✉♥✐✈❡rs✐❞❛❞❡ ❢❡❞❡r❛❧ ❞♦ r✐♦ ❣r❛♥❞❡ ❞♦ ♥♦rt❡
❝❡♥tr♦ ❞❡ ❝✐ê♥❝✐❛s ❡①❛t❛s ❡ ❞❛ t❡rr❛
❞❡♣❛rt❛♠❡♥t♦ ❞❡ ❢ís✐❝❛ t❡ór✐❝❛ ❡ ❡①♣❡r✐♠❡♥t❛❧
♣r♦❣r❛♠❛ ❞❡ ♣ós✲❣r❛❞✉❛çã♦ ❡♠ ❢ís✐❝❛
Pr♦♣r✐❡❞❛❞❡s ❈rít✐❝❛s ❞❡ s✐st❡♠❛s ❢♦r❛ ❞♦
❡q✉✐❧í❜r✐♦ ✈✐❛ s✐♠✉❧❛çã♦ ▼♦♥t❡ ❈❛r❧♦
♠❛r❝❡❧♦ ❜r✐t♦ ❞❛ s✐❧✈❛
♠❛r❝❡❧♦ ❜r✐t♦ ❞❛ s✐❧✈❛
Pr♦♣r✐❡❞❛❞❡s ❈rít✐❝❛s ❞❡ s✐st❡♠❛s ❢♦r❛ ❞♦
❡q✉✐❧í❜r✐♦ ✈✐❛ s✐♠✉❧❛çã♦ ▼♦♥t❡ ❈❛r❧♦
❚❡s❡ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ❋ís✐❝❛ ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛ ❚❡ór✐❝❛ ❡ ❊①✲ ♣❡r✐♠❡♥t❛❧ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❘✐♦ ●r❛♥❞❡ ❞♦ ◆♦rt❡ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ❉♦✉t♦r ❡♠ ❋ís✐❝❛✳
❈♦♠✐ssã♦ ❊①❛♠✐♥❛❞♦r❛✿
Pr♦❢✳ ❉r✳ ❯♠❜❡rt♦ ▲❛✐♥♦ ❋✉❧❝♦ ✭❖r✐❡♥t❛❞♦r✮
Pr♦❢✳ ❉r✳ ❊✉❞❡♥✐❧s♦♥ ▲✐♥s ❞❡ ❆❧❜✉q✉❡rq✉❡ ✭❯❋❘◆✮
Pr♦❢✳ ❉r✳ ▲✉❝✐❛♥♦ ❘♦❞r✐❣✉❡s ❞❛ ❙✐❧✈❛ ✭❯❋❘◆✮
Pr♦❢✳ ❉r✳ ❆♥t♦♥✐♦ ❞❡ ▼❛❝❡❞♦ ❋✐❧❤♦ ✭❈♦♦r✐❡♥t❛❞♦r ✲ ❯❊❙P■✮
Pr♦❢✳ ❉r✳ ▼❛✉r✐③✐♦ ❙❡r✈❛ ✭❯♥✐✈❡rs✐tà ❞❡❧❧ ➪q✉✐❧❛ ✮
❊st❛ t❡s❡ é ❞❡❞✐❝❛❞❛ às ♣❡ss♦❛s q✉❡ ❞✐r❡t❛✱ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡✱ ❝♦♥tr✐❜✉✐r❛♠ ♣❛r❛ s✉❛ ♠❛t❡r✐❛❧✐③❛çã♦✳
❆❣r❛❞❡ç♦
❆ ❉❡✉s ♣❡❧❛s ❣r❛ç❛s ❝♦♥❝❡❞✐❞❛s✱ ❞❛s q✉❛✐s ❝✐t♦ ❛ ♠❛✐♦r − ♠✐♥❤❛ ✈✐❞❛✳
❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ Pr♦❢❡ss♦r ❉r✳ ❯♠❜❡rt♦ ▲❛✐♥♦ ❋✉❧❝♦✱ ♣❡❧❛ ♦r✐❡♥t❛çã♦ ❡ ♣❡❧♦ ❛♣♦✐♦ ❧♦❣íst✐❝♦ ❡ ✐♥t❡❧❡❝t✉❛❧ ❞✉r❛♥t❡ t♦❞♦ ♦ ♣❡rí♦❞♦ ❞♦ ❞♦✉t♦r❛❞♦✳ ❙❡♠❡❧❤❛♥t❡♠❡♥t❡ ❛ ✉♠ ✈ír✉s ❞❡ ❝♦♠♣✉t❛❞♦r✱ ❤♦❥❡ ❡①✐st❡ ✉♠❛ ❜❛♥❛❧✐③❛çã♦ ❞❛s ♣r♦✜ssõ❡s ❞❡ ♠♦❞♦ ❡♠ ❣❡r❛❧✱ t❛❧✈❡③ ♠❛s❝❛r❛❞❛ ♣❡❧♦ ♣r♦✜ss✐♦♥❛❧✐s♠♦✳ ❆❧❣✉é♠ ❝♦♥t❛❣✐❛❞♦ ♣♦r ❡st❡ ✏✈ír✉s s♦❝✐❛❧✑ ❞✐r✐❛ q✉❡ ♦ ♦r✐✲ ❡♥t❛❞♦r ♥ã♦ ❢❛③ ♠❛✐s ❞♦ q✉❡ s✉❛ ♦❜r✐❣❛çã♦ ❛♦ ♦r✐❡♥t❛r ✉♠ ❛❧✉♥♦✱ ❡ ♦s ♠❛✐s ❛tr❡✈✐❞♦s ❛✐♥❞❛ ❞✐r✐❛♠ q✉❡ ♦ ♦r✐❡♥t❛❞♦r r❡❝❡❜❡ ❜❡♥❡❢í❝✐♦s ❡♠ tr♦❝❛✱ ♦✉ s❡❥❛✱ ♦ ❢❛③❡s ♣♦r ♣✉r♦ ✐♥t❡r❡ss❡✳ ▼❛s ♣r♦❝✉r♦ ♥ã♦ ✈❡r ❞❡st❛ ❢♦r♠❛✱ ❡ ♣❛r❛ r❡❢♦rç❛r ♠✐♥❤❛ ❣r❛t✐❞ã♦ ❜❛st❛ ❞✐③❡r q✉❡ s❡♠ ♦ ❛♣♦✐♦ ❞♦ Pr♦❢✳ ❯♠❜❡rt♦✱ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ❡st❡ tr❛❜❛❧❤♦ ♥ã♦ s❡r✐❛ ♣♦ssí✈❡❧✳ ▼✉✐t♦ ♦❜r✐❣❛❞♦ ❯♠❜❡rt♦✦
❆❣r❛❞❡ç♦ ❛♦ Pr♦❢❡ss♦r ❉r✳ ❊✉❞❡♥✐❧s♦♥ ▲✐♥s ❞❡ ❆❧❜✉q✉❡rq✉❡ q✉❡✱ ♠❡s♠♦ ♥ã♦ t❡♥❞♦ s✐❞♦ ♠❡✉ ♣r♦❢❡ss♦r ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✱ ❡st❡✈❡ s❡♠♣r❡ ♣r❡s❡♥t❡ ❡♠ ♠✐♥❤❛ ❢♦r♠❛çã♦✱ ❞♦ ✐♥í❝✐♦✱ ♥❛ ❝♦♥❞✐çã♦ ❞❡ ❝❤❡❢❡ ❞❛ ♣ós✲❣r❛❞✉❛çã♦✱ ❛té ♦ ✜♠✱ ❝♦♠♦ ♠❡♠❜r♦ ❞❛ ❜❛♥❝❛ ❞❡ t❡s❡✳
❆♦ ❉r✳ ❆♥t♦♥✐♦ ❞❡ ▼❛❝❡❞♦ ❋✐❧❤♦✱ ❤♦❥❡ Pr♦❢❡ss♦r ♠❛s ❤á ❜❡♠ ♣♦✉❝♦ t❡♠♣♦ ❝♦❧❡❣❛ ❞❡ ♣ós✲❣r❛❞✉❛çã♦✱ ♣❡ss♦❛ ✐♥t❡❧✐❣❡♥t❡ ❡ ❞❡ ✉♠❛ s❡♥s✐❜✐❧✐❞❛❞❡ ❤✉♠❛♥❛ ✐♥✈❡❥á✈❡❧✱ ♣♦r s❡r ❞❡ ❢❛t♦ ❡ ❞❡ ❞✐r❡✐t♦✱ ✉♠ ✈❡r❞❛❞❡✐r♦ ❝♦♦r✐❡♥t❛❞♦r✱ t❛♥t♦ ❞✉r❛♥t❡ ♦ ♠❡str❛❞♦ q✉❛♥t♦ ♥♦ ❞♦✉t♦r❛❞♦✳
❆♦s Pr♦❢❡ss♦r❡s ❞♦ ❉❋❚❊ ♣♦r s❡r❡♠ r❡s♣♦♥sá✈❡✐s ♣♦r ♠✐♥❤❛ ❢♦r♠❛çã♦✳ ❊♠ ❡s♣❡❝✐❛❧✱ ♣❡❧♦ ❡①❡♠♣❧♦ ❞❡ ♣r♦✜ss✐♦♥❛❧✐s♠♦ ✭♥♦ s❡♥t✐❞♦ ♠❛✐s ❛♠♣❧♦ ❞♦ t❡r♠♦✮✱ ❛❣r❛❞❡ç♦ ❛♦s Pr♦❢❡ss♦r❡s✿ ❉r✳ ➪❧✈❛r♦ ❋❡rr❛③ ❋✐❧❤♦✱ ❉r✳ ❆♥❛♥✐❛s ▼♦♥t❡✐r♦ ▼❛r✐③✱ ❉r✳ ❈❧❛✉❞✐♦♥♦r ●♦♠❡s ❇❡③❡rr❛✱ ❉r✳ ▲✐❛❝✐r ❞♦s ❙❛♥t♦s ▲✉❝❡♥❛ ❡ ♦ ❉r✳ ▲✉❝✐❛♥♦ ❘♦❞r✐❣✉❡s ❞❛ ❙✐❧✈❛✳
❚❛♠❜é♠ ❛♦s ❛♠✐❣♦s ❡ ❝♦❧❡❣❛s q✉❡ ❝♦♥q✉✐st❡✐ ♥♦ ❉❋❚❊✲❯❋❘◆✱ ❡ss❡s ♦s ♠❛✐s ♣r❡✲ s❡♥t❡s✱ ❡ ♣♦r ✐ss♦ ♥ã♦ ♠❡♥♦s ✐♠♣♦rt❛♥t❡ ♣♦r ❡st❛r❡♠ ❥✉st❛♠❡♥t❡ ♥♦ ❞✐❛ ❛ ❞✐❛ ❞❛ ❧❛❜✉t❛ ❞❡ ❡st✉❞❛♥t❡ ❞❡ ♣ós✲❣r❛❞✉❛çã♦✳ ◆❛s ♠❛❞r✉❣❛❞❛s ❡ ✜♥s ❞❡ s❡♠❛♥❛ ❞❡ ❡st✉❞♦✱ ♥♦s ♣❛♣♦s ❞❡s❝♦♥tr❛í❞♦s ❞❡ ❝♦rr❡❞♦r❡s ❡ ❧❛♥❝❤♦♥❡t❡s✱ ♥♦ ❛✉①í❧✐♦ ♣❛r❛ ❛s ❝♦✐s❛s ✏❢♦r❛ ❞❛ ❋ís✐❝❛✑✱ ♥♦ ❡ss❡♥❝✐❛❧íss✐♠♦ ❢✉t❡❜♦❧ ♣ós✲❛✉❧❛✱ ❝❛❢❡③✐♥❤♦s ❡ ♠✉✐t♦s ♦✉tr♦s ♠♦♠❡♥t♦s q✉❡ t♦r♥❛♠ ♦s ❢✉t✉r♦s ✏❋ís✐❝♦s✑ ♣❡ss♦❛s ♠❛✐s ❤✉♠❛♥❛s✳
❆♦ ❈◆Pq ❡ ❛♦ ■♥st✐t✉t♦ ◆❛t✉r❡③❛ ❡ ❈✉❧t✉r❛ ✭■◆❈✮✱ ❈❛♠♣✉s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❆♠❛③♦♥❛s ✭❯❋❆▼✮✱ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳ ❆♦s ❝♦❧❡❣❛s ❞❡ tr❛❜❛❧❤♦ ❞♦ ■◆❈ q✉❡ ❢❛❝✐❧✐t❛r❛♠ ♣❛r❛ q✉❡ ❛q✉✐ ✜❝❛ss❡ ❡ t❡r♠✐♥❛ss❡ ♠❡✉ ❞♦✉t♦r❛❞♦✳
➚q✉❡❧❛s q✉❡ ❤♦❥❡✱ ❢♦r♠❛♠ ❛ ❜❛s❡ ❞❡ ♠✐♥❤❛ ❢❛♠í❧✐❛✳ ▼✐♥❤❛ ❛♠❛❞❛ ❡s♣♦s❛ ❉❡♥✐③❡✱ ❡ ♠✐♥❤❛s ❧✐♥❞❛s ✜❧❤❛s ❉❡②s❡ ❡ ❨❛s♠✐♥✳ ❆♦s ♠❡✉s ♣❛✐s ✭❙r✳ ❋r❛♥❝✐s❝♦ ❡ ❙r❛✳ ❋r❛♥❝✐s❝❛✮✱ q✉❡ ❝♦♠ ♠✉✐t♦ s❛❝r✐❢í❝✐♦✱ ❝♦♥s❡❣✉✐r❛♠ ❧✐✈r❛r ❡✉ ❡ ♠❡✉s ❞♦✐s ✐r♠ã♦s ✭▼❛r❝✐❧✐♦ ❡ ▼❛r❞♦♥✐♦✮ ❞❡ ✉♠❛ ✈✐❞❛ ❝r✉❡❧✳ ❆♦s ♠❡✉s s♦❣r♦s ✭❉❡✉s❞❡t❡ ❡ ❙r✳ ▲✐♠❛✮ ♣♦r s❡r❡♠ ♣❛r❛ ♠✐♠ ♠❡✉s s❡❣✉♥❞♦s ♣❛✐s✳ ❊ ❛♦s ❣r❛♥❞❡s ❛♠✐❣♦s✱ ❞✐st❛♥t❡s ❡ ❞❛q✉✐ ❞❡ ◆❛t❛❧✱ ♣♦r s❡r❡♠ ✏❢♦♥t❡ ❞❡ ❡♥❡r❣✐❛✑ ♥❡st❛ ❢❛s❡ ❞❡s❛✜❛♥t❡ ❞❡ ♠✐♥❤❛ ✈✐❞❛✳
❉❡ t♦❞❛s ❛s ár✈♦r❡s q✉❡ t❡♥❤♦ ❝✉❧t✐✈❛❞♦ ♥❛ ✈✐❞❛✱ ❛ ❞♦ ❛♠♦r é ❛ q✉❡ ♠❡ t❡♠ ❞❛❞♦ ♦s ♠❡❧❤♦r❡s ❢r✉t♦s✳ ➱ ♦ ❛♠♦r q✉❡♠ ♠❡ ❡♥s✐♥❛ ❛ ✈❛❧♦r✐③❛r ♦ ♣♦❞❡r ❞❛ ✉♥✐ã♦ à ❡s♣♦s❛✱ ✜❧❤❛s✱ ♣❛✐s✱ ✐r♠ã♦s✱ ❛♠✐❣♦s✱ ❞❡♠❛✐s ❢❛♠✐❧✐❛r❡s✱ ❝♦❧❡❣❛s ❞♦ tr❛❜❛❧❤♦ ❡ ❞❛s r❡❞❡s s♦❝✐❛✐s✳
✭ ▼❛r❝❡❧♦ ❇r✐t♦ ❞❛ ❙✐❧✈❛✮
❘❡s✉♠♦
◆♦s ú❧t✐♠♦s ❛♥♦s✱ ♣r♦♣❛❣❛çõ❡s ❡♣✐❞ê♠✐❝❛s tê♠ s✐❞♦ ❛❧✈♦ ❞❡ ♠✉✐t♦s ❡st✉❞♦s ❜❛s❡❛❞♦s ♥♦s ♠ét♦❞♦s ❞❛ ❋ís✐❝❛ ❊st❛tíst✐❝❛✳ ❆s ❞✐♥â♠✐❝❛s ❞❡ss❡s ♣r♦❝❡ss♦s ❡♣✐❞ê♠✐❝♦s✱ t✐♣✐❝❛♠❡♥t❡ ❞❡ ♥ã♦ ❡q✉✐❧í❜r✐♦✱ r❡s✉❧t❛♠ ♥❛ ❝♦♠♣❡t✐çã♦ ❡♥tr❡ ✐♥❞✐✈í❞✉♦s ✐♥❢❡❝t❛❞♦s ✭❛t✐✈♦s✮ ❡ ✐♥❞✐✈í❞✉♦s s❛✉❞á✈❡✐s ✭✐♥❛t✐✈♦✮✳ ❊st❡s s✐st❡♠❛s ❞❡ ♥ã♦ ❡q✉✐❧í❜r✐♦ ♣♦ss✉❡♠ ✉♠ ❡st❛❞♦ ❛t✐✈♦ q✉❡ r❡♣r❡✲ s❡♥t❛ ❛ ♣❡rs✐stê♥❝✐❛ ❞❛ ❡♣✐❞❡♠✐❛✱ ❡ ✉♠ ❡st❛❞♦ ❛❜s♦r✈❡♥t❡ q✉❡ r❡✢❡t❡ ♦ ✜♠ ❞❛ ❡♣✐❞❡♠✐❛✳ ➱ ❛ tr❛♥s✐çã♦ ❡♥tr❡ ❡st❡s ❡st❛❞♦s ✭❛t✐✈♦ ❡ ❛❜s♦r✈❡♥t❡✮ q✉❡ ♥♦s ♣❡r♠✐t❡ ❛ ❛♥á❧✐s❡ ❝rít✐❝❛ ❞❡ss❡s s✐st❡♠❛s✳ ◆❡st❡ ❝♦♥t❡①t♦✱ ❡st❛ t❡s❡ ✐♥✈❡st✐❣❛ ❞♦✐s ❞❡st❡s ♣r♦❝❡ss♦s✱ s❡♥❞♦ q✉❡ ♦ ♣r✐♠❡✐r♦ ❞❡❧❡s ❝♦rr❡s♣♦♥❞❡ ❛ ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ♣❛r❛ ♦ ♣r♦❝❡ss♦ ❞❡ ❝♦♥t❛t♦ ❡♠ ✉♠❛ ❝❛❞❡✐❛ ❧✐♥❡❛r✳ ◆❡st❡ ♠♦❞❡❧♦✱ ❝❛❞❛ ♣❛r ❞❡ sít✐♦s ❡stá ❝♦♥❡❝t❛❞♦ ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡P(r)q✉❡ ❞❡❝❛✐ ❝♦♠ ❛ ❞✐stâ♥❝✐❛
r ❡♥tr❡ ♦s sít✐♦s ❞❛ ❢♦r♠❛ 1/rα✳ ❖ ♠♦❞❡❧♦ ♣❡r♠✐t❡ ✉♠❛ ✈❛r✐❛çã♦ ❡♥tr❡ ❛ ❝❛❞❡✐❛ ✉♥✐❞✐♠❡♥✲
s✐♦♥❛❧ ♣❛❞rã♦✱ ❝❛r❛❝t❡r✐③❛❞❛ ♣♦r ❧✐❣❛çõ❡s ❛♣❡♥❛s ❡♥tr❡ ♣r✐♠❡✐r♦s ✈✐③✐♥❤♦s ✭α → ∞✮✱ ❛té ✉♠❛ r❡❞❡ ❝♦♠♣❧❡t❛♠❡♥t❡ ❝♦♥❡❝t❛❞❛ ✭α = 0✮ ❝❛r❛❝t❡r✐③❛❞❛ ♣♦r ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ❝❛♠♣♦ ♠é❞✐♦✳
❉❡s❡♥✈♦❧✈❡♠♦s ❛♥á❧✐s❡ ❞❡ ❡s❝❛❧❛ ❞❡ t❛♠❛♥❤♦ ✜♥✐t♦ ♣❛r❛ ♦❜t❡r ♦ ♣♦♥t♦ ❝rít✐❝♦✳ ❊ ❞✐♥â♠✐❝❛ ❞❡ r❡❧❛①❛çã♦ ❝rít✐❝❛ ♣❛r❛ ❡♥❝♦♥tr❛r ♦s ❡①♣♦❡♥t❡s ❝rít✐❝♦s ♣❛r❛ ❞✐st✐♥t♦s ✈❛❧♦r❡s ❞♦ ❡①♣♦❡♥t❡ ❞❡ ❧✐❣❛çã♦α✳ ❉❛❞♦s ❞♦ ♣❛râ♠❡tr♦ ❞❡ ♦r❞❡♠ ❝♦❧❛♣s❛♠ ❡♠ ✉♠❛ ❝✉r✈❛ ✉♥✐✈❡rs❛❧✱ ♠♦str❛♠♦s t❛♠❜é♠ q✉❡ ♦s ❡①♣♦❡♥t❡s ❝rít✐❝♦s ✈❛r✐❛♠ ❝♦♠ α✳ ◆♦ s❡❣✉♥❞♦ tr❛❜❛❧❤♦✱ ✐♥tr♦❞✉③✐♠♦s ♦ ♠♦❞❡❧♦ ♣r♦❝❡ss♦ ❡♣✐❞ê♠✐❝♦ s✉♣❡r❞✐❢✉s✐✈♦✱ ♦♥❞❡ ♣❛rtí❝✉❧❛s ❞♦ t✐♣♦ A ✭✐♥❞✐✈í❞✉♦s s❛✉❞á✈❡✐s✮ ❡ ♣❛rtí❝✉❧❛s ❞♦ t✐♣♦ B ✭✐♥❞✐✈í❞✉♦s ✐♥❢❡❝t❛❞♦s✮ ♣♦❞❡♠ ❞❛r s❛❧t♦s ♥❛ r❡❞❡✱ ❝♦♠ ❞✐st✐♥t❛s ♣r♦✲ ❜❛❜✐❧✐❞❛❞❡s ✭DA ❡ DB r❡s♣❡❝t✐✈❛♠❡♥t❡✮✱ ❞❡ ✉♠❛ ❞✐stâ♥❝✐❛ ℓ ❞✐str✐❜✉í❞❛ ❞❡ ❛❝♦r❞♦ ❝♦♠ ✉♠❛
♣r♦❜❛❜✐❧✐❞❛❞❡ t✐♣♦ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛ P(ℓ) ∝ 1/ℓµ✳ P❛r❛ µ ≥ 3 ❛ ♣r♦♣❛❣❛çã♦ ❝♦rr❡s♣♦♥❞❡ ❛
❞✐❢✉sã♦ ♥♦r♠❛❧✱ ❡ ♣❛r❛ µ < 3 ❡q✉✐✈❛❧❡ ❛♦s ✈♦♦s ❞❡ ▲é✈②✳ ◆♦ r❡❣✐♠❡ ❞❡ ❞✐❢✉sã♦ DA > DB✱
r❡s✉❧t❛❞♦s ❞❛ t❡♦r✐❛ ❞❡ ❝❛♠♣♦ tê♠ s✉❣❡r✐❞♦ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ ❞❡s❝♦♥tí♥✉❛✱ ❝♦♥❥❡❝t✉r❛ ❡st❛ ♥ã♦ ❡♥❞♦ss❛❞❛ ♣♦r ✈ár✐♦s ❡st✉❞♦s ♥✉♠ér✐❝♦s✳ ◆❡st❡ ❝❡♥ár✐♦✱ r❡❛❧✐③❛♠♦s ✉♠ ❡①t❡♥s✐✈♦ ❡st✉❞♦ ♥✉♠ér✐❝♦ ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❝rít✐❝♦ ❞❡ ❛♠❜♦s ♦s r❡❣✐♠❡s✱ ❞✐❢✉s✐✈♦ ✭µ ≥ 3✮ ❡ s✉♣❡r❞✐❢✉s✐✈♦
✭µ < 3✮✱ ♣❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡ DA> DB✳ ❆♣❧✐❝❛♠♦s ❛♥á❧✐s❡ ❞❡ ❡s❝❛❧❛ ❞❡ t❛♠❛♥❤♦ ✜♥✐t♦ ♣❛r❛
♦❜t❡r ❛s ♣r♦♣r✐❡❞❛❞❡s ❝rít✐❝❛s ✐♥❡r❡♥t❡s ❛♦ ♠♦❞❡❧♦ ♣❛r❛ ✈ár✐♦s ✈❛❧♦r❡s ❞❡µ✳
P❛❧❛✈r❛s✲❈❤❛✈❡✿ ♣r♦❝❡ss♦ ❡♣✐❞ê♠✐❝♦ ❞✐❢✉s✐✈♦❀ ♣r♦❝❡ss♦ ❞❡ ❝♦♥t❛t♦❀ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ ♣❛r❛ ❡st❛❞♦ ❛❜s♦r✈❡♥t❡❀ ♠ét♦❞♦ ▼♦♥t❡ ❈❛r❧♦✳
❆❜str❛❝t
❖✈❡r t❤❡ ❧❛st ②❡❛rs ❡♣✐❞❡♠✐❝s ♣r♦♣❛❣❛t✐♦♥ ❤❛s ❜❡❡♥ t❤❡ t❛r❣❡t ♦❢ ♠❛♥② st✉❞✐❡s ❜❛s❡❞ ♦♥ st❛t✐st✐❝❛❧ ♣❤②s✐❝s ♠❡t❤♦❞s✳ ❚❤❡ ❞②♥❛♠✐❝s ♦❢ t❤✐s ♣r♦❝❡ss❡s r❡s✉❧ts ❢r♦♠ t❤❡ ❝♦♠♣❡t✐t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ✐♥❢❡❝t✐♦♥ ♦❢ ❤❡❛❧t❤② ✭✐♥❛❝t✐✈❡✮ ✐♥❞✐✈✐❞✉❛❧s ❛♥❞ t❤❡ r❡❝♦✈❡r② ♦❢ ✐♥❢❡❝t❡❞ ✭❛❝t✐✈❡✮ ♦♥❡s✳ ❚❤❡s❡ ♥♦♥❡q✉✐❧✐❜r✐✉♠ s②st❡♠s ❤❛✈❡ ❛ st❛t✐st✐❝❛❧❧② st❛t✐♦♥❛r② ❛❝t✐✈❡ st❛t❡ ✭❡♣✐❞❡♠✐❝ r✉♥♥✐♥❣✮ ❛♥❞ ❛♥ ❛❜s♦r❜✐♥❣ st❛t❡ ✭❡♣✐❞❡♠✐❝ ♦✈❡r✮✳ ❚❤❡ tr❛♥s✐t✐♦♥ ❜❡t✇❡❡♥ ❛❝t✐✈❡ ❛♥❞ ✐♥❛❝t✐✈❡ st❛t❡s ❛❧❧♦✇✐♥❣ ❝r✐t✐❝❛❧ ❛♥❛❧②s✐s ♦❢ t❤❡s❡ ♥♦♥❡q✉✐❧✐❜r✐✉♠ s②st❡♠s✳ ■♥ t❤✐s ❝♦♥t❡①t t❤✐s t❤❡s✐s ✐♥✈❡st✐❣❛t❡s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ❢♦r t✇♦ ♠♦❞❡❧s✿ t❤❡ ❝♦♥t❛❝t ♣r♦❝❡ss ❛♥❞ s✉♣❡r❞✐✛✉s✐✈❡ ❡♣✐❞❡♠✐❝ ♣r♦❝❡ss ❜♦t❤ ✐♥ ❛ ❧✐♥❡❛r ❝❤❛✐♥✳ ■♥ t❤❡ ✜rst ♠♦❞❡❧✱ ❡❛❝❤ ♣❛✐r ♦❢ s✐t❡s ✐s ❝♦♥♥❡❝t❡❞ ✇✐t❤ ❛ ♣r♦❜❛❜✐❧✐t② P(r) t❤❛t ❞❡❝❛②s ✇✐t❤ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ s✐t❡s r ❛s 1/rα✳ ❆s s✉❝❤✱ t❤❡
♠♦❞❡❧ ❛❧❧♦✇s ❢♦r ❛ ❝♦♥t✐♥✉♦✉s t✉♥✐♥❣ ❜❡t✇❡❡♥ ❛ st❛♥❞❛r❞ ♦♥❡❞✐♠❡♥s✐♦♥❛❧ ❝❤❛✐♥ ✇✐t❤ ♦♥❧② ♥❡❛r❡st ♥❡✐❣❤❜♦r ❝♦✉♣❧✐♥❣s (α → ∞) t♦ ❛ ❢✉❧❧② ❝♦♥♥❡❝t❡❞ ♥❡t✇♦r❦ (α → 0) ❝❤❛r❛❝t❡r✐③❡❞
❜② ❛ ♠❡❛♥✲✜❡❧❞ ❜❡❤❛✈✐♦r✳ ❲❡ ❞❡✈❡❧♦♣ ❛ ✜♥✐t❡✲s✐③❡ s❝❛❧✐♥❣ ❛♥❛❧②s✐s t♦ ♦❜t❛✐♥ t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥t ❛♥❞ ❛ s❡t ♦❢ ❞②♥❛♠✐❝❛❧ ❡①♣♦♥❡♥ts ❢♦r ❞✐st✐♥❝t ✈❛❧✉❡s ♦❢ t❤❡ ❜✐♥❞✐♥❣ ❡①♣♦♥❡♥t α✳ ❉❛t❛ ❢♦r t❤❡ ♦r❞❡r ♣❛r❛♠❡t❡r ❝♦❧❧❛♣s❡ ♦✈❡r ❛ ✉♥✐✈❡rs❛❧ ❝✉r✈❡✳ ❲❡ s❤♦✇ ❢✉rt❤❡r t❤❛t t❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥ts ✈❛r② ❝♦♥t✐♥✉♦✉s❧② ✇✐t❤α✳ ■♥ t❤❡ s❡❝♦♥❞ ✇♦r❦✱ ✇❡ ❤❛✈❡ ✐♥tr♦❞✉❝❡❞ ❛ s✉♣❡r❞✐✛✉s✐✈❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❡♣✐❞❡♠✐❝ ♣r♦❝❡ss ♠♦❞❡❧ ♦♥ ✇❤✐❝❤ ❤❡❛❧t❤② ✭❆✮ ❛♥❞ ✐♥❢❡❝t❡❞ ✭❇✮ ✐♥❞✐✈✐❞✉❛❧s ❝❛♥ ❥✉♠♣ ✇✐t❤ ❞✐st✐♥❝t ♣r♦❜❛❜✐❧✐t✐❡s DA ❛♥❞ DB ♦✈❡r ❛ ❞✐st❛♥❝❡ ℓ ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦
❛ ♣♦✇❡r✲❧❛✇ ♣r♦❜❛❜✐❧✐t② P(ℓ) ∝ 1/ℓµ✳ ❋♦r µ≥ 3 t❤❡ ♣r♦♣❛❣❛t✐♦♥ ✐s ❡q✉✐✈❛❧❡♥t t♦ ❞✐✛✉s✐♦♥✱
✇❤✐❧❡ µ <3 ❝♦rr❡s♣♦♥❞s t♦ ▲é✈② ✢✐❣❤ts✳ ■♥ t❤❡ DA > DB ❞✐✛✉s✐♦♥ r❡❣✐♠❡✱ ✜❡❧❞✲t❤❡♦r❡t✐❝❛❧
r❡s✉❧ts ❤❛✈❡ s✉❣❣❡st❡❞ ❛ ✜rst✲♦r❞❡r tr❛♥s✐t✐♦♥✱ ❛ ♣r❡❞✐❝t✐♦♥ ♥♦t s✉♣♣♦rt❡❞ ❜② s❡✈❡r❛❧ ♥✉♠❡r✐❝❛❧ st✉❞✐❡s✳ ❆♥ ❡①t❡♥s✐✈❡ ♥✉♠❡r✐❝❛❧ st✉❞② ♦❢ t❤❡ ❝r✐t✐❝❛❧ ❜❡❤❛✈✐♦r ✐♥ ❜♦t❤ t❤❡ ❞✐✛✉s✐✈❡ (µ ≥3)
❛♥❞ s✉♣❡r❞✐✛✉s✐✈❡µ <3DA> DB r❡❣✐♠❡s ✐s ❛❧s♦ r❡♣♦rt❡❞✳ ❲❡ ❡♠♣❧♦②❡❞ ❛ ✜♥✐t❡✲s✐③❡ s❝❛❧✐♥❣
❛♥❛❧②s✐s t♦ ♦❜t❛✐♥ t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥t ❛s ✇❡❧❧ ❛s t❤❡ st❛t✐❝ ❛♥❞ ❞②♥❛♠✐❝ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥ts ❢♦r s❡✈❡r❛❧ ✈❛❧✉❡s ♦❢µ✳
❑❡②✇♦r❞s✿ ❞✐✛✉s✐✈❡ ❡♣✐❞❡♠✐❝ ♣r♦❝❡ss❀ ❝♦♥t❛❝t ♣r♦❝❡ss❀ ❛❜s♦r❜✐♥❣✲st❛t❡ ♣❤❛s❡ tr❛♥✲ s✐t✐♦♥❀ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞✳
❙✉♠ár✐♦
❆❣r❛❞❡❝✐♠❡♥t♦ ✐✐
❘❡s✉♠♦ ✈
❆❜str❛❝t ✈✐
▲✐st❛ ❞❡ ❋✐❣✉r❛s ①✐✐✐
▲✐st❛ ❞❡ ❚❛❜❡❧❛s ①✐✈
✶ ■♥tr♦❞✉çã♦ ✶
✷ ❚❡♦r✐❛ ♣❛r❛ ●❡r❛çã♦ ❡ ❆♥á❧✐s❡ ❞❡ ❉❛❞♦s ✺
✷✳✶ ❚r❛♥s✐çõ❡s ❞❡ ❋❛s❡s ❡ ❈r✐t✐❝❛❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻
✷✳✷ ❙✐♠✉❧❛çã♦ ◆✉♠ér✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽
✷✳✷✳✶ Pr♦❝❡ss♦s ❊st♦❝ást✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽
✷✳✷✳✷ ▼ét♦❞♦ ▼♦♥t❡ ❈❛r❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸
✷✳✸ ❊①♣♦❡♥t❡s ❈rít✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
✷✳✸✳✶ ❍✐♣ót❡s❡ ❞❡ ❊s❝❛❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼
✷✳✸✳✷ ❈❧❛ss❡ ❞❡ ❯♥✐✈❡rs❛❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✷✳✸✳✸ ❚❛♠❛♥❤♦ ❋✐♥✐t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✷✳✸✳✹ ❘❡❧❛①❛çã♦ ❈rít✐❝❛ ❡ ❉✐♥â♠✐❝❛ ❞❡ ❚❡♠♣♦s ❈✉rt♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✷✳✹ ❚é❝♥✐❝❛s ♣❛r❛ ❉❡t❡r♠✐♥❛r P♦♥t♦ ❈rít✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
✷✳✹✳✶ ❋✉♥çã♦ g ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✷✳✹✳✷ ❘❛③ã♦ ❞❡ ▼♦♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✷✳✹✳✸ ❈✉♠✉❧❛♥t❡ ❞❡ ❇✐♥❞❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻
✷✳✹✳✹ ▼ét♦❞♦ ❞❛ ❈✉r✈❛t✉r❛ ◆✉❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻
✸ ◆ã♦ ❯♥✐✈❡rs❛❧✐❞❛❞❡ ❞♦ Pr♦❝❡ss♦ ❞❡ ❈♦♥t❛t♦ ❡♠ ✉♠❛ ❈❛❞❡✐❛ ▲✐♥❡❛r ❝♦♠
❈♦♥❡①õ❡s ❞❡ ▲♦♥❣♦ ❆❧❝❛♥❝❡ ✷✾
✸✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾
✸✳✷ ▼♦❞❡❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸
✸✳✸ ❘❡s✉❧t❛❞♦s ❞❛ ❙✐♠✉❧❛çã♦ ❡ ❉✐s❝✉ssã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹
✸✳✹ ❈♦♥❝❧✉sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸
✹ ❈♦♠♣♦rt❛♠❡♥t♦ ❈rít✐❝♦ ❞❡ ✉♠ Pr♦❝❡ss♦ ❊♣✐❞ê♠✐❝♦ ❙✉♣❡r❉✐❢✉s✐✈♦ ✹✺
✹✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺
✹✳✷ ▼♦❞❡❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼
✹✳✸ ❘❡s✉❧t❛❞♦s ❞❛ ❙✐♠✉❧❛çã♦ ❡ ❉✐s❝✉ssã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾
✹✳✹ ❈♦♥❝❧✉sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵
✺ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ❡ P❡rs♣❡❝t✐✈❛s ✻✷
❘❡❢❡rê♥❝✐❛s ❜✐❜❧✐♦❣rá✜❝❛s ✻✹
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✷✳✶ ■❧✉str❛çã♦ ❞❛ ❝❛♠✐♥❤❛❞❛ ❛❧❡❛tór✐❛ ❡♠ ✉♠❛ ❞✐♠❡♥sã♦1d✳ ❆ ♣❛rt✐r ❞♦ ♣❛ss♦ ❛❧❡❛tór✐❛
x ❡①✐st❡♠ ❞✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s✱ ♣❛r❛ ❞✐r❡✐t❛x+ 1♦✉ ♣❛r❛ ❡sq✉❡r❞❛x−1✳ ✳ ✳ ✳ ✳ ✳ ✾
✷✳✷ ■❧✉str❛çã♦ ❞❛ ❝❛♠✐♥❤❛❞❛ ❛❧❡❛tór✐❛ ❡♠ ✉♠ t❡♠♣♦t ❡ ❧♦❣♦ ❡♠ s❡❣✉✐❞❛ ❡♠ ✉♠ t❡♠♣♦
t+ 1✳ ❆ ♣♦s✐çã♦ x ❡♠t+ 1 ♣♦❞❡ ♣r♦✈✐r t❛♥t♦ ❞❛ ♣♦s✐çã♦ x+ 1q✉❛♥t♦ ❞❛ ♣♦s✐çã♦
x−1♥♦ t❡♠♣♦ ❛♥t❡r✐♦r t✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✸ ❯♠ q✉❛rt♦ ❞❡ ✉♠ ❝ír❝✉❧♦ ✐♥s❝r✐t♦ ❡♠ ✉♠ q✉❛❞r❛❞♦ ❬✷✶❪✱ ♣❛r❛ r❡❛❧✐③❛çã♦ ❞♦ ❝á❧❝✉❧♦ ❞❛
❝♦♥st❛♥t❡π✉s❛♥❞♦ ♦ ♠ét♦❞♦ ▼♦♥t❡ ❈❛r❧♦✳ ❖ ▼▼❈ ♣❡r♠✐t❡ ✏❧❛♥ç❛r✑ ❛❧❡❛t♦r✐❛♠❡♥t❡
♣♦♥t♦s q✉❡ ♣♦❞❡♠ ❛t✐♥❣✐r ❛ ár❡❛ ❛❜❛✐①♦ ❞❛ ❝✉r✈❛ ✭❞❡♥tr♦ ❞♦ q✉❛rt♦ ❞❡ ❝ír❝✉❧♦✮ ♦✉ ❞❡♥tr♦ ❞♦ r❡st❛♥t❡ ❞♦ q✉❛❞r❛❞♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺
✷✳✹ ❚❡♠♣❡r❛t✉r❛ r❡❞✉③✐❞❛ ✈❡rs✉s ❞❡♥s✐❞❛❞❡ r❡❞✉③✐❞❛✱ ♥❛ r❡❣✐ã♦ ❞❡ ❝♦❡①✐stê♥❝✐❛ ❣ás✲ ❧íq✉✐❞♦ ♣❛r❛ ♦✐t♦ s✉❜stâ♥❝✐❛s ❞✐❢❡r❡♥t❡s ❛❥✉st❛❞♦ ♣♦r ✉♠ ♣♦❧✐♥ô♠✐♦ ❝ú❜✐❝♦✱ ♦❜t✐❞❛ ❞❛ ❘❡❢✳ ❬✷✺❪✳ ❖s ❞❛❞♦s ♣❛r❛ ❛s ♦✐t♦ s✉❜stâ♥❝✐❛s ❝♦❧❛♣s❛♠ ♣❛r❛ ✉♠❛ ú♥✐❝❛ ❢✉♥çã♦✳ ✳ ✷✵
✷✳✺ P❛râ♠❡tr♦ ❞❡ ♦r❞❡♠ ρ ✈❡rs✉s ♣❛râ♠❡tr♦ ❞❡ ❝♦♥tr♦❧❡ p✳ ❆ r❡t❛ tr❛❝❡❥❛❞❛✱ ❝r✉③❛♥❞♦
❛s ❝✉r✈❛s✱ ♣♦♥t✉❛ ❡ r❡ss❛❧t❛ ♦ ✈❛❧♦r ❞♦ρ ♥♦ ♣♦♥t♦ ❝rít✐❝♦p∗c✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✻ ❖ q✉❡ ❝❤❛♠❛♠♦s ❞❡ ❝✉r✈❛t✉r❛✱ ♥♦ ❡✐①♦ ❞❛s ♦r❞❡♥❛❞❛s✱ ❡q✉✐✈❛❧❡ ❛♦s ✈❛❧♦r❡s ❞❡dq✱ ❡♥✲
❝♦♥tr❛❞♦s ♥❛ ❚❛❜❡❧❛ ✷✳✷✳ ❆ ❧✐♥❤❛ q✉❡ ❧✐❣❛ ♦s ♣♦♥t♦s é ❞❛❞❛ ♣♦r ✉♠ ❛❥✉st❡ q✉❛❞rát✐❝♦✳ ❖ ✈❛❧♦r ❞♦ ♣♦♥t♦ ❝rít✐❝♦ pc r❡✜♥❛❞♦ ❡q✉✐✈❛❧❡ ❛♦ ❧♦❝❛❧ ♦♥❞❡ ❛ ❝✉r✈❛ ❝r✉③❛ ♦ ③❡r♦ ❞❛
♦r❞❡♥❛❞❛ ✭dq= 0✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽
✸✳✶ ❘❡❞❡ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ ❞❡ ❝♦♠♣r✐♠❡♥t♦ L r❡♣r❡s❡♥t❛♥❞♦ ♦ ♠♦❞❡❧♦ P C✳ ❆ ❞✐♥â♠✐❝❛
s♦rt❡✐❛ ✉♠ sít✐♦ ❞❡ ❝❛❞❛ ✈❡③✿ ♥♦ t❡♠♣♦ t✱ s❡ ❛♠❜♦s ♦s ✈✐③✐♥❤♦s ❡st✐✈❡r❡♠ ❝♦♥t❛♠✐✲
♥❛❞♦s✱ ❝♦♥t❛♠✐♥❛ ♦ sít✐♦ s♦rt❡❛❞♦❀ ❡♠ t′✱ s❡ ♦ sít✐♦ ❡st✐✈❡r ❝♦♥t❛♠✐♥❛❞♦✱ ❡①✐st❡ ✉♠❛
♣r♦❜❛❜✐❧✐❞❛❞❡λ❞❡ ❝✉r❛❀ ❡♠t′′ t❡♠♦s ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❞♦ ❡st❛❞♦ ❛❜s♦r✈❡♥t❡ ❞❡st❡
♠♦❞❡❧♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵
✸✳✷ ❉✐❛❣r❛♠❛ ❞❡ ❢❛s❡ ♣❛r❛ ♦ ♠♦❞❡❧♦ ♣r♦❝❡ss♦ ❞❡ ❝♦♥t❛t♦✳ P❛râ♠❡tr♦ ❞❡ ♦r❞❡♠ ρ✱ r❡♣r❡✲
s❡♥t❛♥❞♦ ❛ ❞❡♥s✐❞❛❞❡ ❞❡ sít✐♦s ❛t✐✈♦s✱ ✈❡rs✉s ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❝✉r❛λq✉❡ r❡♣r❡s❡♥t❛
♦ ♣❛râ♠❡tr♦ ❞❡ ❝♦♥tr♦❧❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✸✳✸ ✭❛✮ P❛râ♠❡tr♦ ❞❡ ♦r❞❡♠ ρ ❝♦♠♦ ❢✉♥çã♦ ❞♦ ♣❛râ♠❡tr♦ ❞❡ ❝♦♥tr♦❧❡ λ✱ ♣❛r❛α = 3.0
✭❝ír❝✉❧♦s✮ ❡ α = 5.0 ✭tr✐â♥❣✉❧♦s✮✳ ❆q✉✐ ✉s❛♠♦s L = 800 ♣❛r❛ t♦❞♦s ♦s ✈❛❧♦r❡s ❞♦
♣❛râ♠❡tr♦ ❞❡ ❧✐❣❛çã♦ α✳ ❖❜s❡r✈❡ q✉❡ ✈❛❧♦r❡s ♣❡q✉❡♥♦s ❞❡ α ❢❛✈♦r❡❝❡♠ ❛ ❛t✐✈❛çã♦
❞♦ sít✐♦✳ ✭❜✮ ●rá✜❝♦ ❞♦ ❡①♣♦❡♥t❡ ❞❡ ❧✐❣❛çã♦α✈❡rs✉s ♦ ✈❛❧♦r ❝rít✐❝♦ ❞♦ ♣❛râ♠❡tr♦ ❞❡
❝♦♥tr♦❧❡ λc✳ ◗✉❛♥❞♦ α ❞✐♠✐♥✉✐✱ λc t❡♥❞❡ ♣❛r❛ ♦ ✈❛❧♦r ♣❛❞ã♦ ❝♦♠♦ ♥❛ tr❛♥s✐çã♦ ❞♦
P❈ ❡♠ ✉♠❛ ❝❛❞❡✐❛ ❧✐♥❡❛r ❝♦♠ ❧✐❣❛çã♦ ❛♣❡♥❛s ❡♥tr❡ ♣r✐♠❡✐r♦s ✈✐③✐♥❤♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺
✸✳✹ ❘❛③ã♦ ❞❡ ♠♦♠❡♥t♦s ♠ ❝♦♠♦ ❢✉♥çã♦ ❞♦ ♣❛râ♠❡tr♦ ❞❡ ❝♦♥tr♦❧❡ λ ♣❛r❛ ❞✐❢❡r❡♥t❡s
t❛♠❛♥❤♦s ❞❡ r❡❞❡ L✳ ❆s ❝✉r✈❛s ✐♥t❡r❝❡♣t❛♠ ♥♦ ♣♦♥t♦ ❝rít✐❝♦ ♣❡r♠✐t✐♥❞♦ ❢❛③❡r ✉♠❛
❡st✐♠❛t✐✈❛ ♣❛r❛ λc✿ ✭❛✮ ♣❛r❛ α = 2.0 ♦❜t✐✈❡♠♦s λc = 0.507❀ ✭❜✮ ♣❛r❛ α = 5.0
♦❜t✐✈❡♠♦sλc = 0.306✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✸✳✺ P❛râ♠❡tr♦ ❞❡ ♦r❞❡♠ ρ ❝♦♠♦ ❢✉♥çã♦ ❞♦ t❡♠♣♦ t ♥♦ ♣♦♥t♦ ❝rít✐❝♦✳ ❖ ♠❡❧❤♦r ❛❥✉st❡
❢♦r♥❡❝❡ ♦s ❡①♣♦❡♥t❡s ❝rít✐❝♦s β/zν ♣❛r❛ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s ❞❡ α✳ ❖ t❛♠❛♥❤♦ ❞♦
s✐st❡♠❛ ❝♦♥s✐❞❡r❛❞♦ é L= 10000✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✸✳✻ ●rá✜❝♦ ❧♦❣✲❧♦❣ ❞❛ r❛③ã♦ ❞❡ ♠♦♠❡♥t♦s U ✈❡rs✉s ♦ t❡♠♣♦ t✱ ♥♦ ♣♦♥t♦ ❝rít✐❝♦✱ ♣❛r❛
✉♠ s✐st❡♠❛ ❞❡ t❛♠❛♥❤♦ L= 10000✳ ❖ ♠❡❧❤♦r ❛❥✉st❡ ❞❛s ❝✉r✈❛s ❢♦r♥❡❝❡ ♦ ❡①♣♦❡♥t❡
❝rít✐❝♦ 1/z ♣❛r❛ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s ❞❡α✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✸✳✼ ●rá✜❝♦ ❧♦❣✲❧♦❣ ❞❛ ❞❡r✐✈❛❞❛ d|ln(ρ)|/dλ ✈❡rs✉s ♦ t❡♠♣♦t ♥♦ ♣♦♥t♦ ❝rít✐❝♦ ♣❛r❛ ✉♠
s✐st❡♠❛ ❞❡ t❛♠❛♥❤♦L= 10000✳ ❖ ♠❡❧❤♦r ❛❥✉st❡ ❧✐♥❡❛r ❢♦r♥❡❝❡ ❛ r❛③ã♦ ❞❡ ❡①♣♦❡♥t❡s
❝rít✐❝♦s 1/zν✳ P❛r❛ α = 2.0 t❡♠♦s 1/zν = 0.61(1)✱ α = 2.5 t❡♠♦s 1/zν = 0.46(1)✱
α = 3.0 t❡♠♦s 1/zν = 0.52(1)✱ α = 4.0 t❡♠♦s 1/zν = 0.57(2) ❡ α = 5.0 t❡♠♦s
1/zν = 0.56(1)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
✸✳✽ ❈♦❧❛♣s♦ ❞❡ ❞❛❞♦s ❞♦ ♣❛râ♠❡tr♦ ❞❡ ♦r❞❡♠✱ ❝❛❧❝✉❧❛❞♦ ♣❛r❛ ❞✐❢❡r❡♥t❡s t❛♠❛♥❤♦s ❞❡ r❡❞❡s✳ ✭❛✮ α = 2.0✱ ♠❡❧❤♦r ❝♦❧❛♣s♦ ❝♦♠ λc = 0.507✱ β/ν = 0.43 ❡ 1/ν = 0.58❀ ✭❜✮ α = 5.0✱ ♠❡❧❤♦r ❝♦❧❛♣s♦ ❝♦♠ λc = 0.306✱β/ν = 0.24 ❡ 1/ν = 0.90✳ ❆ s♦❜r❡♣♦s✐çã♦
❞♦s ❞❛❞♦s ♣ró①✐♠♦ à ❝r✐t✐❝❛❧✐❞❛❞❡ (λ = λc ⇒ (λ−λc)L1/ν = 0) ❛❢❡r❡ q✉❡ ♥♦ss♦s
♣♦♥t♦s ❡ ❡①♣♦❡♥t❡s ❝rít✐❝♦s ♥ã♦ ❡stã♦ s♦❜r❡ ❡❢❡✐t♦ ❞❡ t❛♠❛♥❤♦ ✜♥✐t♦✳ ❖❜s❡r✈❛çã♦✿ ❝♦♠♦ ❥á ✈✐st♦✱ ❛s s❡❣✉✐♥t❡s ♥♦t❛çõ❡s sã♦ ❡q✉✐✈❛❧❡♥t❡sν⊥≡ν✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶
✸✳✾ ❊①♣♦❡♥t❡s ❝rít✐❝♦s β✱ β/zν ❡ 1/z ❝♦♠♦ ❢✉♥çã♦ ❞♦ ❡①♣♦❡♥t❡ ❞❡ ❧✐❣❛çã♦ α✳ ❚♦❞♦s
♦s ✈❛❧♦r❡s ❝❛❧❝✉❧❛❞♦s ❞♦s ❡①♣♦❡♥t❡s ❛♣♦♥t❛♠ ♣❛r❛ ♦s ✈❛❧♦r❡s ❝rít✐❝♦s ❞❛ ❝❧❛ss❡ ❞❡ ✉♥✐✈❡rs❛❧✐❞❛❞❡ ❞❛ P❉ q✉❛♥❞♦ α ❛✉♠❡♥t❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
✹✳✶ ■❧✉str❛çã♦ ❞♦ ♠♦❞❡❧♦ P❊❙❉✳ ❆s ✏❝❛✐①❛s✑ r❡♣r❡s❡♥t❛♠ ♦s sít✐♦s ♥❛ r❡❞❡ ✉♥✐❞✐♠❡♥✲ s✐♦♥❛❧✱ ♣r❡❡♥❝❤✐❞❛s ♣♦r ♣❛rtí❝✉❧❛sA✭❝❛r❛s s♦rr✐❞❡♥t❡s✮ ❡ ♣❛rtí❝✉❧❛sB ✭❝❛r❛s tr✐st❡s✮✳
❖s s❛❧t♦s ♥❛ r❡❞❡ sã♦ ❞❛❞♦s ♣♦r ℓ(x)✱ s❡♥❞♦x ✉♠ ♥ú♠❡r♦ ❛❧❡❛tór✐♦ ✈❛r✐❛♥❞♦ ❡♥tr❡
0 < x < 1✱ ❡ µ ♦ ❡①♣♦❡♥t❡ q✉❡ ❝❛r❛❝t❡r✐③❛ ♦ t✐♣♦ ❞❡ s❛❧t♦✱ ❞❡ ❧♦♥❣♦ ♦✉ ❝✉rt♦ ❛❧✲
❝❛♥❝❡✳ ❊♠ (a) t❡♠♦s ❛ ♣❛rtí❝✉❧❛ A tr❛♥s❢♦r♠❛♥❞♦✲s❡ ♥❛ ♣❛rtí❝✉❧❛ B ✭♣r♦❝❡ss♦ ❞❡
❝♦♥t❛♠✐♥❛çã♦✮✱ ❡ ❡♠(b)❛ ♣❛rtí❝✉❧❛Btr❛♥s❢♦r♠❛♥❞♦✲s❡ ♥❛ ♣❛rtí❝✉❧❛ A✭♣r♦❝❡ss♦ ❞❡
❝✉r❛✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽
✹✳✷ ❖ ♣❛râ♠❡tr♦ ❞❡ ♦r❞❡♠ ρB ❝♦♠♦ ❢✉♥çã♦ ❞❛ ❞❡♥s✐❞❛❞❡ ρ ♣❛r❛ ✉♠❛ r❡❞❡ ❞❡ t❛♠❛♥❤♦ L = 640✳ ▼❡s♠♦ s❡♥❞♦ DA > DB✱ t♦❞❛s ❛s ❝✉r✈❛s ❝❧❛r❛♠❡♥t❡ ♠♦str❛♠ ✉♠❛ tr❛♥✲
s✐çã♦ ❞❡ ❢❛s❡ ❝♦♥tí♥✉❛ ♣❛r❛ ✉♠ ❡st❛❞♦ ❛❜s♦r✈❡♥t❡✱ ❡♠ t♦❞♦s ♦s ✈❛❧♦r❡s ❞♦ ❡①♣♦❡♥t❡
µ ✐♥✈❡st✐❣❛❞♦✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵
✹✳✸ ❈♦♥❥✉♥t♦ ❞❡ ❢✉♥çõ❡s ❛✉①✐❧✐❛r❡s g(L, L′, ρ) ✈❡rs✉s à ❞❡♥s✐❞❛❞❡ ρ✱ ♣❛r❛ ✈ár✐♦s ♣❛r❡s
✭L, L′✮✳ ◆❛ ✐♥t❡rs❡çã♦ ❡st✐♠❛✲s❡ ♦ ♣♦♥t♦ ❝rít✐❝♦ρc ❡ ♦ ❡①♣♦❡♥t❡β/ν ♣❛r❛ ❝❛❞❛ ✈❛❧♦r
❞❡µ✳ ◆❡st❛ ✜❣✉r❛ ♠♦str❛♠♦s s♦♠❡♥t❡ ♦s ❞♦✐s ❝❛s♦s ❡①tr❡♠♦s ✭❛✮µ= 1.1✭ρc = 2.72
❡ β/ν = 0.72✮ ❡ µ= 8.0 ✭ρc = 4.72 ❛♥❞ β/ν = 0.34✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶
✹✳✹ ❉❡♥s✐❞❛❞❡ ❞❡ ♣❛rtí❝✉❧❛s ❛t✐✈❛s ρB ♣❡❧♦ t❛♠❛♥❤♦ ❞❛ r❡❞❡ L ♥♦ ♣♦♥t♦ ❝rít✐❝♦✳ ❖
❛❥✉st❡ ❧✐♥❡❛r ❞❡ ρB(L, ρc)∝L−β/ν ❢♦r♥❡❝❡ ❞✐r❡t❛♠❡♥t❡ β/ν ♣❛r❛ ♦s s❡t❡ ✈❛❧♦r❡s ❞❡ µ ✐♥✈❡st✐❣❛❞♦✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸
✹✳✺ P❛râ♠❡tr♦ ❞❡ ♦r❞❡♠ ❡s❝❛❧❛❞♦✱ ❞❛ ❢♦r♠❛ ρBLβ/ν ♣❡❧♦ t❛♠❛♥❤♦ ❞♦ s✐st❡♠❛ L✱ s❡♥❞♦
✉s❛❞❛ ❛ ♠❡❧❤♦r ❡st✐♠❛t✐✈❛ ❞❡ β/ν ♦❜t✐❞❛ ♣r❡✈✐❛♠❡♥t❡✳ ◆♦ ♣♦♥t♦ ρ = ρc ♦s ❞❛❞♦s
♣❡r♠❛♥❡❝❡♠ ❣r♦ss❡✐r❛♠❡♥t❡ ❝♦♥st❛♥t❡s✱ ❡♥q✉❛♥t♦ q✉❡✱ ❡♠ρ > ρc ❝✉r✈❛♠ ♣❛r❛ ❝✐♠❛
❡ ♣❛r❛ ❜❛✐①♦ ❡♠ ρ < ρc✳ ◆❡st❛ ✜❣✉r❛✱ ♠♦str❛♠♦s ♥♦ss♦s r❡s✉❧t❛❞♦s ♣❛r❛ ✭❛✮ µ= 1.1
✭♥♦ss❛ ♠❡❧❤♦r ❡st✐♠❛t✐✈❛ ❢♦✐ ρc = 2.72 ❡ β/ν = 0.72✮ ❡ ✭❜✮ µ = 8.0 ✭♥♦ss❛ ♠❡❧❤♦r
❡st✐♠❛t✐✈❛ ❢♦✐ ρc = 4.72 ❡β/ν = 0.34✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹
✹✳✻ ❉❡✜♥✐♥❞♦ ψ ❝♦♠♦ ❛ ❞❡r✐✈❛❞❛ ❞♦ lnρB(L, ρc) ❝♦♠ r❡s♣❡✐t♦ ❛ ρ ♥♦ ♣♦♥t♦ ❝rít✐❝♦✱
♥❡st❛ ✜❣✉r❛ t❡♠✲s❡ ♦ ❣rá✜❝♦ ❧♦❣✲❧♦❣ ❞❡ ψ ❝♦♥tr❛ ♦ t❛♠❛♥❤♦ ❞♦ s✐st❡♠❛L✳ ❖ ❛❥✉st❡
❧✐♥❡❛r ❢♦r♥❡❝❡ ❞✐r❡t❛♠❡♥t❡ 1/ν ♣❛r❛ ❝❛❞❛ ✉♠ ❞♦s s❡t❡s ✈❛❧♦r❡s ❞❡µ ✭❛ ✜♠ ❞❡ ❡✈✐t❛r
s♦❜r❡♣♦s✐çã♦ ❡s❝❛❧❛♠♦s ♥♦ss♦s ❞❛❞♦s ❡♠ ❞✉❛s ❞✐❢❡r❡♥t❡s ✜❣✉r❛s✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺
✹✳✼ ❈♦❧❛♣s♦ ❞❡ ❞❛❞♦s ❞♦ ♣❛râ♠❡tr♦ ❞❡ ♦r❞❡♠ ♣❛r❛ ✭❛✮ µ= 1.1 ✭ρc = 2.72✱β/ν = 0.72 ❡ 1/ν = 0.65✮ ❡ ✭❜✮µ= 8.0 ✭ρc = 4.72✱β/ν = 0.34❡ 1/ν = 0.50✮✳ ❆ s♦❜r❡♣♦s✐çã♦ ❞♦s
❞❛❞♦s ♣ró①✐♠♦ à ❝r✐t✐❝❛❧✐❞❛❞❡ (ρ=ρc ⇒(ρ−ρc)L1/ν = 0)❛❢❡r❡ q✉❡ ♥♦ss♦s ♣♦♥t♦s
❡ ❡①♣♦❡♥t❡s ❝rít✐❝♦s ♥ã♦ ❡stã♦ s♦❜r❡ ❡❢❡✐t♦ ❞❡ t❛♠❛♥❤♦ ✜♥✐t♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻
✹✳✽ ❚❡♠♣♦ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛τ ✈❡rs✉s ♦ t❛♠❛♥❤♦ ❞♦ s✐st❡♠❛L♥♦ ♣♦♥t♦ ❝rít✐❝♦✳ ❖ ❛❥✉st❡
❧✐♥❡❛r ♣❛r❛ τ ∝Lz✱ ♣r♦❞✉③z ✭❡ ♣♦rt❛♥t♦ 1/z✮ ♣❛r❛ ❝❛❞❛ ✉♠ ❞♦s s❡t❡ ✈❛❧♦r❡s ❞❡ µ✳
❖s três ♠❛✐♦r❡s ✈❛❧♦r❡s ❞❡ µ ❢♦r♥❡❝❡♠ ✈❛❧♦r❡s ♣❛r❛ 1/z t❡♥❞❡♥❞♦ ♣❛r❛ 0.5✱ q✉❡ é ♦
❡s♣❡r❛❞♦ ♣❛r❛ ❛ ❞✐❢✉sã♦ ♥♦r♠❛❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼
✹✳✾ ❉❡♥s✐❞❛❞❡ ❝rít✐❝❛ρc ✭❛❝✐♠❛✮ ❡ ❡①♣♦❡♥t❡s ❝rít✐❝♦s ✭❛❜❛✐①♦✮ ❝♦♠♦ ❢✉♥çã♦ ❞❡µ✳ ❖s três
♠❡♥♦r❡s ✈❛❧♦r❡s ❝♦rr❡s♣♦♥❞❡♠ ❛♦s ✈♦♦s ❞❡ ▲é✈②✱ ❡♥q✉❛♥t♦ ♦s q✉❛tr♦ ♠❛✐♦r❡s ✈❛❧♦r❡s ❝♦rr❡s♣♦♥❞❡♠ ❛ ❞✐❢✉sã♦ ♥♦r♠❛❧✳ ❚♦❞♦s ♦s ♣❛râ♠❡tr♦s ❝rít✐❝♦s sã♦ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ❝♦♥st❛♥t❡s ♥♦ r❡❣✐♠❡ ❞❡ ❞✐❢✉sã♦ ♥♦r♠❛❧ µ ≥ 3✱ ❝♦♠ ✉♠❛ ♣❡q✉❡♥❛ ❞✐s❝r❡♣â♥❝✐❛
s♦♠❡♥t❡ ♣❛r❛ 1/z ❝♦rr❡s♣♦♥❞❡♥t❡ ❛ µ = 3 ✭❞✐❢✉sã♦ ♥♦r♠❛❧ ❝♦♠ ✈❛r✐â♥❝✐❛ ✐♥✜♥✐t❛✮✳
❖ ♣♦♥t♦ ❝rít✐❝♦ ρc ❛♣r❡s❡♥t❛ ✉♠❛ ❝♦♥✈❡r❣ê♥❝✐❛ q✉❛♥❞♦ t❡♠♦s ❣r❛♥❞❡s ✈❛❧♦r❡s ❞❡ µ✳ ✺✽
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
✷✳✶ ❱❛❧♦r❡s ❞♦s ❡①♣♦❡♥t❡s ❝rít✐❝♦s ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ ♣❡r❝♦❧❛çã♦ ❞✐r❡❝✐♦♥❛❞❛✳ ❖ ❡①♣♦❡♥t❡
ν⊥ ≡ ν ❡stá ❛ss♦❝✐❛❞♦ ❛♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❝♦rr❡❧❛çã♦ ❡s♣❛❝✐❛❧ ❡ ν|| à ❝♦rr❡❧❛çã♦
t❡♠♣♦r❛❧✱ ❡♥❝♦♥tr❛❞♦s ♥❛ r❡❧❛çã♦ ❝♦♠ ♦ ❡①♣♦❡♥t❡ ❞✐♥â♠✐❝♦z=ν||/ν⊥✳ ❆ ❞✐♠❡♥sã♦
❞♦ s✐st❡♠❛ é ✐♥❞✐❝❛❞❛ ♣♦r d✱ ❡CM ❢❛③ r❡❢❡rê♥❝✐❛ ❛♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ✈✐❛ ❝❛♠♣♦
♠é❞✐♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✷✳✷ ❚❛❜❡❧❛ ❝♦♠ ❞❛❞♦s ✉t✐❧✐③❛❞♦s ♥❛ té❝♥✐❝❛ ❞❛ ❝✉r✈❛t✉r❛ ♥✉❧❛✳ ❆s ✈❛r✐á✈❡✐s dq ❡ er(dq)
r❡♣r❡s❡♥t❛♠ ♦ ❞❡s✈✐♦ q✉❛❞át✐❝♦ ❡ s❡✉ ❡rr♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆s ❞❡♠❛✐s ✈❛r✐á✈❡✐s sã♦
neg =dq−er(dq)❡ pos=dq+er(dq)✳ ❖s ✈❛❧♦r❡s ♥❛ t❛❜❡❧❛ sã♦ ❛♣❡♥❛s ✐❧✉str❛t✐✈♦s✳ ✷✼
✸✳✶ ❊st✐♠❛t✐✈❛ ❞♦ ♣♦♥t♦ ❝rít✐❝♦ ❡ s❡✉s ❝♦rr❡s♣♦♥❞❡♥t❡s ❡①♣♦❡♥t❡s ❝rít✐❝♦s ♣❛r❛ ✈ár✐♦s ✈❛❧♦r❡s ❞❡ α✳ ❖s ✈❛❧♦r❡s ❞♦s ❡①♣♦❡♥t❡s ♣❛r❛ ❝❧❛ss❡ P❉ ❢♦r❛♠ r❡t✐r❛❞♦s ❞❡ ❏❡♥s❡♥ ❡t
❛❧✳ ❬✻✼❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✹✳✶ ❊st✐♠❛t✐✈❛ ❞♦ ♣♦♥t♦ ❝rít✐❝♦ ❡ s❡✉s ❝♦rr❡s♣♦♥❞❡♥t❡s ❡①♣♦❡♥t❡s ❝rít✐❝♦s ♣❛r❛ ♦ ♣r♦❝❡ss♦ ❡♣✐❞ê♠✐❝♦ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ ❝♦♠ ✈♦♦s ❞✐str✐❜✉í❞♦s ♣♦r ❧❡✐ ❞❡ ♣♦tê♥❝✐❛ ♣❛r❛ ✈ár✐♦s ✈❛✲ ❧♦r❡s ❞♦ ❡①♣♦❡♥t❡ ❝❛r❛❝t❡ríst✐❝♦ µ✱ ✈❛r✐❛♥❞♦ ❞❡ s✉♣❡r❞✐❢✉sã♦ (µ < 3) ♣❛r❛ ❞✐❢✉sã♦
♥♦r♠❛❧(µ≥3)✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷
Cap´ıtulo
1
■♥tr♦❞✉çã♦
❆♦ ❞✐❣✐t❛r ▼♦♥t❡ ❈❛r❧♦ ♠❡t❤♦❞ ♥❛ ❲❡❜ ♦❢ ❑♥♦✇❧❡❞❣❡ ✭❜❛s❡ ❞❡ ❞❛❞♦s ♣❛r❛ ♣❡sq✉✐s❛s ❝✐❡♥tí✜❝❛s✮ ❛♣❛r❡❝❡♠ ♠❛✐s ❞❡ 73.000 tr❛❜❛❧❤♦s ♣✉❜❧✐❝❛❞♦s✳ ❙ó ♥♦ ❛♥♦ ❞❡ 2013 ❛❝✉s❛✈❛♠
♠❛✐s ❞❡ 290 ♣✉❜❧✐❝❛çõ❡s ♥❡ss❡ t❡♠❛ ✭❡♠25/07/2013✱ ❞❛t❛ ❞❛ ♣❡sq✉✐s❛✮✳ ❊ q✉❛♥t♦ ❛♦ tó♣✐❝♦
❈r✐t✐❝❛❧ ♣❤❡♥♦♠❡♥❛ ❥á ❡r❛♠ ♠❛✐s ❞❡ 27.400 ♣✉❜❧✐❝❛çõ❡s✳ ❙❡♥❞♦ q✉❡ ♦ tr❛❜❛❧❤♦✱ ♣♦r tít✉❧♦
◆♦♥❡q✉✐❧✐❜r✐✉♠ ❝r✐t✐❝❛❧ ♣❤❡♥♦♠❡♥❛ ❛♥❞ ♣❤❛s❡ tr❛♥s✐t✐♦♥s ✐♥t♦ ❛❜s♦r❜✐♥❣ st❛t❡s ❬✶❪✱ ♣✉❜❧✐❝❛❞♦ ❡♠ ♥♦✈❡♠❜r♦ ❞❡ 2000✱ ❥á ♣♦ss✉✐❛ ♠❛✐s ❞❡ 700 ❝✐t❛çõ❡s✳ ❊ss❡s ❞❛❞♦s sã♦ ♣❛r❛ ❝❤❛♠❛r ❛
❛t❡♥çã♦ ❞❛ ✐♠♣♦rtâ♥❝✐❛ ❡ ❛t✉❛❧✐❞❛❞❡ ❞❡ ♥♦ss❛s ♣❡sq✉✐s❛s✳
❊ ♥♦ ✐♥t✉✐t♦ ❞❡ ❝♦♥t❡①t✉❛❧✐③❛r✱ ♠❛✐s ❛✐♥❞❛✱ ♥♦ss❛s ♣❡sq✉✐s❛s ♥❡st❛ ❜❛s❡ ❞❡ ❞❛❞♦s✱ ❝✐t❛♠♦s ❆❧❜❡rt✲▲ás③❧ó ❇❛r❛❜ás✐ q✉❡ ❡♠ ✉♠ ❞❡ s❡✉s ❛rt✐❣♦s✱ ✐♥t✐t✉❧❛❞♦ ❚❤❡ ♦r✐❣✐♥ ♦❢ ❜✉rsts ❛♥❞ ❤❡❛✈② t❛✐❧s ✐♥ ❤✉♠❛♥ ❞②♥❛♠✐❝s✱ ♣✉❜❧✐❝❛❞♦ ♥❛ r❡✈✐st❛ ◆❛t✉r❡ ❬✷❪ ❝♦♠❡♥t❛✿
❚❤❡ ❞②♥❛♠✐❝s ♦❢ ♠❛♥② s♦❝✐❛❧✱ t❡❝❤♥♦❧♦❣✐❝❛❧ ❛♥❞ ❡❝♦♥♦♠✐❝ ♣❤❡♥♦♠❡♥❛ ❛r❡ ❞r✐✈❡♥ ❜② ✐♥❞✐✈✐❞✉❛❧ ❤✉♠❛♥ ❛❝t✐♦♥s✱ t✉r♥✐♥❣ t❤❡ q✉❛♥t✐t❛t✐✈❡ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ ❤✉♠❛♥ ❜❡❤❛✈✐♦✉r ✐♥t♦ ❛ ❝❡♥tr❛❧ q✉❡st✐♦♥ ♦❢ ♠♦❞❡r♥ s❝✐❡♥❝❡✳
❊st❛ t❡s❡ ♣r♦❝✉r❛ ❛❧✐♥❤❛r✲s❡ ❛ ❡ss❡ ♣r♦♣ós✐t♦✱ ❝✐t❛❞♦ ♣♦r ❇❛r❛❜ás✐✱ ❛♦ ❡st✉❞❛r ❛s ♣r♦♣r✐❡❞❛❞❡s ❝rít✐❝❛s ❞❡ s✐st❡♠❛s ❢♦r❛ ❞♦ ❡q✉✐❧í❜r✐♦ ✈✐❛ s✐♠✉❧❛çã♦ ▼♦♥t❡ ❈❛r❧♦✳ ■ss♦ ♣♦r q✉❡✱ ♦s ♠♦❞❡❧♦s ❛q✉✐ ❡st✉❞❛❞♦s tr❛t❛♠ ❞❡ ♣r♦♣❛❣❛çã♦ ❡♣✐❞ê♠✐❝❛✱ q✉❡ ♣♦❞❡♠♦s ❢❛❝✐❧♠❡♥t❡ ❛ss♦❝✐❛r ❛ ♠♦❜✐❧✐❞❛❞❡ ❤✉♠❛♥❛✳ Pr✐♥❝✐♣❛❧♠❡♥t❡ ❛tr❛✈és ❞❛s r❡❞❡s ❞❡ ❝♦♥❡①õ❡s✱ ❝♦♠✉♥s t❛♥t♦ ❛♦s s✐st❡♠❛s ❡♣✐❞ê♠✐❝♦s q✉❛♥t♦ ❛ ♠♦❜✐❧✐❞❛❞❡ ❤✉♠❛♥❛✳
❈❛♣ít✉❧♦ ✶✳ ■♥tr♦❞✉çã♦ ✷
❖ ❡st✉❞♦ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❝rít✐❝❛s ❞♦s ♠♦❞❡❧♦s ❛q✉✐ ❛❜♦r❞❛❞♦s ❝♦♠❡ç❛♠ ♣♦r tr❛ç❛r ♦s ❞✐❛❣r❛♠❛s ❞❡ ❢❛s❡s ❞❡ ❝❛❞❛ ♠♦❞❡❧♦✱ q✉❡ s❡r✈❡ ❝♦♠♦ ✉♠❛ ♣r✐♠❡✐r❛ ✐♥❢♦r♠❛çã♦ s♦❜r❡ ❛s tr❛♥s✐çõ❡s ❞❡ ❢❛s❡✱ s❡ sã♦ ❝♦♥tí♥✉❛s ♦✉ ❞❡s❝♦♥tí♥✉❛s✳ ❊♠ s❡❣✉✐❞❛ sã♦ ✉s❛❞❛s ❛❧❣✉♠❛s té❝♥✐❝❛s ♣❛r❛ ❝❛❧❝✉❧❛r ♦ ♣♦♥t♦ ❝rít✐❝♦✳ ❈♦♠ ❡ss❡s ♣♦♥t♦s ❝rít✐❝♦s✱ ❡ ✉t✐❧✐③❛♥❞♦✲s❡ ❞❡ ❧❡✐s ❞❡ ❡s❝❛❧❛s ❜❡♠ ❡st❛❜❡❧❡❝✐❞❛s ♥❛ ❧✐t❡r❛t✉r❛✱ sã♦ ❝❛❧❝✉❧❛❞♦s ♦s ❡①♣♦❡♥t❡s ❝rít✐❝♦s✳ P❛r❛ ❡♥tã♦ ❛✈❛❧✐❛r s✉❛s ❝❧❛ss❡s ❞❡ ✉♥✐✈❡rs❛❧✐❞❛❞❡s✳
❖s ❝♦♥❝❡✐t♦s ❡♠ q✉❡ s❡ ❜❛s❡✐❛♠ ♦ ❡st✉❞♦ ❞❡ s✐st❡♠❛s ♥❛ ❝r✐t✐❝❛❧✐❞❛❞❡ s✉r❣✐r❛♠ ♥♦ â♠❜✐t♦ ❞♦s s✐st❡♠❛s ❡♠ ❡q✉✐❧í❜r✐♦✳ ❆s ❛❜♦r❞❛❣❡♥s ❞❡ ❝❛♠♣♦ ♠é❞✐♦ ❞❡ ❱❛♥ ❞❡r ❲❛❛❧s ♣❛r❛ ♦s ✢✉✐❞♦s ❡ ❞❡ ❈✉r✐❡✲❲❡✐ss ♣❛r❛ ♦ ♠❛❣♥❡t✐s♠♦ sã♦ ❝♦♥s✐❞❡r❛❞♦s ♥❛ ❧✐t❡r❛t✉r❛ ❬✸❪ ❝♦♠♦ ♦s ♣r✐♠❡✐r♦s ❡st✉❞♦s ❞❛ ❝r✐t✐❝❛❧✐❞❛❞❡✱ s❡♥❞♦ q✉❡ t❛✐s s✐st❡♠❛s ❡♥❝♦♥tr❛♠✲s❡ ❡♠ ❡q✉✐❧í❜r✐♦✳ ▼❛s ♥ã♦ ❞❡♠♦r♦✉ ♣❛r❛ s❡r ♦❜s❡r✈❛❞♦ q✉❡ ♦s ❝♦♥❝❡✐t♦s ❞❡ tr❛♥s✐çã♦ ❞❡ ❢❛s❡✱ ♣♦♥t♦ ❝rít✐❝♦✱ ❡①♣♦❡♥t❡s ❝rít✐❝♦s✱ ❧❡✐s ❞❡ ❡s❝❛❧❛✱ ✉♥✐✈❡rs❛❧✐❞❛❞❡✱ ❡♥tr❡ ♦✉tr♦s ❝♦♥❝❡✐t♦s✱ ❡r❛♠ ❛♣❧✐❝á✈❡✐s ❛ s✐st❡♠❛s ❢♦r❛ ❞♦ ❡q✉✐❧í❜r✐♦✱ ♠❡s♠♦ s❡♥❞♦ ✐♠♣♦ssí✈❡❧ ❞❡ss❡s s✐t❡♠❛s ❛t✐♥❣✐r❡♠ ♦ ❡q✉✐❧í❜r✐♦✱ ❡ ❛té ♠❡s♠♦ ♣♦r ♠✉✐t❛s ✈❡③❡s ♥ã♦ tr❛t❛r✲s❡ ❞❡ s✐st❡♠❛s t❡r♠♦❞✐♥â♠✐❝♦s✳
❆s ❧❡✐s q✉❡ r❡❣❡♠ ❡ss❡s s✐t❡♠❛s✱ ❞❡ ❡q✉✐❧í❜r✐♦ ❡ ❢♦r❛ ❞♦ ❡q✉✐❧í❜r✐♦✱ ♣♦❞❡♠ s❡r s✉❜s✲ t✐t✉í❞❛s ♣♦r ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ♦❝♦rrê♥❝✐❛ ♦✉ ❞❡ ♥ã♦ ♦❝♦rrê♥❝✐❛ ❞❡ ✉♠ ❝❡rt♦ ❡✈❡♥t♦✱ q✉❡ ♣♦❞❡♠ s❡r tr❛❞✉③✐❞❛s ♣♦r ❛❧❣♦r✐t♠♦s✳ ❖ ✉s♦ ❞❛s s✐♠✉❧❛çõ❡s ♥✉♠ér✐❝❛s tê♠ s✐❞♦ ❝r❡s❝❡♥t❡✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣❡❧♦s ❛✈❛♥ç♦s ♥❛ ✈❡❧♦❝✐❞❛❞❡ ❞❡ ♣r♦❝❡ss❛♠❡♥t♦ ❞♦s ♥♦✈♦s ❝♦♠♣✉t❛❞♦r❡s✱ ♠❛s t❛♠❜é♠ ♣❡❧❛s té❝♥✐❝❛s ❞❡ s✐♠✉❧❛çõ❡s ♥✉♠ér✐❝❛s✱ ❛ss♦❝✐❛❞❛s à ❋ís✐❝❛✱ ❝❛❞❛ ✈❡③ ♠❛✐s só❧✐❞❛✳ ❉❡♥tr❡ ❡ss❛s té❝♥✐❝❛s ❞❡ s✐♠✉❧❛çã♦ ♥✉♠ér✐❝❛s ❡stá ❛ s✐♠✉❧❛çã♦ ▼♦♥t❡ ❈❛r❧♦ ✉s❛❞❛ ♥❡st❛ t❡s❡✳
❈❛♣ít✉❧♦ ✶✳ ■♥tr♦❞✉çã♦ ✸
♣r♦❝❡ss♦ ❞❡ ❝♦♥t❛t♦ ❡ ♦✉tr❛ ♣❛r❛ ♦ ♣r♦❝❡ss♦ ❡♣✐❞ê♠✐❝♦ ❞✐❢✉s✐✈♦✳
❖ ♠♦❞❡❧♦ ❣❡♥❡r❛❧✐③❛❞♦ ♣❛r❛ ♦ ♣r♦❝❡ss♦ ❞❡ ❝♦♥t❛t♦ ❞✐❢❡r❡ ❞♦ ♠♦❞❡❧♦ ♣❛❞rã♦✱ ♣♦r ❝♦♥s✐❞❡r❛r q✉❡ ♦ ♣r♦❝❡ss♦ ❞❡ ❝♦♥t❛♠✐♥❛çã♦ é ❣♦✈❡r♥❛❞♦ ♣♦r ✐♥t❡r❛çõ❡s ❝♦♠ ♦✉tr♦s ✐♥❞✐✈í❞✉♦s ❛t✐✈♦s ❧♦❝❛❧✐③❛❞♦s à ❧♦♥❣❛s ❞✐stâ♥❝✐❛s✳ ❖✉ s❡❥❛✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ✐♥❢❡❝çã♦ ❞❡❝❛✐ ❝♦♠ ✉♠❛ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛ ❞❛ ❞✐stâ♥❝✐❛ ♣❛r❛ ♦ ✐♥❞✐✈í❞✉♦ ❛t✐✈♦✳ ◆❡ss❡ ♠♦❞❡❧♦✱ ✉♠ ❡①♣♦❡♥t❡ α ❣♦✈❡r♥❛ ♦ ❞❡❝❛✐♠❡♥t♦ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❝♦♥❡①ã♦✱ ♣❡r♠✐t✐♥❞♦ ✉♠❛ ✈❛r✐❛çã♦ ❞❡s❞❡ ♦ ♣r♦❝❡ss♦ ❞❡ ❝♦♥t❛t♦ ♣❛❞rã♦✱ ❝♦♠ ❧✐❣❛çõ❡s s♦♠❡♥t❡ ❡♥tr❡ ♣r✐♠❡✐r♦s ✈✐③✐♥❤♦s✱ ❛té ♦ r❡❣✐♠❡ ❞❡ ❝❛♠♣♦ ♠é❞✐♦ ❝♦♥s✐❞❡r❛♥❞♦ ❛ r❡❞❡ t♦t❛❧♠❡♥t❡ ❝♦♥❡❝t❛❞❛✳ P❛r❛ ✐ss♦✱ ❞❡s❡♥✈♦❧✈❡♠♦s s✐♠✉❧❛çõ❡s ♥✉♠ér✐❝❛s ❡♠ r❡❞❡s ✜♥✐t❛s✱ ❛♣❧✐❝❛♠♦s ❛♥á❧✐s❡ ❞❡ ❡s❝❛❧❛ ❞❡ t❛♠❛♥❤♦ ✜♥✐t♦ ♣❛r❛ ❞❡t❡r♠✐♥❛r ♦ ♣♦♥t♦ ❝rít✐❝♦ ❡ ✉t✐❧✐③❛♠♦s ❛ ❞✐♥â♠✐❝❛ ❝rít✐❝❛ ♣❛r❛ ❡♥❝♦♥tr❛r ♦s ❡①♣♦❡♥t❡s ❝rít✐❝♦s✱ ♣❛r❛ ❞✐st✐♥t♦s ✈❛❧♦r❡s ❞♦ ❡①♣♦❡♥t❡ ❞❡ ❧✐❣❛çã♦α✳
◆♦ s❡❣✉♥❞♦ tr❛❜❛❧❤♦ ❞❡st❛ t❡s❡ é ❝♦♥s✐❞❡r❛❞♦ ✉♠ ♣r♦❝❡ss♦ ❡♣✐❞ê♠✐❝♦ s✉♣❡r❞✐❢✉s✐✈♦✱ ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❧❛♥ç❛r ✉♠❛ ♥♦✈❛ ❧✉③ s♦❜r❡ ♦ ❞❡❜❛t❡ ❛ r❡s♣❡✐t♦ ❞♦ t✐♣♦ ❞❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡✱ ♣❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡ ❛s ♣❛rtí❝✉❧❛s ❞♦ t✐♣♦ A ♣♦ss✉❡♠ ✉♠❛ ♠♦❜✐❧✐❞❛❞❡ ♠❛✐♦r q✉❡ ❛s ♣❛tí❝✉❧❛s ❞♦ t✐♣♦ B✳ ❆ ♣r♦♣♦st❛ ❞♦ ♣r♦❝❡ss♦ ❡♣✐❞ê♠✐❝♦ s✉♣❡r❞✐❢✉s✐✈♦ é ♣❡r♠✐t✐r q✉❡ ❛s ♣❛rtí❝✉❧❛s A ✭s❛✉❞á✈❡❧✮ ❡ B ✭✐♥❢❡❝t❛❞❛✮ ♣♦ss✉❛♠ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❛r s❛❧t♦s✱ r❡❣✐❞❛ ♣♦r ✉♠❛ ❞✐str✐❜✉✐çã♦ t✐♣♦ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛✱ ♣❡r♠✐t✐♥❞♦ s❛❧t♦s ♣❛r❛ ❛❧é♠ ❞♦s ♣r✐♠❡✐r♦s ✈✐③✐♥❤♦s✳ ❊ ❛✈❛❧✐❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ ❞❛❞❛ ❡ss❛ ❛❧t❡r❛çã♦ ❛♦ ♠♦❞❡❧♦ ♣❛❞rã♦✳ ❖❜s❡r✈❛❞♦ q✉❡ ❛ ❝✉r❛ ❡ ❛ ❝♦♥t❛♠✐♥❛çã♦ ❝♦♥t✐♥✉❛♠ ♦❝♦rr❡♥❞♦ ❧♦❝❛❧♠❡♥t❡✳ ❚❛♠❜é♠ ❝♦♠ ♦ r❡❝✉rs♦ ❞❛s s✐♠✉❧❛çõ❡s ♥✉♠ér✐❝❛s✱ ❞❡t❡r♠✐♥❛♠♦s ❛s ♣r♦♣r✐❡❞❛❞❡s ❝rít✐❝❛s ❛♦ ❝❛❧❝✉❧❛r ♦s ♣♦♥t♦s ❝rít✐❝♦s ❡ s❡✉s ❝♦rr❡s♣♦♥❞❡♥t❡s ❡①♣♦❡♥t❡s ❝rít✐❝♦s✱ ❛tr❛✈és ❞❛ ❛♥á❧✐s❡ ❞❡ ❡s❝❛❧❛ ❞❡ t❛♠❛♥❤♦ ✜♥✐t♦✱ ♣❛r❛ ❞✐st✐♥t♦s ✈❛❧♦r❡s ❞♦ ❡①♣♦❡♥t❡µ❞❛ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛ q✉❡ ❣♦✈❡r♥❛ ♦s s❛❧t♦s ♥❛ r❡❞❡✳
❈❛♣ít✉❧♦ ✶✳ ■♥tr♦❞✉çã♦ ✹
Cap´ıtulo
2
❚❡♦r✐❛ ♣❛r❛ ●❡r❛çã♦ ❡ ❆♥á❧✐s❡ ❞❡ ❉❛❞♦s
❖ ♣r❡s❡♥t❡ ❝❛♣ít✉❧♦ ♦❢❡r❡❝❡ ✉♠❛ ✈✐sã♦ ❣❡r❛❧✱ ❡ ♣♦♥t✉❛❧♠❡♥t❡ ✉♠❛ ❞❡s❝r✐çã♦ ♠❛✐s ❛♣r♦❢✉♥❞❛❞❛✱ ❞♦s ❝♦♥❝❡✐t♦s ✉t✐❧✐③❛❞♦s ♥❡st❛ t❡s❡✱ ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ♣r♦♣♦r❝✐♦♥❛r ✉♠❛ ♠❡❧❤♦r ❝♦♠♣r❡❡♥sã♦ ❞❡ ♥♦ss♦s ♠♦❞❡❧♦s ❡st✉❞❛❞♦s✱ ♥♦ q✉❡ t❛♥❣❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❢ís✐❝♦ q✉❡ t❛✐s ♠♦❞❡❧♦s ♣♦❞❡♠ ❛♣r❡s❡♥t❛r✳ ❖ ❡st✉❞♦ ❞❡ss❡s s✐st❡♠❛s✱ ❢♦r❛ ❞♦ ❡q✉✐❧í❜r✐♦✱ ♣♦❞❡ s❡r r❡❛❧✐③❛❞♦ ❜❛s✐❝❛♠❡♥t❡ ♦✉ ♣♦r té❝♥✐❝❛s ❞♦ ❣r✉♣♦ ❞❡ r❡♥♦r♠❛❧✐③❛çã♦✱ ♦✉ ♣♦r s✐♠✉❧❛çã♦ ❝♦♠♣✉t❛❝✐♦♥❛❧✱ ♦✉ ❛♠❜♦s✳ ◆♦ss❛ ❛❜♦r❞❛❣❡♠ s❡ ❝♦♥❝❡♥tr❛ ♥♦ ❝❛♠♣♦ ❞❛ s✐♠✉❧❛çã♦✱ ❢❛③❡♥❞♦ ✉s♦ ❞♦ ♠ét♦❞♦ ▼♦♥t❡ ❈❛r❧♦ ✭▼▼❈✮✱ q✉❡ t❡♠ ❝♦♠♦ ♣❛♥♦ ❞❡ ❢✉♥❞♦ ♦s ♣r♦❝❡ss♦s ❡st♦❝ást✐❝♦s✱ ❡ q✉❡ ❡stá ❛ss♦❝✐❛❞♦ ❛ s✐st❡♠❛s ❞❡ ♠✉✐t❛s ♣❛rtí❝✉❧❛s ❡ ❞❡ t❛♠❛♥❤♦s ✜♥✐t♦s✳ ❆ss✐♠✱ ❣❡r❛♠♦s ♦s ❞❛❞♦s ♣❛r❛ ❡st✉❞♦ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❝rít✐❝❛s ❞♦s ♠♦❞❡❧♦s ❞❡ ♥ã♦ ❡q✉✐❧í❜r✐♦ ❛q✉✐ ❡st✉❞❛❞♦s✳ ❈♦♠ r❡s♣❡✐t♦ à ❛♥á❧✐s❡ ❞♦s ❞❛❞♦s✱ ❛♣❡s❛r ❞❛s tr❛♥s✐çõ❡s ❞❡ ❢❛s❡ ✭❚❋✮ ♦❝♦rr❡r❡♠ ♥♦ ❧✐♠✐t❡ t❡r♠♦❞✐♥â♠✐❝♦ ✭▲❚✮✱ ❛ ❤✐♣ót❡s❡ ❞❡ ❡s❝❛❧❛ ❞❡ t❛♠❛♥❤♦ ✜♥✐t♦ ♣❡r♠✐t❡ q✉❡ ❛s ❧❡✐s ❞❡ ❡s❝❛❧❛✱ ❝♦♠ s❡✉s ❡①♣♦❡♥t❡s ❝rít✐❝♦s✱ ❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❝❧❛ss❡ ❞❡ ✉♥✐✈❡rs❛❧✐❞❛❞❡✱ ❛♣❧✐❝á✈❡✐s ♥♦ ▲❚✱ ♣♦ss❛♠ s❡r ❛❞♦t❛❞♦s ♣❛r❛ s✐st❡♠❛s ✜♥✐t♦s✳
❈❛♣ít✉❧♦ ✷✳ ❚❡♦r✐❛ ♣❛r❛ ●❡r❛çã♦ ❡ ❆♥á❧✐s❡ ❞❡ ❉❛❞♦s ✻
✷✳✶ ❚r❛♥s✐çõ❡s ❞❡ ❋❛s❡s ❡ ❈r✐t✐❝❛❧✐❞❛❞❡
◆❛ ♥❛t✉r❡③❛✱ ❛❧❣✉♥s s✐st❡♠❛s sã♦ ❜❡♠ ❝♦♥❤❡❝✐❞♦s ♣♦r ♠✉❞❛r❡♠ ❞❡ ❢❛s❡ q✉❛♥❞♦ ✈❛r✐❛♠♦s ❝♦♥t✐♥✉❛♠❡♥t❡ ♦s ✈❛❧♦r❡s ❞♦s s❡✉s ♣❛râ♠❡tr♦s ❡①t❡r♥♦s✱ t❛✐s ❝♦♠♦✿ t❡♠♣❡r❛t✉r❛✱ ♣r❡ssã♦✱ ♦✉ ❝❛♠♣♦s ❡❧étr✐❝♦s ❡ ♠❛❣♥ét✐❝♦s✳ ❚❛❧ ♠✉❞❛♥ç❛ ❞❡✜♥❡ ✉♠❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ q✉❡ s❡ ❝❛r❛❝t❡r✐③❛ ♣❡❧❛ ♠✉❞❛♥ç❛ ♥♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❣❧♦❜❛❧ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❢ís✐❝❛s ❞❛ s✉❜stâ♥❝✐❛✳ ❊ ♦ ♣♦♥t♦ ♦♥❞❡ ♦❝♦rr❡ ❛ ♠✉❞❛♥ç❛ ❞❡ ❝♦♠♣♦rt❛♠❡♥t♦ é ❝❤❛♠❛❞♦ ❞❡ ♣♦♥t♦ ❝rít✐❝♦✳
P❛r❛ ❞❡s❝r❡✈❡r ♦ ❢❡♥ô♠❡♥♦ ❞❛ ❚❋✱ ▲❛♥❞❛✉ ✐♥tr♦❞✉③✐✉ ✉♠ ♥♦✈♦ ❝♦♥❝❡✐t♦−♦ ♣❛râ♠❡tr♦ ❞❡ ♦r❞❡♠✳ P❛r❛ ❛❧❣✉♥s ❝❛s♦s✱ ❝♦♠♦ ♥❛ ♠❛❣♥❡t✐③❛çã♦✱ ♦ ♣❛râ♠❡tr♦ ❞❡ ♦r❞❡♠ é ✐❣✉❛❧ ❛ ③❡r♦ ❛❝✐♠❛ ❞♦ ♣♦♥t♦ ❞❡ tr❛♥s✐çã♦✱ ❡ ❛ss✉♠❡ ✉♠ ✈❛❧♦r ✜♥✐t♦ ❛❜❛✐①♦ ❞❡ss❡ ♠❡s♠♦ ♣♦♥t♦✳ ❖ ❝♦♥✲ ❝❡✐t♦✱ ♣❛râ♠❡tr♦ ❞❡ ♦r❞❡♠✱ ♣❡r♠✐t✐✉ ❛ ▲❛♥❞❛✉ ❡s❝r❡✈❡r ❛ ❡♥❡r❣✐❛ ❧✐✈r❡ ❞❡ q✉❛❧q✉❡r s✐st❡♠❛ ♥❛s ♣r♦①✐♠✐❞❛❞❡s ❞♦ ♣♦♥t♦ ❝rít✐❝♦ s♦❜ ❛ ❢♦r♠❛ ❞❡ ✉♠❛ ❡①♣❛♥sã♦ ❡♠ sér✐❡ ❞❡ ♣♦tê♥❝✐❛ ❞♦ ♣❛râ♠❡tr♦ ❞❡ ♦r❞❡♠ ❬✸❪✳
❆❧❣✉♥s s✐st❡♠❛s ❢♦r❛ ❞♦ ❡q✉✐❧í❜r✐♦ ♣♦❞❡♠✱ ❞✉r❛♥t❡ s✉❛ ❡✈♦❧✉çã♦✱ ✜❝❛r ❛♣r✐s✐♦♥❛❞♦s ✐♥❞❡✜♥✐❞❛♠❡♥t❡ ❡♠ ✉♠❛ ❝♦♥✜❣✉r❛çã♦ ✐♥❛t✐✈❛ ✭❡st❛❞♦ ❛❜s♦r✈❡♥t❡✮✳ ◆❛ ♠♦❞❡❧❛❣❡♠ ❡st♦❝ást✐❝❛ ✉♠ ❡st❛❞♦ ❛❜s♦r✈❡♥t❡ é ❛q✉❡❧❡ q✉❡ ♣♦❞❡ s❡r ❛t✐♥❣✐❞♦ ❛ ♣❛rt✐r ❞❡ ♦✉tr♦s ❡st❛❞♦s✱ ♠❛s ❛ tr❛♥s✐çã♦ ❛ ♣❛rt✐r ❞❡❧❡ ♣❛r❛ ♦✉tr♦s ❡st❛❞♦s é ♣r♦✐❜✐❞❛✳ ❈♦♠♦ ❡①❡♠♣❧♦✱ ✐♠❛❣✐♥❡ ✉♠ ♠♦❞❡❧♦ ❡♠ q✉❡ ✐♥❞✐✈í❞✉♦s s❛✉❞á✈❡✐s ✭sít✐♦s ✐♥t❛t✐✈♦s✮ sã♦ ❝♦♥t❛♠✐♥❛❞♦s ♣♦r ✐♥❞✐✈í❞✉♦s ✐♥❢❡❝t❛❞♦s ✭sít✐♦s ❛t✐✈♦s✮✱ ❝♦♠ ❝❡rt❛ ♣r♦❜❛❜✐❧✐❞❛❞❡✱ ❡ q✉❡ ❡ss❡s ✐♥❞✐✈í❞✉♦s ✐♥❢❡❝t❛❞♦s ♣♦❞❡♠ s❡r ❝✉r❛❞♦s t❛♠❜é♠ ❝♦♠ ❝❡rt❛ ♣r♦❜❛❜✐❧✐❞❛❞❡✳ ❙❡ ❡♠ ✉♠ ❞❛❞♦ ♠♦♠❡♥t♦✱ ❞❡✈✐❞♦ à ❛❧❡❛t♦r✐❡❞❛❞❡✱ t♦❞♦s ♦s ✐♥❞✐✈í❞✉♦s ❡st✐✈❡r❡♠ s❛✉❞á✈❡✐s✱ ♥ã♦ t❡rá ♠❛✐s ❝♦♠♦ ❤❛✈❡r ❝♦♥t❛♠✐♥❛çã♦✱ ❥á q✉❡ ♥ã♦ ❤á ♠❛✐s ✐♥❞✐✈í❞✉♦s ✐♥❢❡❝t❛❞♦s ♣❛r❛ t❛❧✳ ◗✉❛♥❞♦ ♦ s✐st❡♠❛ ❛t✐♥❣❡ ❡st❡ ❡st❛❞♦ ❞✐③❡♠♦s q✉❡ ❡❧❡ ❡♥tr♦✉ ♥♦ ❡st❛❞♦ ❛❜s♦r✈❡♥t❡✳
❈❛♣ít✉❧♦ ✷✳ ❚❡♦r✐❛ ♣❛r❛ ●❡r❛çã♦ ❡ ❆♥á❧✐s❡ ❞❡ ❉❛❞♦s ✼
❞❛s ❢❛s❡s q✉❡ t❡♥❞❡♠ ❛ ❧❡✈❛r ♦ s✐st❡♠❛ ♣❛r❛ ✉♠ ❧❛❞♦ ♦✉ ♦✉tr♦✱ ❞❡ ♠❛♥❡✐r❛ q✉❡ ♠❡s♠♦ s❡♠ ♣♦❞❡r♠♦s ❛ss♦❝✐❛r q✉❛❧q✉❡r ❣r❛♥❞❡③❛ t❡r♠♦❞✐♥â♠✐❝❛ ♣❛r❛ ♦ ♣r♦❝❡ss♦ ❞❡ ❝♦♥t❛t♦✱ ♦ ♠❡❝❛♥✐s♠♦ ❞❡ ❝♦♠♣❡t✐çã♦ ♦r❞❡♠✲❞❡s♦r❞❡♠✱ ♣r♦✈❡♥✐❡♥t❡ ❞❛ ✐♥t❡r❛çã♦ ❞❛s ♠✉✐t❛s ♣❛rtí❝✉❧❛s✱ ❝♦♥t✐♥✉❛ ♣r❡s❡♥t❡ ❡ ♣r♦✈♦❝❛ ✉♠❛ q✉❡❜r❛ ❡s♣♦♥tâ♥❡❛ ❞❡ s✐♠❡tr✐❛ ❬✺❪ ❡ ❛ ❝♦♥s❡q✉❡♥t❡ ❚❋ ❬✻✱ ✼❪✳
❆✐♥❞❛ ❝♦♠♣❛r❛♥❞♦ ❝♦♠ ♦s s✐st❡♠❛s ❞❡ ❡q✉✐❧í❜r✐♦✱ ♦❜s❡r✈❛♠♦s q✉❡ ♥♦s s✐st❡♠❛s ❢♦r❛ ❞♦ ❡q✉✐❧í❜r✐♦ ♦s ❡st❛❞♦s ❛❜s♦r✈❡♥t❡ ❡ ❛t✐✈♦ ♣♦❞❡♠ s❡r ✈✐st♦s ❝♦♠♦ ❢❛s❡s✱ ✉♠❛ ✈❡③ q✉❡ ❝❛r✲ ❛❝t❡r✐③❛♠ ✉♠ ❛rr❛♥❥♦ ❜❡♠ ❞❡t❡r♠✐♥❛❞♦ ❞♦s ❝♦♥st✐t✉✐♥t❡s ♠✐❝r♦s❝ó♣✐❝♦s ❞♦ s✐st❡♠❛✳ ❆❧é♠ ❞✐ss♦✱ ♦✉tr❛s ❛ss♦❝✐❛çõ❡s sã♦ ❞✐r❡t❛♠❡♥t❡ ♣❧❛✉sí✈❡✐s✱ ❝♦♠♦ ❛ ✐❞❡♥t✐✜❝❛çã♦ ❞❛ ❞❡♥s✐❞❛❞❡ ❞❡ sít✐♦s ❛t✐✈♦s ✭✐♥❞✐✈í❞✉♦s ✐♥❢❡❝t❛❞♦s✮ ❝♦♠ ♦ ♣❛râ♠❡tr♦ ❞❡ ♦r❞❡♠✱ ❡ ❛ t❛①❛ ❞❡ ✐♥❢❡❝çã♦ ❝♦♠ ♦ ♣❛râ♠❡tr♦ ❞❡ ❝♦♥tr♦❧❡ ♣❛r❛ ♦ ♠♦❞❡❧♦ P❈✳ ❖❜s❡r✈❡ q✉❡ ❡ss❛s ❛ss♦❝✐❛çõ❡s ♣r❡❡♥❝❤❡♠ ♦s r❡q✉✐s✐t♦s ♥❡❝❡ssár✐♦s ♣❛r❛ s❡ tr❛t❛r ❞❡ ✉♠❛ ❚❋✳
❆s tr❛♥s✐çõ❡s ❞❡ ❢❛s❡ ♣♦❞❡♠ s❡r ❝❧❛ss✐✜❝❛❞❛s ❝♦♠♦✿ ❞❡s❝♦♥tí♥✉❛s✱ q✉❡ ♦❝♦rr❡♠ q✉❛♥❞♦ ❤á ❞❡s❝♦♥t✐♥✉✐❞❛❞❡ ♥❛ ♣r✐♠❡✐r❛ ❞❡r✐✈❛❞❛ ❞♦s ♣♦t❡♥❝✐❛✐s t❡r♠♦❞✐♥â♠✐❝♦s❀ ♦✉ ❝♦♥✲ tí♥✉❛s✱ q✉❛♥❞♦ ❛ ❞❡s❝♦♥t✐♥✉✐❞❛❞❡ é ♥❛ s❡❣✉♥❞❛ ❞❡r✐✈❛❞❛✳ ◆♦ s❡❣✉♥❞♦ ❝❛s♦✱ ❛s s✐♥❣✉❧❛r✐❞❛❞❡s sã♦ ❝♦♥❤❡❝✐❞❛s ❝♦♠♦ ♣♦♥t♦s ❝rít✐❝♦s ❡ ❡♠ s✉❛s ✈✐③✐♥❤❛♥ç❛s ❡♥❝♦♥tr❛♠♦s ❧❡✐s ❞❡ ♣♦tê♥❝✐❛ ❝♦♠ ❡①♣♦❡♥t❡s ✭❡①♣♦❡♥t❡s ❝rít✐❝♦s✮ ❞❡♣❡♥❞❡♥t❡s ❛♣❡♥❛s ❞❡ ❛❧❣✉♠❛s ❝❛r❛❝t❡ríst✐❝❛s ❞♦ s✐st❡♠❛✱ ❝♦♠♦ ❛ ❞✐♠❡♥s✐♦♥❛❧✐❞❛❞❡ ❞♦ ❡s♣❛ç♦ ❡ ❛s s✐♠❡tr✐❛s ❡♥✈♦❧✈✐❞❛s✳
❈❛♣ít✉❧♦ ✷✳ ❚❡♦r✐❛ ♣❛r❛ ●❡r❛çã♦ ❡ ❆♥á❧✐s❡ ❞❡ ❉❛❞♦s ✽
✷✳✷ ❙✐♠✉❧❛çã♦ ◆✉♠ér✐❝❛
❆s s✐♠✉❧❛çõ❡s ♥✉♠ér✐❝❛s tê♠ t♦r♥❛❞♦✲s❡ ✉♠❛ ♠❛♥❡✐r❛ ❞❡ ❢❛③❡r ♣❡sq✉✐s❛ ❡①t❡♥s❛✲ ♠❡♥t❡ ✉t✐❧✐③❛❞❛✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ❡♠ ❋ís✐❝❛✳ ❆♦ ♣♦♥t♦ ❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦ ✉♠ t❡r❝❡✐r♦ r❛♠♦ ❞❛ ❋ís✐❝❛✱ ❛❧é♠ ❞❛ ❢ís✐❝❛ t❡ór✐❝❛ ❡ ❡①♣❡r✐♠❡♥t❛❧ ❬✾❪✳ ❯s❛♠♦s s✐♠✉❧❛çõ❡s ♥✉♠ér✐❝❛s ♣❛r❛ r❡♣✲ r❡s❡♥t❛r q✉❛♥t✐t❛t✐✈❛♠❡♥t❡ ❛ ❡✈♦❧✉çã♦ ❞❡ ✉♠ s✐st❡♠❛ ❢ís✐❝♦✱ ❜✐♦❧ó❣✐❝♦ ❞❡♥tr❡ ✈ár✐♦s ♦✉tr♦s✳ Ó❜✈✐♦ q✉❡ ♣❛r❛ ✐ss♦✱ ❞❡✈❡♠♦s ❣❛r❛♥t✐r q✉❡ ❛ ❡✈♦❧✉çã♦ ❞♦ s✐st❡♠❛✱ ♥❛ s✐♠✉❧❛çã♦✱ ♦❜❡❞❡ç❛ ❡str✐t❛♠❡♥t❡ às ❧❡✐s ❢ís✐❝❛s ♣♦r ❡①❡♠♣❧♦✱ q✉❡ ❣♦✈❡r♥❛ ♦ ♣r♦❝❡ss♦ ❢ís✐❝♦ r❡❛❧✳ ❆ss✐♠✱ ♦ r❡s✉❧✲ t❛❞♦ ❞❛s s✐♠✉❧❛çõ❡s ♣♦❞❡♠ r❡♣r❡s❡♥t❛r ❜❡♠ ♦ ❛♠❜✐❡♥t❡ r❡❛❧✱ ❡ ❝♦♠ ✐ss♦ ❢❛③❡r ✐♥❢❡rê♥❝✐❛s ♠❛✐s ♣r❡❝✐s❛s ❞❛ ♥❛t✉r❡③❛✳ ➱ ♥❡st❡ ❝♦♥t❡①t♦✱ ❞❛ ♦❜s❡r✈❛çã♦ ❞♦s r❡s✉❧t❛❞♦s ❢♦r♥❡❝✐❞♦s ♣❡❧❛ s✐♠✉❧❛çã♦ ♥✉♠ér✐❝❛✱ q✉❡ r❡❛❧✐③❛♠♦s ♥♦ss♦s ❡st✉❞♦s✳ ❊ ❝♦♠ ❛ ✜♥❛❧✐❞❛❞❡ ❞❡ ❢♦r♥❡❝❡r ✉♠ ❡♠✲ ❜❛s❛♠❡♥t♦ ❢ís✐❝♦ ♣❛r❛ ❛s s✐♠✉❧❛çõ❡s ♥✉♠é♠✐❝❛s✱ ❛♣r❡s❡♥t❛✲s❡ ❡♠ s❡❣✉✐❞❛ ✉♠ ♣♦✉❝♦ ❞❛s ❜❛s❡s ❞♦s ♣r♦❝❡ss♦s ❡st♦❝ást✐❝♦s✳
✷✳✷✳✶ Pr♦❝❡ss♦s ❊st♦❝ást✐❝♦s
❆ t❡♦r✐❛ ❞♦s ♣r♦❝❡ss♦s ❡st♦❝ást✐❝♦s s✉r❣✐✉ ❞♦s ❡s❢♦rç♦s ♣❛r❛ ❞❡s❝r❡✈❡r q✉❛♥t✐t❛t✐✈❛✲ ♠❡♥t❡ ♦ ♠♦✈✐♠❡♥t♦ ❇r♦✇♥✐❛♥♦✳ ❊ ♣❛r❛ tr❛t❛r ❞❡ ♣r♦❝❡ss♦s ❡st♦❝ást✐❝♦s ✉♠ ❜♦♠ ❡①❡♠♣❧♦ é ❛ ❝❛♠✐♥❤❛❞❛ ❛❧❡❛tór✐❛✱ q✉❡ ❝♦♥s✐st❡ ❞❡ ✉♠❛ s✉❝❡ssã♦ ❞❡ ♣❛ss♦s ❛❧❡❛tór✐♦s✳ ❆ ❝❛♠✐♥❤❛❞❛ ❛❧❡❛tór✐❛ ❡stá ♥❛s ❜❛s❡s ❞❡ ♠✉✐t♦s ♣r♦❝❡ss♦s ❢✉♥❞❛♠❡♥t❛✐s ❡♠ ❋ís✐❝❛ ❊st❛tíst✐❝❛ ❞❡ ♥ã♦ ❡q✉✐✲ ❧í❜r✐♦✳ P❛r❛ ✐❧✉str❛r à ❝❛♠✐♥❤❛❞❛ ❛❧❡❛tór✐❛✱ ❢❛r❡♠♦s ❛ s❡❣✉✐r✱ ❛ ♣❛rt✐r ❞❛ ❝❛♠✐♥❤❛❞❛✱ ❛ ❞❡❞✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❛ ❞✐❢✉sã♦ ❬✶✵❪✳ ❈♦♥s✐❞❡r❛♥❞♦ ❛ ❋✐❣✳ ✷✳✶✱ s❡♥❞♦x✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛✱ q✉❡ r❡♣r❡s❡♥t❛ ✉♠ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛ ❞♦ ❝❛♠✐♥❤❛♥t❡✱ ❡ x+ 1 ✭x−1✮ r❡♣r❡s❡♥t❛ ✉♠ ♣❛ss♦ ❛
❈❛♣ít✉❧♦ ✷✳ ❚❡♦r✐❛ ♣❛r❛ ●❡r❛çã♦ ❡ ❆♥á❧✐s❡ ❞❡ ❉❛❞♦s ✾
1
x
x
x
1
❋✐❣✉r❛ ✷✳✶✿ ■❧✉str❛çã♦ ❞❛ ❝❛♠✐♥❤❛❞❛ ❛❧❡❛tór✐❛ ❡♠ ✉♠❛ ❞✐♠❡♥sã♦1d✳ ❆ ♣❛rt✐r ❞♦ ♣❛ss♦ ❛❧❡❛tór✐❛x
❡①✐st❡♠ ❞✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s✱ ♣❛r❛ ❞✐r❡✐t❛x+ 1♦✉ ♣❛r❛ ❡sq✉❡r❞❛x−1✳
❆ r❡♣r❡s❡♥t❛çã♦✱ ♥❛ ❋✐❣✳ ✷✳✶✱ ♣♦❞❡ ✐♥❞✉③✐r à ♣❡r❣✉♥t❛✿ q✉❛❧ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ P(x, t)
❞♦ ❝❛♠✐♥❤❛♥t❡✱ ❡♠ ✉♠ t❡♠♣♦ s❡❣✉✐♥t❡✱ ✐r ♣❛r❛ ❞✐r❡✐t❛ ♦✉ ♣❛r❛ ❡sq✉❡r❞❛❄ P❛r❛ ❜✉s❝❛r ❡st❛ r❡s♣♦st❛✱ ❢❛ç❛♠♦s ❛ s❡❣✉✐♥t❡ ❝♦♥str✉çã♦✱ ❝♦♥s✐❞❡r❛♥❞♦ ❛ ❋✐❣✳ ✷✳✷✳
1
x
x
1
x
t
1
t
❋✐❣✉r❛ ✷✳✷✿ ■❧✉str❛çã♦ ❞❛ ❝❛♠✐♥❤❛❞❛ ❛❧❡❛tór✐❛ ❡♠ ✉♠ t❡♠♣♦ t ❡ ❧♦❣♦ ❡♠ s❡❣✉✐❞❛ ❡♠ ✉♠ t❡♠♣♦ t+ 1✳ ❆ ♣♦s✐çã♦ x ❡♠t+ 1♣♦❞❡ ♣r♦✈✐r t❛♥t♦ ❞❛ ♣♦s✐çã♦ x+ 1q✉❛♥t♦ ❞❛ ♣♦s✐çã♦ x−1♥♦ t❡♠♣♦
❛♥t❡r✐♦rt✳
❆ ♣r♦❜❛❜✐❧✐❞❛❞❡ q✉❡ ❞❡s❝r❡✈❡ ♦ ♣❛ss♦ ❞♦ ❝❛♠✐♥❤❛♥t❡✱ ❞♦ t❡♠♣♦t ♣❛r❛ ♦ t❡♠♣♦t+ 1✱
é ❞❛❞❛ ♣❡❧❛ s❡❣✉✐♥t❡ ❡q✉❛çã♦✿
P(x, t+ 1) = 1
2P(x−1, t) + 1
2P(x+ 1, t). ✭✷✳✶✮
◆♦ ❧✐♠✐t❡ ❞♦ ❝♦♥tí♥✉♦ ❛ ❡q✉❛çã♦ ✭✷✳✶✮ t♦r♥❛✲s❡✿
P(x, t+δt) = 1
2P(x−δx, t) + 1
❈❛♣ít✉❧♦ ✷✳ ❚❡♦r✐❛ ♣❛r❛ ●❡r❛çã♦ ❡ ❆♥á❧✐s❡ ❞❡ ❉❛❞♦s ✶✵
❡①♣❛♥❞✐♥❞♦ ♦s três t❡r♠♦s ❞❛ ❡q✉❛çã♦ ✭✷✳✷✮✱ t❡♠✲s❡✿
P(x, t+δt) =P(x, t) +δt∂P
∂t +... ✭✷✳✸✮
P(x−δx, t) = P(x, t)−δx∂P ∂x +
1 2!(−δx)
2∂2P
∂x2 +... ✭✷✳✹✮
P(x+δx, t) = P(x, t) +δx∂P ∂x +
1 2!(+δx)
2∂2P
∂x2 +.... ✭✷✳✺✮
❙✉❜st✐t✉✐♥❞♦ ❛s ❡q✉❛çõ❡s ✭✷✳✸✮✱ ✭✷✳✹✮ ❡ ✭✷✳✺✮ ♥❛ ❡q✉❛çã♦ ✭✷✳✷✮ ❡ ❞❡s♣r❡③❛♥❞♦ ♦s t❡r♠♦s ❞❡ ♦r❞❡♠ s✉♣❡r✐♦r ❛1♥❛ ❡q✉❛çã♦ ✭✷✳✸✮ ❡ ♦r❞❡♠ s✉♣❡r✐♦r ❛2♥❛s ❡q✉❛çõ❡s ✭✷✳✹✮ ❡ ✭✷✳✺✮✱ r❡s✉❧t❛✿
P(x, t) +δt∂P ∂t =
1 2
P(x, t)−δx∂P ∂x +
1 2(δx)
2∂2P ∂x2
+1 2
P(x, t) +δx∂P ∂x +
1 2(δx)
2∂2P ∂x2
, ✭✷✳✻✮
q✉❡ ❛♣ós ✉♠❛ á❧❣❡❜r❛ ❢♦r♥❡❝❡✱
∂P(x, t)
∂t =
1 2
(δx)2 δt
∂2P(x, t)
∂x2 . ✭✷✳✼✮
❋❛③❡♥❞♦ ❛ s❡❣✉✐♥t❡ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧✱
D= 1 2
(δx)2
δt , ✭✷✳✽✮
s❡♥❞♦D ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❞✐❢✉sã♦✱ ✜♥❛❧♠❡♥t❡ ♦❜t❡♠♦s✿
∂P(x, t)
∂t =D
∂2P(x, t)
∂x2 , ✭✷✳✾✮
❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❡q✉❛çã♦ ❞❡ ❞✐❢✉sã♦ ❞❛ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ P(x, t)✳ ❆ ❡q✉❛çã♦ ✭✷✳✾✮
