2 0

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

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


  1. 𝑂 𝑂 ~∫ β…†πœ : π‘Œ 2 0, 𝜏 : 𝑆 2 𝑂𝛽 β€² ~Length 𝑂 ~ 𝑂 = 𝑂 R β„“ 𝑑 𝑇string β‰… 𝑇BH 𝑆 β‰… (β…† = 3) 2 𝑕 𝑑 𝑂 β„“ 𝑇

  2. β€’ β€’ β€’ β€’ 3 1 ~𝑂 Θ ~𝑂 5 3

  3. β„“ 𝑑 β„“ 𝑇 𝑆 0 β‰… β„“ 𝑂 β„“ β„“ 3 𝑆 β‰… ℓ𝑂 𝑒+2 𝑆 π‘ˆ 𝑀 ∝ 𝑓 𝑒 𝑆 π‘ˆ ∝ 𝑓 𝑒 𝑆 π‘ˆ ∝ 𝑒

  4. 𝑂 πœ–π‘Ί 2 𝑂 𝑂 𝛾𝐼 = β…† 2β„“ 2 β…†πœ + β…†πœ 1 β…†πœ 2 π‘Š 𝑺 𝜏 1 , 𝑺 𝜏 2 πœ–πœ 0 0 0 βˆ’π‘• 2 β„“ π‘’βˆ’2 π‘’βˆ’2 + 𝑣 β„“ 𝑒 πœ€ 𝑒 𝑺 𝜏 1 βˆ’ 𝑺(𝜏 2 ) π‘Š = 𝑺 𝜏 1 βˆ’ 𝑺 𝜏 2 𝑺 2 π‘Š=0 = β„“ 2 𝑂 ≑ 𝑆 0 2 𝑆 0 = β„“ 𝑂

  5. 𝛾𝐺 ~ βˆ’ β…† βˆ’ 1 ln 𝑆 + 𝑆 2 βˆ’ 𝑕 2 β„“ π‘’βˆ’2 𝑂 2 + 𝑣ℓ 𝑒 𝑂 2 𝑂ℓ 2 𝑆 𝑒 𝑆 π‘’βˆ’2 gravity diffusion elasticity repulsive (excluded-volume effect) Entropic force 𝑆 0 = β„“ 𝑂 𝑕 2 β„“ π‘’βˆ’2 𝑂 2 π‘’βˆ’6 ~𝑃(1) 𝑕 𝑝 ~𝑂 4 π‘’βˆ’2 𝑆 0 π‘’βˆ’4 𝑣ℓ 𝑒 𝑂 2 𝑣 𝑝 ~𝑂 β…† = 4 2 ~𝑃(1) 𝑒 𝑆 0

  6. 𝑣 Repulsive + Gravity Entropic + Repulsive π‘’βˆ’4 𝑣 𝑝 ~𝑂 2 1 𝑆 𝑑 β‰… β„“ 𝑕 2 𝑂 π‘’βˆ’2 Entropic + Gravity Entropic π‘’βˆ’6 𝑕 𝑑 ~𝑂 βˆ’1 𝑕 𝑕 𝑝 ~𝑂 4 2

  7. β€’ β€’ β€’ β€’

  8. 2 𝑂 + π‘Ÿ 2 β…† 𝑂 𝛾𝐼 0 = β…† πœ–π‘Ί β…†πœ 𝑺 𝜏 2 2β„“ 2 β…†πœ 2β„“ 2 πœ–πœ 0 0 𝛾𝐺 ≀ 𝛾𝐺 0 π‘Ÿ + 𝛾(𝐼 βˆ’ 𝐼 0 ) 0 𝑒 𝑒 2βˆ’1 + 𝑂 2 π‘£π‘Ÿ 𝛾𝐺 ≀ π‘Ÿπ‘‚ βˆ’ 𝑂 2 𝑕 2 π‘Ÿ π‘Ÿ 2 (π‘Ÿ 0 𝑂 β‰ͺ 1 β„“ 2 𝑂 𝑆 2 0 = β„“ 2 tanh π‘Ÿ 0 𝑂 π‘Ÿ 0 β„“ 2 π‘Ÿ 0 𝑂 β‰₯ 𝑃(1) π‘Ÿ 0

  9. β„“ 2 𝑂 (π‘Ÿ 0 𝑂 β‰ͺ 1 𝑺 2 0 = β„“ 2 𝑒 𝑒 2βˆ’1 + 𝑂 2 π‘£π‘Ÿ 𝛾𝐺 ≀ π‘Ÿπ‘‚ βˆ’ 𝑂 2 𝑕 2 π‘Ÿ tanh π‘Ÿ 0 𝑂 2 β„“ 2 π‘Ÿ 0 π‘Ÿ 0 𝑂 β‰₯ 𝑃(1) π‘Ÿ 0 (2 < β…† < 4) 𝑆 0 = β„“ 𝑂 𝑕 = 0, 𝑣 β‰₯ 0 π‘Ÿ 0 = 0 𝑕 > 0, 𝑣 > 0 1 𝑆 𝑑 β‰… β„“ 𝑣𝑂 𝑒 stable shrink shrink 𝑕 BH 𝑆 β‰… β„“ 𝑣 1 1 𝑆 β‰… β„“ 𝑕 2 𝑂 𝑆 0 = β„“ 𝑂 π‘’βˆ’4 𝑆 𝑑 β‰… β„“ 𝑕 2 𝑂 π‘’βˆ’2 𝑕 π‘’βˆ’6 π‘’βˆ’4 𝑕 = 0 π‘’βˆ’2 2𝑒 𝑂 βˆ’1 2 π‘’βˆ’2 𝑂 βˆ’ 1 𝑕 𝑝 ~𝑂 4 𝑕′ 𝑑 ~𝑣 𝑒 𝑕 𝑝 β‰… 𝑣 π‘’βˆ’2 𝑒 𝑒 2βˆ’1 + 𝑂 2 π‘£π‘Ÿ 𝛾𝐺 ≀ π‘Ÿπ‘‚ βˆ’ 𝑂 2 𝑕 2 π‘Ÿ 2 π‘’βˆ’4 π‘’βˆ’2 2 + π‘‚π‘£π‘Ÿ 0 0 = 1 βˆ’ 𝑂𝑕 2 π‘Ÿ 0 2 π‘’βˆ’4 π‘’βˆ’2 2 + π‘‚π‘£π‘Ÿ 0 0 = 1 βˆ’ 𝑂𝑕 2 π‘Ÿ 0 2

  10. β…† = 3 𝑆 log 𝑂 𝑆 𝑇 β„“ log 𝑂 β„“ 𝑆 𝑑 β‰… ℓ𝑕 2 𝑂 𝑂 βˆ’1 < 𝑣 < 𝑣 𝑝 1 2 (𝑆 0 ) 𝑆 β‰… β„“ 𝑕 2 𝑂 βˆ’1 𝑣 < 𝑂 βˆ’1 𝑆 β‰… β„“ 𝑣 𝑕 𝑆 𝑑 0 log 𝑂 𝑕 𝑕 𝑝 𝑕 𝑝 𝑕 𝑑 𝑕′ 𝑑

  11. β„“ β†’ 𝑏ℓ (𝑏 > 0) β„“ 𝑏ℓ β€’ 𝑆 = 𝑏𝑆 0 = 𝑏ℓ 𝑂 β€’ 𝑏ℓ 2 𝑂 β…† πœ–π‘Ί 𝛾𝐼′ = 2𝑏 2 β„“ 2 β…†πœ πœ–πœ 0 𝐡 β€² ≑ 1 π‘Ž β€² ∫ 𝐡𝑓 βˆ’π›ΎπΌ β€² 2 β€² 𝑓 βˆ’π›Ύ πΌβˆ’πΌ β€² 2 𝑓 βˆ’π›ΎπΌ 𝑺 𝑂 βˆ’ 𝑺 0 𝑺 2 = ∫ 𝑺 𝑂 βˆ’ 𝑺 0 = β€² ∫ 𝑓 βˆ’π›ΎπΌ 𝑓 βˆ’π›Ύ πΌβˆ’πΌ β€²

  12. 2 β€² 𝑓 βˆ’π›Ύ πΌβˆ’πΌ β€² 2 𝑓 βˆ’π›ΎπΌ 𝑺 𝑂 βˆ’ 𝑺 0 𝑺 2 = ∫ 𝑺 𝑂 βˆ’ 𝑺 0 = β€² ∫ 𝑓 βˆ’π›ΎπΌ 𝑓 βˆ’π›Ύ πΌβˆ’πΌ β€² 2 β€² 2 β€² 1 + 𝛾 𝐼 βˆ’ 𝐼 β€² β€² βˆ’ 𝛾 𝐼 βˆ’ 𝐼 β€² β‰… 𝑺 𝑂 βˆ’ 𝑺 0 𝑺 𝑂 βˆ’ 𝑺 0 +𝑃 𝛾 𝐼 βˆ’ 𝐼 β€² 2 4βˆ’π‘’ 6βˆ’π‘’ β‰… 𝑂𝑏 2 β„“ 2 + 𝑏 𝑒 1 βˆ’ 𝑏 2 + 𝐷 1 𝑣𝑂 2 𝑏 2 𝑂ℓ 2 𝑏 2βˆ’π‘’ 2 βˆ’ 𝐷 2 𝑕 2 𝑂 = 0 𝐷 1 , 𝐷 2 : Positive 𝑂 independent constants 4βˆ’π‘’ 6βˆ’π‘’ 𝑏 𝑒 βˆ’ 𝑏 𝑒+2 + 𝑣𝑂 2 𝑏 2 = 0 2 βˆ’ 𝑕 2 𝑂 𝑆 = β„“ 𝑏 𝑂

  13. 4βˆ’π‘’ 6βˆ’π‘’ 𝑏 𝑒 βˆ’ 𝑏 𝑒+2 + 𝑣𝑂 2 𝑏 2 = 0 2 βˆ’ 𝑕 2 𝑂 𝑆 = β„“ 𝑏 𝑂 𝑕 2 = 0, 𝑣 > 0 Puff-up 𝑣 stable 1 3 𝑆 β‰… ℓ𝑣 𝑒+2 𝑂 (expanded configuration) 𝑒+2 𝑆 0 = β„“ 𝑂 𝑣 = 0 π‘’βˆ’4 𝑣 𝑝 ~𝑂 2 4βˆ’π‘’ 4βˆ’π‘’ 𝑏 𝑒 βˆ’ 𝑏 𝑒+2 + 𝑣𝑂 𝑏 𝑒 βˆ’ 𝑏 𝑒+2 + 𝑣𝑂 = 0 2 3 = 0 2 (𝑣 β‰… 𝑂 0 ) 𝑆 β‰… ℓ𝑂 𝑒+2 (𝑣 > 𝑣 𝑝 ) 𝑕, 𝑣 > 0 1 𝑆 𝑑 β‰… β„“ 𝑣𝑂 𝑒 𝑕 Puff-up shrink BH 𝑆 β‰… β„“ 𝑣 1 3 1 𝑆 𝑑 β‰… β„“ 𝑕 2 𝑂 𝑆 β‰… ℓ𝑣 𝑒+2 𝑂 𝑒+2 𝑕 π‘’βˆ’2 𝑕 = 0 𝑒 π‘’βˆ’2 2𝑒 𝑂 βˆ’1 2 𝑒+2 𝑂 βˆ’ 3 β€²β€² β‰… 𝑣 𝑕′ 𝑑 ~𝑣 𝑒 𝑕 𝑝 𝑒+2 4βˆ’π‘’ 6βˆ’π‘’ 4βˆ’π‘’ 6βˆ’π‘’ 𝑏 𝑒 βˆ’ 𝑏 𝑒+2 + 𝑣𝑂 2 𝑏 2 = 0 𝑏 𝑒 βˆ’ 𝑏 𝑒+2 + 𝑣𝑂 2 𝑏 2 = 0 2 βˆ’ 𝑕 2 𝑂 2 βˆ’ 𝑕 2 𝑂 𝑣 < 𝑣 0 𝑕

  14. β…† = 3 𝑆 log 𝑂 𝑆 𝑇 β„“ log 𝑂 β„“ 3 𝑣~𝑃(1) 𝑆 β‰… β„“ 𝑣 𝑆 𝑑 β‰… ℓ𝑕 2 𝑂 𝑕 5 𝑂 βˆ’1 < 𝑣 < 𝑣 𝑝 1 2 (𝑆 0 ) 𝑆 β‰… β„“ 𝑕 2 𝑂 βˆ’1 𝑣 < 𝑂 βˆ’1 𝑆 β‰… β„“ 𝑣 𝑕 0 log 𝑂 𝑕 𝑕′′ 𝑝 𝑕′ 𝑑

  15. 2 < β…† < 4 β„“ = 1 log 𝑂 𝑆 𝑑 log 𝑂 𝑣 1 β…† βˆ’ 1 puff-up β€²β€² 𝑕 𝑝 1 0 1 3 β…† 𝑆~𝑣 𝑒+2 𝑂 𝑒+2 𝑕′ 𝑑 𝑣 𝑆~ 𝑕 β…† βˆ’ 4 β…† βˆ’ 2 (𝑣 𝑝 ) 2 2β…† 𝑕 𝑝 free Black hole βˆ’1 π‘’βˆ’2 2𝑒 𝑂 βˆ’1 𝑕′ 𝑑 β‰… 𝑣 𝑒 𝑆 0 ~ 𝑂 1 0 𝑆~ 𝑕 2 𝑂 π‘’βˆ’4 π‘’βˆ’4 2 π‘’βˆ’2 𝑂 βˆ’ 1 𝑕 𝑝 β‰… 𝑣 π‘’βˆ’2 β…† βˆ’ 6 βˆ’ 1 log 𝑂 𝑕 𝑒 2 (𝑕 𝑑 ) (𝑕 𝑝 ) 2 𝑒+2 𝑂 βˆ’ 3 β€²β€² β‰… 𝑣 4 𝑕 𝑝 𝑒+2

  16. β…† = 4 β„“ = 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 βˆ’1 𝑆 0 ~ 𝑂 0 βˆ’ 1 log 𝑂 𝑕 2 (𝑕 𝑑 ) β…† > 4

  17. β€’ β†’ β€’ β€’ β€’ β€’ β€’

  18. 2 𝑂 + π‘Ÿ 2 β…† 𝑂 𝛾𝐼 0 = β…† πœ–π‘Ί β…†πœ 𝑺 𝜏 2 2β„“ 2 β…†πœ 2β„“ 2 πœ–πœ 0 0 𝑓 βˆ’π›ΎπΊ = ∫ ⅆ𝑺 𝑓 βˆ’π›ΎπΌ = ∫ ⅆ𝑺 𝑓 βˆ’π›ΎπΌ 0 𝑓 βˆ’π›Ύ πΌβˆ’πΌ 0 β‰₯ e βˆ’π›Ύ πΌβˆ’πΌ 0 0 e βˆ’π›ΎπΊ 0 𝛾𝐺 ≀ 𝛾𝐺 0 π‘Ÿ + 𝛾(𝐼 βˆ’ 𝐼 0 ) 0 π‘Ÿ d exp βˆ’ π‘Ÿβ…† [𝑺 𝜏 2 + 𝑺 𝜏 β€² 2 ] cosh π‘Ÿ 𝜏 βˆ’ πœβ€² βˆ’ 2𝑺(𝜏) βˆ™ 𝑺(𝜏 β€² ) π‘Ÿβ…† 2 𝐻 0 𝜏, 𝜏 β€² = 2πœŒβ„“ 2 sinh π‘Ÿ 𝜏 βˆ’ 𝜏 β€² 2β„“ 2 sinh π‘Ÿ 𝜏 βˆ’ πœβ€² 𝑂 𝑂 β…†πœ β€² π‘Š βˆ’ π‘Ÿ 2 β…† 𝑂 𝛾𝐺 0 = βˆ’log π‘Ž 0 β…†πœ 𝑺 𝜏 2 𝛾(𝐼 βˆ’ 𝐼 0 ) 0 = β…†πœ 2β„“ 2 0 0 0

  19. 𝛾𝐺 ≀ β…† 2 ln cosh π‘Ÿπ‘‚ βˆ’ π‘Ÿβ…†π‘‚ tanh π‘Ÿπ‘‚ 4 π‘’βˆ’2 𝑒 𝜏 β€² 𝑂 𝑕 2 π‘Ÿβ…† 2 π‘Ÿβ…† 2 β…†πœ β€² βˆ’2 β…†πœ βˆ’ 𝑣 Ξ“ β…† 1 𝜏, 𝜏 β€² ; π‘Ÿ 2𝐺 2 𝜏, 𝜏 β€² ; π‘Ÿ 2𝐺 0 0 2 1 𝜏, 𝜏 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’ 𝜏 β€² βˆ’ 2 sinh π‘Ÿπœ cosh π‘Ÿ 𝑂 βˆ’ 𝜏 β€² 𝐺 cosh π‘Ÿπ‘‚ + cosh π‘Ÿ 𝑂 βˆ’ 𝜏 β€² sinh π‘Ÿπœβ€² 2 2 𝜏, 𝜏 β€² ; π‘Ÿ = sinh π‘Ÿπœ sinh π‘Ÿ πœβ€² βˆ’ 𝜏 1 βˆ’ sinh π‘Ÿπœ 𝐺 sinh π‘Ÿπœ β€² sinh π‘Ÿπœβ€² cosh π‘Ÿπ‘‚

  20. 𝑓 βˆ’π‘Ÿπ‘‚ 𝑓 βˆ’π‘Ÿ(π‘‚βˆ’πœ β€² ) 𝑓 βˆ’π‘Ÿ(π‘‚βˆ’πœ) β‰ͺ 1 𝑓 βˆ’π‘Ÿπœ β€² 𝑓 βˆ’π‘Ÿ 𝜏 β€² βˆ’πœ 𝑓 βˆ’π‘Ÿπœ 𝑒 𝑒 2βˆ’1 + 𝑂 2 π‘£π‘Ÿ 𝛾𝐺 ≀ π‘Ÿπ‘‚ βˆ’ 𝑂 2 𝑕 2 π‘Ÿ 2 π‘’βˆ’4 π‘’βˆ’2 2 + π‘‚π‘£π‘Ÿ 0 0 = 1 βˆ’ 𝑂𝑕 2 π‘Ÿ 0 2 β„“ 2 𝑂 (π‘Ÿ 0 𝑂 β‰ͺ 1 𝑺 2 0 = β„“ 2 tanh π‘Ÿ 0 𝑂 β„“ 2 π‘Ÿ 0 π‘Ÿ 0 𝑂 β‰₯ 𝑃(1) π‘Ÿ 0

  21. 𝑏ℓ 2 𝑂 β…† πœ–π‘Ί 𝛾𝐼′ = 2𝑏 2 β„“ 2 β…†πœ πœ–πœ 0 d β…† 2 β…† 2 𝐻′ 𝜏, 𝜏 β€² = 𝑺 𝜏 βˆ’ 𝑺 𝜏 β€² exp βˆ’ 2πœŒπ‘ 2 β„“ 2 𝜏 βˆ’ πœβ€² 2𝑏 2 β„“ 2 𝜏 βˆ’ πœβ€² 𝐡 β€² ≑ 1 2 β€² π‘Ž β€² ∫ 𝐡𝑓 βˆ’π›ΎπΌ β€² 𝑓 βˆ’π›Ύ πΌβˆ’πΌ β€² 2 𝑓 βˆ’π›ΎπΌ 𝑺 𝑂 βˆ’ 𝑺 0 𝑺 2 = ∫ 𝑺 𝑂 βˆ’ 𝑺 0 = β€² ∫ 𝑓 βˆ’π›ΎπΌ 𝑓 βˆ’π›Ύ πΌβˆ’πΌ β€² 2 β€² 2 β€² 1 + 𝛾 𝐼 βˆ’ 𝐼 β€² β€² βˆ’ 𝛾 𝐼 βˆ’ 𝐼 β€² +𝑃 𝛾 𝐼 βˆ’ 𝐼 β€² 2 β‰… 𝑺 𝑂 βˆ’ 𝑺 0 𝑺 𝑂 βˆ’ 𝑺 0 4βˆ’π‘’ 6βˆ’π‘’ β‰… 𝑂𝑏 2 β„“ 2 + 𝑏 𝑒 1 βˆ’ 𝑏 2 + 𝐷 1 𝑣𝑂 2 𝑏 2 𝑂ℓ 2 𝑏 2βˆ’π‘’ 2 βˆ’ 𝐷 2 𝑕 2 𝑂 = 0 𝐷 1 , 𝐷 2 : Positive 𝑂 independent constants

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