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slide-1
SLIDE 1

❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ❙②st❡♠s ✇✐t❤ ●❧❛ss② ❉②♥❛♠✐❝s

❙t❡❛❞② st❛t❡ ❛♥❞ ✢✉❝t✉❛t✐♦♥ ♣r♦♣❡rt✐❡s ▼❛✉r♦ ❙❡❧❧✐tt♦

❉✐♣❛rt✐♠❡♥t♦ ❞✐ ■♥❣❡❣♥❡r✐❛ ■♥❞✉str✐❛❧❡ ❡ ❞❡❧❧✬■♥❢♦r♠❛③✐♦♥❡ ❙❡❝♦♥❞❛ ❯♥✐✈❡rs✐tà ❞✐ ◆❛♣♦❧✐

  • ●■ ✲ ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹

▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✶ ✴ ✸✺

slide-2
SLIDE 2

▼♦t✐✈❛t✐♦♥s

❯♥❞❡rst❛♥❞✐♥❣ t❤❡ ✐♥t❡r♣❧❛② ♦❢ ❞r✐✈✐♥❣ ❛♥❞ ❣❧❛ss✐♥❡ss✱ ✐♥ ◆♦♥✲♠♦♥♦t♦♥✐❝ ✭♦r ♥❡❣❛t✐✈❡ ❞✐✛❡r❡♥t✐❛❧✮ r❡s♣♦♥s❡ t♦ ❛ ❞r✐✈✐♥❣ ❢♦r❝❡

◮ ❙❤❡❛r✲t❤✐♥♥✐♥❣ ❛♥❞ s❤❡❛r✲t❤✐❝❦❡♥✐♥❣ ✭r❤❡♦❧♦❣②✮ ◮ ◆❡❣❛t✐✈❡ r❡s✐st❛♥❝❡ ✭✐♦♥ ❝❤❛♥♥❡❧s✱ ❤♦♠❡♦st❛t✐❝ ❜❛❧❛♥❝❡✳✳✳✮ ◮ ▼❛❝r♦♠♦❧❡❝✉❧❛r ❝r♦✇❞✐♥❣✿ ❛♥♦♠❛❧♦✉s ❞✐✛✉s✐♦♥

❋❧✉❝t✉❛t✐♦♥ r❡❧❛t✐♦♥s

◮ ❆♣♣r♦❛❝❤✐♥❣ t❤❡ ❧❛r❣❡✲❞❡✈✐❛t✐♦♥ r❡❣✐♠❡ ◮ ❚✐♠❡✲r❡✈❡rs❛❧ s②♠♠❡tr② ♦❢ ❝✉rr❡♥t ✢✉❝t✉❛t✐♦♥s ◮ ❊✛❡❝t✐✈❡ ✢✉❝t✉❛t✐♥❣ ❤②❞r♦❞②♥❛♠✐❝s ❞❡s❝r✐♣t✐♦♥

❆ ❞♦✉❜❧❡ ❝❤❛❧❧❡♥❣❡ ✲ ◆♦ ❣❡♥❡r❛❧ ●✐❜❜s✲❇♦❧t③♠❛♥♥ ❢r❛♠❡✇♦r❦ ✐♥ ◆❊❙❙ ✲ ❲❡ st✐❧❧ ❞♦ ♥♦t ❦♥♦✇ ✇❤❛t ❛♥ ❡q✉✐❧✐❜r✐✉♠ ❣❧❛ss ✐s

▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✷ ✴ ✸✺

slide-3
SLIDE 3

❚❤❡ ♠❛✐♥ ✐♥❣r❡❞✐❡♥t

❈❛❣❡ ❡✛❡❝t ✐♥ ✈✐s❝♦✉s ❧✐q✉✐❞s

❍✐❣❤❡r✴▲♦✇❡r ❞❡♥s✐t② r❡❣✐♦♥s ♣r❡✈❡♥t✴❢❛❝✐❧✐t❛t❡ ♣❛rt✐❝❧❡ r❡❛rr❛♥❣❡♠❡♥ts

❊①✳✿ ❛ ♣❛rt✐❝❧❡ r❛♥❞♦♠❧② ❥✉♠♣s t♦ ❛ ◆◆ ❤♦❧❡ ✐✛ ✐t ❤❛s ❛t ❧❡❛st ✷ ◆◆ ❤♦❧❡s ❜❡❢♦r❡ ❛♥❞ ❛❢t❡r t❤❡ ♠♦✈❡ ✭❞❡t❛✐❧❡❞ ❜❛❧❛♥❝❡✮✳ ◆♦ st❛t✐❝ ✐♥t❡r❛❝t✐♦♥ H = ✵ ❆t ❤✐❣❤ ❞❡♥s✐t② ❦✐♥❡t✐❝ ❝♦♥str❛✐♥ts ❛r❡ ❤❛r❞❧② s❛t✐s✜❡❞✱ s♦ ❞②♥❛♠✐❝s ✐s s❧♦✇ ◆♦ ♣❛rt✐❝❧❡ ✐s ♣❡r♠❛♥❡♥t❧② ❜❧♦❝❦❡❞✱ ✭✉♥❧❡ss ✐t ✇❛s s♦ ✐♥ t❤❡ ✐♥✐t✐❛❧ st❛t❡✮ ❯♥❧✐❦❡ ❣❡♦♠❡tr✐❝ r❡str✐❝t✐♦♥s✱ ✇❤❡r❡ st❛t❡s ✭✐♥st❡❛❞ ♦❢ ♠♦✈❡s✮ ❛r❡ ❢♦r❜✐❞❞❡♥✱ t❤❡ ❦✐♥❡t✐❝ r❡str✐❝t✐♦♥s ✐♠♣❧② ❛ tr✐✈✐❛❧ t❤❡r♠♦❞②♥❛♠✐❝s

▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✸ ✴ ✸✺

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SLIDE 4

P❧❛♥ ♦❢ t❛❧❦

❇♦✉♥❞❛r②✲❞r✐✈❡♥ tr❛♥s♣♦rt

◆❡❣❛t✐✈❡ ✭❞✐✛❡r❡♥t✐❛❧✮ r❡s✐st❛♥❝❡ ❛♥❞ ❞✐r❡❝t❡❞ ♠♦t✐♦♥ ❆❣✐♥❣ ❛♥❞ st❡❛❞② st❛t❡ r❡❣✐♠❡

❈♦♥str❛✐♥❡❞ ❡①❝❧✉s✐♦♥ ♣r♦❝❡ss❡s

❚r❛♥s♣♦rt ❛♥❞ r❡❧❛①❛t✐♦♥ ♣r♦♣❡rt✐❡s ❙♦♠❡ ❢❡❛t✉r❡s ♦❢ t❤❡ ◆❊❙❙ ♠❡❛s✉r❡

❋❧✉❝t✉❛t✐♦♥ s②♠♠❡tr② ❙t❡❛❞② st❛t❡ ✢✉❝t✉❛t✐♦♥ r❡❧❛t✐♦♥

❈✉rr❡♥t ✢✉❝t✉❛t✐♦♥s st❛t✐st✐❝s

▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✹ ✴ ✸✺

slide-5
SLIDE 5

❇♦✉♥❞❛r②✲❞r✐✈❡♥ tr❛♥s♣♦rt

❜♦✉♥❞❛r②✲✐♥❞✉❝❡❞ ❞✐ss✐♣❛t✐♦♥ ❞❡♥s✐t②✲❞❡♣❡♥❞❡♥t ❞✐✛✉s✐♦♥✱ ❉(ρ) ✈❛♥✐s❤✐♥❣ ❞✐✛✉s✐♦♥ ❛t ❤✐❣❤ ❞❡♥s✐t②

ρ+ − → ρ− ρ(③, t) ❧♦❝❛❧ ❞❡♥s✐t②✱ |③| ≤ ▲ ✳ P❛rt✐❝❧❡ r❡s❡r✈♦✐rs ❛t ρ(±▲, t) = ρ±✳

∂ρ ∂t = ∂ ∂③

  • ❉(ρ) ∂ρ

∂③

  • ❆♥ ❡①❛❝t❧② s♦❧✈❛❜❧❡ ❝❛s❡ ✐s ♦❜t❛✐♥❡❞ ❢♦r✿

❉(ρ) ∼ (ρ❝ − ρ)φ − → P♦r♦✉s ▼❡❞✐✉♠ ❊q✉❛t✐♦♥

▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✻ ✴ ✸✺

slide-6
SLIDE 6

◆♦♥✲❡q✉✐❧✐❜r✐✉♠ st❡❛❞② st❛t❡

❚❤✐s ✐s ♦❜t❛✐♥❡❞ ❜② s❡tt✐♥❣ ∂tρ = ✵ ✇✐t❤ ρ− < ρ+ < ρ❝ ❏(ρ+, ρ−) ∼ (ρ❝ − ρ−)✶+φ − (ρ❝ − ρ+)✶+φ ρ❝ − ρ(③) =

  • ❛+ − ❛−

③ ▲ ✶/(✶+φ)

✇✐t❤ ❛± = ✶

  • (ρ❝ − ρ−)✶+φ ± (ρ❝ − ρ+)✶+φ

  • ❞❡♥s✐t② ♣r♦✜❧❡ ✐s ♥♦♥❧✐♥❡❛r
  • ❝✉rr❡♥t ❞❡♣❡♥❞s ♦♥ ρ❝ − ρ±✱ ♥♦t s✐♠♣❧② ρ+ − ρ−

▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✼ ✴ ✸✺

slide-7
SLIDE 7

◆❡❣❛t✐✈❡ ❞✐✛❡r❡♥t✐❛❧ r❡s✐st❛♥❝❡

❚♦ ❦❡❡♣ t❤✐♥❣s s✐♠♣❧❡✱ ❝♦♥s✐❞❡r δ = ρ−/ρ+ ✜①❡❞✳

0.02 0.04 0.06 0.08 0.1 0.1 0.2 0.3 0.4 0.5 0.6

J(ρ+, ρ-) ρ+ - ρ-

❈✉rr❡♥t ♠❛①✐♠✉♠ ❛♥❞ ❝♦rr❡s♣♦♥❞✐♥❣ ❞r✐✈✐♥❣ ❢♦r❝❡ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❜② ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡rs ♠❡t❤♦❞✿ ❏♠❛① = ρ✶+φ

✶ + φ (✶ − δ)✶+φ

  • ✶ − δ

✶ ✶+φ

φ , (ρ+ − ρ−)♠❛① = ρ❝(✶ − δ) ✶ − δ

✶ φ

✶ − δ

✶+φ φ

. ▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✽ ✴ ✸✺

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SLIDE 8

❉✐r❡❝t❡❞ ♠♦t✐♦♥

❆s②♠♠❡tr✐❝ ♣✐❡❝❡✇✐s❡✲❝♦♥st❛♥t ♣❡r✐♦❞✐❝ ❢♦r❝✐♥❣

  • 0.4
  • 0.2

0.2 0.4 0.6 0.8 1 1 2 3 4 5

ρ+ - ρ- t α = 4

ρ±(t) = ✶ − τ✵/τ ♦r ✵✱ t ∈ [✵, τ✵] ✵ ♦r τ✵/τ✱ t ∈ [τ✵, τ]

③❡r♦ ❛✈❡r❛❣❡ ❜✐❛s

τ

✵ ∆ρ(t)❞t = ✵

  • 0.3
  • 0.2
  • 0.1

0.1 0.2 0.3 0.01 0.1 1 10 100

Jav α

❏❛✈(α) = ✶ τ τ

❏[ρ+(t), ρ−(t)]❞t

❆s②♠♠❡tr② ♣❛r❛♠❡t❡r α = τ/τ✵ − ✶

♦♣t✐♠❛❧ ♣✉♠♣✐♥❣ ❝♦♥❞✐t✐♦♥

▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✾ ✴ ✸✺

slide-9
SLIDE 9

◆♦♥✲st❛t✐♦♥❛r② ❛❣✐♥❣ r❡❣✐♠❡

❙❧♦✇ ❣❧❛ss② ❞②♥❛♠✐❝s ✐s ♦❜t❛✐♥❡❞ ❜② s❡tt✐♥❣ ρ− = ρ+ = ρ❝

▲♦♦❦✐♥❣ ❢♦r s♦❧✉t✐♦♥ ♦❢ t❤❡ ❢♦r♠ ρ❝ − ρ(③, t) = ❢ (③) ❣(t)✱ ✇❡ ❣❡t✿

❢ (③) =

  • ❢ φ(③)❢ ′(③)

✇✐t❤ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥✿ ❢ (±▲) = ✵,

❣′(t) = ❣✶+φ(t).

❙♦♠❡ ✐♥t❡r❡st✐♥❣ ❢❡❛t✉r❡s

P♦✇❡r✲❧❛✇ ❝r✐t✐❝❛❧ r❡❧❛①❛t✐♦♥✿ ρ❝ − ρ(③, t) ∼ t−✶/φ ❙✐♠♣❧❡ ❛❣✐♥❣ ❜❡❤❛✈✐♦✉r✿ ❇(t, t✇) =

t

t✇

❉(s) ❞s ∼ ❧♦❣ t − ❧♦❣ t✇

❚r✐❛♥❣❧❡ r❡❧❛t✐♦♥✿ ❇(t, t✇) = ❇(t, s) + ❇(s, t✇) ❲❡❛❦✲❡r❣♦❞✐❝✐t② ❜r❡❛❦✐♥❣✿

❧✐♠

t→∞ ❧✐♠ t✇→∞ ❇(t + t✇, t✇) =

❧✐♠

t✇→∞ ❧✐♠ t→∞ ❇(t + t✇, t✇)

▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✶✵ ✴ ✸✺

slide-10
SLIDE 10

❈♦♥str❛✐♥❡❞ ❧❛tt✐❝❡ ❣❛s❡s✿ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ❝❛s❡

✷❞ ❑❆ ♠♦❞❡❧✿ ❛ ♣❛rt✐❝❧❡ ♦♥ ❛ sq✉❛r❡ ❧❛tt✐❝❡ ❝❛♥ ❥✉♠♣ t♦ ❛ ◆◆ ✈❛❝❛♥❝② ✐✛ ✐t ❤❛s ❧❡ss t❤❛♥ ✸ ◆◆ ♣❛rt✐❝❧❡s ❜❡❢♦r❡ ❛♥❞ ❛❢t❡r ✐t ❤❛s ❥✉♠♣❡❞✳ ❙✐♥❣✉❧❛r ❞✐✛✉s✐♦♥ ❛t ❤✐❣❤✲❞❡♥s✐t② ✭✈✐❛ ❛ ❜♦♦tstr❛♣ ♣❡r❝♦❧❛t✐♦♥ ❛r❣✉♠❡♥t✮✿

❉(ρ) ∼ ❡①♣ ❝ ✶ − ρ

❛♥❞ s❡✈❡r❛❧ ❣❧❛ss② ❢❡❛t✉r❡s✱ ❡✳❣✳✿ ❊①t❡♥s✐✈❡ ❡♥tr♦♣② ♦❢ ♠❡t❛st❛❜❧❡ ✭= ♣❡r♠❛♥❡♥t❧② ❜❧♦❝❦❡❞✮ st❛t❡s ❙tr❡t❝❤❡❞ ❡①♣♦♥❡♥t✐❛❧ r❡❧❛①❛t✐♦♥✱ ❛❣✐♥❣ ❞②♥❛♠✐❝s ❛♥❞ ❤❡t❡r♦❣❡♥❡✐t② ❊r❣♦❞✐❝✐t② ❜r❡❛❦✐♥❣ ♦♥ ❇❡t❤❡ ❧❛tt✐❝❡ s✐♠✐❧❛r t♦ ▼♦❞❡✲❈♦✉♣❧✐♥❣ ❚❤❡♦r② ✭❤②❜r✐❞ tr❛♥s✐t✐♦♥ ❛♥❞ ❤✐❣❤❡r✲♦r❞❡r ❣❧❛ss s✐♥❣✉❧❛r✐t✐❡s✮

▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✶✷ ✴ ✸✺

slide-11
SLIDE 11

❈♦♥str❛✐♥❡❞ ❡①❝❧✉s✐♦♥ ♣r♦❝❡ss❡s✿ t❤❡ s②♠♠❡tr✐❝ ❝❛s❡

✷❉ ❑❆ ♠♦❞❡❧ ❜♦✉♥❞❛r②✲❞r✐✈❡♥ ❜② t✇♦ r❡s❡r✈♦✐rs ❛t ✉♥❡q✉❛❧ ❞❡♥s✐t✐❡s✳ ◆♦ ♠✐❝r♦s❝♦♣✐❝ ❜✐❛s✳ ❙t❡❛❞② ❝✉rr❡♥t ❞r✐✈❡♥ ❜② t❤❡ ❞❡♥s✐t② ❣r❛❞✐❡♥t✳

❞❡♥s✐t② ♣r♦✜❧❡ ❧♦❝❛❧ ❞❡♥s✐t② ✢✉❝t✉❛t✐♦♥s

0.54 0.61 0.68 0.75 0.82

  • 1
  • 0.6
  • 0.2

0.2 0.6 1

<> z /L

L = 50

0.15 0.2 0.25

  • 1
  • 0.6
  • 0.2

0.2 0.6 1

(<

2> - <> 2) L 2

z /L

L = 50

  • r❡❡♥ ❧✐♥❡s r❡❢❡r t♦ t❤❡ ✉♥❝♦♥str❛✐♥❡❞ ❞②♥❛♠✐❝s✱ t❤❛t ✐s t♦ t❤❡ ❙②♠♠❡tr✐❝ ❊①❝❧✉s✐♦♥ Pr♦❝❡ss ✭❙❊P✮

▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✶✸ ✴ ✸✺

slide-12
SLIDE 12

❆s②♠♠❡tr✐❝ ❝♦♥str❛✐♥❡❞ ❡①❝❧✉s✐♦♥ ♣r♦❝❡ss❡s

✷❞ ❑❆ ♠♦❞❡❧ ❜✉❧❦✲❞r✐✈❡♥ ❜② ❛ ❝♦♥st❛♥t ❛♥❞ ✉♥✐❢♦r♠ ❛♣♣❧✐❡❞ ❢♦r❝❡ ❊✳ P❛rt✐❝❧❡s ❤♦♣ t♦ ❛ ♥❡❛r❜② ❡♠♣t② s✐t❡ ✇✐t❤ ♣r♦❜❛❜✐❧✐t②✿

♣ = δ(❝♦♥str❛✐♥t) × ♠✐♥

  • ✶, ❡

− → ❊ ·− → ❞r

❉❡t❛✐❧❡❞ ❜❛❧❛♥❝❡ ❤♦❧❞s ❧♦❝❛❧❧② ❜✉t ♥♦t ❣❧♦❜❛❧❧② ❞✉❡ t♦ t❤❡ ♣❡r✐♦❞✐❝ ❜♦✉♥❞❛r②

external field E

▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✶✹ ✴ ✸✺

slide-13
SLIDE 13

P♦s✐t✐✈❡ r❡s✐st❛♥❝❡ r❡❣✐♠❡

▼♦♥♦t♦♥✐❝ ❝✉rr❡♥t ❜❡❤❛✈✐♦✉r ✐s ♦❜s❡r✈❡❞ ❢♦r ♣❛rt✐❝❧❡ ❞❡♥s✐t② ρ < ✵.✼✾

0.1 0.2 2 4 6 8

J E

(a)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1 2 3 4 5 0.2 0.4 0.6 0.8 1

J/Jsat E

(b)

1 - e-E = 0.1 = 0.2 = 0.3 = 0.4 = 0.5 = 0.6 = 0.7

❖❤♠✐❝ tr❛♥s♣♦rt ❛t s♠❛❧❧ ✜❡❧❞✱ s❛t✉r❛t✐♦♥ ❛t ❧❛r❣❡ ✜❡❧❞s✱ ❥✉st ❛s ✐♥ ❆❙❊P ❚r✐✈✐❛❧ ✜❡❧❞ ❞❡♣❡♥❞❡♥❝❡ ♦❢ r❡s❝❛❧❡❞ ❝✉rr❡♥t ❏(ρ, ❊)/❏(ρ, ∞) = ✶ − ❡−❊

❋♦r st❛♥❞❛r❞ ❆❙❊P ❏(ρ, ❊) = ρ(✶ − ρ)(✶ − ❡−❊ )/✹. ▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✶✺ ✴ ✸✺

slide-14
SLIDE 14

❚❤❡ t♦t❛❧❧② ❛s②♠♠❡tr✐❝ ❝❛s❡✿ ❊ → ∞

❙❛t✉r❛t✐♦♥ ❝✉rr❡♥t ❝❛♥ ❜❡ ❛♣♣r♦①✐♠❛t❡❞ ✈✐❛ ❛ s✐♠♣❧❡ ♠❡❛♥✲✜❡❧❞ ❛r❣✉♠❡♥t ❏(ρ) ≈ ✶

✹ ρ(✶ − ρ) (✶ − ρ✸)✷

2 4 6 0.25 0.5 0.75 1

J x 10-2

  • MEAN-FIELD

TASEP TACEP

▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✶✻ ✴ ✸✺

slide-15
SLIDE 15

■♥t❡r♣❧❛② ♦❢ ❞r✐✈✐♥❣ ❛♥❞ ❝♦♥str❛✐♥ts ❛t ❧❛r❣❡ ρ ❛♥❞ ❊

◗✉❛❧✐t❛t✐✈❡❧②✿ P❛rt✐❝❧❡ ♠♦✈❡s ❛❣❛✐♥st t❤❡ ✜❡❧❞ ❛r❡ r❛r❡ ⇓ P❛rt✐❝❧❡s ❛r❡ ❣❡♥❡r❛❧❧② ♠♦r❡ ❝❛❣❡❞ ⇓ ❘❡❛rr❛♥❣❡♠❡♥ts ❛r❡ ♠♦r❡ ❞✐✣❝✉❧t ⇓ ❋❧♦✇ ✐s ♠♦r❡ ♦❜str✉❝t❡❞

▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✶✼ ✴ ✸✺

slide-16
SLIDE 16

◆❡❣❛t✐✈❡ r❡s✐st❛♥❝❡ ❛♥❞ ❥❛♠♠✐♥❣

◆♦♥✲♠♦♥♦t♦♥✐❝ ❝✉rr❡♥t ❜❡❤❛✈✐♦✉r ✐s ♦❜s❡r✈❡❞ ❢♦r ρ ≥ ✵.✼✾

1 2 3 4 2 4 6 8 10

J x 10-3 E

L = 400

ρ = 0.79 ρ = 0.80 ρ = 0.81 ρ = 0.82

2 4 6 8 10 2 4 6 8 10

J x 10-4 E

L = 400

ρ = 0.83 ρ = 0.84 ρ = 0.85 ρ = 0.86

❖❤♠✐❝ r❡❣✐♠❡ s❤r✐♥❦s ❛t ❧❛r❣❡r ❞❡♥s✐t② ❆♣♣❛r❡♥t ❥❛♠♠✐♥❣ ❛t ❧❛r❣❡ ✜❡❧❞s ❚✇♦ r❡❣✐♠❡s ♦❢ ✈❛♥✐s❤✐♥❣❧② s♠❛❧❧ ❝✉rr❡♥t✴❡♥tr♦♣②✲♣r♦❞✉❝t✐♦♥

▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✶✽ ✴ ✸✺

slide-17
SLIDE 17

✏◆♦♥✲◆❡✇t♦♥✐❛♥✑ ❢❡❛t✉r❡s

◆♦♥✲♠♦♥♦t♦♥✐❝ ✜❡❧❞✲❞❡♣❡♥❞❡♥❝❡ ♦❢ str✉❝t✉r❛❧ r❡❧❛①❛t✐♦♥ t✐♠❡ ∼ ✈✐s❝♦s✐t②

2 3 4 5 6 1 2 3 4 5 6

log10 τrel

E

ρ = 0.86 ρ = 0.84 ρ = 0.82 ρ = 0.80 ρ = 0.78

✏❚❤✐♥♥✐♥❣✑ r❡❣✐♠❡ ❛t s♠❛❧❧ ❊ ✏❚❤✐❝❦❡♥✐♥❣✑ ❛t ❧❛r❣❡r ❊

▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✶✾ ✴ ✸✺

slide-18
SLIDE 18

❇❧♦❝❦❡❞ ♣❛rt✐❝❧❡s ❛♥❞ ❢r❡❡ ✈♦❧✉♠❡

❆t ✈❛r✐❛♥❝❡ ✇✐t❤ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ♠❡❛s✉r❡ ❛♥❞ ✇✐t❤ t❤❡ ❆❙❊P✱ ❝♦♥st❛♥t✲❞❡♥s✐t② ❝♦♥✜❣✉r❛t✐♦♥s ❛r❡ ♥♦t ❡q✉✐♣r♦❜❛❜❧❡ ✐♥ ◆❊❙❙

❞❡♥s✐t② ♦❢ ❜❧♦❝❦❡❞ ♣❛rt✐❝❧❡s ❞❡♥s✐t② ♦❢ ❛❝❝❡ss✐❜❧❡ ❤♦❧❡s

0.7 0.74 0.78 0.82 1 2 3 4 5 6

ρblocked E

ρ = 0.80 ρ = 0.81 ρ = 0.83 0.01 0.03 0.05 0.07 0.09 1 2 3 4 5 6

ρholes E

ρ = 0.80 ρ = 0.81 ρ = 0.83

❚❤❡ ◆❊❙❙ ♠❡❛s✉r❡ ❞❡♣❡♥❞s ✐♥ ❛ ♥♦♥tr✐✈✐❛❧ ♠❛♥♥❡r ♦♥ t❤❡ ❛♣♣❧✐❡❞ ✜❡❧❞

▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✷✵ ✴ ✸✺

slide-19
SLIDE 19

P❛✐r ❝♦rr❡❧❛t✐♦♥

▲♦♥❣✐t✉❞✐♥❛❧ ♣❛✐r ❝♦rr❡❧❛t✐♦♥

  • 0.08
  • 0.06
  • 0.04
  • 0.02

0.02 1 2 3 4 5 6 7 8

gL r

ρ = 0.86 E = 0.4 E = 0.8 E = 1.2 E = 1.6 E = 2.0

▲✐q✉✐❞✲❧✐❦❡ s❤♦rt✲r❛♥❣❡ ❧♦♥❣✐t✉❞✐♥❛❧ r❡♣✉❧s✐♦♥ ❋❧✉❝t✉❛t✐♦♥✲✐♥❞✉❝❡❞ tr❛♥s✈❡rs❡ ❛ttr❛❝t✐♦♥

❚r❛♥s✈❡rs❡ ♣❛✐r ❝♦rr❡❧❛t✐♦♥

  • 0.04
  • 0.03
  • 0.02
  • 0.01

0.01 0.02 0.03 0.04 1 2 3 4 5 6 7 8 9 10

gT r

ρ = 0.86 E = 0.4 E = 0.8 E = 1.2 E = 1.6 E = 2.0

▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✷✶ ✴ ✸✺

slide-20
SLIDE 20

❆ ♠❡❛♥✲✜❡❧❞ ❛tt❡♠♣t

■♥ ❛♥❛❧♦❣② ✇✐t❤ t❤❡ ❆❙❊P ✇r✐t❡ ❏(ρ, ❊) ∼ (✶ − ❡−❊) (✶ − ρ❜❧♦❝❦❡❞) ρ❤♦❧❡

1 2 1 2 3 4 5 rescaled current

E

= 0.70 = 0.81

❚❤❡ tr❛♥s✐t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ♣♦s✐t✐✈❡ ❛♥❞ ◆❘ r❡❣✐♠❡s ✐s ✇❡❧❧ ❞❡s❝r✐❜❡❞✱ ❤♦✇❡✈❡r t❤❡ ❛♣♣r♦❛❝❤ ❢❛✐❧s ❛s s♦♦♥ ❛s ρ > ✵.✽✶✱ ❢r❡❡ ♣❛rt✐❝❧❡s ❛♥❞ ❤♦❧❡s ❛r❡ str♦♥❣❧② ❝♦rr❡❧❛t❡❞

▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✷✷ ✴ ✸✺

slide-21
SLIDE 21

❆♥♦♠❛❧♦✉s ❞✐✛✉s✐♦♥

❚✐♠❡ ❛✈❡r❛❣❡❞ ❧♦♥❣✐t✉❞✐♥❛❧ ♠❡❛♥✲sq✉❛r❡ ❞✐s♣❧❛❝❡♠❡♥t

0.02 0.04 0.06 0.08 0.1 0.12

  • 1

1 2 3 4 5 6 7

∆r❘ ❘

2 /t

log10 t

ρ = 0.8, L = 400 E = 0.1 E = 0.3 E = 0.6 E = 1.2 E = 2.8

0.02 0.04 0.06 0.08 0.1

  • 1

1 2 3 4 5 6 7

∆r❘ ❘

2 /t

log10 t

ρ = 0.8, E = 2.8 L = 50 100 200 400

▲♦♥❣✐t✉❞✐♥❛❧ ❞✐✛✉s✐♦♥ ✐s ❣❡♥❡r❛❧❧② ❡♥❤❛♥❝❡❞ ❛t ❧❛t❡ t✐♠❡s ❙✉❜✲❞✐✛✉s✐♦♥ r❡❣✐♠❡ ❛t ❡❛r❧② t✐♠❡s ❢♦r s♠❛❧❧ ❛♣♣❧✐❡❞ ✜❡❧❞s ▲♦♥❣✲❧✐✈❡❞ ❧♦♥❣✐t✉❞✐♥❛❧ s✉♣❡r✲❞✐✛✉s✐♦♥ r❡❣✐♠❡ ❛t ❧❛r❣❡r ❊

▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✷✸ ✴ ✸✺

slide-22
SLIDE 22

✸✳ ❙t❡❛❞② st❛t❡ ✢✉❝t✉❛t✐♦♥ r❡❧❛t✐♦♥

❆ s②♠♠❡tr② ♣r♦♣❡rt② ♦❢ t❤❡ P❉❋ ♦❢ ❡♥tr♦♣② ♣r♦❞✉❝t✐♦♥ ❲τ ♦✈❡r ❧♦♥❣ t✐♠❡ τ

Πτ(+❲τ) Πτ(−❲τ) = ❡❲τ

✐✳❡✳✱ ✇❤❡♥ Πτ s❛t✐s✜❡s t❤❡ t✐♠❡ s❝❛❧✐♥❣ ♦❢ t❤❡ ❧❛r❣❡✲❞❡✈✐❛t✐♦♥ r❡❣✐♠❡ ✭τ → ∞✮

Πτ(❲τ) = ❡τπ(❲τ/τ)

❋♦r ✈❛♥✐s❤✐♥❣ ❞r✐✈✐♥❣ ❢♦r❝❡s ②♦✉ ❣❡t ✢✉❝t✉❛t✐♦♥✲❞✐ss✐♣❛t✐♦♥ t❤❡♦r❡♠

✷ ❲τ = ❲ ✷

τ − ❲τ✷

❚✇♦ s❡r✐♦✉s ♣r♦❜❧❡♠s ❲τ ✐s ❡①t❡♥s✐✈❡ ✐♥ t✐♠❡ ❛♥❞ s♣❛❝❡✱ ❛♥❞ ♠♦♥♦t♦♥✐❝ ✐♥ t❤❡ ❞r✐✈✐♥❣ ❢♦r❝❡ ❍♦✇ ❧❛r❣❡ τ ♠✉st ❜❡❄ t❤❡ ❧❛r❣❡✲❞❡✈✐❛t✐♦♥ r❡❣✐♠❡ ♠❛② ❜❡ ♦✉t ♦❢ r❡❛❝❤

▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✷✺ ✴ ✸✺

slide-23
SLIDE 23

❊♥tr♦♣② ♣r♦❞✉❝t✐♦♥

❈♦♥s✐❞❡r t❤❡ ❛❝t✐♦♥ ❢✉♥❝t✐♦♥❛❧

❲τ({σ}) = ❧♦❣ ❦(σ✵, σ✶) · · · ❦(στ−✶, στ) ❦(στ, στ−✶) · · · ❦(σ✶, σ✵)

✇❤❡r❡ ❦(σ, σ′) ≥ ✵ ❛r❡ t❤❡ tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ❢♦r σ → σ′✳ ❉❡t❛✐❧❡❞ ❜❛❧❛♥❝❡ ❧♦❝❛❧❧② ❤♦❧❞s✱ ❛♥❞ ❢♦r ♠♦❜✐❧❡ ♣❛rt✐❝❧❡s✿

❧♦❣ ❦(σ, σ′) ❦(σ′, σ) = ❧♦❣ ♠✐♥

  • ✶, ❡

− → ❊ ·− → ❞r

♠✐♥

  • ✶, ❡−−

→ ❊ ·− → ❞r

= ❊ · ❞r = ✵, ±❊

❙♦✱ t❤❡ ❛❝t✐♦♥ ❢✉♥❝t✐♦♥❛❧ ❲τ r❡♣r❡s❡♥ts t❤❡ t❤❡r♠♦❞②♥❛♠✐❝ ❡♥tr♦♣② ♣r♦❞✉❝t✐♦♥✿

❲τ = ❊ ❏τ

▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✷✻ ✴ ✸✺

slide-24
SLIDE 24

❈✉rr❡♥t ✢✉❝t✉❛t✐♦♥s ■✿ ▲♦✇✲❞❡♥s✐t② ❖❤♠✐❝ r❡❣✐♠❡

❈✉rr❡♥t ✢✉❝t✉❛t✐♦♥s ❛r❡ ❣❡♥❡r❛❧❧② ●❛✉ss✐❛♥

  • 10
  • 8
  • 6
  • 4
  • 2
  • 8
  • 6
  • 4
  • 2

2 4 6 8 log10 (στ Πτ)

(Jτ-J)/στ E = 0.1, ρ = 0.4

τ = 1 τ = 2 τ = 4 τ = 8 1 2 3 4 1 2 3 4 log[Πτ(Jτ)/Πτ(-Jτ)]

E Jτ E = 0.1, ρ = 0.4

τ = 1

❋❘ ✐s ❛❧✇❛②s s❛t✐s✜❡❞

▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✷✼ ✴ ✸✺

slide-25
SLIDE 25

❈✉rr❡♥t ✢✉❝t✉❛t✐♦♥s ■■✿ ❍✐❣❤✲❞❡♥s✐t② ❖❤♠✐❝ r❡❣✐♠❡

  • 10
  • 8
  • 6
  • 4
  • 2
  • 8
  • 6
  • 4
  • 2

2 4 6 8 log10 (στ Πτ)

(Jτ-J)/στ E = 0.1, ρ = 0.8 L = 25 τ = 8

16 32 64

s♠❛❧❧ ♥♦♥✲●❛✉ss✐❛♥ t❛✐❧ ❞❡✈❡❧♦♣s ♥♦ τ✲❞❡♣❡♥❞❡♥❝❡ ✐♥ r❡s❝❛❧❡❞ P❉❋ ❘❡❧❛t✐✈❡ ✢✉❝t✉❛t✐♦♥s ❞❡❝r❡❛s❡ ❙♦♠❡ ✜♥✐t❡✲s✐③❡ ❡✛❡❝ts

20 25 30 35 40 45 1 10 100 1000 10000

στ

2/J

τ E = 0.1, ρ = 0.8 L = 10 L = 15 L = 25 L = 40 ▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✷✽ ✴ ✸✺

slide-26
SLIDE 26

❋❘ ✐♥ t❤❡ ❤✐❣❤✲❞❡♥s✐t② ❖❤♠✐❝ r❡❣✐♠❡

❖❜s❡r✈❛❜❧❡ ❞❡✈✐❛t✐♦♥s ❢r♦♠ ❋❘

0.4 0.8 1.2 1.6 2 0.4 0.8 1.2 1.6 2 log[Πτ(Jτ)/Πτ(-Jτ)]

E Jτ E = 0.1, ρ = 0.8, L = 25 τ = 2 τ = 8 τ = 32 τ = 128 τ = 512

❋❘ ✐s ❡✈❡♥t✉❛❧❧② r❡❝♦✈❡r❡❞✱ ♣r♦✈✐❞❡❞ s②st❡♠ s✐③❡ ✐s ♥♦t t♦♦ ❧❛r❣❡

▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✷✾ ✴ ✸✺

slide-27
SLIDE 27

❈✉rr❡♥t ✢✉❝t✉❛t✐♦♥s ✐♥ t❤❡ ◆❘ r❡❣✐♠❡

  • 10
  • 8
  • 6
  • 4
  • 2
  • 6
  • 4
  • 2

2 4 6 8 10 log10 (στ Πτ)

(Jτ-J)/στ E = 2.8, ρ = 0.8 L = 45, τ = 1

♥♦♥✲●❛✉ss✐❛♥ ✢✉❝t✉❛t✐♦♥s ❣❡♥❡r❛❧❧② s❦❡✇❡❞ P❉❋ str♦♥❣ τ✲❞❡♣❡♥❞❡♥❝❡ ♦❢ P❉❋ r❡❧❛t✐✈❡ ✢✉❝t✉❛t✐♦♥s ✐♥❝r❡❛s❡ s❧♦✇ ❛♣♣r♦❛❝❤ t♦ ❛s②♠♣t♦♣②

  • 0.4
  • 0.2

0.2 0.4 0.6 0.8 1 1 10 100 1000 10000

τ E = 2.8, ρ = 0.8, L = 45

στ

2/J

skewness kurtosis

▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✸✵ ✴ ✸✺

slide-28
SLIDE 28

❙t❡❛❞②✲st❛t❡ ❋❘ ✐♥ t❤❡ ◆❘ r❡❣✐♠❡

❚✐♠❡✲r❡✈❡rs❛❧ s②♠♠❡tr② ♦❢ ❋❘ ✐s r❡s♣❡❝t❡❞✱ ❜✉t✳✳✳

5 10 15 20 25 5 10 15 20 25 log[Πτ(Jτ)/Πτ(-Jτ)]

E Jτ E = 2.8, ρ = 0.8

τ = 1 τ = 4 τ = 8 τ = 16

❞❡✈✐❛t✐♦♥s ❢r♦♠ ❋❘ t❡♥❞ t♦ ✐♥❝r❡❛s❡ ❛t ❧♦♥❣❡r t✐♠❡s

▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✸✶ ✴ ✸✺

slide-29
SLIDE 29

▲❛r❣❡ ❉❡✈✐❛t✐♦♥ ❋✉♥❝t✐♦♥

❚✉r❝✐ P✐t❛r❞✱ ❊P▲ ✭✷✵✶✶✮

❆s②♠♠❡tr✐❝ ❛♥❞ s✐♥❣✉❧❛r ▲❉❋ ❛t ❧❛r❣❡ ❞❡♥s✐t② ❛♥❞ ✜❡❧❞

▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✸✷ ✴ ✸✺

slide-30
SLIDE 30

❈♦♥❝❧✉s✐♦♥s ❛♥❞ s♦♠❡ ♦♣❡♥ ♣r♦❜❧❡♠s

❉r✐✈❡♥ ❞✐✛✉s✐✈❡ s②st❡♠s ✇✐t❤ ❣❧❛ss② ❞②♥❛♠✐❝s ❣❡♥❡r❛❧❧② ❡①❤✐❜✐t✿

◆♦♥✲♠♦♥♦t♦♥✐❝ tr❛♥s♣♦rt ◆♦♥✲◆❡✇t♦♥✐❛♥ ❢❡❛t✉r❡s ◆♦♥✲●❛✉ss✐❛♥ ✢✉❝t✉❛t✐♦♥s ❆♥♦♠❛❧♦✉s ✭s✉❜ ❛♥❞ s✉♣❡r✮ ❞✐✛✉s✐♦♥ ❖❜s❡r✈❛❜❧❡ ❞❡✈✐❛t✐♦♥s ❢r♦♠ ❋❘ P❤②s✐❝❛❧ ✐rr✐❧❡✈❛♥❝❡ ♦❢ ❧❛r❣❡✲❞❡✈✐❛t✐♦♥ r❡❣✐♠❡ ✕❃ ●♦✐♥❣ ❜❡②♦♥❞ t❤❡ ♥♦♥✲❧✐♥❡❛r ❞✐✛✉s✐♦♥ ♠♦❞❡❧ ✕❃ ❊✛❡❝t✐✈❡ ✢✉❝t✉❛t✐♥❣ ❤②❞r♦❞②♥❛♠✐❝ ❞❡s❝r✐♣t✐♦♥ ✕❃ P❤❛s❡ ❞✐❛❣r❛♠ ♦❢ ❚❆❈❊P ✇✐t❤ ♦♣❡♥ ❜♦✉♥❞❛r✐❡s ✕❃ ❊①♣❧♦r✐♥❣ s✉r❢❛❝❡ ❣r♦✇t❤ ♣r♦❜❧❡♠s ✭❆❙❊P ↔ ❑P❩✱ ❆❈❊P ↔ ❄✮ ✕❃ ❈♦♥str❛✐♥ts ✇✐t❤ ♥♦♥✉♥✐❢♦r♠ ❞r✐✈❡ ✭r❤❡♦❧♦❣②✮

▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✸✸ ✴ ✸✺

slide-31
SLIDE 31

❘❡❧❛t❡❞ ♣❛♣❡rs

▲ P❡❧✐t✐ ❛♥❞ ▼ ❙❡❧❧✐tt♦✱ ❆❣✐♥❣ ✐♥ ❛ s✐♠♣❧❡ ♠♦❞❡❧ ♦❢ str✉❝t✉r❛❧ ❣❧❛ss✱ ❏✳ P❤②s✐q✉❡ ❱■ ✽✱ ✹✾ ✭✶✾✾✽✮ ▼✳ ❙❡❧❧✐tt♦✱ ❋❧✉❝t✉❛t✐♦♥s ♦❢ ❡♥tr♦♣② ♣r♦❞✉❝t✐♦♥ ✐♥ ❞r✐✈❡♥ ❧❛tt✐❝❡ ❣❧❛ss❡s✱ ❝♦♥❞✲♠❛t✴✾✽✵✾✶✽✻ ✭✉♥♣✉❜❧✐s❤❡❞✮ ▼ ❙❡❧❧✐tt♦✱ ❉r✐✈❡♥ ❧❛tt✐❝❡✲❣❛s ❛s ❛ r❛t❝❤❡t ❛♥❞ ♣❛✇❧ ♠❛❝❤✐♥❡✱ P❤②s✳ ❘❡✈✳ ❊ ✻✺ ✵✷✵✶✵✶ ✭✷✵✵✷✮ ▼ ❙❡❧❧✐tt♦✱ ❆s②♠♠❡tr✐❝ ❡①❝❧✉s✐♦♥ ♣r♦❝❡ss❡s ✇✐t❤ ❝♦♥str❛✐♥❡❞ ❞②♥❛♠✐❝s✱ P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✶✵✶✱ ✵✹✽✸✵✶ ✭✷✵✵✽✮ ▼ ❙❡❧❧✐tt♦✱ ❋❧✉❝t✉❛t✐♦♥ r❡❧❛t✐♦♥ ❛♥❞ ❤❡t❡r♦❣❡♥❡♦✉s s✉♣❡r❞✐✛✉s✐♦♥ ✐♥ ❣❧❛ss② tr❛♥s♣♦rt✱ P❤②s✳ ❘❡✈✳ ❊ ✽✵✱ ✵✶✶✶✸✹ ✭✷✵✵✾✮ ❋ ❚✉r❝✐✱ ❊ P✐t❛r❞✱ ▼ ❙❡❧❧✐tt♦✱ ❉r✐✈✐♥❣ ❦✐♥❡t✐❝❛❧❧② ❝♦♥str❛✐♥❡❞ ♠♦❞❡❧s ✐♥t♦ ♥♦♥✲❡q✉✐❧✐❜r✐✉♠ st❡❛❞② st❛t❡s✿ str✉❝t✉r❛❧ ❛♥❞ s❧♦✇ tr❛♥s♣♦rt ♣r♦♣❡rt✐❡s✱ P❤②s✳ ❘❡✈✳ ❊ ✽✻✱ ✵✸✶✶✶✷ ✭✷✵✶✷✮

▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✸✹ ✴ ✸✺

slide-32
SLIDE 32

❘❡❧❡✈❛♥t s♣❛t✐❛❧ str✉❝t✉r❡s

E ≫ 1

1) 2) 3) 4)

▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✸✺ ✴ ✸✺