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

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

May 02, 2023 583 likes β€’823 views

~ : 2 0, : 2 ~Length ~ = R string BH ( = 3) 2 3 1 ~ ~ 5 3 0

slide-1
SLIDE 1
slide-2
SLIDE 2

𝑆2

𝑂 ~∫ β…†πœ : π‘Œ2 0, 𝜏 : 𝑂~

𝑂𝛽′~Length

𝑂 𝑂 = 𝑂 R 𝑇string β‰… 𝑇BH ℓ𝑇

𝑆 β‰… ℓ𝑑 𝑕𝑑

2

𝑂

(β…† = 3)

slide-3
SLIDE 3
  • ~𝑂

3 5

~𝑂

1 3

Θ

slide-4
SLIDE 4

ℓ𝑑 ℓ𝑇 β„“ β„“ 𝑆 β‰… ℓ𝑂

3 𝑒+2

π‘†π‘ˆ ∝ 𝑒 π‘†π‘ˆ ∝ 𝑓𝑒 𝑀 ∝ 𝑓𝑒 𝑆0 β‰… β„“ 𝑂

π‘†π‘ˆ

slide-5
SLIDE 5

𝛾𝐼 = β…† 2β„“2

𝑂 πœ–π‘Ί

πœ–πœ

2

β…†πœ +

𝑂

β…†πœ1

𝑂

β…†πœ2 π‘Š 𝑺 𝜏1 , 𝑺 𝜏2

π‘Š = βˆ’π‘•2 β„“π‘’βˆ’2 𝑺 𝜏1 βˆ’ 𝑺 𝜏2

π‘’βˆ’2 + 𝑣 β„“π‘’πœ€ 𝑒 𝑺 𝜏1 βˆ’ 𝑺(𝜏2)

𝑺2 π‘Š=0 = β„“2𝑂 ≑ 𝑆0

2 𝑆0 = β„“ 𝑂

slide-6
SLIDE 6

𝛾𝐺 ~ βˆ’ β…† βˆ’ 1 ln 𝑆 + 𝑆2 𝑂ℓ2 + 𝑣ℓ𝑒𝑂2 𝑆𝑒 βˆ’ 𝑕2β„“π‘’βˆ’2𝑂2 π‘†π‘’βˆ’2

diffusion elasticity repulsive

(excluded-volume effect)

gravity Entropic force 𝑆0 = β„“ 𝑂

𝑕2β„“π‘’βˆ’2𝑂2 𝑆0

π‘’βˆ’2

~𝑃(1) 𝑣ℓ𝑒𝑂2 𝑆0

𝑒

~𝑃(1) 𝑕𝑝~𝑂

π‘’βˆ’6 4

𝑣𝑝~𝑂

π‘’βˆ’4 2

β…† = 4

slide-7
SLIDE 7

𝑕𝑝~𝑂

π‘’βˆ’6 4

𝑕𝑑~π‘‚βˆ’1

2

𝑣𝑝~𝑂

π‘’βˆ’4 2

Entropic Entropic + Repulsive Entropic + Gravity Repulsive + Gravity 𝑣 𝑕 𝑆𝑑 β‰… β„“ 𝑕2𝑂

1 π‘’βˆ’2

slide-8
SLIDE 8
slide-9
SLIDE 9

𝛾𝐼0 = β…† 2β„“2

𝑂

β…†πœ πœ–π‘Ί πœ–πœ

2

+ π‘Ÿ2β…† 2β„“2

𝑂

β…†πœ 𝑺 𝜏 2 𝛾𝐺 ≀ 𝛾𝐺0 π‘Ÿ + 𝛾(𝐼 βˆ’ 𝐼0) 0

π‘Ÿ

𝑆2 0 = β„“2 π‘Ÿ0 tanh π‘Ÿ0𝑂 β„“2𝑂

(π‘Ÿ0𝑂 β‰ͺ 1

β„“2 π‘Ÿ0

π‘Ÿ0𝑂 β‰₯ 𝑃(1)

𝛾𝐺 ≀ π‘Ÿπ‘‚ βˆ’ 𝑂2𝑕2π‘Ÿ

𝑒 2βˆ’1 + 𝑂2π‘£π‘Ÿ 𝑒 2

slide-10
SLIDE 10

shrink

β„“2𝑂

(π‘Ÿ0𝑂 β‰ͺ 1

β„“2 π‘Ÿ0

π‘Ÿ0𝑂 β‰₯ 𝑃(1)

𝑺2 0 = β„“2 π‘Ÿ0 tanh π‘Ÿ0𝑂

𝑆𝑑 β‰… β„“ 𝑕2𝑂

1 π‘’βˆ’2

𝑕𝑝~𝑂

π‘’βˆ’6 4

(2 < β…† < 4)

𝑕 = 0, 𝑣 β‰₯ 0 π‘Ÿ0 = 0

𝑆0 = β„“ 𝑂

𝑕 > 0, 𝑣 > 0 𝑕 = 0 𝛾𝐺 ≀ π‘Ÿπ‘‚ βˆ’ 𝑂2𝑕2π‘Ÿ

𝑒 2βˆ’1 + 𝑂2π‘£π‘Ÿ 𝑒 2

𝑕

𝑆 β‰… β„“ 𝑕2𝑂

1 π‘’βˆ’4

𝑆𝑑 β‰… β„“ 𝑣𝑂

1 𝑒

𝑕′𝑑~𝑣

π‘’βˆ’2 2𝑒 π‘‚βˆ’1 𝑒

𝑆 β‰… β„“ 𝑣 𝑕

𝑕𝑝 β‰… 𝑣

π‘’βˆ’4 2 π‘’βˆ’2 π‘‚βˆ’ 1 π‘’βˆ’2

stable shrink BH 𝑆0 = β„“ 𝑂 𝛾𝐺 ≀ π‘Ÿπ‘‚ βˆ’ 𝑂2𝑕2π‘Ÿ

𝑒 2βˆ’1 + 𝑂2π‘£π‘Ÿ 𝑒 2

0 = 1 βˆ’ 𝑂𝑕2π‘Ÿ0

π‘’βˆ’4 2 + π‘‚π‘£π‘Ÿ0 π‘’βˆ’2 2

0 = 1 βˆ’ 𝑂𝑕2π‘Ÿ0

π‘’βˆ’4 2 + π‘‚π‘£π‘Ÿ0 π‘’βˆ’2 2

slide-11
SLIDE 11

β…† = 3

log𝑂 𝑆 β„“ log𝑂 𝑕

𝑆 β‰… β„“ 𝑣 𝑕

1 2 (𝑆0)

𝑆 β‰… β„“ 𝑕2𝑂 βˆ’1

log𝑂 𝑆𝑇 β„“

𝑣 < π‘‚βˆ’1 π‘‚βˆ’1 < 𝑣 < 𝑣𝑝

𝑆𝑑 𝑆𝑑 β‰… ℓ𝑕2𝑂

𝑕𝑝 𝑕𝑝 𝑕𝑑 𝑕′𝑑

slide-12
SLIDE 12

β„“ β†’ 𝑏ℓ (𝑏 > 0)

β„“

𝑏ℓ

  • 𝑆 = 𝑏𝑆0 = 𝑏ℓ 𝑂

𝛾𝐼′ = β…† 2𝑏2β„“2

𝑂

β…†πœ πœ–π‘Ί πœ–πœ

2

𝑏ℓ

𝑺2 = ∫ 𝑺 𝑂 βˆ’ 𝑺 0

2π‘“βˆ’π›ΎπΌ

∫ π‘“βˆ’π›ΎπΌ = π‘“βˆ’π›Ύ πΌβˆ’πΌβ€² 𝑺 𝑂 βˆ’ 𝑺 0

2 β€²

π‘“βˆ’π›Ύ πΌβˆ’πΌβ€²

β€²

𝐡 β€² ≑ 1 π‘Žβ€² ∫ π΅π‘“βˆ’π›ΎπΌβ€²

slide-13
SLIDE 13

𝑺2 = ∫ 𝑺 𝑂 βˆ’ 𝑺 0

2π‘“βˆ’π›ΎπΌ

∫ π‘“βˆ’π›ΎπΌ = π‘“βˆ’π›Ύ πΌβˆ’πΌβ€² 𝑺 𝑂 βˆ’ 𝑺 0

2 β€²

π‘“βˆ’π›Ύ πΌβˆ’πΌβ€²

β€²

β‰… 𝑺 𝑂 βˆ’ 𝑺 0

2 β€²

1 + 𝛾 𝐼 βˆ’ 𝐼′ β€² βˆ’ 𝛾 𝐼 βˆ’ 𝐼′ 𝑺 𝑂 βˆ’ 𝑺 0

2 β€²

+𝑃 𝛾 𝐼 βˆ’ 𝐼′

2

β‰… 𝑂𝑏2β„“2 + 𝑏𝑒 1 βˆ’ 𝑏2 + 𝐷1𝑣𝑂

4βˆ’π‘’ 2 βˆ’ 𝐷2𝑕2𝑂 6βˆ’π‘’ 2 𝑏2 𝑂ℓ2𝑏2βˆ’π‘’

= 0

𝑏𝑒 βˆ’ 𝑏𝑒+2 + 𝑣𝑂

4βˆ’π‘’ 2 βˆ’ 𝑕2𝑂 6βˆ’π‘’ 2 𝑏2 = 0

𝑆 = β„“ 𝑏 𝑂

𝐷1, 𝐷2: Positive 𝑂 independent constants

slide-14
SLIDE 14

𝑏𝑒 βˆ’ 𝑏𝑒+2 + 𝑣𝑂

4βˆ’π‘’ 2 βˆ’ 𝑕2𝑂 6βˆ’π‘’ 2 𝑏2 = 0

𝑆 = β„“ 𝑏 𝑂

𝑕2 = 0, 𝑣 > 0 𝑣𝑝~𝑂

π‘’βˆ’4 2

𝑆 β‰… ℓ𝑣

1 𝑒+2𝑂 3 𝑒+2

𝑆0 = β„“ 𝑂

stable Puff-up (expanded configuration)

𝑣

(𝑣 β‰… 𝑂0)

𝑆 β‰… ℓ𝑂

3 𝑒+2

𝑏𝑒 βˆ’ 𝑏𝑒+2 + 𝑣𝑂

4βˆ’π‘’ 2

= 0

𝑏𝑒 βˆ’ 𝑏𝑒+2 + 𝑣𝑂

4βˆ’π‘’ 2

= 0

𝑣 = 0

𝑕, 𝑣 > 0

𝑕𝑝

β€²β€² β‰… 𝑣 𝑒 2 𝑒+2 π‘‚βˆ’ 3 𝑒+2

(𝑣 > 𝑣𝑝)

Puff-up

𝑆 β‰… ℓ𝑣

1 𝑒+2𝑂 3 𝑒+2

𝑏𝑒 βˆ’ 𝑏𝑒+2 + 𝑣𝑂

4βˆ’π‘’ 2 βˆ’ 𝑕2𝑂 6βˆ’π‘’ 2 𝑏2 = 0

𝑏𝑒 βˆ’ 𝑏𝑒+2 + 𝑣𝑂

4βˆ’π‘’ 2 βˆ’ 𝑕2𝑂 6βˆ’π‘’ 2 𝑏2 = 0

shrink

𝑆𝑑 β‰… β„“ 𝑕2𝑂

1 π‘’βˆ’2

𝑕

𝑆𝑑 β‰… β„“ 𝑣𝑂

1 𝑒

𝑕′𝑑~𝑣

π‘’βˆ’2 2𝑒 π‘‚βˆ’1 𝑒

𝑆 β‰… β„“ 𝑣 𝑕 BH

𝑕 = 0

𝑣 < 𝑣0 𝑕

slide-15
SLIDE 15

β…† = 3

log𝑂 𝑆 β„“ log𝑂 𝑕

𝑆 β‰… β„“ 𝑣 𝑕

1 2 (𝑆0)

𝑆 β‰… β„“ 𝑕2𝑂 βˆ’1

log𝑂 𝑆𝑇 β„“

𝑣 < π‘‚βˆ’1 π‘‚βˆ’1 < 𝑣 < 𝑣𝑝

𝑆𝑑 β‰… ℓ𝑕2𝑂

𝑕′′𝑝 𝑕′𝑑

𝑆 β‰… β„“ 𝑣 𝑕

𝑣~𝑃(1)

3 5

slide-16
SLIDE 16

2 < β…† < 4

log𝑂 𝑕 log𝑂 𝑣 𝑆0~ 𝑂

β„“ = 1

𝑆~𝑣

1 𝑒+2𝑂 3 𝑒+2

𝑆~ 𝑣 𝑕

𝑆~ 𝑕2𝑂

1 π‘’βˆ’4

free puff-up Black hole log𝑂 𝑆𝑑

βˆ’ 1 2 (𝑕𝑑) β…† βˆ’ 6 4 (𝑕𝑝) β…† βˆ’ 4 2 (𝑣𝑝) β…† βˆ’ 1 1 1 β…† β…† βˆ’ 2 2β…† βˆ’1 𝑕𝑝

β€²β€² β‰… 𝑣 𝑒 2 𝑒+2 π‘‚βˆ’ 3 𝑒+2

𝑕𝑝

β€²β€²

𝑕𝑝 β‰… 𝑣

π‘’βˆ’4 2 π‘’βˆ’2 π‘‚βˆ’ 1 π‘’βˆ’2

𝑕𝑝 𝑕′𝑑 β‰… 𝑣

π‘’βˆ’2 2𝑒 π‘‚βˆ’1 𝑒

𝑕′𝑑

slide-17
SLIDE 17

β…† = 4

log𝑂 𝑕 log𝑂 𝑣 𝑆0~ 𝑂

β„“ = 1

𝑆~𝑣

1 𝑒+2𝑂 3 𝑒+2

𝑆~ 𝑣 𝑕 free puff-up Black hole log𝑂 𝑆𝑑

βˆ’ 1 2 (𝑕𝑑) 3 1 1 4 βˆ’1

β…† > 4

slide-18
SLIDE 18
  • β†’
slide-19
SLIDE 19

𝛾𝐼0 = β…† 2β„“2

𝑂

β…†πœ πœ–π‘Ί πœ–πœ

2

+ π‘Ÿ2β…† 2β„“2

𝑂

β…†πœ 𝑺 𝜏 2

π‘“βˆ’π›ΎπΊ = ∫ ⅆ𝑺 π‘“βˆ’π›ΎπΌ = ∫ ⅆ𝑺 π‘“βˆ’π›ΎπΌ0π‘“βˆ’π›Ύ πΌβˆ’πΌ0 β‰₯ e βˆ’π›Ύ πΌβˆ’πΌ0

0eβˆ’π›ΎπΊ

𝛾𝐺 ≀ 𝛾𝐺0 π‘Ÿ + 𝛾(𝐼 βˆ’ 𝐼0) 0

π‘Ÿ

𝐻0 𝜏, πœβ€² = π‘Ÿβ…† 2πœŒβ„“2 sinh π‘Ÿ 𝜏 βˆ’ πœβ€²

d 2

exp βˆ’ π‘Ÿβ…† [𝑺 𝜏 2 + 𝑺 πœβ€² 2] cosh π‘Ÿ 𝜏 βˆ’ πœβ€² βˆ’ 2𝑺(𝜏) βˆ™ 𝑺(πœβ€²) 2β„“2 sinh π‘Ÿ 𝜏 βˆ’ πœβ€²

𝛾𝐺0 = βˆ’log π‘Ž0

𝛾(𝐼 βˆ’ 𝐼0) 0 =

𝑂

β…†πœ

𝑂

β…†πœβ€² π‘Š βˆ’ π‘Ÿ2β…† 2β„“2

𝑂

β…†πœ 𝑺 𝜏 2

slide-20
SLIDE 20

𝛾𝐺 ≀ β…† 2 ln cosh π‘Ÿπ‘‚ βˆ’ π‘Ÿβ…†π‘‚ 4 tanh π‘Ÿπ‘‚ βˆ’2

𝑂

β…†πœβ€²

πœβ€²

β…†πœ 𝑕2 Ξ“ β…† 2 π‘Ÿβ…† 2𝐺

1 𝜏, πœβ€²; π‘Ÿ π‘’βˆ’2 2

βˆ’ 𝑣 π‘Ÿβ…† 2𝐺2 𝜏, πœβ€²; π‘Ÿ

𝑒 2

𝐺

1 𝜏, πœβ€²; π‘Ÿ = sinh π‘Ÿπœ cosh π‘Ÿ 𝑂 βˆ’ 𝜏 + sinh π‘Ÿπœβ€² cosh π‘Ÿ 𝑂 βˆ’ πœβ€² βˆ’ 2 sinh π‘Ÿπœ cosh π‘Ÿ 𝑂 βˆ’ πœβ€²

cosh π‘Ÿπ‘‚ 𝐺

2 𝜏, πœβ€²; π‘Ÿ = sinh π‘Ÿπœ sinh π‘Ÿ πœβ€² βˆ’ 𝜏

sinh π‘Ÿπœβ€² + cosh π‘Ÿ 𝑂 βˆ’ πœβ€² sinh π‘Ÿπœβ€² cosh π‘Ÿπ‘‚ 1 βˆ’ sinh π‘Ÿπœ sinh π‘Ÿπœβ€²

2

slide-21
SLIDE 21

π‘“βˆ’π‘Ÿπ‘‚ π‘“βˆ’π‘Ÿ(π‘‚βˆ’πœβ€²) π‘“βˆ’π‘Ÿ(π‘‚βˆ’πœ) π‘“βˆ’π‘Ÿ πœβ€²βˆ’πœ π‘“βˆ’π‘Ÿπœβ€² π‘“βˆ’π‘Ÿπœ

β‰ͺ 1

𝛾𝐺 ≀ π‘Ÿπ‘‚ βˆ’ 𝑂2𝑕2π‘Ÿ

𝑒 2βˆ’1 + 𝑂2π‘£π‘Ÿ 𝑒 2

0 = 1 βˆ’ 𝑂𝑕2π‘Ÿ0

π‘’βˆ’4 2 + π‘‚π‘£π‘Ÿ0 π‘’βˆ’2 2

𝑺2 0 = β„“2 π‘Ÿ0 tanh π‘Ÿ0𝑂 β„“2𝑂

(π‘Ÿ0𝑂 β‰ͺ 1

β„“2 π‘Ÿ0

π‘Ÿ0𝑂 β‰₯ 𝑃(1)

slide-22
SLIDE 22

𝛾𝐼′ = β…† 2𝑏2β„“2

𝑂

β…†πœ πœ–π‘Ί πœ–πœ

2

𝑏ℓ

𝐻′ 𝜏, πœβ€² = β…† 2πœŒπ‘2β„“2 𝜏 βˆ’ πœβ€²

d 2

exp βˆ’ β…† 2𝑏2β„“2 𝜏 βˆ’ πœβ€² 𝑺 𝜏 βˆ’ 𝑺 πœβ€²

2

𝑺2 = ∫ 𝑺 𝑂 βˆ’ 𝑺 0

2π‘“βˆ’π›ΎπΌ

∫ π‘“βˆ’π›ΎπΌ = π‘“βˆ’π›Ύ πΌβˆ’πΌβ€² 𝑺 𝑂 βˆ’ 𝑺 0

2 β€²

π‘“βˆ’π›Ύ πΌβˆ’πΌβ€²

β€²

𝐡 β€² ≑ 1 π‘Žβ€² ∫ π΅π‘“βˆ’π›ΎπΌβ€²

β‰… 𝑺 𝑂 βˆ’ 𝑺 0

2 β€²

1 + 𝛾 𝐼 βˆ’ 𝐼′ β€² βˆ’ 𝛾 𝐼 βˆ’ 𝐼′ 𝑺 𝑂 βˆ’ 𝑺 0

2 β€²

+𝑃 𝛾 𝐼 βˆ’ 𝐼′

2

β‰… 𝑂𝑏2β„“2 + 𝑏𝑒 1 βˆ’ 𝑏2 + 𝐷1𝑣𝑂

4βˆ’π‘’ 2 βˆ’ 𝐷2𝑕2𝑂 6βˆ’π‘’ 2 𝑏2 𝑂ℓ2𝑏2βˆ’π‘’

= 0

𝐷1, 𝐷2: Positive 𝑂 independent constants