✶✻t❤� ❊✉r♦♣❡❛♥� ❚✉r❜✉❧❡♥❝❡� ❈♦♥❢❡r❡♥❝❡� ❆✉❣✉st� ✷✶✲✷✹� ✷✵✶✼✱� ❙t♦❝❦❤♦❧♠✱� ❙✇❡❞❡♥� ❋❧✉✐❞✴❡❧❛st✐❝� st❛t✐♦♥❛r②� ♠♦❞❡s� ♦❢� ❛� ✢❡①✐❜❧❡� ♣❧❛t❡� ❝❧❛♠♣❡❞� ♦♥� ❛� ❝②❧✐♥❞❡r� ❏❡❛♥✲▲♦✉� P❋■❙❚❊❘� ✶ ✱� ▼❛r❝♦� ❈❆❘■◆■� ✷ ✱� ❖❧✐✈✐❡r� ▼❆❘◗❯❊❚� ✷ ✶ P❤❉� st✉❞❡♥t� ❛t� ❖◆❊❘❆� ✭▼❡✉❞♦♥✱� ❋r❛♥❝❡✮� ✷ ❖◆❊❘❆� ✭▼❡✉❞♦♥✱� ❋r❛♥❝❡✮�
❙♣♦♥t❛♥❡♦✉s� ♣❧❛t❡� ❞❡✈✐❛t✐♦♥� ♥♦♥✲③❡r♦� ❧✐❢t� ❢♦r❝❡✳� ❍♦✇� t♦� ♣r❡❞✐❝t� t❤❡� ✐♥st❛❜✐❧✐t✐②� ♦♥s❡t� ❄� ❲❤❛t� ❛r❡� t❤❡� ✉♥❞❡r❧②✐♥❣� ♠❡❝❤❛♥✐s♠s� ❄� ■♥tr♦❞✉❝t✐♦♥� Pr♦❜❧❡♠� st❛t❡♠❡♥t� ✹� ✷� ✵� − ✷� − ✹� − ✷� ✵� ✷� ✹� ✻� ✽� ✶✵� ✶✷� ✶✹� ✶✻� ✶✽� ✷✵� ❉✐r❡❝t� ♥♦♥✲❧✐♥❡❛r� ✢✉✐❞✴❡❧❛st✐❝� s✐♠✉❧❛t✐♦♥ ✿� str❡❛♠✇✐s❡� ✈❡❧♦❝✐t②� ❛♥❞� ♣❧❛t❡� ❞❡❢♦r♠❛t✐♦♥� ❏❡❛♥✲▲♦✉� P❋■❙❚❊❘� ✷� ✴� ✶✼� ❊❚❈✶✻✱� ❙t♦❝❦❤♦❧♠�
■♥tr♦❞✉❝t✐♦♥� Pr♦❜❧❡♠� st❛t❡♠❡♥t� ✹� ✷� ✵� − ✷� − ✹� − ✷� ✵� ✷� ✹� ✻� ✽� ✶✵� ✶✷� ✶✹� ✶✻� ✶✽� ✷✵� ❉✐r❡❝t� ♥♦♥✲❧✐♥❡❛r� ✢✉✐❞✴❡❧❛st✐❝� s✐♠✉❧❛t✐♦♥ ✿� str❡❛♠✇✐s❡� ✈❡❧♦❝✐t②� ❛♥❞� ♣❧❛t❡� ❞❡❢♦r♠❛t✐♦♥� ❙♣♦♥t❛♥❡♦✉s� ♣❧❛t❡� ❞❡✈✐❛t✐♦♥� → ♥♦♥✲③❡r♦� ❧✐❢t� ❢♦r❝❡✳� ❍♦✇� t♦� ♣r❡❞✐❝t� t❤❡� ✐♥st❛❜✐❧✐t✐②� ♦♥s❡t� ❄� ❲❤❛t� ❛r❡� t❤❡� ✉♥❞❡r❧②✐♥❣� ♠❡❝❤❛♥✐s♠s� ❄� ❏❡❛♥✲▲♦✉� P❋■❙❚❊❘� ✷� ✴� ✶✼� ❊❚❈✶✻✱� ❙t♦❝❦❤♦❧♠�
❍❡r❡ ✿� ❛♣♣r♦❛❝❤� ❜❛s❡❞� ♦♥� ❧✐♥❡❛r� st❛❜✐❧✐t②� ❛♥❛❧②s✐s✳� ■♥tr♦❞✉❝t✐♦♥� ▲✐t❡r❛t✉r❡� r❡✈✐❡✇� ✕� s♣♦♥t❛♥❡♦✉s� ❞❡✈✐❛t✐♦♥s� ✐♥� ✢✉✐❞✴s♦❧✐❞� s②st❡♠s� ▼♦t✐✈❛t✐♦♥ ✿� ✢♦✇� ❝♦♥tr♦❧✱� ❜✐♦✲✐♥s♣✐r❡❞� s②st❡♠s✳✳✳� ▲♦✇✲❘❡②♥♦❞s� ✢♦✇✱� ❢r❡❡❧②� r♦t❛t❛❜❧❡� ♣❧❛t❡� ✶� ❍✐❣❤❡r✲❘❡②♥♦❞s� ✢♦✇✱� ❢r❡❡❧②� r♦t❛t❛❜❧❡� ♣❧❛t❡� ✷� ❘✐❣✐❞� r♦t❛t❛❜❧❡� ♣❧❛t❡� ✭♥✉♠❡r✐❝� st✉❞②✮� ✸� ❋❧❡①✐❜❧❡� ❛tt❛❝❤❡❞� ♣❧❛t❡� ✭♥✉♠❡r✐❝� st✉❞②✮� ✹� ✶�✳� ▲❛❝✐s✱� ❡t� ❛❧ ✱� ◆❛t✉r❡✱� ✷✵✶✹� ✸�✳� ❳✉✱� ❏✳❈✳� ❡t� ❛❧ ✳✱� P❤②s✳� ❋❧✉✐❞s✱� ✶✾✾✵� ✷�✳� ❈✐♠❜❛❧❛✱� ❏✳� ❛♥❞� ●❛r❣✱� ❙✳✱� ❆■❆❆✱� ✶✾✾✶� ✹�✳� ❇❛❣❤❡r✐✱� ❙✳� ❡t� ❛❧ ✳✱� P❤②s✳� ❘❡✈✳� ▲❡tt✳✱� ✷✵✶✷� ❏❡❛♥✲▲♦✉� P❋■❙❚❊❘� ✸� ✴� ✶✼� ❊❚❈✶✻✱� ❙t♦❝❦❤♦❧♠�
■♥tr♦❞✉❝t✐♦♥� ▲✐t❡r❛t✉r❡� r❡✈✐❡✇� ✕� s♣♦♥t❛♥❡♦✉s� ❞❡✈✐❛t✐♦♥s� ✐♥� ✢✉✐❞✴s♦❧✐❞� s②st❡♠s� ▼♦t✐✈❛t✐♦♥ ✿� ✢♦✇� ❝♦♥tr♦❧✱� ❜✐♦✲✐♥s♣✐r❡❞� s②st❡♠s✳✳✳� ▲♦✇✲❘❡②♥♦❞s� ✢♦✇✱� ❢r❡❡❧②� r♦t❛t❛❜❧❡� ♣❧❛t❡� ✶� ❍✐❣❤❡r✲❘❡②♥♦❞s� ✢♦✇✱� ❢r❡❡❧②� r♦t❛t❛❜❧❡� ♣❧❛t❡� ✷� ❘✐❣✐❞� r♦t❛t❛❜❧❡� ♣❧❛t❡� ✭♥✉♠❡r✐❝� st✉❞②✮� ✸� ❋❧❡①✐❜❧❡� ❛tt❛❝❤❡❞� ♣❧❛t❡� ✭♥✉♠❡r✐❝� st✉❞②✮� ✹� ❍❡r❡ ✿� ❛♣♣r♦❛❝❤� ❜❛s❡❞� ♦♥� ❧✐♥❡❛r� st❛❜✐❧✐t②� ❛♥❛❧②s✐s✳� ✶�✳� ▲❛❝✐s✱� ❡t� ❛❧ ✱� ◆❛t✉r❡✱� ✷✵✶✹� ✸�✳� ❳✉✱� ❏✳❈✳� ❡t� ❛❧ ✳✱� P❤②s✳� ❋❧✉✐❞s✱� ✶✾✾✵� ✷�✳� ❈✐♠❜❛❧❛✱� ❏✳� ❛♥❞� ●❛r❣✱� ❙✳✱� ❆■❆❆✱� ✶✾✾✶� ✹�✳� ❇❛❣❤❡r✐✱� ❙✳� ❡t� ❛❧ ✳✱� P❤②s✳� ❘❡✈✳� ▲❡tt✳✱� ✷✵✶✷� ❏❡❛♥✲▲♦✉� P❋■❙❚❊❘� ✸� ✴� ✶✼� ❊❚❈✶✻✱� ❙t♦❝❦❤♦❧♠�
✵ ✵✻ ✷ ❋❧✉✐❞✴s♦❧✐❞� st❛❜✐❧✐t②� ❛♥❛❧②s✐s� ▼❡t❤♦❞s� ❋❧✉✐❞✲s♦❧✐❞� ✐♥t❡r❛❝t✐♦♥� ♠♦❞❡❧� ♣r♦❜❧❡♠� ✺� U ∞ ✺�✳� ▲❡❡✱� ❏✳� ❛♥❞� ❨♦✉✱� ❉✳✱� P❤②s✳� ❋❧✉✐❞s✱� ✷✵✶✸� ❏❡❛♥✲▲♦✉� P❋■❙❚❊❘� ✹� ✴� ✶✼� ❊❚❈✶✻✱� ❙t♦❝❦❤♦❧♠�
✵ ✵✻ ✷ ❋❧✉✐❞✴s♦❧✐❞� st❛❜✐❧✐t②� ❛♥❛❧②s✐s� ▼❡t❤♦❞s� ❋❧✉✐❞✲s♦❧✐❞� ✐♥t❡r❛❝t✐♦♥� ♠♦❞❡❧� ♣r♦❜❧❡♠� ✺� D U ∞ ✺�✳� ▲❡❡✱� ❏✳� ❛♥❞� ❨♦✉✱� ❉✳✱� P❤②s✳� ❋❧✉✐❞s✱� ✷✵✶✸� ❏❡❛♥✲▲♦✉� P❋■❙❚❊❘� ✹� ✴� ✶✼� ❊❚❈✶✻✱� ❙t♦❝❦❤♦❧♠�
❋❧✉✐❞✴s♦❧✐❞� st❛❜✐❧✐t②� ❛♥❛❧②s✐s� ▼❡t❤♦❞s� ❋❧✉✐❞✲s♦❧✐❞� ✐♥t❡r❛❝t✐♦♥� ♠♦❞❡❧� ♣r♦❜❧❡♠� ✺� D H = ✵ . ✵✻ D L = ✷ D U ∞ P❛r❛♠❡t❡rs� ✿� ◮ ρ s♦❧✐❞� /ρ ✢✉✐❞� = ✶�✱� ◮ P♦✐ss♦♥� ❝♦❡✣❝✐❡♥t� ν s = ✵ . ✹�✱� ✈❛r✐❛❜❧❡� r✐❣✐❞✐t②� ❝♦❡✣❝✐❡♥t� K ❇� ✱� ◮ ❱❛r✐❛❜❧❡� ❘❡②♥♦❧❞s� ♥✉♠❜❡r� R ❡� = U ∞ D /ν ✳� P❤②s✐❝s� ♠♦❞❡❧✐♥❣� ✿� ◮ ■♥❝♦♠♣r❡ss✐❜❧❡� ❧❛♠✐♥❛r� ✢♦✇✱� ◮ ◆♦♥✲❧✐♥❡❛r� ❡❧❛st✐❝� s♦❧✐❞� ✭❙❛✐♥t✲❱❡♥❛♥t� ❑✐r❝❤❤♦✛✮✳� ✺�✳� ▲❡❡✱� ❏✳� ❛♥❞� ❨♦✉✱� ❉✳✱� P❤②s✳� ❋❧✉✐❞s✱� ✷✵✶✸� ❏❡❛♥✲▲♦✉� P❋■❙❚❊❘� ✹� ✴� ✶✼� ❊❚❈✶✻✱� ❙t♦❝❦❤♦❧♠�
❚❤❡� ❝♦✉♣❧❡❞� ♣r♦❜❧❡♠� ✐s� t❤❡♥� ✇r✐tt❡♥� ❛s� q q ✵ ✐♥� ❛� ✜①❡❞� ❞♦♠❛✐♥� ✵ ◆❛✈✐❡r✲❙t♦❦❡s� ❡q✉❛t✐♦♥s� ✭✢✉✐❞✮� ✈❡❧♦❝✐t②� ✭s♦❧✐❞✴✢✉✐❞✮� ❊❧❛st✐❝✐t②� ❡q✉❛t✐♦♥s� ✭s♦❧✐❞✮� q ❞✐s♣❧❛❝❡♠❡♥t� ✭s♦❧✐❞✴�✢✉✐❞�✮� ❙tr❡ss✴✈❡❧♦❝✐t②� ❝♦♥t✐♥✉✐t②� ✭✐♥t❡r❢❛❝❡✮� ♣r❡ss✉r❡� ✭✢✉✐❞✮� ❋❧✉✐❞� ♠❡s❤� ❡q✉❛t✐♦♥� ❋❧✉✐❞✴s♦❧✐❞� st❛❜✐❧✐t②� ❛♥❛❧②s✐s� ▼❡t❤♦❞s� ❍♦✇� t♦� ❤❛♥❞❧❡� t❤❡� ✢✉✐❞✲str✉❝t✉r❡� ❝♦✉♣❧✐♥❣� ❚♦� ❤❛♥❞❧❡� t❤❡� ♠♦✈✐♥❣� ❞♦♠❛✐♥� ✿� ❆r❜✐tr❛r②� ▲❛❣r❛♥❣✐❛♥� ❊✉❧❡r✐❛♥� ❢♦r♠✉❧❛t✐♦♥� ✻� ❛❝t✉❛❧� ❞♦♠❛✐♥� Ω t ✭t✐♠❡✲❞❡♣❡♥❞❛♥t✮� r❡❢❡r❡♥❝❡� ❞♦♠❛✐♥� Ω ✵ ✭✜①❡❞✮� ✻�✳� ❉♦♥❡❛✱� ❏✳� ❡t� ❛❧✳� ❊♥❝✳� ❈♦♠♣✳� ▼❡❝❤✳✱� ✷✵✵✹� ❏❡❛♥✲▲♦✉� P❋■❙❚❊❘� ✺� ✴� ✶✼� ❊❚❈✶✻✱� ❙t♦❝❦❤♦❧♠�
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