❙■▼❯▲❆➬➹❖ ❉❊ ▼❖◆❚❊ ❈❆❘▲❖ ◆❖ ▼❖❉❊▲❖ ❉❊ ■❙■◆● ◆❆ ❘❊❉❊ ◗❯❆❉❘❆❉❆
▼✉r✐❧♦ ▲❛❝❡r❞❛ ❙❛♥t♦s
❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s ✲ ❯❋▼● ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ✲ ■❈❊① Pr♦❣r❛♠❛ ❞❡ Pós ●r❛❞✉❛çã♦ ❡♠ ❋ís✐❝❛
❙■▼❯▲❆➬➹❖ ❉❊ ▼❖◆❚❊ ❈❆❘▲❖ ◆❖ ▼❖❉❊▲❖ ❉❊ ■❙■◆● ◆❆ ❘❊❉❊ ◗❯❆❉❘❆❉❆
▼✉r✐❧♦ ▲❛❝❡r❞❛ ❙❛♥t♦s
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❏♦ã♦ ❆♥tô♥✐♦ P❧❛s❝❛❦ ❈♦✲♦r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❇✐s♠❛r❝❦ ❱❛③ ❞❛ ❈♦st❛
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❞❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛ ❞❛ ❯♥✐✈❡r✲ s✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s✱ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞❡ ❚ít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ❋ís✐❝❛
➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ▼❡❝â♥✐❝❛ ❊st❛tíst✐❝❛✳
✏❈♦♠❡❝❡ ♣♦r ❢❛③❡r ♦ q✉❡ é ♥❡❝❡ssár✐♦✱ ❞❡♣♦✐s ❢❛ç❛ ♦ q✉❡ é ♣♦ssí✈❡❧ ❡ ❡♠ ❜r❡✈❡ ❡st❛rá ❢❛③❡♥❞♦ ♦ q✉❡
❆❣r❛❞❡❝✐♠❡♥t♦s
❆♦ ♣r♦❢❡ss♦r P❧❛s❝❛❦ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛✱ ♦r✐❡♥t❛çã♦ ❡ ❞❡❞✐❝❛çã♦✳ ◗✉❡ ♥ã♦ s♦♠❡♥t❡ ♦r✐❡♥t♦✉✱ ♠❛s q✉❡ ♠❡ ❛❝♦♥s❡❧❤♦✉ ❞✉r❛♥t❡ ❡st❛ ❢❛s❡ tã♦ ✐♠♣♦rt❛♥t❡ ❞❛ ♠✐♥❤❛ ✈✐❞❛✳ ❆♦ ♣r♦❢❡ss♦r ❇✐s✲ ♠❛r❝❦ ♣❡❧❛ ❝♦✲♦r✐❡♥t❛çã♦ ❡ ❛❥✉❞❛ ♥❛ ❞❡❢❡s❛ ❞❡ss❛ ❞✐ss❡rt❛çã♦✳ ❆♦s ❞❡♠❛✐s ♣r♦❢❡ss♦r❡s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛ ❞♦ ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s✱ q✉❡ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ♠✐♥❤❛ ❢♦r✲ ♠❛çã♦ ❛❝❛❞ê♠✐❝❛✳ ❆♦s ♠❡✉s ♣❛✐s✱ ✐r♠ã✱ ❡ t♦❞♦s ♦s ❞❡♠❛✐s ❢❛♠✐❧✐❛r❡s ❡ ❛♠✐❣♦s q✉❡ ❡st✐✈❡r❛♠ ❝♦♠✐❣♦ ❞✉r❛♥t❡ ♦ ♠❡str❛❞♦✳ ❊♠ ❡s♣❡❝✐❛❧ ❛♦s ❛♠✐❣♦s ❞♦ ❞❡♣❛rt❛♠❡♥t♦ q✉❡ ❛❣r❡❣❛r❛♠ ✉♠ ❝r❡s❝✐♠❡♥t♦ t❛♥t♦ ♣❡ss♦❛❧✱ ❝♦♠♦ ♣r♦✜ss✐♦♥❛❧ ❡♠ ♠✐♥❤❛ ✈✐❞❛✿ ❉✐❡❣♦✱ ❚❛ss✐✉s ❡ ▼❛r❝♦s✳ ❆❧é♠ ❞❡ ♦✉tr♦s ❛♠✐❣♦s q✉❡ ❡st✐✈❡r❛♠ ♣r❡s❡♥t❡ ♥❡ss❡ ♣r♦❝❡ss♦✿ ❆❞❛í❧t♦♥✱ ❏♦✐❧s♦♥✱ ■❛♥❛✱ ❘❛❢❛❡❧✱ ➱r✐❝♦✱ ❆❧✐ss♦♥✱ ❊r✐❦✱ ●✉st❛✈♦✱ P❡❞r♦✱ ❊❧✐❡❧ ❡ ❉❛✈✐✳ ❆ ❈❆P❊❙ ♣❡❧♦ ❛✉①í❧✐♦ ✜♥❛♥❝❡✐r♦✳
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦✱ ❡st✉❞❛✲s❡✱ ❝♦♠♣✉t❛❝✐♦♥❛❧♠❡♥t❡✱ ❛s ♣r♦♣r✐❡❞❛❞❡s ♠❛❣♥ét✐❝❛s ❞♦ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣ ❢❡rr♦♠❛❣♥ét✐❝♦ ♥✉♠❛ r❡❞❡ q✉❛❞r❛❞❛ ❡ ♥❛ ❛✉sê♥❝✐❛ ❞❡ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❡①t❡r♥♦✳ ❆ ♣❛rt❡ ✐♥✐❝✐❛❧ ❞♦ tr❛❜❛❧❤♦ ❡♥✈♦❧✈❡ ❛ ❢✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛ ❡ ♦ ♣r♦❝❡ss♦ ❞❡ ❝♦♥str✉çã♦ ❞♦s ♣r♦❣r❛♠❛s ❞❡ s✐♠✉❧❛çã♦✱ t❛✐s ❝♦♠♦✿ ♦ ♣r♦❣r❛♠❛ ❞❡ ▼♦♥t❡ ❈❛r❧♦✱ té❝♥✐❝❛ ❞♦ ❤✐st♦❣r❛♠❛ s✐♠♣❧❡s ❡ ❝á❧❝✉❧♦ ❞❡ ❡rr♦s✳ P❛r❛ ♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❡ ✉♠ ❜❧♦❝♦ ❞❡ q✉❛tr♦ s♣✐♥s ❢♦✐ ♦❜t✐❞♦✱ ❛♥❛❧✐t✐❝❛♠❡♥t❡✱ ♦s r❡s✉❧t❛❞♦s ❡①❛t♦s✱ ♦s q✉❛✐s ❢♦r❛♠ ❝♦♠♣❛r❛❞♦s ❝♦♠ ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ❝♦♠ ❛ s✐♠✉❧❛çã♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦✳ ❊ss❛ ❝♦♠♣❛r❛çã♦ ❣❛r❛♥t✐✉ ✉♠ ♠❡❧❤♦r ❡♥t❡♥❞✐♠❡♥t♦ s♦❜r❡ ♦ ♣r♦❝❡ss♦ ❞❡ ❡st✐♠❛t✐✈❛ ❞♦s r❡s✉❧t❛❞♦s ❣❡r❛❞♦s ♣❡❧❛ s✐♠✉❧❛çã♦ ♥✉♠ér✐❝❛✱ ❛❧é♠ ❞❡ ❝♦♥✜r♠❛r q✉❡ ♦ ❝ó❞✐❣♦ ❡stá ❝♦rr❡t♦✳ ❈♦♠ ♦ ❛✉①í❧✐♦ ❞❛ té❝♥✐❝❛ ❞♦ ❤✐st♦❣r❛♠❛ s✐♠♣❧❡s ❡ ❞❛ t❡♦r✐❛ ❞❡ ❡s❝❛❧❛ ❞❡ t❛♠❛♥❤♦ ✜♥✐t♦ ✭❋❙❙✮ ❢♦✐ ♣♦ssí✈❡❧ ✉♠ tr❛t❛♠❡♥t♦ ❡st❛tíst✐❝♦ ♠❛✐s ❞❡t❛❧❤❛❞♦ ♣❡rt♦ ❞❛ tr❛♥s✐çã♦✳ ❖❜t❡♠♦s ❛s ❣r❛♥❞❡③❛s ❞❡s❡❥❛❞❛s✿ ♠❛❣♥❡t✐③❛çã♦✱ s✉s❝❡♣t✐❜✐❧✐❞❛❞❡ ♠❛❣♥ét✐❝❛✱ ❡♥❡r❣✐❛✱ ❝❛❧♦r ❡s♣❡❝í✜❝♦ ❡ ❝✉♠✉❧❛♥t❡ ❞❡ ❇✐♥❞❡r ❛ ♣❛rt✐r ❞❛s ♠é❞✐❛s✱ ♥♦ ❞❡❝♦rr❡r ❞❛ s✐♠✉✲ ❧❛çã♦✳ ❖❜t✐✈❡♠♦s ❛s q✉❛♥t✐❞❛❞❡s t❡r♠♦❞✐♥â♠✐❝❛s ❞♦ s✐st❡♠❛ ♣❛r❛ ❞✐❢❡r❡♥t❡s t❛♠❛♥❤♦s ❞❡ r❡❞❡s ❡ t❡♠♣❡r❛t✉r❛s✳ ❱❡r✐✜❝❛♠♦s ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡✱ ❜❡♠ ❝♦♠♦ ♦ ❝♦♠✲ ♣♦rt❛♠❡♥t♦ ♣❡❝✉❧✐❛r ❞❡ss❛s ❣r❛♥❞❡③❛s ♣ró①✐♠❛s à r❡❣✐ã♦ ❝rít✐❝❛✳ ❖s ✈❛❧♦r❡s ❞♦s ❡①♣♦❡♥t❡s ❝rít✐❝♦s ❡ ❞❛ t❡♠♣❡r❛t✉r❛ ❞❡ tr❛♥s✐çã♦ ❡st✐♠❛❞♦s ♥❡ss❡ tr❛❜❛❧❤♦ ❝♦✐♥❝✐❞❡♠ ❝♦♠ ♦s ✈❛❧♦r❡s ❡♥❝♦♥tr❛❞♦s ♥❛ ❧✐t❡r❛t✉r❛✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ▼♦❞❡❧♦ ❞❡ ■s✐♥❣✱ ▼♦♥t❡ ❈❛r❧♦✱ ❊①♣♦❡♥t❡s ❝rít✐❝♦s ❡ ❚❡♦r✐❛ ❞❡ ❊s❝❛❧❛ ❞❡ ❚❛♠❛♥❤♦
❋✐♥✐t♦
❆❜str❛❝t
■♥ t❤✐s ✇♦r❦ ✇❡ st✉❞② t❤❡ ♠❛❣♥❡t✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❢❡rr♦♠❛❣♥❡t✐❝ ■s✐♥❣ ♠♦❞❡❧ ♦♥ ❛ sq✉❛r❡ ❧❛tt✐❝❡✱ ✐♥ ③❡r♦ ❡①t❡r♥❛❧ ♠❛❣♥❡t✐❝ ✜❡❧❞✱ t❤r♦✉❣❤ ▼♦♥t❡ ❈❛r❧♦ s✐♠✉❧❛t✐♦♥s✳ ❚❤❡ ✐♥✐t✐❛❧ ♣❛rt ❝♦♠♣r✐s❡s t❤❡ t❤❡♦r❡t✐❝❛❧ ❜❛❝❦❣r♦✉♥❞ ❛♥❞ t❤❡ s✐♠✉❧❛t✐♦♥s ❛s t❤❡ ▼♦♥t❡ ❈❛r❧♦ ❝♦❞❡✱ t❤❡ s✐♥❣❧❡ ❤✐st♦❣r❛♠ t❡❝❤♥✐q✉❡ ❛♥❞ t❤❡ ❡st✐♠❛t❡ ♦❢ t❤❡ ❡rr♦r ❜❛rs✳ ❋♦r t❤❡ ♣❛rt✐❝✉❧❛r ❢♦✉r✲s♣✐♥ ❝❧✉st❡r t❤❡ ❡①❛❝t r❡s✉❧ts ❤❛✈❡ ❜❡❡♥ ♦❜t❛✐♥❡❞ ❛♥❞ ❝♦♠♣❛r❡❞ t♦ t❤❡ s✐♠✉❧❛t✐♦♥❛❧ ♦♥❡s✳ ❙✉❝❤ ❝♦♠♣❛r✐s♦♥ ❣r❛♥t❡❞ ❛ ❜❡tt❡r ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ ♣r♦❝❡ss ♦❢ ❣❡♥❡r❛t✐♥❣ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❞❛t❛ ❜❡s✐❞❡s ❛ ✇❛② ♦❢ ❝♦♥✜r♠✐♥❣ t❤❡ ❝♦rr❡❝t♥❡ss ♦❢ t❤❡ ❝♦❞❡✳ ❚♦❣❡t❤❡r ✇✐t❤ t❤❡ s✐♥❣❧❡ ❤✐st♦❣r❛♠ t❡❝❤♥✐q✉❡ ❛♥❞ t❤❡ ✜♥✐t❡ s✐③❡ s❝❛❧✐♥❣ ❤②♣♦t❤❡s✐s✳ ❲❡ ❤❛✈❡ ♦❜t❛✐♥❡❞ t❤❡ ♠❡❛♥ ✈❛❧✉❡s ♦❢ t❤❡ ♠❛❣♥❡t✐③❛t✐♦♥✱ ❡♥❡r❣②✱ s♣❡❝✐✜❝ ❤❡❛t✱ s✉s❝❡♣t✐❜✐❧✐t② ❛♥❞ t❤❡ ❇✐♥❞❡r ❝✉♠✉❧❛♥t✳ ■t ✇❛s ♣♦ss✐❜❧❡ t♦ ♦❜t❛✐♥ t❤❡ t❤❡r♠♦❞②♥❛♠✐❝ q✉❛♥t✐t✐❡s ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❡♠♣❡r❛t✉r❡ ❛♥❞ t❤❡ s②st❡♠ s✐③❡✳ ■t ✇❛s ❡✈✐❞❡♥t t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ♣❤❛s❡ tr❛♥s✐t✐♦♥ ✇✐t❤ ✐ts ♣❡❝✉❧✐❛r ❜❡❤❛✈✐♦r✳ ❆❝❝♦r❞✐♥❣❧②✱ ✇❡ ❤❛✈❡ ♦❜t❛✐♥❡❞ t❤❡ ❝r✐t✐❝❛❧ t❡♠♣❡r❛t✉r❡ ❛s ✇❡❧❧ ❛s t❤❡ ❝r✐t✐❝❛❧ ❡①♣♦♥❡♥ts ♦❢ t❤❡ ♠♦❞❡❧✳ ❚❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❝r✐t✐❝❛❧ ✐♥❞✐❝❡s ❛♥❞ t❤❡ ❝r✐t✐❝❛❧ t❡♠♣❡r❛t✉r❡ ❛❣r❡❡ ✇❡❧❧ ✇✐t❤ t❤❡ ❡①❛❝t ♦♥❡s ❛♥❞ t❤♦s❡ ❝♦♠✐♥❣ ❢r♦♠ ♦t❤❡r ❝♦♠♣✉t❛t✐♦♥❛❧ ♠❡t❤♦❞s✳
❑❡②✇♦r❞s✿ ■s✐♥❣ ▼♦❞❡❧✱ ▼♦♥t❡ ❈❛r❧♦✱ ❈r✐t✐❝❛❧ ❊①♣♦♥❡♥ts ❛♥❞ ❋✐♥✐t❡ ❙✐③❡ ❙❝❛❧✐♥❣
❙✉♠ár✐♦
❘❡s✉♠♦ ■
❆❜str❛❝t ■■
▲✐st❛ ❞❡ ❋✐❣✉r❛s ❱■
▲✐st❛ ❞❡ ❚❛❜❡❧❛s ❱■■
✶ ■♥tr♦❞✉çã♦ ✶
✷ ❙✐♠✉❧❛çã♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ✹
✷✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✷✳✷ ❖ ▼ét♦❞♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✷✳✸ Pr✐♥❝í♣✐♦ ❞♦ ❜❛❧❛♥ç♦ ❞❡t❛❧❤❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✹ ❆❧❣♦r✐t♠♦ ❞❡ ▼❡tr♦♣♦❧✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✺ ◆ú♠❡r♦s Ps❡✉❞♦✲❛❧❡❛tór✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✻ ❊st✐♠❛t✐✈❛s ❞❡ ❡rr♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✼ ❚é❝♥✐❝❛s ❞❡ ❤✐st♦❣r❛♠❛s s✐♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷
✸ ❙✐♠✉❧❛çã♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ♥♦ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣ ✶✹
✸✳✶ ▼♦❞❡❧♦ ❞❡ ■s✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✸✳✷ ❙✐♠✉❧❛çã♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✸✳✸ ❈❛♣❛❝✐❞❛❞❡ ❚ér♠✐❝❛ ❡ ❙✉s❝❡♣t✐❜✐❧✐❞❛❞❡ ▼❛❣♥ét✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✸✳✹ ❊q✉✐❧í❜r✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✸✳✺ ❈♦♠♣❛r❛♥❞♦ ♦ r❡s✉❧t❛❞♦ ❞❛ s✐♠✉❧❛çã♦ ♥✉♠ér✐❝❛ ❝♦♠ ❛ s♦❧✉çã♦ ❡①❛t❛ ♣❛r❛ ♦
♠♦❞❡❧♦ ❞❡ ■s✐♥❣ ❝♦♠L= 2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
❙❯▼➪❘■❖ ■❱
✹ ❚r❛♥s✐çõ❡s ❞❡ ❋❛s❡s ❡ ❋❡♥ô♠❡♥♦s ❈rít✐❝♦s ✷✹
✹✳✶ ❋❡♥ô♠❡♥♦s ❈rít✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✹✳✷ ❚❡♦r✐❛ ❞❡ ❊s❝❛❧❛ ❞❡ ❚❛♠❛♥❤♦ ❋✐♥✐t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽
✺ ❘❡s✉❧t❛❞♦s ❡ ❉✐s❝✉sõ❡s ✸✶
✺✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✺✳✷ ❖ ❝✉♠✉❧❛♥t❡ ❞❡ ❇✐♥❞❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✺✳✸ ❊st✐♠❛t✐✈❛ ❞♦ ❡①♣♦❡♥t❡ ν ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸
✺✳✸✳✶ ◆♦✈❛ ❡st✐♠❛t✐✈❛ ❞❡Tc ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✺✳✹ ❊①♣♦❡♥t❡s β ❡ γ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✻ ❈♦♥❝❧✉sã♦ ❡ P❡rs♣❡❝t✐✈❛s ✸✾
✻✳✶ ❈♦♥❝❧✉sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✻✳✷ P❡rs♣❡❝t✐✈❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
❆ ❯♠ ❛♣ê♥❞✐❝❡ ✹✶
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✸✳✶ ▼❛❣♥❡t✐③❛çã♦ ❡♠ ❢✉♥çã♦ ❞♦s ♣❛ss♦s ❞❡ ▼♦♥t❡ ❈❛r❧♦✳ ❚ r❡♣r❡s❡♥t❛ ❛s t❡♠♣❡r✲ ❛t✉r❛s ♥❛s q✉❛✐s ❛s s✐♠✉❧❛çõ❡s ❢♦r❛♠ ❢❡✐t❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✸✳✷ ❊♥❡r❣✐❛ ❡♠ ❢✉♥çã♦ ❞♦ ♥ú♠❡r♦s ❞❡ ♣❛ss♦s ❞❡ ▼♦♥t❡ ❈❛r❧♦✳ ❚ r❡♣r❡s❡♥t❛ ❛s
t❡♠♣❡r❛t✉r❛s ♥❛s q✉❛✐s ❛s s✐♠✉❧❛çõ❡s ❢♦r❛♠ ❢❡✐t❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✸✳✸ ❘❡s✉❧t❛❞♦s ♣❛r❛ ❛ ❡♥❡r❣✐❛✱ ❝❛❧♦r ❡s♣❡❝✐✜❝♦✱ ♠♦❞✉❧♦ ❞❛ ♠❛❣♥❡t✐③❛çã♦ ❡ s✉s✲
❝❡♣t✐❜✐❧✐❞❛❞❡ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣ ♥❛ r❡❞❡ ❝♦♠ L = 2✳ ❆s ❧✐♥❤❛s sã♦ ♦s
r❡s✉❧t❛❞♦s ❡①❛t♦s ❡ ♦s ♣♦♥t♦s ♦s r❡s✉❧t❛❞♦s ❞❡ ▼♦♥t❡ ❈❛r❧♦ ✉s❛♥❞♦ ♦ ♠ét♦❞♦ ❞♦ ❤✐st♦❣r❛♠❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✺✳✶ ❍✐st♦❣r❛♠❛ ❞♦ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣ ❝❧áss✐❝♦ ❝♦♠T = 3.0❡ t❛♠❛♥❤♦ ❞❡ r❡❞❡L= 16✳ ✸✷ ✺✳✷ ❈✉♠✉❧❛♥t❡ ❞❡ ❇✐♥❞❡r ❞❡ q✉❛rt❛ ♦r❞❡♠ ❡♠ ❢✉♥çã♦ ❞❛ t❡♠♣❡r❛t✉r❛ ♣❛r❛ ❞✐❢❡r✲
❡♥t❡s ✈❛❧♦r❡s ❞❡ L✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸
✺✳✸ ln[U4′]❝♦♠♦ ❢✉♥çã♦ ❞❡ln[L]✳ ❆ r❡t❛ ❛♣r❡s❡♥t❛ ♦ ♠❡❧❤♦r ❛❥✉st❡ ♣❛r❛1/ν✳ ✳ ✳ ✳ ✸✸
✺✳✹ ln[Q]❝♦♠♦ ❢✉♥çã♦ ❞❡ln[L]✳ ❆ r❡t❛ ❛♣r❡s❡♥t❛ ♦ ♠❡❧❤♦r ❛❥✉st❡ ♣❛r❛1/ν✳ ✳ ✳ ✳ ✸✹
✺✳✺ ❈✉r✈❛s ❞♦ ❝❛❧♦r ❡s♣❡❝í✜❝♦ ❡♠ ❢✉♥çã♦ ❞❛ t❡♠♣❡r❛t✉r❛ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣ ✷❉ ♦❜t✐❞♦ ❝♦♠ ❛ té❝♥✐❝❛ ❞❡ ❤✐st♦❣r❛♠❛ ♣❛r❛ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s ❞❡L✳ ✳ ✳ ✳ ✳ ✳ ✸✺
✺✳✻ ❈✉r✈❛s ❞❛ s✉s❝❡♣t✐❜✐❧✐❞❛❞❡ ❡♠ ❢✉♥çã♦ ❞❛ t❡♠♣❡r❛t✉r❛ ♣❛r❛ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s ❞❡ L✳ ❖ ❣rá✜❝♦ ♠❡♥♦r ♠♦str❛ ♦s ♠á①✐♠♦s ♣❛r❛ ❛s r❡❞❡s ♠❡♥♦r❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺
✺✳✼ ❚❡♠♣❡r❛t✉r❛ ♣s❡✉❞♦✲❝rít✐❝❛✱ ♦❜t✐❞❛ ❞♦ ♠á①✐♠♦ ❞♦ ❝❛❧♦r ❡s♣❡❝í✜❝♦✱ ❝♦♠♦ ❢✉♥çã♦ ❞❡L−ν1 ♣❛r❛ 1
ν = 1.01✳ ❆s r❡t❛s ❛♣r❡s❡♥t❛♠ ♦ ♠❡❧❤♦r ❛❥✉st❡ ♣❛r❛T c
c(L)✱ ❝♦♠ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦Tc = 2,2724(30)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✺✳✽ ❚❡♠♣❡r❛t✉r❛ ♣s❡✉❞♦✲❝rít✐❝❛✱ ♦❜t✐❞❛ ❞♦ ♠á①✐♠♦ ❞❛ s✉s❝❡♣t✐❜✐❧✐❞❛❞❡✱ ❝♦♠♦
❢✉♥çã♦ ❞❡ L−ν1 ♣❛r❛ 1
ν = 1.01✳ ❆s r❡t❛s ❛♣r❡s❡♥t❛♠ ♦ ♠❡❧❤♦r ❛❥✉st❡ ♣❛r❛
Tcχ(L)✱ ❈♦♠ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦ Tcχ = 2,2706(20)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
▲■❙❚❆ ❉❊ ❋■●❯❘❆❙ ❱■
✺✳✾ ❈✉r✈❛s ❞❛ ♠❛❣♥❡t✐③❛çã♦ ❡♠ ❢✉♥çã♦ ❞❛ t❡♠♣❡r❛t✉r❛ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣ ❝❧áss✐❝♦ ✷❉ ♦❜t✐❞♦ ❝♦♠ ❛ té❝♥✐❝❛ ❞❡ ❤✐st♦❣r❛♠❛ ♣❛r❛ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s ❞❡L✳ ✳ ✸✼
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
✸✳✶ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ✐♥✈❡rsã♦ ❞❡ ✉♠ s♣✐♥ ❞❡ ↑ ♣❛r❛ ↓ ♥✉♠❛ r❡❞❡ q✉❛❞r❛❞❛✳
❆ss✉♠✐♠♦s q✉❡ ✉♠ sít✐♦ ❞❛ r❡❞❡ r❡♣r❡s❡♥t❛❞♦ ♥❛ t❛❜❡❧❛ ❡stá ❛ss♦❝✐❛❞♦ ❝♦♠ s❡✉s ♣r✐♠❡✐r♦s ✈✐③✐♥❤♦s✱ ♦♥❞❡ ❛ s❡t❛↑r❡♣r❡s❡♥t❛Si= +1✱ ❡ ❛ s❡t❛↓r❡♣r❡s❡♥t❛ ♦ s♣✐♥Si=−1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✸✳✷ ❆ ❡♥❡r❣✐❛ ❞❡ ✐♥t❡r❛çã♦ ❡♥tr❡ ♣r✐♠❡✐r♦s ✈✐③✐♥❤♦s ♥❛ ❛✉sê♥❝✐❛ ❞❡ ❝❛♠♣♦ ♠❛❣✲
♥ét✐❝♦ ❡①t❡r♥♦ ♥♦ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣ ✭❊q✳ ✸✳✶✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✸✳✸ ❆ ❡♥❡r❣✐❛ ❡ ♠❛❣♥❡t✐③❛çã♦ ❞❡ 24 ❡st❛❞♦s ❞♦ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣ ❛ ❝❛♠♣♦ ♥✉❧♦
s♦❜r❡ ✉♠❛ r❡❞❡ q✉❛❞r❛❞❛ 2×2✳ ❖s sít✐♦s ❞♦ q✉❛❞r❛❞♦✱ ❡♠ s❡♥t✐❞♦ ❤♦rár✐♦✱ ❡stã♦ ❞✐s♣♦st♦s ❧✐♥❡❛r♠❡♥t❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✸✳✹ ❆ ❡♥❡r❣✐❛ ❡ ♠❛❣♥❡t✐③❛çã♦ ❞❡ 24 ❡st❛❞♦s ❞♦ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣ ❛ ❝❛♠♣♦ ♥✉❧♦
s♦❜r❡ ✉♠❛ r❡❞❡ q✉❛❞r❛❞❛ 2×2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✻✳✶ ❚❡♠♣❡r❛t✉r❛ ❝rít✐❝❛ Tc ❡ ❡①♣♦❡♥t❡s ❝rít✐❝♦s 1ν, βν ❡ γν ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣ ❝❧áss✐❝♦✱ ❜✐✲❞✐♠❡♥s✐♦♥❛❧ ♦❜t✐❞♦s ♣❡❧❛s s✐♠✉❧❛çõ❡s ❞❡st❡ tr❛❜❛❧❤♦ ♣❛r❛ ❝♦♠✲ ♣❛r❛çã♦ sã♦ ♠♦str❛❞♦s ♦s r❡s✉❧t❛❞♦s ❡①❛t♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
❈❛♣ít✉❧♦ ✶
■♥tr♦❞✉çã♦
❍á ✉♠❛ ❣r❛♥❞❡ ✈❛r✐❡❞❛❞❡ ❞❡ s✐st❡♠❛s ❢ís✐❝♦s q✉❡ ❡①✐❜❡♠ ❝♦♠♣♦rt❛♠❡♥t♦ ❝rít✐❝♦✱ t❛✐s ❝♦♠♦✿ ♠❛t❡r✐❛✐s ♠❛❣♥ét✐❝♦s✱ ❧✐❣❛s ♠❡tá❧✐❝❛s✱ ♠❛t❡r✐❛✐s ❢❡rr♦❡❧étr✐❝♦s✱ s✉♣❡r✢✉✐❞♦s✱ s✉♣❡r✲ ❝♦♥❞✉t♦r❡s✱ ❝r✐st❛✐s ❧íq✉✐❞♦s✱ ❡♥tr❡ ♦✉tr♦s❬✶❪✳ ❊ss❡s ❢❡♥ô♠❡♥♦s✱ q✉❡ ♦❝♦rr❡♠ ♥❛ ♣r♦①✐♠✐❞❛❞❡ ❞❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡✱ sã♦ ❝❛r❛❝t❡r✐③❛❞♦s ♣♦r ❝♦♠♣♦rt❛♠❡♥t♦s ❞❡s❝♦♥tí♥✉♦s ♦✉ s✐♥❣✉❧❛r✐❞❛❞❡s ❡♠ ❛❧❣✉♠❛s ❢✉♥çõ❡s t❡r♠♦❞✐♥â♠✐❝❛s ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♣♦r ♠✉❞❛♥ç❛s r❡♣❡♥t✐♥❛s ♥❡ss❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡✈✐❞♦ à ✈❛r✐❛çã♦ ❝♦♥tí♥✉❛ ❞❡ ✉♠ ♣❛râ♠❡tr♦ ❡①t❡r♥♦✱ ❝♦♠♦ ❛ t❡♠♣❡r❛t✉r❛✱ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦✱ ♣r❡ssã♦ ❡ ❡t❝✳ ❯♠ ❡①❡♠♣❧♦ ❜❡♠ ❝♦♥❤❡❝✐❞♦ é ❛ tr❛♥s✐çã♦ ❧íq✉✐❞♦✲✈❛♣♦r✱ ♥♦ q✉❛❧ ❛ á❣✉❛✱ q✉❛♥❞♦ ❛q✉❡❝✐❞❛ ❛ ✉♠❛ t❡♠♣❡r❛t✉r❛ ❡s♣❡❝í✜❝❛ ❡ ❝♦♠ ♣r❡ssã♦ ❝♦♥st❛♥t❡✱ ❡♥tr❛ ❡♠ ❡❜✉❧✐çã♦✱ tr❛♥s❢♦r♠❛♥❞♦✲s❡ ❡♠ ✈❛♣♦r✳ ❖✉tr♦ ❡①❡♠♣❧♦✱ r❡❧❛❝✐♦♥❛❞♦ ❛ ♠❛t❡r✐❛✐s ♠❛❣♥ét✐❝♦s✱ é ❛ tr❛♥s✐çã♦ ❢❡rr♦♠❛❣♥ét✐❝❛ ✲ ♣❛r❛♠❛❣♥ét✐❝❛✳ ❯♠ ✐♠ã✱ q✉❛♥❞♦ ❛q✉❡❝✐❞♦ ❛❝✐♠❛ ❞❡ ✉♠❛ ❝❡rt❛ t❡♠♣❡r❛t✉r❛✱ s♦❢r❡ ✉♠❛ ❞❡s♠❛❣♥❡t✐③❛çã♦✳ ▲✐❣❛s ❜✐♥ár✐❛s s♦❢r❡♠ tr❛♥s✐çõ❡s ❞❡ ♦r❞❡♥❛♠❡♥t♦✱ ❝♦♠♦ ♥❛ ❧✐❣❛ ♠❡tá❧✐❝❛ ③✐♥❝♦✲❝♦❜r❡✳
❚r❛♥s✐çõ❡s ❞❡ ❢❛s❡ sã♦ ♦❜❥❡t♦ ❞❡ ❡st✉❞♦ ❤á ♠❛✐s ❞❡ ✉♠ sé❝✉❧♦ ❬✶❪✳ ❊♠ ✶✽✼✸✱ ✉♠ tr❛❜❛❧❤♦ ❞❡ ❱❛♥ ❞❡r ❲❛❛❧s ❝♦♥té♠ ❛ ♣r✐♠❡✐r❛ t❡♦r✐❛ q✉❡ ❝❛r❛❝t❡r✐③❛ ❛ tr❛♥s✐çã♦ ❡♥tr❡ ❛s ❢❛s❡s ❧íq✉✐❞❛ ❡ ❣❛s♦s❛ ❡♠ ✢✉✐❞♦s✱ ❧❡✈❛♥❞♦ ❡♠ ❝♦♥t❛ ❛ r❡♣✉❧sã♦ ❞❡ ❝❛r♦ç♦ ❞✉r♦ ❡♥tr❡ ❛s ♠♦❧é❝✉❧❛s✳ ❆s tr❛♥s✐çõ❡s ❢❡rr♦♠❛❣♥ét✐❝❛s✱ q✉❡ ✐♥✐❝✐❛❧♠❡♥t❡ ❢♦r❛♠ ❡①♣❧✐❝❛❞❛s ♣♦r ✉♠❛
❈❛♣ít✉❧♦ ✶✳ ■♥tr♦❞✉çã♦ ✷
t❡♦r✐❛ ❢❡♥♦♠❡♥♦❧ó❣✐❝❛ ♣r♦♣♦st❛ ♣♦r ❈✉r✐❡✲❲❡✐ss ❬✶❪✱ ❛♣r❡s❡♥t❛r❛♠ ♠✉✐t♦s ♣♦♥t♦s ❡♠ ❝♦♠✉♠ ❝♦♠ ❛ t❡♦r✐❛ ❞❡ ❱❛♥ ❞❡r ❲❛❛❧s❬✶✱ ✷❪✳ ❊ss❛s t❡♦r✐❛s ♣❛r❛ ❛ ❞❡s❝r✐çã♦ q✉❛❧✐t❛t✐✈❛ ❞❡ ✈ár✐♦s t✐♣♦s ❞❡ tr❛♥s✐çã♦ ❞❡ ❢❛s❡s ✜❝❛r❛♠ ❝♦♥❤❡❝✐❞❛s ❝♦♠♦ t❡♦r✐❛s ❝❧áss✐❝❛s✳ P♦st❡r✐♦r♠❡♥t❡✱ ♥❛ t❡♦r✐❛ ♣r♦♣♦st❛ ♣♦r ▲❛♥❞❛✉✱ ✉♥✐✜❝♦✉✲s❡ ❛s t❡♦r✐❛s ❝❧áss✐❝❛s ♣❛r❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡✱ ♦♥❞❡ ✐♥tr♦❞✉③✐✉✲s❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ♣❛râ♠❡tr♦ ❞❡ ♦r❞❡♠ ❡ ❢♦r♠✉❧♦✉✲s❡ ✉♠❛ ❡①♣❛♥sã♦ ❞❛ ❡♥❡r❣✐❛ ❧✐✈r❡ ❝♦♠♦ ✉♠❛ sér✐❡ ❞❡ ♣♦tê♥❝✐❛s ❡♠ t❡r♠♦s ❞♦s ✐♥✈❛r✐❛♥t❡s ❞♦s ♣❛râ♠❡tr♦s ❞❡ ♦r❞❡♠✳ ❊①✐❣✐✉✲s❡ q✉❡ ❛ ❡♥❡r❣✐❛ ❧✐✈r❡ ❢♦ss❡ ❛♥❛❧ít✐❝❛ ♥❛s ✈✐③✐♥❤❛♥ç❛s ❞❛ ❝r✐t✐❝❛❧✐❞❛❞❡✱ r❡s✉❧t❛♥❞♦ ❡♠ ✈❛❧♦r❡s ❝❧áss✐❝♦s ♣❛r❛ ♦s ❡①♣♦❡♥t❡s ❝rít✐❝♦s✳ ❊♥tr❡t❛♥t♦✱ ❛ ♥ã♦ ❛♥❛❧✐t✐❝✐❞❛❞❡ ❞❛ ❡♥❡r❣✐❛ ❧✐✈r❡ ♥♦ ❧✐♠✐t❡ t❡r♠♦❞✐♥â♠✐❝♦ é ✉♠❛ ❝❛r❛❝t❡r✐st✐❝❛ ❞❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡✳
❋♦r❛ ❞❛ ❝r✐t✐❝❛❧✐❞❛❞❡✱ ❛s ❝♦rr❡❧❛çõ❡s ❡s♣❛❝✐❛✐s ❞❡ ❣r❛♥❞❡③❛s t❡r♠♦❞✐♥â♠✐❝❛s ❞♦ s✐st❡♠❛ ❢ís✐❝♦ ❞❡❝❛❡♠ ❡①♣♦♥❡♥❝✐❛❧♠❡♥t❡ ❝♦♠ ❛ ❞✐stâ♥❝✐❛✳ ◆♦ ❡♥t❛♥t♦✱ ♥❛ ❝r✐t✐❝❛❧✐❞❛❞❡ ❛s ❝♦rr❡❧❛çõ❡s ❞❡❝❛❡♠ ♦❜❡❞❡❝❡♥❞♦ ✉♠❛ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛✳ ❖ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❝♦rr❡❧❛çã♦ ❝r❡s❝❡ ❡ ❞✐✈❡r❣❡ à ♠❡❞✐❞❛ q✉❡ s❡ ❛♣r♦①✐♠❛ ❞❛ t❡♠♣❡r❛t✉r❛ ❝rít✐❝❛✱ t♦r♥❛♥❞♦ ♦ s✐st❡♠❛ ❛❧t❛♠❡♥t❡ ❝♦rr❡❧❛❝✐♦♥❛❞♦✳ ❊ss❛ ❞✐✈❡r❣ê♥❝✐❛ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❝♦rr❡❧❛çã♦✱ ♥❛ ❝r✐t✐❝❛❧✐❞❛❞❡✱ ❛♣r❡s❡♥t❛✲ s❡ ❝♦♠♦ ✉♠❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❛s tr❛♥s✐çõ❡s ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ❬✸❪✳ ❖ ❝♦♠♣♦rt❛♠❡♥t♦ ❝rít✐❝♦ ♣♦❞❡ s❡r ❞❡s❝r✐t♦ ♣♦r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❡①♣♦❡♥t❡s ❝rít✐❝♦s q✉❡ ❞❡♣❡♥❞❡♠ ❞❡ ♣♦✉❝♦s ❢❛t♦r❡s✱ ❝♦♠♦ ❛ ❞✐♠❡♥s✐♦♥❛❧✐❞❛❞❡ ❞♦ ♣❛râ♠❡tr♦ ❞❡ ♦r❞❡♠✱ ❛ ❞✐♠❡♥s✐♦♥❛❧✐❞❛❞❡ ❞♦ s✐st❡♠❛✱ ❡ ♦ ❛❧❝❛♥❝❡ ❞❛s ✐♥t❡r❛çõ❡s✱ ❝❛r❛❝t❡r✐③❛♥❞♦ ✉♠ ❝♦♠♣♦rt❛♠❡♥t♦ ❝♦♠✉♠ ❛ s✐st❡♠❛s✱ ❡♠ ♣r✐♥❝í♣✐♦✱ ♠✉✐t♦ ❞✐❢❡r❡♥t❡s✳ ❉✐③❡♠♦s ❡♥tã♦ q✉❡ ❡ss❡s s✐st❡♠❛s ❡stã♦ ❡♠ ✉♠❛ ♠❡s♠❛ ❝❧❛ss❡ ❞❡ ✉♥✐✈❡rs❛❧✐❞❛❞❡✳ ❖ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ té❝♥✐❝❛s ❞❡ s✐♠✉❧❛çã♦ ❝♦♠♣✉t❛❝✐♦♥❛❧ ♠♦str♦✉✲s❡ ✐♠♣♦r✲ t❛♥t❡ ♥♦ ❡st✉❞♦ ❞❡ ♠♦❞❡❧♦s ❡♠ ♠❡❝â♥✐❝❛ ❡st❛tíst✐❝❛✱ ♣♦ss✐❜✐❧✐t❛♥❞♦ ✉♠❛ ❛❜♦r❞❛❣❡♠ ❛❧t❡r✲ ♥❛t✐✈❛ ♥❛ ❞❡s❝r✐çã♦ ❛❞❡q✉❛❞❛ ❞❡ ❢❡♥ô♠❡♥♦s ❢ís✐❝♦s ♥❛s ✈✐③✐♥❤❛♥ç❛s ❞♦s ♣♦♥t♦s ❝rít✐❝♦s✳ ❖ ♠ét♦❞♦ ❞❡ s✐♠✉❧❛çã♦ ♥✉♠ér✐❝❛✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ▼ét♦❞♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦✱ q✉❡ s❡rá ❞✐s❝✉t✐❞♦ ♥❡ss❡ tr❛❜❛❧❤♦✱ ♠♦str♦✉✲s❡ ✉♠❛ ❢❡rr❛♠❡♥t❛ ♣♦❞❡r♦s❛ ♣❛r❛ ♦❜t❡r r❡s✉❧t❛❞♦s ♦♥❞❡ ♦s ♠ét♦❞♦s ❞❡ ❛♣r♦①✐♠❛çã♦ ❛♥❛❧ít✐❝❛ ❢❛❧❤❛✈❛♠❬✹❪✳ ❆♣❧✐❝❛❞♦ ❛ ♠♦❞❡❧♦s ♠❛❣♥ét✐❝♦s✱ ♣❛r❛ tr❛t❛r tr❛♥s✐çõ❡s ❞❡ ❢❛s❡✱ ♣♦ss✐❜✐❧✐t♦✉ ❛ ❡st✐♠❛t✐✈❛ ❞❡ ❡①♣♦❡♥t❡s ❡ t❡♠♣❡r❛t✉r❛s ❝rít✐❝❛s✳
❈❛♣ít✉❧♦ ✶✳ ■♥tr♦❞✉çã♦ ✸
❈❛♣ít✉❧♦ ✷
❙✐♠✉❧❛çã♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦
✷✳✶ ■♥tr♦❞✉çã♦
❆ ♠❡❝â♥✐❝❛ ❡st❛tíst✐❝❛ ❞❡ ❡q✉✐❧í❜r✐♦ ♣♦ss✐❜✐❧✐t❛ ♦❜t❡r ♣r♦♣r✐❡❞❛❞❡s ♠❛❝r♦s❝ó♣✐❝❛s ❞♦ s✐st❡♠❛ ❝♦♥❤❡❝❡♥❞♦✲s❡ s❡✉s ❡st❛❞♦s ♠✐❝r♦s❝ó♣✐❝♦s ❬✶❪✳ ❆ss✐♠✱ ♥♦ ❡♥s❡♠❜❧❡ ❝❛♥ô♥✐❝♦✱ ♦ ✈❛❧♦r ♠é❞✐♦ ❞❡ ✉♠❛ q✉❛♥t✐❞❛❞❡A é ❝❛❧❝✉❧❛❞♦ ❝♦♠♦
< A >= 1
Z
X
{X}
A({X})e−βE({X}), ✭✷✳✶✮
♦♥❞❡ ❛ s♦♠❛ ♣❡r❝♦rr❡ t♦❞❛s ❛s ❝♦♥✜❣✉r❛çõ❡s ❞♦ s✐st❡♠❛{X}✱A({X})é ♦ ✈❛❧♦r ❞❛ q✉❛♥t✐❞❛❞❡ ❢ís✐❝❛A ♥❛ ❝♦♥✜❣✉r❛çã♦ {X}✱ ❡Z é ❛ ❢✉♥çã♦ ❞❡ ♣❛rt✐çã♦ ♥♦ ❡♥s❡♠❜❧❡ ❝❛♥ô♥✐❝♦ ❞❛❞❛ ♣♦r
Z =X
{X}
e−βE({X}). ✭✷✳✷✮
❖ s✙♠❜♦❧♦ < ... > ♥❛ ❡q✉❛çã♦ ✭✷✳✶✮ ❞❡♥♦t❛ ♠é❞✐❛ tér♠✐❝❛✳ ❖ s✐st❡♠❛ é ❝♦♥s✐❞❡r❛❞♦ ❡st❛r
❡♠ ❝♦♥t❛t♦ tér♠✐❝♦ ❝♦♠ ✉♠ r❡s❡r✈❛tór✐♦ à t❡♠♣❡r❛t✉r❛T✳ ❖ ❝♦♥❥✉♥t♦✱ s✐st❡♠❛ ♠❛✐s r❡s❡r✲
✈❛tór✐♦✱ ❡♥❝♦♥tr❛✲s❡ ✐s♦❧❛❞♦ ❡ ❡♠ ❡q✉✐❧í❜r✐♦✱ ❞❡ ♠♦❞♦ q✉❡ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡
❈❛♣ít✉❧♦ ✷✳ ❙✐♠✉❧❛çã♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ✺
♣❛r❛ ✉♠ ♠✐❝r♦❡st❛❞♦{X} é ❞❛❞❛ ♣❡❧❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❇♦❧t③♠❛♥♥
Peq(E({X})) = 1
Ze
−βE({X}). ✭✷✳✸✮
▲♦❣♦✱ ❛ ❡q✉❛çã♦ ✭✷✳✸✮ r❡♣r❡s❡♥t❛ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ q✉❡ ♦❝♦rr❛ ❛ ❝♦♥✜❣✉r❛çã♦ {X} ❡♠
❡q✉✐❧í❜r✐♦ tér♠✐❝♦ à t❡♠♣❡r❛t✉r❛ T✳
❆ s♦♠❛ ♥❛ ❡q✉❛çã♦ ✭✷✳✶✮ só ♣♦❞❡ s❡r ❢❡✐t❛ ❞❡ ♠❛♥❡✐r❛ ❡①❛t❛ ❡♠ ❛❧❣✉♥s ❝❛s♦s ❡s♣❡❝✐❛✐s✱ ♣❛r❛ ♠♦❞❡❧♦s ♣❛rt✐❝✉❧❛r♠❡♥t❡ s✐♠♣❧❡s✱ ❞❡✈✐❞♦ ❛♦ ♥ú♠❡r♦ ♠✉✐t♦ ❣r❛♥❞❡ ❞❡ ❝♦♥✜✲ ❣✉r❛çõ❡s ❡ ❛ ❝♦♠♣❧❡①✐❞❛❞❡ ❞❛s ✐♥t❡r❛çõ❡s ❡♥✈♦❧✈✐❞❛s✳ P♦r ❡①❡♠♣❧♦✱ ♥♦ ❝❛s♦ ❞♦ ♠♦❞❡❧♦ ■s✐♥❣ ❝♦♠ N sít✐♦s✱ q✉❡ s❡rá ❡①♣♦st♦ ♥♦ ❝❛♣ít✉❧♦ ✸✱ ❛ s♦♠❛ é s♦❜r❡ 2N ❝♦♥✜❣✉r❛çõ❡s✳ ❙❡ N ❢♦r ♠✉✐t♦ ❣r❛♥❞❡ ♦ ❝á❧❝✉❧♦ ❞❛s ♠é❞✐❛s é ✉♠❛ t❛r❡❢❛ ♣r❛t✐❝❛♠❡♥t❡ ✐♠♣♦ssí✈❡❧ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❝♦♠♣✉t❛❝✐♦♥❛❧✳
❊♠ ❞❡❝♦rrê♥❝✐❛ ❞✐ss♦✱ ✉♠ ♥ú♠❡r♦ ❞❡ té❝♥✐❝❛s ❡ ♠ét♦❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ❢♦r❛♠ ❞❡s❡♥✈♦❧✈✐❞♦s ❛♦ ❧♦♥❣♦ ❞❛ ❤✐stór✐❛✳ ❊ss❡ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❛♣r❡s❡♥t♦✉✲s❡ ❞❡ ♠❛♥❡✐r❛ ❡✜❝✐❡♥t❡ ♣❛r❛ s✐t✉❛çõ❡s ♦♥❞❡ é ❞✐❢í❝✐❧✱ ♦✉ ❛✐♥❞❛ ✐♠♣r❛t✐❝á✈❡❧✱ ♦❜t❡r✲s❡ r❡s✉❧t❛❞♦s ❞❡ ❢♦r♠❛ ❛♥❛❧ít✐❝❛✳ ❖ ✉s♦ ❞❛ s✐♠✉❧❛çã♦ ❝♦♠♣✉t❛❝✐♦♥❛❧ ♣♦ss✐❜✐❧✐t♦✉ ✉♠❛ ❡①♣❛♥sã♦ ❞❛ ♠❡❝â♥✐❝❛ ❡st❛tíst✐❝❛✱ ❡s✲ t❡♥❞❡♥❞♦ ♦ ❡st✉❞♦ ♣❛r❛ ♦✉tr❛s ár❡❛s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✱ t❛✐s ❝♦♠♦ s✐st❡♠❛s ❜✐♦❧ó❣✐❝♦s ✭♣♦r ❡①❡♠♣❧♦✱ ❤❡♠♦❣❧♦❜✐♥❛✱ ❡♥③✐♠❛s ❛❧♦stér✐❝❛s ❡ ❉◆❆✮✱ r❡❞❡s s♦❝✐❛✐s ❡ s✐st❡♠❛s ♠❛❣♥ét✐❝♦s ❬✺✱ ✹❪✳ ❆ ✉t✐❧✐③❛çã♦ ❞❡ t❛✐s s✐♠✉❧❛çõ❡s t♦r♥♦✉✲s❡ ✉♠ ✐♥str✉♠❡♥t♦ ✐♠♣♦rt❛♥t❡ ❡♥tr❡ ❛ t❡♦r✐❛ ❡ ♦ ❡①✲ ♣❡r✐♠❡♥t♦✱ ❝♦rr♦❜♦r❛♥❞♦ r❡s✉❧t❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s ❡ t❛♠❜é♠ ❛❥✉❞❛♥❞♦ ♥❛ ✐♥t❡r♣r❡t❛çã♦ ❞❡ ❞❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s ❬✻❪ ❡ ❛♥❛❧ít✐❝♦s✳
✷✳✷ ❖ ▼ét♦❞♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦
❈❛♣ít✉❧♦ ✷✳ ❙✐♠✉❧❛çã♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ✻
❖ ✉s♦ ❞♦ ♠ét♦❞♦ ❞❡ ▼❈ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ ✐♥t❡❣r❛✐s t❡♠ ✉♠❛ ❝♦♥✈❡r❣ê♥❝✐❛ q✉❡ é ♣r♦♣♦r❝✐♦♥❛❧ ❛ √ 1
NM CS✱ ❝♦♠NM CS r❡♣r❡s❡♥t❛♥❞♦ ♦ ♥ú♠❡r♦ ❞❡ ❝♦♥✜❣✉r❛çõ❡s ✉s❛❞❛s ♣❛r❛
❡st✐♠❛r ❛ ♠é❞✐❛✱ ❡♥q✉❛♥t♦ ♠ét♦❞♦s ❞❡ ❞✐❢❡r❡♥ç❛ ✜♥✐t❛ ❝♦♥✈❡r❣❡♠ ❝♦♠ 1
d √
N✱ ♦♥❞❡ d é ❛ ❞✐♠❡♥sã♦ ❡s♣❛❝✐❛❧ ♦♥❞❡ s❡ r❡❛❧✐③❛ ❛ ✐♥t❡❣r❛❧ ❡ N ❢♦r♥❡❝❡ ♦ ♥ú♠❡r♦ ❞❡ ♣♦♥t♦s ♥✉♠ ❡s♣❛ç♦ m✲❞✐♠❡♥s✐♦♥❛❧✳ P♦rt❛♥t♦✱ ♦ ❡rr♦ ♥♦ ♠ét♦❞♦ ❞❡ ▼❈ ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ❞✐♠❡♥sã♦✱ t♦r♥❛♥❞♦
❡st❡ ♠ét♦❞♦ ❜❛st❛♥t❡ ❡✜❝❛③ q✉❛♥❞♦ d > 4 ❬✾❪✳ ❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛✱ ✐ss♦ ♣r♦❞✉③ ✉♠❛ ❜♦❛ ❡st✐♠❛t✐✈❛ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ ♠é❞✐❛s ❡♠ s✐st❡♠❛s ❞❡ ♠✉✐t❛s ♣❛rtí❝✉❧❛s ❬✶✵❪✳
❊♠ ❣❡r❛❧✱ ♦ ♠ét♦❞♦ ❞❡ ▼❈ ❝♦♥s✐st❡ ❡♠ ❣❡r❛r ❝♦♥✜❣✉r❛çõ❡s ❞❡ ✉♠ s✐st❡♠❛ ❢ís✐❝♦✱ ❢✉♥❞❛♠❡♥t❛❞♦ s♦❜r❡ ✉♠❛ ❝❡rt❛ ❞✐str✐❜✉✐çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✱ ♣❛r❛ ❡♠ s❡❣✉✐❞❛ ❡st✐♠❛r ♦ ✈❛❧♦r ❡s♣❡r❛❞♦< A >M C✱ ❞❡ ❛❧❣✉♠❛ q✉❛♥t✐❞❛❞❡ A✳
❈♦♠♦ ❞✐s❝✉t✐❞♦ ♥❛ ú❧t✐♠❛ s❡çã♦✱ ♦ ❝á❧❝✉❧♦ ❞✐r❡t♦ ❞❛s s♦♠❛s ♥❛ ❡q✉❛çã♦ ✭✷✳✶✮ só é ♣♦ssí✈❡❧ ❡♠ s✐st❡♠❛s ♠✉✐t♦ ♣❡q✉❡♥♦s✳ ❈❛s♦ ❝♦♥trár✐♦✱ ✐ss♦ s❡ t♦r♥❛ ✐♠♣r❛t✐❝á✈❡❧✱ ✉♠❛ ✈❡③ q✉❡ ♦ ♥ú♠❡r♦ ❞❡ ❝♦♥✜❣✉r❛çõ❡s ❛✉♠❡♥t❛ ❡①♣♦♥❡♥❝✐❛❧♠❡♥t❡ ❝♦♠ ♦ t❛♠❛♥❤♦ N ❞♦ s✐st❡♠❛✳
❈♦♥t✉❞♦✱ ❛ s✐♠✉❧❛çã♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦✱ ❛❧✐❛❞❛ ❛♦ ❛❧❣♦r✐t♠♦ ❞❛ s❡çã♦ ✭✷✳✹✮ ❛ s❡❣✉✐r✱ ♣r♦♣õ❡ ✉♠❛ s♦❧✉çã♦ ♣❛r❛ ❡st❡ ♣r♦❜❧❡♠❛✳ ❖ ♣r♦❝❡❞✐♠❡♥t♦ ❝♦♥s✐st❡ ❡♠ ♦❜t❡r ❛♠♦str❛s ♥♦ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çõ❡s ❡ ❝❛❧❝✉❧❛r ❛s ♠é❞✐❛s ✉s❛♥❞♦ ❡st❛ ❛♠♦str❛❣❡♠✳ ❈♦♠♦ s❡rá ♠♦str❛❞♦ ❛❞✐❛♥t❡✱ ❡st❛ ❛♠♦str❛❣❡♠ ♣♦❞❡ s❡r ❢❡✐t❛ ❞❡ ❢♦r♠❛ ✐♥t❡❧✐❣❡♥t❡ s❡❣✉✐♥❞♦ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❇♦❧t③♠❛♥♥✱ ✐st♦ é✱ s❡❧❡❝✐♦♥❛♥❞♦ ✉♠❛ ♣♦rçã♦ ❞♦ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çõ❡s ❞❡ ♦♥❞❡ ✈ê♠ ❛s ❝♦♥tr✐❜✉✐çõ❡s ♠❛✐s ✐♠♣♦rt❛♥t❡s ♣❛r❛ ♦ ❡❢❡✐t♦ ❞♦ ❝á❧❝✉❧♦ ❞♦ ✈❛❧♦r ♠é❞✐♦✳
❙❡ ❛ ❛♠♦str❛ ❢♦r s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✱ ♣♦❞❡✲s❡ ❣❛r❛♥t✐r q✉❡ ❛ ❡st✐♠❛t✐✈❛ ❝♦♥✲ ✈❡r❣❡ ♣❛r❛ ♦ r❡s✉❧t❛❞♦ ❡①❛t♦✳ ❊ss❛ é ❛ ✐❞❡✐❛ ❜ás✐❝❛ ❞❛ ❛♠♦str❛❣❡♠ ♣♦r ✐♠♣♦rtâ♥❝✐❛ ✭✐♠♣♦r✲ t❛♥❝❡ s❛♠♣❧✐♥❣✮✳ ❆ ❡st✐♠❛t✐✈❛ ❞❛ ♠é❞✐❛ ❞❡ ✉♠❛ q✉❛♥t✐❞❛❞❡A é ❞❛❞❛ ♣♦r
< A >= 1
Z
X
{X}
A({X})e−βE({X})=. < A >M C, ✭✷✳✹✮
♦♥❞❡< A >M Cé ❛ ❡st✐♠❛t✐✈❛ ♦❜t✐❞❛ ❝♦♠ ♦ ♠ét♦❞♦ ❞❡ ▼❈✱ ❜❛s❡❛❞♦ ♥❛ ✐❞é✐❛ ❞❡ ❛♠♦str❛❣❡♠ ♣♦r ✐♠♣♦rtâ♥❝✐❛✳
❙❡ t✐✈❡r♠♦s M ❝♦♥✜❣✉r❛çõ❡s {X}1,{X}2,{X}3, ...,{X}M ♦❜t✐❞❛s s❡❣✉♥❞♦ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❇♦❧t③♠❛♥♥✱ ❛ ❡st✐♠❛t✐✈❛ < A >M C ❞♦ ✈❛❧♦r ♠é❞✐♦ ❞❡ A ♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦
hAiM C =
PM
i=1A({X}i)
M
.
❈❛♣ít✉❧♦ ✷✳ ❙✐♠✉❧❛çã♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ✼
❆ss✐♠✱ s✉❜st✐t✉✐♠♦s ❛ ♠é❞✐❛ ❝❛♥ô♥✐❝❛ ✭♠é❞✐❛ ❞❡ ❡♥s❡♠❜❧❡✮< A >♣♦r ✉♠❛ ♠é❞✐❛< A >M C s♦❜r❡ ❛ s❡q✉ê♥❝✐❛ ❞❡ ♠✐❝r♦❡st❛❞♦s ♣❡❧♦ q✉❛❧ ♦ s✐st❡♠❛ ♣❛ss❛ ❛♦ s❡ ❣❡r❛r ❛s ❝♦♥✜❣✉r❛çõ❡s ❞❡ ▼❈✳
❊♠ s✉♠❛✱ ❛ ✈❛♥t❛❣❡♠ ❞♦ ♠ét♦❞♦ ❡stá ♥❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ♦❜t❡r ❡st✐♠❛t✐✈❛s ❞❡ ❣r❛♥❞❡③❛s ❢ís✐❝❛s ❛♣❡♥❛s s♦rt❡❛♥❞♦ ✉♠❛ ❛♠♦str❛ ❞❡ ❝♦♥✜❣✉r❛çõ❡s ❞♦ s✐st❡♠❛✱ ❞✐str✐❜✉í❞♦s ♣❡❧♦s ♠✐❝r♦❡st❛❞♦s s❡❣✉♥❞♦ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❡q✉✐❧í❜r✐♦ ✭❞✐str✐❜✉✐çã♦ ❞❡ ❇♦❧t③♠❛♥♥✮✳ ❙✉❜st✐t✉í♠♦s ❛ s♦♠❛ s♦❜r❡ t♦❞♦s ♦s ♣♦ssí✈❡✐s ❡st❛❞♦s ❞♦ ❡♥s❡♠❜❧❡ ♣❡❧❛ s♦♠❛ s♦❜r❡ ♦s ♠✐❝r♦❡st❛❞♦s ❞❛ ❛♠♦str❛❣❡♠✳ ➱ ❡✈✐❞❡♥t❡ q✉❡ ♦ ✉s♦ ❞❡ ✉♠❛ r❡❣✐ã♦ r❡str✐t❛ ❞♦ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çõ❡s ❞♦ s✐st❡♠❛ ✐♥tr♦❞✉③ ❡rr♦s s✐st❡♠át✐❝♦s ♥❛ ❡st✐♠❛t✐✈❛✳ ❖ ❝á❧❝✉❧♦ ❞❡ ❛❧❣✉♥s ❞❡ss❡s ❡rr♦s s❡rá ❞✐s❝✉t✐❞♦ ♠❛✐s ❛❞✐❛♥t❡✳
P❛r❛ ❣❛r❛♥t✐r q✉❡ ❛ ❡q✉❛çã♦ ✭✷✳✺✮ s❡❥❛ r❡❛❧✐③❛❞❛ s♦❜r❡ ♦s ❡st❛❞♦s ❣❡r❛❞♦s ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❇♦❧t③♠❛♥♥✱ ❞❡✈❡♠♦s ❣❛r❛♥t✐r q✉❡ ♦ ♣r♦❝❡ss♦ ❞❡ ❡s❝♦❧❤❛ ❞❛s ❝♦♥✜❣✉r❛çõ❡s s❡❥❛ ♠❛r❦♦✈✐❛♥♦ ❡ ♦❜❡❞❡ç❛✱ ♣❡❧♦ ♠❡♥♦s✱ ❛ ❝♦♥❞✐çã♦ ❞❡ ❜❛❧❛♥ç♦ ❞❡t❛❧❤❛❞♦ ❝♦♠♦ ❞✐s❝✉t✐❞♦ ♥❛ ♣ró①✐♠❛ s❡çã♦✳
✷✳✸ Pr✐♥❝í♣✐♦ ❞♦ ❜❛❧❛♥ç♦ ❞❡t❛❧❤❛❞♦
❆tr❛✈és ❞❡ ✉♠ ♣r♦❝❡ss♦ ❞❡ ▼❛r❦♦✈ ❛ s✐♠✉❧❛çã♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ❡s❝♦❧❤❡ ❝♦♥✜✲ ❣✉r❛çõ❡s ❝♦♠ ❝❡rt❛ ♣r♦❜❛❜✐❧✐❞❛❞❡✱ ❣❡r❛♥❞♦ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❝♦♥✜❣✉r❛çõ❡s ✐♥❞❡♣❡♥❞❡♥t❡s✳ P♦r ❡①❡♠♣❧♦✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞♦ s✐st❡♠❛ s❡r ❡♥❝♦♥tr❛❞♦ ❡♠ q✉❛❧q✉❡r ❡st❛❞♦ ❡s♣❡❝í✜❝♦ µ é P(Eµ)✱ q✉❡ ❞❡♣❡♥❞❡ ❛♣❡♥❛s ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❛ ❝♦♥✜❣✉r❛çã♦ ✐♠❡❞✐❛t❛♠❡♥t❡ ❛♥t❡r✐♦r ♦♥❞❡ ♦ s✐st❡♠❛ s❡ ❡♥❝♦♥tr❛✈❛✳
❈❛♣ít✉❧♦ ✷✳ ❙✐♠✉❧❛çã♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ✽
✖✖✖✕
✖✖✖✕µ⇒P(Eµ, t) ✖✖✖✕↓↑
✖✖✖✕ν ⇒P(Eν, t) ✖✖✖✕
◆♦ ❡sq✉❡♠❛ ❛❝✐♠❛✱ ❛s s❡t❛s ✐♥❞✐❝❛♠ ❛s tr❛♥s✐çõ❡s ❞♦s ❡st❛❞♦sµ ♣❛r❛ν✱ ❡ ✈✐❝❡✲
✈❡rs❛✳ ❊ss❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ tr❛♥s✐çã♦ ωµν ❡ ωνµ ❞❡✈❡♠ s❛t✐s❢❛③❡r ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿
✶✳ ◆ã♦ ✈❛r✐❛r ❝♦♠ ♦ t❡♠♣♦❀
✷✳ ❉❡♣❡♥❞❡r s♦♠❡♥t❡ ❞♦s ❡st❛❞♦sµ ❡ν✱ ❡ ♥ã♦ ❞♦s ❡st❛❞♦s ❛♥t❡r✐♦r❡s✳
❙❛t✐s❢❛③❡r ❡ss❡s ❝r✐tér✐♦s s✉❣❡r❡ q✉❡ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞♦ s✐st❡♠❛ ✐r ♣❛r❛ ♦ ❡st❛❞♦ ν✱ ✉♠❛ ✈❡③
❡st❛♥❞♦ ♥♦ ❡st❛❞♦ µ✱ é s❡♠♣r❡ ❛ ♠❡s♠❛✱ ♥ã♦ ✐♠♣♦rt❛♥❞♦ ♦ q✉❡ t❡♥❤❛ ❛❝♦♥t❡❝✐❞♦ ❛♥t❡r✐♦r✲
♠❡♥t❡✳
❆ s✐♠✉❧❛çã♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ❝♦♠ ♦ ❛❧❣♦r✐t♠♦ ❛❝✐♠❛ ♥ã♦ é ✉♠❛ ❞✐♥â♠✐❝❛✳ ❊❧❛ ❣❡r❛ ❡st❛❞♦sµ ❡ν ❛tr❛✈és ❞❡ ✉♠ ♣r♦❝❡ss♦ ❡st♦❝ást✐❝♦✱ ❞❡ ♠♦❞♦ q✉❡ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡P(Eµ) é ❞❛❞❛ ♣❡❧❛ ❞✐str✐❜✉✐çã♦ ❝❛♥ô♥✐❝❛✳ P❛r❛ ❛t✐♥❣✐r ❡ss❡ ♦❜❥❡t✐✈♦✱ ❡s❝♦❧❤❡✲s❡ ✉♠❛ ❝✐♥ét✐❝❛ q✉❡ ❣♦✈❡r♥❛ ❛ ♠✉❞❛♥ç❛ ❞❡ ✉♠ ❡st❛❞♦ ♣❛r❛ ♦✉tr♦✳ ❊st❡ ♣r♦❝❡ss♦ é ❣♦✈❡r♥❛❞♦ ♣♦r ✉♠❛ ❡q✉❛çã♦ ♠❡str❛ q✉❡ ❞❡s❝r❡✈❡ ❛ ❝✐♥ét✐❝❛ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ♦❝♦rrê♥❝✐❛ ❞♦s ❡st❛❞♦s P(Eµ, t)✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ❞❡s❝r❡✈❡ ❛ ❝✐♥ét✐❝❛ ❞❛s ❞✐str✐❜✉✐çõ❡s ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞♦s ♣r♦❝❡ss♦s ❡s✲ t♦❝ást✐❝♦s ▼❛r❦♦✈✐❛♥♦s ❞✐s❝r❡t♦s✳ ❆ ❡q✉❛çã♦ ♠❡str❛ é
d
dtP(Eµ, t) =
X
ν
[P(Eν)ωνµ−P(Eµ)ωµν], ✭✷✳✻✮
♦♥❞❡ P(Eµ, t) é ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞♦ s✐st❡♠❛ ❡st❛r ♥♦ ❡st❛❞♦ µ♥♦ ✐♥st❛♥t❡ t ❝♦♠ ωµν s❡♥❞♦ ❛ t❛①❛ ❞❡ tr❛♥s✐çã♦ ❡♥tr❡ ♦s ❡st❛❞♦s ❝♦rr❡s♣♦♥❞❡♥t❡s✳ ❆s ❡♥❡r❣✐❛s ❞♦s ❡st❛❞♦s µ ❡ ν sã♦
r❡s♣❡❝t✐✈❛♠❡♥t❡Eµ ❡ Eν✳
❆s t❛①❛s ❞❡ tr❛♥s✐çã♦ ωµν sã♦ ❡s❝♦❧❤✐❞❛s ❞❡ ♠♦❞♦ q✉❡ ❛ s♦❧✉çã♦ ❞❡ ❡q✉✐❧í❜r✐♦ ♣❛r❛ ❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❡q✉❛çã♦ ♠❡str❛ s❡❥❛ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❇♦❧t③♠❛♥♥✳ ❆ ♣❛rt✐r ❞♦s ❡st❛❞♦s ❣❡r❛❞♦s ♦❜t❡♠♦s ❛s ❡st✐♠❛t✐✈❛s ♣❛r❛ ♦ ♦❜s❡r✈á✈❡❧ ❞❡ ✐♥t❡r❡ss❡✳ ❊♠ ❡q✉✐❧í❜r✐♦ d
dtP(Eµ, t) = 0✱
✐st♦ é
X
ν
Peq(Eν)ωνµ−
X
ν
❈❛♣ít✉❧♦ ✷✳ ❙✐♠✉❧❛çã♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ✾
❆ ❡q✉❛çã♦ ❛❝✐♠❛ ♣♦ss✉✐ ✈ár✐❛s s♦❧✉çõ❡s✳ ◆♦ ❡♥t❛♥t♦✱ ❡ss❛ ❝♦♥❞✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❡st❛rá ❣❛r❛♥t✐❞❛ s❡ ♦s t❡r♠♦s ❞❛ s♦♠❛tór✐❛ ❝❛♥❝❡❧❛r❡♠✲s❡ t❡r♠♦ ❛ t❡r♠♦✱ ❧❡✈❛♥❞♦ ❛ ❝♦♥❝❧✉✐r q✉❡ ✉♠❛ s♦❧✉çã♦ ♣❛rt✐❝✉❧❛r é ❞❛❞❛ ♣♦r
Peq(Eν)ωνµ=Peq(Eµ)ωµν. ✭✷✳✽✮
❆ ❝♦♥❞✐çã♦ ❛❝✐♠❛ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ♣r✐♥❝í♣✐♦ ❞♦ ❜❛❧❛♥ç♦ ❞❡t❛❧❤❛❞♦ ✭P❇❉✮✳ ❈♦♠ ♦ ú♥✐❝♦ r❡q✉✐s✐t♦ q✉❡ ❛ t❛①❛ ❞❡ tr❛♥s✐çã♦ s❛t✐s❢❛ç❛ ❛ ❝♦♥❞✐çã♦ ❞❡ ❜❛❧❛♥ç♦ ❞❡t❛❧❤❛❞♦✱ ♣❛r❛ q✉❡ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s s❡❥❛ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦✱ ♣♦❞❡♠♦s ❡s❝♦❧❤❡rP(Eµ) ❝♦♠♦ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❇♦❧t③♠❛♥♥✳ ❯s❛♥❞♦ ❛ ❡q✉❛çã♦ ✭✷✳✸✮ ❡ ♦ ♣r✐♥❝í♣✐♦ ❞♦ ❜❛❧❛♥ç♦ ❞❡t❛❧❤❛❞♦ ✭✷✳✽✮ t❡♠♦s
ωνµ
ωµν = e−
βEµ e−βEν =e
−β(Eµ−Eν). ✭✷✳✾✮
❖ P❇❉ é ✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ♣❛r❛ ❣❛r❛♥t✐r q✉❡ ♦ s✐st❡♠❛ ❛t✐♥❥❛ ❛ s✐t✉❛çã♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❛♣ós ❝❡rt❛ q✉❛♥t✐❞❛❞❡ ❞❡ tr❛♥s✐çõ❡s✳ ❈♦♠♦ ♦s ❞❡♥♦♠✐♥❛❞♦r❡s Z s❡ ❝❛♥❝❡❧❛♠✱
s♦♠❡♥t❡ ❛ ❞✐❢❡r❡♥ç❛ ❞❡ ❡♥❡r❣✐❛ ❡♥tr❡ ♦s ❞♦✐s ❡st❛❞♦s é ♥❡❝❡ssár✐❛✱ ❛ss✐♠ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ tr❛♥s✐çã♦ ❞❡♣❡♥❞❡ ❞❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❛s ❡♥❡r❣✐❛s ❞♦s r❡s♣❡❝t✐✈♦s ❡st❛❞♦sµ❡ ν
δEµν =Eν −Eµ. ✭✷✳✶✵✮
◗✉❛❧q✉❡r t❛①❛ ❞❡ tr❛♥s✐çã♦ q✉❡ s❛t✐s❢❛ç❛ ❛ ❝♦♥❞✐çã♦ ❞❡ ❜❛❧❛♥ç♦ ❞❡t❛❧❤❛❞♦ é s✉✜❝✐❡♥t❡ ♣❛r❛ s❡ ✉t✐❧✐③❛r ❡♠ ✉♠ ❛❧❣♦r✐t♠♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦✳ ❆❧é♠ ❞♦ ♠❛✐s✱ ❛ ❡s❝♦❧❤❛ ✭✷✳✾✮ ❞❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ tr❛♥s✐çã♦ s❛t✐s❢❛③❡♥❞♦ ♦ P❇❉✱ ♣❛r❛ q✉❛❧q✉❡r µ✱ ❧❡✈❛ ❛ ✉♠❛ t❛①❛ ωµν ♣♦s✐t✐✈❛ ❡✱ ♣♦rt❛♥t♦✱ ♣♦❞❡♠♦s ❛t✐♥❣✐r q✉❛❧q✉❡r ❝♦♥✜❣✉r❛çã♦ ❞♦ s✐st❡♠❛ ❛ ♣❛rt✐r ❞❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ♣❛ss♦s✳ ❊ss❡ ❝r✐tér✐♦ ❛❞✐❝✐♦♥❛❧ é ❝❤❛♠❛❞♦ ❞❡ ❡r❣♦❞✐❝✐❞❛❞❡✳ ❆ ❡r❣♦❞✐❝✐❞❛❞❡ ❡stá ❣❛r❛♥t✐❞❛✱ ♣♦✐s ❡①✐st❡ ♣❛r❛ ❡ss❡ ❛❧❣♦r✐t♠♦ ✉♠❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦ ♣❛r❛ q✉❡ q✉❛❧q✉❡r ♣♦ssí✈❡❧ ❝♦♥✜❣✉r❛çã♦ ❞♦ s✐st❡♠❛ s❡❥❛ ❛t✐♥❣✐❞❛✳ ❖ ❝♦♥❝❡✐t♦ ❞❡ ❡r❣♦❞✐❝✐❞❛❞❡ ❬✶✶❪ é ♠❛✐s ❛♠♣❧♦✱ ♥ã♦ s❡♥❞♦ ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ P❇❉✳ ❆ ❡r❣♦❞✐❝✐❞❛❞❡ ❣❛r❛♥t❡ q✉❡ ♠é❞✐❛s ❡♠ ❡♥s❡♠❜❧❡ sã♦ ✐❣✉❛✐s ❛ ♠é❞✐❛s ♥♦ t❡♠♣♦✱ ✐st♦ é✱ q✉❛❧q✉❡r ♠✐❝r♦❡st❛❞♦ ❞♦ s✐st❡♠❛ ❢ís✐❝♦ ♣♦❞❡ s❡r ❛t✐♥❣✐❞♦ ❛ ♣❛rt✐r ❞❡ q✉❛❧q✉❡r ♣❛rt❡ ❞♦ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çõ❡s✳ ▲♦❣♦✱ ♦ ♣r✐♥❝í♣✐♦ ❞♦ ❜❛❧❛♥ç♦ ❞❡t❛❧❤❛❞♦ é ✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛✱ ♠❛s ♥ã♦ s✉✜❝✐❡♥t❡✱ s❡♥❞♦ ♣r❡❝✐s♦ t❛♠❜é♠ ❣❛r❛♥t✐r ♦ ❝r✐tér✐♦ ❞❛ ❡r❣♦❞✐❝✐❞❛❞❡ ♣❛r❛ q✉❡ ♦ s✐st❡♠❛ ❛t✐♥❥❛ ♦ ❡st❛❞♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❞❡ ❇♦❧t③♠❛♥♥✳
❈❛♣ít✉❧♦ ✷✳ ❙✐♠✉❧❛çã♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ✶✵
✷✳✹ ❆❧❣♦r✐t♠♦ ❞❡ ▼❡tr♦♣♦❧✐s
❯♠ ❞♦s ❛❧❣♦r✐t♠♦s ✉s❛❞♦s ♥♦ ♠ét♦❞♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ é ♦ ❆❧❣♦r✐t♠♦ ❞❡ ▼❡tr♦♣♦✲ ❧✐s ❬✶✶❪✱ q✉❡ ❢♦✐ ✐♥tr♦❞✉③✐❞♦ ♣♦r ▼❡tr♦♣♦❧✐s ❡ ❝♦❧❛❜♦r❛❞♦r❡s ❡♠ ✶✾✺✸ ❬✽❪✳ ◆♦ ❛❧❣♦r✐t♠♦ ❞❡ ▼❡tr♦♣♦❧✐s✱ ❝♦♥✜❣✉r❛çõ❡s sã♦ ❣❡r❛❞❛s ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❇♦❧t③♠❛♥♥✱ ❡❢❡t✉❛♥❞♦ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ s♦rt❡✐♦s ❛❧❡❛tór✐♦s ❞❡ tr❛♥s✐çõ❡s ❡♥tr❡ ❡st❛❞♦s✱ ❣❛r❛♥t✐♥❞♦ q✉❡ ❛ ❝♦♥✜❣✉✲ r❛çã♦ ✜♥❛❧ s❡❥❛ ❛ ❞❡ ❡q✉✐❧í❜r✐♦✳ ❆ t❛①❛ ❞❡ tr❛♥s✐çã♦ ❡♥tr❡ ❡st❛❞♦sµ❡ν s❛t✐s❢❛③❡♠ ♦ ♣r✐♥❝í♣✐♦
❞❡ ❜❛❧❛♥ç♦ ❞❡t❛❧❤❛❞♦ ✭✷✳✾✮ ❡ sã♦ ❡s❝♦❧❤✐❞❛s ❞❡ ♠♦❞♦ ❛ ♣❡r♠✐t✐r q✉❡ ❛s ❝♦♥✜❣✉r❛çõ❡s ♠❛✐s ♣r♦✈á✈❡✐s t❡♥❤❛♠ ♠❛✐s ❝❤❛♥❝❡ ❞❡ ♦❝♦rr❡r ♥❛ ♠é❞✐❛ ❞❛s ❣r❛♥❞❡③❛s ❞❡ ✐♥t❡r❡ss❡✱ ✐st♦ é✱ r❡s✲ tr✐♥❣✐♥❞♦ ❛ ❜✉s❝❛ ❛ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ ❝♦♥✜❣✉r❛çõ❡s ❡st❛❝✐♦♥ár✐❛s q✉❡ sã♦ ❡♥❡r❣❡t✐❝❛♠❡♥t❡ ❢❛✈♦rá✈❡✐s✳
❈♦♠♦ ❛s t❛①❛s ❞❡ tr❛♥s✐çã♦ ❞❡✈❡♠ s❛t✐s❢❛③❡r ♣r✐♠❡✐r❛♠❡♥t❡ ♦ ♣r✐♥❝í♣✐♦ ❞♦ ❜❛✲ ❧❛♥ç♦ ❞❡t❛❧❤❛❞♦✱ ♥❡st❡ tr❛❜❛❧❤♦ ❛ ❡s❝♦❧❤❛ ❞❡ss❛s t❛①❛s ❞❡ tr❛♥s✐çã♦ ✉s❛❞❛s ♥❛ s✐♠✉❧❛çã♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ❢♦✐ ♣r❡s❝r✐t❛ ♣❡❧♦ ❛❧❣♦r✐t♠♦ ❞❡ ▼❡tr♦♣♦❧✐s✱ ❛ s❛❜❡r
wµν=
1, s❡ Eν < Eµ,
e−β(Eν−Eµ), s❡ E
ν > Eµ.
✭✷✳✶✶✮
❈♦♠♦ ❡①❡♠♣❧♦✱ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ✉♠ ♠♦❞❡❧♦ ❞✐s❝r❡t♦ ❞❡✜♥✐❞♦ ❡♠ ✉♠❛ r❡❞❡ ❝♦♠N sít✐♦s✳ ❆ ❝❛❞❛ sít✐♦i❛ss♦❝✐❛♠♦s ✉♠❛ ✈❛r✐á✈❡❧ ❡st♦❝ást✐❝❛ Si✳ ❯♠ ❡st❛❞♦ ❝♦❧❡t✐✈♦ ❞♦ s✐st❡♠❛ é r❡♣r❡s❡♥t❛❞♦ ❝♦♠♦ µ= (S1, S2, S3, ..., SN) = {Si}µ✱ ♦♥❞❡ Si ❞❡s✐❣♥❛ ♦ ❡st❛❞♦ ❞❛ ✈❛r✐á✈❡❧ ♥♦ sít✐♦ i✳ ❖ ♣r♦❝❡❞✐♠❡♥t♦ ❛ s❡❣✉✐r ✐❧✉str❛ ♦ ♠ét♦❞♦ ❞❡ ▼❈ ✉s❛♥❞♦ ♦ ❛❧❣♦r✐t♠♦ ❞❡
▼❡tr♦♣♦❧✐s✳
✶✳ ●❡r❛♠♦s ✉♠❛ ❝♦♥✜❣✉r❛çã♦ ✐♥✐❝✐❛❧µ={Si}µ❀
✷✳ ●❡r❛♠♦s ❛❧❡❛t♦r✐❛♠❡♥t❡ ✉♠❛ ♥♦✈❛ ❝♦♥✜❣✉r❛çã♦✱ ♦❜t✐❞❛ ❞❡{Si}µ✱ν= (S1, S2, S3, ..., SN) =
{Si}µ+1❀
✸✳ ❈❛❧❝✉❧❛✲s❡ δE=Eµ+1−Eµ❀
✹✳ ❙❡ δE≤0✱ ❛ ❛♥t✐❣❛ ❝♦♥✜❣✉r❛çã♦ é s✉❜st✐t✉✐❞❛ ♣❡❧❛ ♥♦✈❛❀
❈❛♣ít✉❧♦ ✷✳ ❙✐♠✉❧❛çã♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ✶✶
✻✳ ❙❡ ε > p✱ ❛ ❝♦♥✜❣✉r❛çã♦ ❝♦♥t✐♥✉❛rá ❛ ♠❡s♠❛✱ r❡t♦r♥❛✲s❡ ❛♦ ✐t❡♠ ✸✱ r❡♣❡t✐♥❞♦ ♦ ♣r♦✲
❝❡ss♦✳
❆♣ós ✉♠ ♥ú♠❡r♦ s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ ❞❡ r❡♣❡t✐çõ❡s✱ ✉♠❛ ❝❛❞❡✐❛ ▼❛r❦♦✈✐❛♥❛ é ❣❡r❛❞❛✱ ♦❜❡❞❡❝❡♥❞♦ ❛ ❝♦♥❞✐çã♦ ❞❡ ❜❛❧❛♥ç♦ ❞❡t❛❧❤❛❞♦ ❡ ❛ ❤✐♣ót❡s❡ ❞❡ ❡r❣♦❞✐❝✐❞❛❞❡✳ ❉❡ss❛ ❢♦r♠❛✱ ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ♦ ❡st❛❞♦ ❡st❛❝✐♦♥ár✐♦ é ❣❛r❛♥t✐❞❛ ❬✶✷❪✳
✷✳✺ ◆ú♠❡r♦s Ps❡✉❞♦✲❛❧❡❛tór✐♦s
◆❡st❡ tr❛❜❛❧❤♦ ❢♦✐ ✉t✐❧✐③❛❞♦ ♦ ❣❡r❛❞♦r ❞❡ ♥ú♠❡r♦s ♣s❡✉❞♦ ❛❧❡❛tór✐♦s ❧✐♥❡❛r❡s ❝♦♥❣r✉❡♥t❡s✱ q✉❡ ❣❡r❛ ✉♠ ♥ú♠❡r♦ ♣s❡✉❞♦✲❛❧❡❛tór✐♦ rand ❡♥tr❡ ♦ ✐♥t❡r✈❛❧♦ 0 < rand <
1✳ ❖ ❣❡r❛❞♦r ❝♦♥❣r✉❡♥❝✐❛❧ ✉s❛❞♦ ♥ã♦ ❢♦✐ ♦ ♠❛✐s s♦✜st✐❝❛❞♦ ❞❡ s✉❛ ❢❛♠✐❧✐❛✳ P♦ré♠✱ é ♦ ♠❛✐s s✐♠♣❧❡s ❡ ♦ q✉❡ ❝♦♥s♦♠❡ ♠❡♥♦s t❡♠♣♦ ❞❡ ❝♦♠♣✉t❛çã♦ ♣❛r❛ ❛ s✉❛ ✐♠♣❧❡♠❡♥t❛çã♦ ♥❛s s✐♠✉❧❛çõ❡s ♥✉♠ér✐❝❛s✳ P♦r ❡ss❛ r❛③ã♦ ♦ ✉t✐❧✐③❛♠♦s ♥❡ss❡ ❡st✉❞♦✱ ❝♦♠ ♦ ✐♥t✉✐t♦ s✐♠♣❧❡s♠❡♥t❡ ❞❡ r❡♣r♦❞✉③✐r♠♦s ♦s r❡s✉❧t❛❞♦s ❞❛ ❝r✐t✐❝❛❧✐❞❛❞❡ ❞♦ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣ ❜✐❞✐♠❡♥s✐♦♥❛❧ ❞❡ s♣✐♥✲1/2✳
✷✳✻ ❊st✐♠❛t✐✈❛s ❞❡ ❡rr♦s
❆♣ós ✉♠❛ s✐♠✉❧❛çã♦ ❝♦♠ ✭❞♦ ✐♥❣❧ês ▼♦♥t❡ ❈❛r❧♦ st❡♣s ❞✐s❝❛r❞❡❞✮M CSD♣❛ss♦s
✐♥✐❝✐❛✐s ❞❡ ▼♦♥t❡ ❈❛r❧♦✱ ❡s♣❡r❛✲s❡ q✉❡ ♦ s✐st❡♠❛ s❡❥❛ ❝♦♥❞✉③✐❞♦ ❛♦ ❡q✉✐❧í❜r✐♦ ❡✱ ❛ ♣❛rt✐r ❞❡ss❡ ❡q✉✐❧í❜r✐♦✱ ✉♠ ♥♦✈♦ ❝♦♥❥✉♥t♦ ❝♦♥t❡♥❞♦M CS❝♦♥✜❣✉r❛çõ❡s sã♦ ❝♦♥str✉í❞❛s ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛s
♠é❞✐❛s✳ ❆ q✉❛♥t✐❞❛❞❡M CSD✐♥✐❝✐❛❧ ❞❡♣❡♥❞❡✱ ❡♠ ❣❡r❛❧✱ ❞❛ ❝✐♥ét✐❝❛ ❞♦ ♠♦❞❡❧♦✱ ❞♦ t❛♠❛♥❤♦
❞❛ r❡❞❡ ❡ ❞❛ t❡♠♣❡r❛t✉r❛ ❞❛ s✐♠✉❧❛çã♦✳ P❛r❡❝✐❞♦ ❝♦♠ ♦s ♣r♦❝❡ss♦s ❡①♣❡r✐♠❡♥t❛✐s✱ ♥♦ ❝á❧❝✉❧♦ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❞❛s q✉❛♥t✐❞❛❞❡s ❞❡s❡❥❛❞❛s✱ s✉r❣❡♠ ❡rr♦s ❡st❛tíst✐❝♦s q✉❡ sã♦ ✐♥❡r❡♥t❡s ❛♦ ♠ét♦❞♦✳ ◆♦ ❝❛s♦ ❞♦ ▼♦♥t❡ ❈❛r❧♦✱ sã♦ ❣❡r❛❞♦s ♣❡❧❛ ♠é❞✐❛ t❡♠♣♦r❛❧ q✉❡ é ❢❡✐t❛ s♦❜r❡ ✉♠ t❡♠♣♦ ✜♥✐t♦ ❞❡ s✐♠✉❧❛çã♦✳ ▲♦❣♦✱ ❛❧é♠ ❞❡ ♠❡❞✐r ♦s ✈❛❧♦r❡s ❡s♣❡r❛❞♦s✱ ❤❛✈❡rá ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❡st✐♠❛r ♦s ❡rr♦s s♦❜r❡ ❡ss❡s ✈❛❧♦r❡s✳ ❍á✱ ♥❛t✉r❛❧♠❡♥t❡✱ ♦✉tr♦s ❡rr♦s ❛ q✉❡ ♦ ▼❈ ❡stá s✉❜♠❡t✐❞♦ ❬✼❪✱ ❝♦♠♦ ❡rr♦s s✐st❡♠át✐❝♦s✳ ❖s ❡rr♦s s✐st❡♠át✐❝♦s ♣♦❞❡♠ s✉r❣✐r ❞❡ ♦✉tr♦s ♠❡✐♦s q✉❡ ❣❡r❛❧♠❡♥t❡ ❛❢❡t❛♠ ❛ s✐♠✉❧❛çã♦✱ ❝♦♠♦ ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞♦s r❡s✉❧t❛❞♦s ❝♦♠ ❛s té❝♥✐❝❛s ❞♦ ❤✐st♦❣r❛♠❛ s✐♠♣❧❡s✱ ❛ q✉❛❧ ❝❛❧❝✉❧❛ ❛s ♠é❞✐❛s ❡♠ ✉♠❛ t❡♠♣❡r❛t✉r❛ ♣ró①✐♠❛ à q✉❡ ❢♦✐ ❣❡r❛❞❛ ❛s ❝♦♥✜❣✉r❛çõ❡s✱ ♦ ✉s♦ ❞❡ ❣❡r❛❞♦r❡s ❞❡ ♥ú♠❡r♦s ❛❧❡❛tór✐♦s r✉✐♥s✱ ❡t❝✳
❈❛♣ít✉❧♦ ✷✳ ❙✐♠✉❧❛çã♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ✶✷
t❡♠♣♦ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❡ r❡♣r❡s❡♥t❛ ✉♠ ♣❛ss♦ ❞❡ ▼❈✳ P❛r❛ ✉♠❛ s✐♠✉❧❛çã♦✱ ❛ ❡st✐♠❛t✐✈❛ ❞❛ ♠é❞✐❛ ❞❡ q✉❛❧q✉❡r q✉❛♥t✐❞❛❞❡ ❞♦ s✐st❡♠❛ ❞✉r❛♥t❡ ❛ s✐♠✉❧❛çã♦ é ❞❛❞❛ ♣❡❧❛ q✉❛çã♦ ✭✷✳✺✮✳ ❙❡ ❛s q✉❛♥t✐❞❛❞❡s ♠❡❞✐❞❛sA(t) sã♦ ❡st❛t✐st✐❝❛♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ✉♠❛ ❞❛ ♦✉tr❛✱ ♦ ♣r♦❝❡ss♦ s✐♠♣❧❡s q✉❡ ❡st✐♠❛ ♦ ❡rr♦ ❡st❛tíst✐❝♦ ♥❡ss❛ q✉❛♥t✐❞❛❞❡ ♠❡❞✐❞❛ é ❞❛❞♦ ♣❡❧♦ s❡✉ ❞❡s✈✐♦ ♣❛❞rã♦
σ=
s
hA2i − hAi2
M CS−1 . ✭✷✳✶✷✮
❊st❛ ❡①♣r❡ssã♦ é ✈á❧✐❞❛ q✉❛♥❞♦ ❛s ❛♠♦str❛s sã♦ ❡st❛t✐st✐❝❛♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✳ ❊♠❜♦r❛ ♥♦ ❝❛s♦ ❞❡ ▼❈ ❡①✐st❛ ❛✐♥❞❛ ✉♠❛ ❝♦rr❡❧❛çã♦ ❡♥tr❡ ❛s ❝♦♥✜❣✉r❛çõ❡s✱ ♣❛r❛ ❛s ❣r❛♥❞❡③❛s q✉❡ sã♦ ♦❜t✐❞❛s ❞✐r❡t❛♠❡♥t❡ ❞❡ ♠❡❞✐❞❛s r❡❛❧✐③❛❞❛s ❞✉r❛♥t❡ ❛ s✐♠✉❧❛çã♦ s✉♣õ❡✲s❡ q✉♦ ❡ ♦ ❡rr♦ ♣♦ss❛ ❛✐♥❞❛ s❡r ♦❜t✐❞♦ ❛tr❛✈és ❞❛ ❡①♣r❡ssã♦ ❛❝✐♠❛✳
✷✳✼ ❚é❝♥✐❝❛s ❞❡ ❤✐st♦❣r❛♠❛s s✐♠♣❧❡s
❆ ❝♦♥str✉çã♦ ❞❡ ✉♠ ❤✐st♦❣r❛♠❛ ❞❡ ❡♥❡r❣✐❛ ❛ ✉♠❛ t❡♠♣❡r❛t✉r❛ ❞❛❞❛ ♣❡r♠✐t❡ ❝❛❧✲ ❝✉❧❛r✱ ❝♦♠ r❛③♦á✈❡❧ ♣r❡❝✐sã♦✱ q✉❛♥t✐❞❛❞❡s t❡r♠♦❞✐♥â♠✐❝❛s ❡♠ t❡♠♣❡r❛t✉r❛s ✈✐③✐♥❤❛s àq✉❡❧❛ t❡♠♣❡r❛t✉r❛✳ ❯s❛♥❞♦ ❛ té❝♥✐❝❛ ❞❡s❡♥✈♦❧✈✐❞❛ ♣♦r ❋❡rr❡♥❜❡r❣ ❬✶✸❪ é ♣♦ssí✈❡❧ ❡①tr❛♣♦❧❛r ♦ ❤✐s✲ t♦❣r❛♠❛ ♣❛r❛ t❡♠♣❡r❛t✉r❛s ♣ró①✐♠❛s à s✐♠✉❧❛❞❛✳ ❘❡❛❧✐③❛♥❞♦ ❛ s✐♠✉❧❛çã♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ❛ ✉♠❛ t❡♠♣❡r❛t✉r❛ ❞❛❞❛T0✱ ❝♦♥✜❣✉r❛çõ❡s sã♦ ❣❡r❛❞❛s ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❞✐str✐❜✉✐çã♦ ❝❛♥ô♥✐❝❛
Pβ0(X) =
e−β0H({X}) Z(β0)
, ✭✷✳✶✸✮
♦♥❞❡ ♦ ❤❛♠✐❧t♦♥✐❛♥♦ H({X}) é ❛ ❡♥❡r❣✐❛ ❞♦ s✐st❡♠❛ ♥♦ ♠✐❝r♦❡st❛❞♦ {X}✱ β0 = 1/kBT0✱
❝♦♠ kB ❛ ❝♦♥st❛♥t❡ ❞❡ ❇♦❧t③♠❛♥♥✱ ❡ Z(β0) = PXe−β0H({X}) é ❛ ❢✉♥çã♦ ❞❡ ♣❛rt✐çã♦✳ ◆♦ ❡♥s❡♠❜❧❡ ❝❛♥ô♥✐❝♦
Pβ0(E) =
g(E)e−β0E Z(β0)
, ✭✷✳✶✹✮
♦♥❞❡Eé ❛ ❡♥❡r❣✐❛ ❡g(E)❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡st❛❞♦s ❝♦♠ ❡♥❡r❣✐❛E✳ ❉❡ss❛ ❢♦r♠❛✱ ♦ ✈❛❧♦r ♠é❞✐♦
❞❡ q✉❛❧q✉❡r ❢✉♥çã♦ ❞❡E ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦
< A >β0=
P
Eg(E)e−β0E
Z(β0)
, ✭✷✳✶✺✮
♦♥❞❡ ❛ s♦♠❛ ❛❝✐♠❛ é s♦❜r❡ t♦❞❛s ❛s ❡♥❡r❣✐❛s ❞♦ s✐st❡♠❛✳ ❈♦♠♦ ❛ s✐♠✉❧❛çã♦ ❣❡r❛ ❝♦♥✜✲ ❣✉r❛çõ❡s s❡❣✉♥❞♦ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❡q✉✐❧í❜r✐♦✱ ✉♠ ❤✐st♦❣r❛♠❛ ❞❛ ❡♥❡r❣✐❛
H(E) ✭♦ ♥ú♠❡r♦ ❞❡ ✈❡③❡s q✉❡ ♦ ❡st❛❞♦ ❞❡ ❡♥❡r❣✐❛E s❡ r❡♣❡t❡✮ ❢♦r♥❡❝❡ ✉♠❛ ❡st✐♠❛t✐✈❛ ♣❛r❛
❈❛♣ít✉❧♦ ✷✳ ❙✐♠✉❧❛çã♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ✶✸
❞❡ ♠❡❞✐❞❛s✱ é ✉♠❛ ❡st✐♠❛t✐✈❛ ❞❛ s✐♠✉❧❛çã♦ ♣❛r❛ ❛ ❞✐str✐❜✉✐çã♦ ♦❜t✐❞❛ ❞♦ r❡s✉❧t❛❞♦ t❡ór✐❝♦
Pβ0
H(E)
N ∼=Pβ0(E) =
g(E)e−β0E Z(β0)
. ✭✷✳✶✻✮
❯t✐❧✐③❛♥❞♦ ❛ ❡q✉❛çã♦ ❛❝✐♠❛ ♣❛r❛ T0 ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r g˜(E) ❝♦♠♦ ✉♠❛ ❡st✐✲
♠❛t✐✈❛ ♣❛r❛ ♦ ✈❛❧♦r ❞❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡st❛❞♦sg(E) ❡ ♦❜t❡♠♦s
˜
g(E) = H(E)Z(β0)e β0E
N . ✭✷✳✶✼✮
P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❛r❛ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❡♠ ✉♠❛ ❝❡rt❛ t❡♠♣❡r❛t✉r❛ ❛r❜✐trár✐❛β t❡♠♦s
Pβ(E) =
g(E)e−βE
Z(β) . ✭✷✳✶✽✮
■♥s❡r✐♥❞♦ ❛ ❡st✐♠❛t✐✈❛ g˜(E)✱ ♥❛ ❡q✉❛çã♦ ❛❝✐♠❛✱ ❡ ✐♠♣♦♥❞♦ ❛ ❝♦♥❞✐çã♦ ❞❡ ♥♦r♠❛❧✐③❛çã♦
P
EPβ(E) = 1✱ ♦❜t❡♠♦s
Pβ(E) =
H(E)e−δβE
P
EH(E)e−δβE
, ✭✷✳✶✾✮
♦♥❞❡ δβ= (β−β0)✳ ❉❡ss❛ ❢♦r♠❛✱ ❛ ♠é❞✐❛ ♣❛r❛ q✉❛❧q✉❡r ❢✉♥çã♦ ❞❡E é ❞❛❞❛ ♣♦r
hf(E)iβ =
P
Ef(E)H(E)e−δβE
P
EH(E)e−δβE
. ✭✷✳✷✵✮
P♦❞❡♠♦s✱ ♣♦rt❛♥t♦✱ ❛♦ r❡❛❧✐③❛r ✉♠❛ s✐♠✉❧❛çã♦ ❡♠T0✱ ❝❛❧❝✉❧❛r H(E) ❡✱ ❞❛ ❡q✉❛çã♦ ❛❝✐♠❛✱
♦❜t❡r ❡st✐♠❛t✐✈❛s ❡♠ t❡♠♣❡r❛t✉r❛sT ♣ró①✐♠❛s ❞❡T0✳
❆ té❝♥✐❝❛ ❞♦ ❤✐st♦❣r❛♠❛ ❡♠ ❝♦♥❥✉♥t♦ ❝♦♠ ❛ s✐♠✉❧❛çã♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ♣❡r♠✐t❡ ✉♠❛ ❞❡s❝r✐çã♦ ❛❝✉r❛❞❛ ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❝rít✐❝♦✱ ♣♦✐s ♣❡r♠✐t❡ r❡✜♥❛r ♦s r❡s✉❧t❛❞♦s ♦❜t✐✲ ❞♦s ❞✉r❛♥t❡ ❛ s✐♠✉❧❛çã♦✳ ❈♦♥❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦s ❤✐st♦❣r❛♠❛s✱ ❛ ❛♥á❧✐s❡ ❞❡ t❛♠❛♥❤♦ ✜♥✐t♦ ♣❡r♠✐t❡ ❛ ♦❜t❡♥çã♦ ❞❡ r❡s✉❧t❛❞♦s ♥♦ ❧✐♠✐t❡ t❡r♠♦❞✐♥â♠✐❝♦✳
❈❛♣ít✉❧♦ ✸
❙✐♠✉❧❛çã♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ♥♦ ♠♦❞❡❧♦
❞❡ ■s✐♥❣
✸✳✶ ▼♦❞❡❧♦ ❞❡ ■s✐♥❣
❖ ♠♦❞❡❧♦ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣ ❢♦✐ ♣r♦♣♦st♦ ♣♦r ❲✐❧❤❡❧♠ ▲❡♥③✱ ❡♠ ✶✾✷✵✱ ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❡st✉❞❛r ♦s ❢❡♥ô♠❡♥♦s ♠❛❣♥ét✐❝♦s ❡♠ ❝❡rt♦s ♠❛t❡r✐❛✐s✳ ❊st❡ ♠♦❞❡❧♦ ❢♦✐ r❡s♦❧✈✐❞♦ ♣♦r s❡✉ ❡st✉❞❛♥t❡ ❞❡ ❞♦✉t♦r❛♠❡♥t♦✱ ❊r♥st ■s✐♥❣ ❬✶✹❪✱ ❡♠ ✶✾✷✺✱ ♣❛r❛ ♦ ❝❛s♦ ✉♥✐❞✐♠❡♥s✐♦♥❛❧✱ ♦ q✉❛❧ ❝❤❡❣♦✉ à ❝♦♥❝❧✉sã♦ q✉❡ ♦ ♠♦❞❡❧♦ ♥ã♦ ❡①✐❜❡ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ ❡♠ t❡♠♣❡r❛t✉r❛ ✜♥✐t❛✳ ◆♦ ❡♥t❛♥t♦✱ r❡s✉❧t❛❞♦s ❞❡ ❝❛♠♣♦ ♠é❞✐♦ s✉❣❡r✐❛♠ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ❡①✐st✐r ♣❡❧♦ ♠❡♥♦s ✉♠❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ ❡♠ ❞✐♠❡♥sõ❡s s✉♣❡r✐♦r❡s✳ P♦st❡r✐♦r♠❡♥t❡✱ ❡♠ ✶✾✹✹✱ ♦ q✉í♠✐❝♦ ♥♦r✉❡❣✉ês ▲❛rs ❖♥s❛❣❡r ❬✶✺❪ ❡♥❝♦♥tr♦✉ ✉♠❛ s♦❧✉çã♦ ❛♥❛❧ít✐❝❛ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❜✐❞✐♠❡♥s✐♦♥❛❧ s❡♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦✱ ♣r♦✈❛♥❞♦ q✉❡ ♦ ♠❡s♠♦ ❛♣r❡s❡♥t❛✈❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✳ P❛r❛ ❞✐♠❡♥sõ❡s s✉♣❡r✐♦r❡s ♥ã♦ ❤á s♦❧✉çã♦ ❡①❛t❛ ❛♥❛❧ít✐❝❛✳ ❱ár✐❛s té❝♥✐❝❛s ❛♣r♦①✐♠❛❞❛s ❢♦r♥❡❝❡♠ r❡s✉❧t❛❞♦s q✉❡ ♣❡r♠✐t❡♠ ❡st✉❞❛r ♦ ❞✐❛❣r❛♠❛ ❞❡ ❢❛s❡ ❞♦ ♠♦❞❡❧♦✳
❖ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣ ❬✶✷✱ ✶✶❪ é ✉♠ ✏t♦② ♠♦❞❡❧✑ ♥♦ ❡st✉❞♦ ❞♦s ❢❡♥ô♠❡♥♦s ❝rít✐❝♦s ❡ tr❛♥s✐çõ❡s ❞❡ ❢❛s❡✳ ❆♣❡s❛r ❞❡ t❡r s✐❞♦✱ ♦r✐❣✐♥❛❧♠❡♥t❡✱ ♣r♦♣♦st♦ ♣❛r❛ ♦ ❡st✉❞♦ ❞❡ ❢❡♥ô♠❡♥♦s
❈❛♣ít✉❧♦ ✸✳ ❙✐♠✉❧❛çã♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ♥♦ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣ ✶✺
♠❛❣♥ét✐❝♦s✱ ♦ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣ t❡♠ s✐❞♦ ✉s❛❞♦ t❛♠❜é♠ ♥♦ ❡st✉❞♦ ❞❡ s✐st❡♠❛s ♥ã♦ ♠❛❣♥ét✐❝♦s✱ ❝✉❥❛s ✈❛r✐á✈❡✐s s❡ ❝❛r❛❝t❡r✐③❛♠ ♣♦r ♣♦❞❡r❡♠ ❛ss✉♠✐r ❛♣❡♥❛s ❞♦✐s ✈❛❧♦r❡s ✭♣✳❡✳ Si=±1✮✳
❖ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣ ❝♦♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❡①t❡r♥♦ é ❞❡s❝r✐t♦ ♣❡❧♦ s❡❣✉✐♥t❡ ❤❛♠✐❧✲ t♦♥✐❛♥♦
HI =−J
X
hi,ji
SiSj+B N
X
i=1
Si, ✭✸✳✶✮
♦♥❞❡J é ❛ ❡♥❡r❣✐❛ ❞❡ ✐♥t❡r❛çã♦ ❡♥tr❡ ♣r✐♠❡✐r♦s ✈✐③✐♥❤♦s ❡B r❡♣r❡s❡♥t❛ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦
❡①t❡r♥♦✳ ❆ ♣r✐♠❡✐r❛ s♦♠❛ é r❡❛❧✐③❛❞❛ s♦❜r❡ ♦s ♣❛r❡s ❞❡ ♣r✐♠❡✐r♦s ✈✐③✐♥❤♦shi, ji✱ ❛ s❡❣✉♥❞❛
s♦♠❛ s♦❜r❡ ♦sN s♣✐♥s ❞❛ r❡❞❡ ❡Si =±1✳
❚❛❜❡❧❛ ✸✳✶✿ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ✐♥✈❡rsã♦ ❞❡ ✉♠ s♣✐♥ ❞❡↑♣❛r❛↓♥✉♠❛ r❡❞❡ q✉❛❞r❛❞❛✳ ❆ss✉♠✲
✐♠♦s q✉❡ ✉♠ sít✐♦ ❞❛ r❡❞❡ r❡♣r❡s❡♥t❛❞♦ ♥❛ t❛❜❡❧❛ ❡stá ❛ss♦❝✐❛❞♦ ❝♦♠ s❡✉s ♣r✐♠❡✐r♦s ✈✐③✐♥❤♦s✱ ♦♥❞❡ ❛ s❡t❛↑ r❡♣r❡s❡♥t❛Si = +1✱ ❡ ❛ s❡t❛↓ r❡♣r❡s❡♥t❛ ♦ s♣✐♥Si =−1✳
↑ ↑
↑ ↑ ↑ ↑ ↓ ↑
↑ ↑
✸✳✷
❙✐♠✉❧❛çã♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦
❆ ❡q✉❛çã♦ ✭✸✳✶✮ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣ ❞❡s❝r❡✈❡ ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞❛ ❡♥❡r❣✐❛ s♦❜r❡ ❛s ❝♦♥✜❣✉r❛çõ❡s ❞❡ s♣✐♥✳ ◆♦ ❡♥t❛♥t♦✱ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❝♦♠♣✉t❛❝✐♦♥❛❧✱ ✐ss♦ ♥ã♦ é s✉✜❝✐❡♥t❡ ♣❛r❛ ❞❡t❡r♠✐♥❛r ❝♦♠♦ ♦ s✐st❡♠❛ ♠✉❞❛ ❞❡ ✉♠❛ ❝♦♥✜❣✉r❛çã♦ ♣❛r❛ ♦✉tr❛✳ ❊ss❛ ❝✐♥ét✐❝❛ é ✐♥tr♦❞✉③✐❞❛ ♥❛ ❢♦r♠❛ ❞❡ ❛❧❣♦r✐t♠♦s ❞❡ ▼♦♥t❡ ❈❛r❧♦✳ P❛r❛ ❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❞♦ ♣r♦❝❡❞✐♠❡♥t♦ ♥✉♠ér✐❝♦ ❞✐s❝✉t✐❞♦ ♥❛ s❡çã♦ ✭✷✳✹✮✱ t❡♠♦s q✉❡ ❡st❛❜❡❧❡❝❡r ✐♥✐❝✐❛❧♠❡♥t❡ ♦ t✐♣♦ ❞❡ r❡❞❡ ✭❝ú❜✐❝❛✱ tr✐❛♥❣✉❧❛r✱ q✉❛❞r❛❞❛✱ r❡t❛♥❣✉❧❛r ❡ ❡t❝✳✳✳✮ ❡ s❡✉ ♥ú♠❡r♦ ❞❡ s♣✐♥s✳ ◆ós ✉s❛♠♦s r❡❞❡s q✉❛❞r❛❞❛s ❝♦♠ ❝♦♥❞✐çõ❡s ♣❡r✐ó❞✐❝❛s ❞❡ ❝♦♥t♦r♥♦✱ ❝♦♠ ✐♥t✉✐t♦ ❞❡ ❡❧✐♠✐♥❛r ♦s ❡❢❡✐t♦s ❞❡ ❜♦r❞❛✱ ♣❡r♠✐t✐♥❞♦ ❛ss✐♠ q✉❡ t♦❞♦s ♦s s♣✐♥s t❡♥❤❛♠ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ✈✐③✐♥❤♦s✳ ❊♠ ❝❛❞❛ sít✐♦ ❞❛ r❡❞❡ t❡♠✲s❡ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❝❛r❛❝t❡r✐③❛❞❛ ♣♦r S=±1✳ ❯♠ ❡st❛❞♦
