Changes between Version 10 and Version 11 of u/JCFeb0713
- Timestamp:
- 07/10/13 11:52:32 (12 years ago)
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u/JCFeb0713
v10 v11 19 19 From the canonical partition function 20 20 21 [[latex( Z(T,F)=\int dE dx \Omega(E,x) e^{-\left(E+Fx)/k_BT\right)})]]21 [[latex($Z(T,F)=\int dE dx \Omega(E,x) e^{-\left(E+Fx)/k_BT\right)}$)]] 22 22 23 23 we can derive 24 24 25 [[latex( F=T\frac{\partial S}{\partial x})]]25 [[latex($F=T\frac{\partial S}{\partial x}$)]] 26 26 27 27 which for polymers gives Hooke's law 28 28 29 [[latex( F = -C T x)]]29 [[latex($F = -C T x$)]] 30 30 31 31 … … 35 35 [http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1103858973 Bardeen, Cater, and Hawking 1973] 36 36 37 || [[latex( M)]] || black hole mass ||38 || [[latex( A)]] || black hole surface area ||39 || [[latex( g)]] || black hole surface gravity ||40 || [[latex( \Omega)]] || horizon angular velocity ||41 || [[latex( J)]] || black hole angular momentum ||42 || [[latex( \phi)]] || electric potential ||43 || [[latex( Q)]] || electric charge ||37 || [[latex($M$)]] || black hole mass || 38 || [[latex($A$)]] || black hole surface area || 39 || [[latex($g$)]] || black hole surface gravity || 40 || [[latex($\Omega$)]] || horizon angular velocity || 41 || [[latex($J$)]] || black hole angular momentum || 42 || [[latex($\phi$)]] || electric potential || 43 || [[latex($Q$)]] || electric charge || 44 44 45 45 Correspondence between Temperature and Entropy with surface gravity and area 46 || Zeroth Law || [[latex( g)]] constant over horizon for stationary black hole || [[latex(T)]] is constant for a body in thermal equilibrium ||47 || First Law || [[latex( d M=\frac{g}{8\pi}dA+\Omega dJ+\phi dQ)]] || [[latex(dU = TdS - pdV + \mu dN)]] ||48 || Second Law || [[latex( \delta A >= 0)]] || [[latex(dS >= 0)]] ||49 || Third Law || [[latex( g > 0)]] || [[latex(T > 0)]] ||46 || Zeroth Law || [[latex($g$)]] constant over horizon for stationary black hole || [[latex($T$)]] is constant for a body in thermal equilibrium || 47 || First Law || [[latex($d M=\frac{g}{8\pi}dA+\Omega dJ+\phi dQ$)]] || [[latex($dU = TdS - pdV + \mu dN$)]] || 48 || Second Law || [[latex($\delta A >= 0$)]] || [[latex($dS >= 0$)]] || 49 || Third Law || [[latex($g > 0$)]] || [[latex($T > 0$)]] || 50 50 51 51 Black hole 'entropy' should be proportional to area. Dimensional analysis gives: 52 52 53 [[latex( S_{BH} = \frac{k_B A}{4 l_p^2})]]53 [[latex($S_{BH} = \frac{k_B A}{4 l_p^2}$)]] 54 54 55 55 where 56 56 57 [[latex( l_p=\sqrt{G\hbar/c^3})]]57 [[latex($l_p=\sqrt{G\hbar/c^3}$)]] 58 58 59 59 and S,,BH,, stands for Bekenstein-Hawking entropy (not Black Hole). … … 64 64 When lowering a particle into a black hole, information about the particle is lost, so entropy of black hole should increase. This corresponds to an minimal increase in area of 65 65 66 [[latex( dA >= 8 \pi \frac{G m r}{c^2})]]66 [[latex($dA >= 8 \pi \frac{G m r}{c^2} $)]] 67 67 68 68 and an entropy increase of 69 69 70 [[latex( d S >= 2 \pi k_B \frac{m r c}{\hbar})]]70 [[latex($d S >= 2 \pi k_B \frac{m r c}{\hbar}$)]] 71 71 72 72 If we use the reduced compton radius for the particle we get 73 73 74 [[latex( d S >= 2 \pi k_B)]]74 [[latex($d S >= 2 \pi k_B$)]] 75 75 76 76 ''1 possible microstate is lost to system as the particle merges with the black hole'' … … 78 78 If we assume that this change in energy occurs linearly over the particle's compton radius then we have 79 79 80 [[latex( dS >= 2 \pi k_B \frac{mc}{\hbar} d x)]]80 [[latex($dS >= 2 \pi k_B \frac{mc}{\hbar} d x $)]] 81 81 82 82 and if we calculate the entropic force: 83 83 84 [[latex( F = T \frac{dS}{dx})]]84 [[latex($F = T \frac{dS}{dx}$)]] 85 85 86 we have [[latex( F = k_B T 2 \pi \frac{mc}{\hbar})]]86 we have [[latex($F = k_B T 2 \pi \frac{mc}{\hbar}$)]] 87 87 88 88 … … 92 92 === Hawking radiation === 93 93 94 [[latex( T = \frac{\hbar g}{1\pi k_B c})]]94 [[latex($T = \frac{\hbar g}{1\pi k_B c}$)]] 95 95 96 96 where 97 97 98 [[latex( g = \frac{GM}{r_s^2})]]98 [[latex($g = \frac{GM}{r_s^2}$)]] 99 99 100 so we have [[latex( F = \frac{mc^4}{4GM} = mg)]]100 so we have [[latex($F = \frac{mc^4}{4GM} = mg$)]] 101 101 102 102 103 103 === Unruh effect: an observer in an accelerated frame experiences a non zero vacuum temperature === 104 104 105 [[latex( T = \frac{\hbar a}{2\pi k_B c})]]105 [[latex($T = \frac{\hbar a}{2\pi k_B c}$)]] 106 106 107 107 108 108 [[Image(JC3.png, width=600)]] 109 109 110 which leads to an entropic force [[latex( F = k_B T 2 \pi \frac{mc}{\hbar} = ma)]]110 which leads to an entropic force [[latex($F = k_B T 2 \pi \frac{mc}{\hbar} = ma$)]] 111 111 112 112 Newton's 2nd law! … … 121 121 If we assume that the number of degrees of freedom is proportional to the area of the enclosed space like an event horizon 122 122 123 [[latex( N=\frac{A}{l_p^2})]]123 [[latex($N=\frac{A}{l_p^2}$)]] 124 124 125 125 and that the temperature is determined by the equipartition rule 126 126 127 127 128 [[latex( E=\frac{1}{2}Nk_BT)]]128 [[latex($E=\frac{1}{2}Nk_BT$)]] 129 129 130 130 and that the total energy is just the enclosed rest mass 131 131 132 [[latex( E=Mc^2)]]132 [[latex($E=Mc^2$)]] 133 133 134 134 we get 135 135 136 [[latex( T=\frac{2 l_p^2 Mc^2}{k_B A} = \frac{G \hbar M}{k_B 2\pi R^2 c})]]136 [[latex($T=\frac{2 l_p^2 Mc^2}{k_B A} = \frac{G \hbar M}{k_B 2\pi R^2 c}$)]] 137 137 138 138 and 139 139 140 [[latex(F=G\frac{Mm}{R^2})]] 140 [[latex($F=G\frac{Mm}{R^2}$)]] 141
