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❉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❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✶ ✴ ✸✺

▼♦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❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✷ ✴ ✸✺

❚❤❡ ♠❛✐♥ ✐♥❣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❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✸ ✴ ✸✺

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❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✹ ✴ ✸✺

❇♦✉♥❞❛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 = ∂ � ❉ ( ρ ) ∂ρ � ∂ ③ ∂ ③ ❉ ( ρ ) ∼ ( ρ ❝ − ρ ) φ ❆♥ ❡①❛❝t❧② s♦❧✈❛❜❧❡ ❝❛s❡ ✐s ♦❜t❛✐♥❡❞ ❢♦r✿ − → P♦r♦✉s ▼❡❞✐✉♠ ❊q✉❛t✐♦♥ ▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✻ ✴ ✸✺

◆♦♥✲❡q✉✐❧✐❜r✐✉♠ st❡❛❞② st❛t❡ ❚❤✐s ✐s ♦❜t❛✐♥❡❞ ❜② s❡tt✐♥❣ ∂ t ρ = ✵ ✇✐t❤ ρ − < ρ + < ρ ❝ ( ρ ❝ − ρ − ) ✶ + φ − ( ρ ❝ − ρ + ) ✶ + φ ❏ ( ρ + , ρ − ) ∼ ③ � ✶ / ( ✶ + φ ) � ρ ❝ − ρ ( ③ ) = ❛ + − ❛ − ▲ ( ρ ❝ − ρ − ) ✶ + φ ± ( ρ ❝ − ρ + ) ✶ + φ � � ✇✐t❤ ❛ ± = ✶ ✳ ✷ • ❞❡♥s✐t② ♣r♦✜❧❡ ✐s ♥♦♥❧✐♥❡❛r • ❝✉rr❡♥t ❞❡♣❡♥❞s ♦♥ ρ ❝ − ρ ± ✱ ♥♦t s✐♠♣❧② ρ + − ρ − ▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✼ ✴ ✸✺

◆❡❣❛t✐✈❡ ❞✐✛❡r❡♥t✐❛❧ r❡s✐st❛♥❝❡ ❚♦ ❦❡❡♣ t❤✐♥❣s s✐♠♣❧❡✱ ❝♦♥s✐❞❡r δ = ρ − /ρ + ✜①❡❞✳ 0.1 0.08 0.06 J( ρ + , ρ - ) 0.04 0.02 0 0 0.1 0.2 0.3 0.4 0.5 0.6 ρ + - ρ - ❈✉rr❡♥t ♠❛①✐♠✉♠ ❛♥❞ ❝♦rr❡s♣♦♥❞✐♥❣ ❞r✐✈✐♥❣ ❢♦r❝❡ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❜② ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡rs ♠❡t❤♦❞✿ ρ ✶ + φ ( ✶ − δ ) ✶ + φ ❏ ♠❛① ❝ = � φ , ✶ + φ ✶ � ✶ + φ ✶ − δ ✶ φ ✶ − δ ( ρ + − ρ − ) ♠❛① = ρ ❝ ( ✶ − δ ) . ✶ + φ ✶ − δ φ ▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✽ ✴ ✸✺

❉✐r❡❝t❡❞ ♠♦t✐♦♥ ❆s②♠♠❡tr✐❝ ♣✐❡❝❡✇✐s❡✲❝♦♥st❛♥t ♣❡r✐♦❞✐❝ ❢♦r❝✐♥❣ 1 � ✶ − τ ✵ /τ ♦r ✵✱ 0.8 t ∈ [ ✵ , τ ✵ ] α = 4 0.6 ρ ± ( t ) = ✵ ♦r τ ✵ /τ ✱ t ∈ [ τ ✵ , τ ] ρ + - ρ - 0.4 0.2 � τ 0 ③❡r♦ ❛✈❡r❛❣❡ ❜✐❛s ✵ ∆ ρ ( t ) ❞t = ✵ -0.2 -0.4 0 1 2 3 4 5 t 0.3 � τ 0.2 ❏ ❛✈ ( α ) = ✶ ❏ [ ρ + ( t ) , ρ − ( t )] ❞t 0.1 τ ✵ J av 0 -0.1 ❆s②♠♠❡tr② ♣❛r❛♠❡t❡r α = τ/τ ✵ − ✶ -0.2 ♦♣t✐♠❛❧ ♣✉♠♣✐♥❣ ❝♦♥❞✐t✐♦♥ -0.3 0.01 0.1 1 10 100 α ▼❛✉r♦ ❙❡❧❧✐tt♦ ✭❙✳❯✳◆✳✮ ❉r✐✈❡♥ ❉✐✛✉s✐✈❡ ●❧❛ss② ❙②st❡♠s ❋❧♦r❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✾ ✴ ✸✺

◆♦♥✲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✿ � ′ � ❢ φ ( ③ ) ❢ ′ ( ③ ) ❢ ( ③ ) = ✇✐t❤ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥✿ ❢ ( ± ▲ ) = ✵ , ❣ ✶ + φ ( t ) . ❣ ′ ( t ) = ❙♦♠❡ ✐♥t❡r❡st✐♥❣ ❢❡❛t✉r❡s P♦✇❡r✲❧❛✇ ❝r✐t✐❝❛❧ r❡❧❛①❛t✐♦♥✿ ρ ❝ − ρ ( ③ , t ) ∼ t − ✶ /φ � t ❙✐♠♣❧❡ ❛❣✐♥❣ ❜❡❤❛✈✐♦✉r✿ ❇ ( t , t ✇ ) = ❉ ( s ) ❞s ∼ ❧♦❣ t − ❧♦❣ 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❡♥❝❡✱ ❏✉♥❡ ✷✵✶✹ ✶✵ ✴ ✸✺