...,qN,v1,v2, ...,vN, t)

= N(N −1)

Z

dq3...dqNdv3...dvNF (q1+a, ...,qN +a,v1, ...,vN, t)

= f2(q1+a,q2+a,v1,v2, t)

⇒ f2(q1,q2,v1,v2, t) = f2(q₁−q₂,v1,v2, t).

❈♦♥s✐❞❡r❡ ✉♠ ❣ás ❝♦♥t❡♥❞♦ ✉♠ ♥ú♠❡r♦ ♠✉✐t♦ ❣r❛♥❞❡ ❞❡ ♣❛rtí❝✉❧❛s ❞❡♥tr♦ ❞❡ ✉♠ ✈♦❧✉♠❡ ♠✉✐t♦ ❣r❛♥❞❡ Ω✱ ♣♦rt❛♥t♦ ♥♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ♠❛t❡♠át✐❝♦ ♣♦❞❡♠♦s ❢❛③❡r ♦ ❧✐♠✐t❡ ◆ −→ ∞❡Ω−→ ∞✳ ❉❡✈❡♠♦s t♦♠❛r ❝✉✐❞❛❞♦ ♥❡st❡ ❧✐♠✐t❡✱ ♣♦rq✉❡ ❛ r❛③ã♦N/Ω❞❡✈❡ s❡r ❝♦♥st❛♥t❡ ❡ ✜♥✐t❛✱ ♦✉ s❡❥❛✱

( N −→ ∞

Ω−→ ∞

N/Ω = c

✭✶✳✻✳✷✺✮

❡st❡ ❧✐♠✐t❡ é ♦ ❧✐♠✐t❡ t❡r♠♦❞✐♥â♥✐❝♦✳

Pr✐❣♦❣✐♥❡ ❛ss✉♠❡ q✉❡ ❛ ❢✉♥çã♦ ❞✐str✐❜✉✐çã♦ r❡❞✉③✐❞❛fs ❞❡✈❡ s❡r ✜♥✐t❛ ♥♦ ❧✐♠✐t❡ ❛❝✐♠❛✱

♦✉ s❡❥❛✱ F(x1, ..., xN, t)❞❡✈❡ ❣❛r❛♥t✐r ❡st❡ r❡q✉❡r✐♠❡♥t♦ ❡ fs ❞❡✈❡ ♣❡r♠❛♥❡❝❡r ❛ss✐♠ ♣❛r❛

q✉❛❧q✉❡r t❡♠♣♦ ✭✈❡r ❋✐❣✉r❛ ✶✳✻✳✼✮✳

■r❡♠♦s ✉s❛r ❡st❡ r❡q✉❡r✐♠❡♥t♦ ♥❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ ❋♦✉r✐❡r ❞❛ ❢✉♥çã♦ ❞✐str✐❜✉✐çã♦✳ P❛r❛ t❛♥t♦ t❡r❡♠♦s q✉❡ ❡♥❝♦♥tr❛r ❛s ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ ♣❛r❛ ♥♦ss♦ s✐st❡♠❛✳ ❱❛♠♦s ❛ss✉♠✐r q✉❡ ♦ ❣ás ❡stá ❡♠ ✉♠❛ ❝❛✐①❛ ❞❡ ✈♦❧✉♠❡ Ω✭✈❡r ✜❣✉r❛ ✶✳✻✳✽✮ ❡ q✉❡ ❡st❡ s✐st❡♠❛ é r❡♣❡t✐❞♦ ♣❡r✐♦❞✐❝❛♠❡♥t❡ ❡♠ ✉♠ ❡s♣❛ç♦ ♥❛ ❞✐r❡çã♦ ✐x✱ ✐y ❡ ✐z✱ t❛❧ q✉❡✿

i_x = (1,0,0)

iy = (0,1,0) ✭✶✳✻✳✷✻✮

✶✳✻✳ ❆◆➪▲■❙❊ ❉❊ P❘■●❖●■◆❊ ✶✽

❋✐❣✉r❛ ✶✳✻✳✼✿ ❋✉♥çã♦ ❞✐str✐❜✉✐çã♦ r❡❞✉③✐❞❛fs(N)♥♦ ❧✐♠✐t❡ t❡r♠♦❞✐♥â♠✐❝♦✳

❋✐❣✉r❛ ✶✳✻✳✽✿ ❱❡t♦r❡s ✐x✱ ✐y ❡ ✐z✳

❚❡♠♦s q✉❡✿

F

q_j+ Ω1/3i_x , v

= F

q_j + Ω1/3i_y , v

= F

qj + Ω1/3iz , v

= F ({q_j}, v). ✭✶✳✻✳✷✼✮

❊①♣❛♥❞✐♥❞♦ ❛ ❢✉♥çã♦ ❞✐str✐❜✉✐çã♦ ✐♥✐❝✐❛❧ ❡♠ sér✐❡ ❞❡ ❋♦✉r✐❡r✿

F (q, v,0) =X

k1

...X

kN

˜

ρk1,...,kN(v1, ...,vN) exp i N

X

j

k_j.qj

!

, ✭✶✳✻✳✷✽✮

❝♦♠ ρ˜k1,...,kN(v1, ...,vN) ❛ ❝♦♠♣♦♥❡♥t❡ ❞❡ ❋♦✉r✐❡r✱ q✉❡ ❞❡♣❡♥❞❡ ❞❛s ✈❡❧♦❝✐❞❛❞❡s✱ ♣♦✐s ❛

❢✉♥çã♦ ❞✐str✐❜✉✐çã♦ é ❡s♣❛❝✐❛❧♠❡♥t❡ ♣❡r✐ó❞✐❝❛✳ ❆ ❝♦♥❞✐çã♦ ❞❡ ❝♦♥t♦r♥♦ ❊q✳ ✶✳✻✳✷✼ r❡q✉❡r q✉❡✿

F

qj + Ω1/3ix , v

= F ({qj}, v)

X

k1

...X

kN

˜

ρk1,...,kN(v1, ...,vN)e( iPN

j=1kj.(qj+Ω1/3ix)) = X

k1

...X

kN

˜

ρk1,...,kN(v1, ...,vN)e( iPN

j=1kj.qj)

k_j.qj = kj. qj + Ω1/3ix

−2πn kj.Ω1/3ix = 2πn

kjx =

2πnx

✶✳✻✳ ❆◆➪▲■❙❊ ❉❊ P❘■●❖●■◆❊ ✶✾

✈❛♠♦s ❝❤❛♠❛r n =nx só ♣♦r ♥♦t❛çã♦✳

P❛r❛ ❛s ♦✉tr❛s ❝♦♥❞✐çõ❡s ❞❛ ❡q✉❛çã♦ ❊q✳ ✶✳✻✳✷✼✱ t❡♠♦s✿

kj.qj = kj. qj + Ω1/3iy

−2πn

kjy =

2πny

Ω1/3

kj.qj = kj. qj + Ω1/3iz

−2πn

kjz =

2πnz

Ω1/3

▲♦❣♦✿

kj =

2πnj

Ω1/3 ✭✶✳✻✳✷✾✮

t❛❧ q✉❡ ❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ nj sã♦ ♥ú♠❡r♦s ✐♥t❡✐r♦s✳

P♦❞❡♠♦s r❡❡s❝r❡✈❡r ✶✳✻✳✷✽✱ t❛❧ q✉❡✿

F (q, v,0) = ˜ρ0...0+

X

j

X

kj

˜

ρ0...kj...0(vj; 0)eikj.qj

+X j X n X kj X kn ˜

ρ0...kj...kn...0(vj,vn; 0)eikj.qj+ikn.qn

+... ✭✶✳✻✳✸✵✮

˜

ρ 0...0 é ♦ t❡r♠♦ ❝♦♠ t♦❞♦s ♦s ❦p ♥✉❧♦s✱ρ˜0...kj...0 é ♦ t❡r♠♦ ❝♦♠ ❛♣❡♥❛s ✉♠ ✈❡t♦r ❞❡ ♦♥❞❛

❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦ ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✳

P❛r❛ ♦ ❝❛s♦ ❡s♣❡❝í✜❝♦ ❞❡ ✉♠ s✐st❡♠❛ ❤♦♠♦❣ê♥❡♦✱ q✉❡ ♦❜❡❞❡❝❡ ❛s ❝♦♥❞✐çõ❡s ❞❛ ❊q✳ ✶✳✻✳✷✸✱ t❡♠♦s✿

F (x1, ...,x_N) = F (x₁+b, ...,x_N +b)

X

k1

...X

kN

˜

ρk1,...,kN(v1, ...,vN)e( iP

jkj.qj) = X

k1

...X

kN

˜

ρk1,...,kN(v1, ...,vN)e( iP

jkj.(qj+b))

X

j

kj = 0. ✭✶✳✻✳✸✶✮

❈♦♠♦ ✈✐♠♦s ❛♥t❡r✐♦r♠❡♥t❡ ❛ ❢✉♥çã♦ ❞✐str✐❜✉✐çã♦ r❡❞✉③✐❞❛ t❡♠ q✉❡ s❡r ✜♥✐t❛ ♥♦ ❧✐♠✐t❡ t❡r♠♦❞✐♥â♠✐❝♦ ✭❊q✳ ✶✳✻✳✷✺✮✱ ❧♦❣♦ ❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ ❋♦✉r✐❡r ❞❡✈❡♠ ❞❡♣❡♥❞❡r ❞♦ ✈♦❧✉♠❡✳ ❙❛❜❡♥❞♦ ❞✐st♦ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ❛ ❊q✳ ✶✳✻✳✸✵ ♥❛ ❢♦r♠❛✿

F (q, v,0) = Ω−N{ρ0(|v; 0) +

8π3

Ω

X

j

X

kj

ρkj(vj|...; 0)e ikj.qj

+

8π3

Ω 2 X j X n X kj X kn

(kj+kn6=0)

ρkj,kn(vj,vn|...; 0)e

ikj.qj+ikn.qn

+

8π3

Ω X j X n X kj

ρkj,k−j(vj,vn|...; 0)e

ikj.(qj−qn)

+

8π3

Ω

r−s X j ...X n X kj ...X kn

δka+...+kb...δkc+...+kdρkj,kn(vj, ...,vn|...; 0)e

ikj.qj+...+ikn.qn