~ : 2 0, : 2 ~Length ~ = R - PowerPoint PPT Presentation

𝑂 𝑂 ~∫ ⅆ𝜏 : 𝑌 2 0, 𝜏 : 𝑆 2 𝑂𝛽 ′ ~Length 𝑂 ~ 𝑂 = 𝑂 R ℓ 𝑡 𝑇string ≅ 𝑇BH 𝑆 ≅ (ⅆ = 3) 2 𝑕 𝑡 𝑂 ℓ 𝑇

• • • • 3 1 ~𝑂 Θ ~𝑂 5 3

ℓ 𝑡 ℓ 𝑇 𝑆 0 ≅ ℓ 𝑂 ℓ ℓ 3 𝑆 ≅ ℓ𝑂 𝑒+2 𝑆 𝑈 𝑀 ∝ 𝑓 𝑢 𝑆 𝑈 ∝ 𝑓 𝑢 𝑆 𝑈 ∝ 𝑢

𝑂 𝜖𝑺 2 𝑂 𝑂 𝛾𝐼 = ⅆ 2ℓ 2 ⅆ𝜏 + ⅆ𝜏 1 ⅆ𝜏 2 𝑊 𝑺 𝜏 1 , 𝑺 𝜏 2 𝜖𝜏 0 0 0 −𝑕 2 ℓ 𝑒−2 𝑒−2 + 𝑣 ℓ 𝑒 𝜀 𝑒 𝑺 𝜏 1 − 𝑺(𝜏 2 ) 𝑊 = 𝑺 𝜏 1 − 𝑺 𝜏 2 𝑺 2 𝑊=0 = ℓ 2 𝑂 ≡ 𝑆 0 2 𝑆 0 = ℓ 𝑂

𝛾𝐺 ~ − ⅆ − 1 ln 𝑆 + 𝑆 2 − 𝑕 2 ℓ 𝑒−2 𝑂 2 + 𝑣ℓ 𝑒 𝑂 2 𝑂ℓ 2 𝑆 𝑒 𝑆 𝑒−2 gravity diffusion elasticity repulsive (excluded-volume effect) Entropic force 𝑆 0 = ℓ 𝑂 𝑕 2 ℓ 𝑒−2 𝑂 2 𝑒−6 ~𝑃(1) 𝑕 𝑝 ~𝑂 4 𝑒−2 𝑆 0 𝑒−4 𝑣ℓ 𝑒 𝑂 2 𝑣 𝑝 ~𝑂 ⅆ = 4 2 ~𝑃(1) 𝑒 𝑆 0

𝑣 Repulsive + Gravity Entropic + Repulsive 𝑒−4 𝑣 𝑝 ~𝑂 2 1 𝑆 𝑡 ≅ ℓ 𝑕 2 𝑂 𝑒−2 Entropic + Gravity Entropic 𝑒−6 𝑕 𝑑 ~𝑂 −1 𝑕 𝑕 𝑝 ~𝑂 4 2

• • • •

2 𝑂 + 𝑟 2 ⅆ 𝑂 𝛾𝐼 0 = ⅆ 𝜖𝑺 ⅆ𝜏 𝑺 𝜏 2 2ℓ 2 ⅆ𝜏 2ℓ 2 𝜖𝜏 0 0 𝛾𝐺 ≤ 𝛾𝐺 0 𝑟 + 𝛾(𝐼 − 𝐼 0 ) 0 𝑒 𝑒 2−1 + 𝑂 2 𝑣𝑟 𝛾𝐺 ≤ 𝑟𝑂 − 𝑂 2 𝑕 2 𝑟 𝑟 2 (𝑟 0 𝑂 ≪ 1 ℓ 2 𝑂 𝑆 2 0 = ℓ 2 tanh 𝑟 0 𝑂 𝑟 0 ℓ 2 𝑟 0 𝑂 ≥ 𝑃(1) 𝑟 0

ℓ 2 𝑂 (𝑟 0 𝑂 ≪ 1 𝑺 2 0 = ℓ 2 𝑒 𝑒 2−1 + 𝑂 2 𝑣𝑟 𝛾𝐺 ≤ 𝑟𝑂 − 𝑂 2 𝑕 2 𝑟 tanh 𝑟 0 𝑂 2 ℓ 2 𝑟 0 𝑟 0 𝑂 ≥ 𝑃(1) 𝑟 0 (2 < ⅆ < 4) 𝑆 0 = ℓ 𝑂 𝑕 = 0, 𝑣 ≥ 0 𝑟 0 = 0 𝑕 > 0, 𝑣 > 0 1 𝑆 𝑑 ≅ ℓ 𝑣𝑂 𝑒 stable shrink shrink 𝑕 BH 𝑆 ≅ ℓ 𝑣 1 1 𝑆 ≅ ℓ 𝑕 2 𝑂 𝑆 0 = ℓ 𝑂 𝑒−4 𝑆 𝑡 ≅ ℓ 𝑕 2 𝑂 𝑒−2 𝑕 𝑒−6 𝑒−4 𝑕 = 0 𝑒−2 2𝑒 𝑂 −1 2 𝑒−2 𝑂 − 1 𝑕 𝑝 ~𝑂 4 𝑕′ 𝑑 ~𝑣 𝑒 𝑕 𝑝 ≅ 𝑣 𝑒−2 𝑒 𝑒 2−1 + 𝑂 2 𝑣𝑟 𝛾𝐺 ≤ 𝑟𝑂 − 𝑂 2 𝑕 2 𝑟 2 𝑒−4 𝑒−2 2 + 𝑂𝑣𝑟 0 0 = 1 − 𝑂𝑕 2 𝑟 0 2 𝑒−4 𝑒−2 2 + 𝑂𝑣𝑟 0 0 = 1 − 𝑂𝑕 2 𝑟 0 2

ⅆ = 3 𝑆 log 𝑂 𝑆 𝑇 ℓ log 𝑂 ℓ 𝑆 𝑡 ≅ ℓ𝑕 2 𝑂 𝑂 −1 < 𝑣 < 𝑣 𝑝 1 2 (𝑆 0 ) 𝑆 ≅ ℓ 𝑕 2 𝑂 −1 𝑣 < 𝑂 −1 𝑆 ≅ ℓ 𝑣 𝑕 𝑆 𝑑 0 log 𝑂 𝑕 𝑕 𝑝 𝑕 𝑝 𝑕 𝑑 𝑕′ 𝑑

ℓ → 𝑏ℓ (𝑏 > 0) ℓ 𝑏ℓ • 𝑆 = 𝑏𝑆 0 = 𝑏ℓ 𝑂 • 𝑏ℓ 2 𝑂 ⅆ 𝜖𝑺 𝛾𝐼′ = 2𝑏 2 ℓ 2 ⅆ𝜏 𝜖𝜏 0 𝐵 ′ ≡ 1 𝑎 ′ ∫ 𝐵𝑓 −𝛾𝐼 ′ 2 ′ 𝑓 −𝛾 𝐼−𝐼 ′ 2 𝑓 −𝛾𝐼 𝑺 𝑂 − 𝑺 0 𝑺 2 = ∫ 𝑺 𝑂 − 𝑺 0 = ′ ∫ 𝑓 −𝛾𝐼 𝑓 −𝛾 𝐼−𝐼 ′

2 ′ 𝑓 −𝛾 𝐼−𝐼 ′ 2 𝑓 −𝛾𝐼 𝑺 𝑂 − 𝑺 0 𝑺 2 = ∫ 𝑺 𝑂 − 𝑺 0 = ′ ∫ 𝑓 −𝛾𝐼 𝑓 −𝛾 𝐼−𝐼 ′ 2 ′ 2 ′ 1 + 𝛾 𝐼 − 𝐼 ′ ′ − 𝛾 𝐼 − 𝐼 ′ ≅ 𝑺 𝑂 − 𝑺 0 𝑺 𝑂 − 𝑺 0 +𝑃 𝛾 𝐼 − 𝐼 ′ 2 4−𝑒 6−𝑒 ≅ 𝑂𝑏 2 ℓ 2 + 𝑏 𝑒 1 − 𝑏 2 + 𝐷 1 𝑣𝑂 2 𝑏 2 𝑂ℓ 2 𝑏 2−𝑒 2 − 𝐷 2 𝑕 2 𝑂 = 0 𝐷 1 , 𝐷 2 : Positive 𝑂 independent constants 4−𝑒 6−𝑒 𝑏 𝑒 − 𝑏 𝑒+2 + 𝑣𝑂 2 𝑏 2 = 0 2 − 𝑕 2 𝑂 𝑆 = ℓ 𝑏 𝑂

4−𝑒 6−𝑒 𝑏 𝑒 − 𝑏 𝑒+2 + 𝑣𝑂 2 𝑏 2 = 0 2 − 𝑕 2 𝑂 𝑆 = ℓ 𝑏 𝑂 𝑕 2 = 0, 𝑣 > 0 Puff-up 𝑣 stable 1 3 𝑆 ≅ ℓ𝑣 𝑒+2 𝑂 (expanded configuration) 𝑒+2 𝑆 0 = ℓ 𝑂 𝑣 = 0 𝑒−4 𝑣 𝑝 ~𝑂 2 4−𝑒 4−𝑒 𝑏 𝑒 − 𝑏 𝑒+2 + 𝑣𝑂 𝑏 𝑒 − 𝑏 𝑒+2 + 𝑣𝑂 = 0 2 3 = 0 2 (𝑣 ≅ 𝑂 0 ) 𝑆 ≅ ℓ𝑂 𝑒+2 (𝑣 > 𝑣 𝑝 ) 𝑕, 𝑣 > 0 1 𝑆 𝑑 ≅ ℓ 𝑣𝑂 𝑒 𝑕 Puff-up shrink BH 𝑆 ≅ ℓ 𝑣 1 3 1 𝑆 𝑡 ≅ ℓ 𝑕 2 𝑂 𝑆 ≅ ℓ𝑣 𝑒+2 𝑂 𝑒+2 𝑕 𝑒−2 𝑕 = 0 𝑒 𝑒−2 2𝑒 𝑂 −1 2 𝑒+2 𝑂 − 3 ′′ ≅ 𝑣 𝑕′ 𝑑 ~𝑣 𝑒 𝑕 𝑝 𝑒+2 4−𝑒 6−𝑒 4−𝑒 6−𝑒 𝑏 𝑒 − 𝑏 𝑒+2 + 𝑣𝑂 2 𝑏 2 = 0 𝑏 𝑒 − 𝑏 𝑒+2 + 𝑣𝑂 2 𝑏 2 = 0 2 − 𝑕 2 𝑂 2 − 𝑕 2 𝑂 𝑣 < 𝑣 0 𝑕

ⅆ = 3 𝑆 log 𝑂 𝑆 𝑇 ℓ log 𝑂 ℓ 3 𝑣~𝑃(1) 𝑆 ≅ ℓ 𝑣 𝑆 𝑡 ≅ ℓ𝑕 2 𝑂 𝑕 5 𝑂 −1 < 𝑣 < 𝑣 𝑝 1 2 (𝑆 0 ) 𝑆 ≅ ℓ 𝑕 2 𝑂 −1 𝑣 < 𝑂 −1 𝑆 ≅ ℓ 𝑣 𝑕 0 log 𝑂 𝑕 𝑕′′ 𝑝 𝑕′ 𝑑

2 < ⅆ < 4 ℓ = 1 log 𝑂 𝑆 𝑑 log 𝑂 𝑣 1 ⅆ − 1 puff-up ′′ 𝑕 𝑝 1 0 1 3 ⅆ 𝑆~𝑣 𝑒+2 𝑂 𝑒+2 𝑕′ 𝑑 𝑣 𝑆~ 𝑕 ⅆ − 4 ⅆ − 2 (𝑣 𝑝 ) 2 2ⅆ 𝑕 𝑝 free Black hole −1 𝑒−2 2𝑒 𝑂 −1 𝑕′ 𝑑 ≅ 𝑣 𝑒 𝑆 0 ~ 𝑂 1 0 𝑆~ 𝑕 2 𝑂 𝑒−4 𝑒−4 2 𝑒−2 𝑂 − 1 𝑕 𝑝 ≅ 𝑣 𝑒−2 ⅆ − 6 − 1 log 𝑂 𝑕 𝑒 2 (𝑕 𝑑 ) (𝑕 𝑝 ) 2 𝑒+2 𝑂 − 3 ′′ ≅ 𝑣 4 𝑕 𝑝 𝑒+2

ⅆ = 4 ℓ = 1 log 𝑂 𝑆 𝑑 log 𝑂 𝑣 𝑣 1 𝑆~ 3 𝑕 puff-up 1 3 𝑆~𝑣 𝑒+2 𝑂 𝑒+2 1 0 4 free Black hole −1 𝑆 0 ~ 𝑂 0 − 1 log 𝑂 𝑕 2 (𝑕 𝑑 ) ⅆ > 4

• → • • • • •

2 𝑂 + 𝑟 2 ⅆ 𝑂 𝛾𝐼 0 = ⅆ 𝜖𝑺 ⅆ𝜏 𝑺 𝜏 2 2ℓ 2 ⅆ𝜏 2ℓ 2 𝜖𝜏 0 0 𝑓 −𝛾𝐺 = ∫ ⅆ𝑺 𝑓 −𝛾𝐼 = ∫ ⅆ𝑺 𝑓 −𝛾𝐼 0 𝑓 −𝛾 𝐼−𝐼 0 ≥ e −𝛾 𝐼−𝐼 0 0 e −𝛾𝐺 0 𝛾𝐺 ≤ 𝛾𝐺 0 𝑟 + 𝛾(𝐼 − 𝐼 0 ) 0 𝑟 d exp − 𝑟ⅆ [𝑺 𝜏 2 + 𝑺 𝜏 ′ 2 ] cosh 𝑟 𝜏 − 𝜏′ − 2𝑺(𝜏) ∙ 𝑺(𝜏 ′ ) 𝑟ⅆ 2 𝐻 0 𝜏, 𝜏 ′ = 2𝜌ℓ 2 sinh 𝑟 𝜏 − 𝜏 ′ 2ℓ 2 sinh 𝑟 𝜏 − 𝜏′ 𝑂 𝑂 ⅆ𝜏 ′ 𝑊 − 𝑟 2 ⅆ 𝑂 𝛾𝐺 0 = −log 𝑎 0 ⅆ𝜏 𝑺 𝜏 2 𝛾(𝐼 − 𝐼 0 ) 0 = ⅆ𝜏 2ℓ 2 0 0 0

𝛾𝐺 ≤ ⅆ 2 ln cosh 𝑟𝑂 − 𝑟ⅆ𝑂 tanh 𝑟𝑂 4 𝑒−2 𝑒 𝜏 ′ 𝑂 𝑕 2 𝑟ⅆ 2 𝑟ⅆ 2 ⅆ𝜏 ′ −2 ⅆ𝜏 − 𝑣 Γ ⅆ 1 𝜏, 𝜏 ′ ; 𝑟 2𝐺 2 𝜏, 𝜏 ′ ; 𝑟 2𝐺 0 0 2 1 𝜏, 𝜏 ′ ; 𝑟 = sinh 𝑟𝜏 cosh 𝑟 𝑂 − 𝜏 + sinh 𝑟𝜏′ cosh 𝑟 𝑂 − 𝜏 ′ − 2 sinh 𝑟𝜏 cosh 𝑟 𝑂 − 𝜏 ′ 𝐺 cosh 𝑟𝑂 + cosh 𝑟 𝑂 − 𝜏 ′ sinh 𝑟𝜏′ 2 2 𝜏, 𝜏 ′ ; 𝑟 = sinh 𝑟𝜏 sinh 𝑟 𝜏′ − 𝜏 1 − sinh 𝑟𝜏 𝐺 sinh 𝑟𝜏 ′ sinh 𝑟𝜏′ cosh 𝑟𝑂

𝑓 −𝑟𝑂 𝑓 −𝑟(𝑂−𝜏 ′ ) 𝑓 −𝑟(𝑂−𝜏) ≪ 1 𝑓 −𝑟𝜏 ′ 𝑓 −𝑟 𝜏 ′ −𝜏 𝑓 −𝑟𝜏 𝑒 𝑒 2−1 + 𝑂 2 𝑣𝑟 𝛾𝐺 ≤ 𝑟𝑂 − 𝑂 2 𝑕 2 𝑟 2 𝑒−4 𝑒−2 2 + 𝑂𝑣𝑟 0 0 = 1 − 𝑂𝑕 2 𝑟 0 2 ℓ 2 𝑂 (𝑟 0 𝑂 ≪ 1 𝑺 2 0 = ℓ 2 tanh 𝑟 0 𝑂 ℓ 2 𝑟 0 𝑟 0 𝑂 ≥ 𝑃(1) 𝑟 0

𝑏ℓ 2 𝑂 ⅆ 𝜖𝑺 𝛾𝐼′ = 2𝑏 2 ℓ 2 ⅆ𝜏 𝜖𝜏 0 d ⅆ 2 ⅆ 2 𝐻′ 𝜏, 𝜏 ′ = 𝑺 𝜏 − 𝑺 𝜏 ′ exp − 2𝜌𝑏 2 ℓ 2 𝜏 − 𝜏′ 2𝑏 2 ℓ 2 𝜏 − 𝜏′ 𝐵 ′ ≡ 1 2 ′ 𝑎 ′ ∫ 𝐵𝑓 −𝛾𝐼 ′ 𝑓 −𝛾 𝐼−𝐼 ′ 2 𝑓 −𝛾𝐼 𝑺 𝑂 − 𝑺 0 𝑺 2 = ∫ 𝑺 𝑂 − 𝑺 0 = ′ ∫ 𝑓 −𝛾𝐼 𝑓 −𝛾 𝐼−𝐼 ′ 2 ′ 2 ′ 1 + 𝛾 𝐼 − 𝐼 ′ ′ − 𝛾 𝐼 − 𝐼 ′ +𝑃 𝛾 𝐼 − 𝐼 ′ 2 ≅ 𝑺 𝑂 − 𝑺 0 𝑺 𝑂 − 𝑺 0 4−𝑒 6−𝑒 ≅ 𝑂𝑏 2 ℓ 2 + 𝑏 𝑒 1 − 𝑏 2 + 𝐷 1 𝑣𝑂 2 𝑏 2 𝑂ℓ 2 𝑏 2−𝑒 2 − 𝐷 2 𝑕 2 𝑂 = 0 𝐷 1 , 𝐷 2 : Positive 𝑂 independent constants