Problem Set 2 for PHYSICS 715: Thermodynamics and Statistical Mechanics, Assignments of Statistical mechanics

Download Problem Set 2 for PHYSICS 715: Thermodynamics and Statistical Mechanics and more Assignments Statistical mechanics in PDF only on Docsity! PHYSICS 715 Problem Set 2 Due Friday, February 3, 2006 Reading: Landau and Lifshitz, Chap. 2; Chap. 3, Secs. 28–31 Huang, Secs. 7.1–7.6 (suggested) LD 4: Hawking radiation and the lifetime of black holes (a) It was observed by J. Bekenstein, Phys. Rev. D 7, 2333 (1973), that the entropy of matter falling into a black hole should increase the entropy of the black hole, and that the entropy of the hole should be proportional to its area. The resulting temperature of a black hole of mass M or energy Mc2 is T = h̄c3/8πkGM , where G is Newton’s constant. Determine the entropy S assuming that S = 0 for a zero-mass black hole, and find the dependence of the area A = 4SL2P l/k on M . Here LP l is the Planck length, Lpl = (h̄G/c 3)1/2 ≈ 1.6 × 10−33 cm. Estimate the size of a black hole of solar mass, M = 2 × 1033 gm. (b) A black hole with T > 0 will radiate photons (and neutrinos) with a thermal spectrum [S.W. Hawking, Nature 248, 30 (1974); Comm. Math. Phys. 43, 199 (1975); Phys. Rev. D 13, 191 (1975)]. The power radiated per unit area in this “Hawking radiation” is given by the expression for blackbody radiation, P = σT 4, up to a factor of order unity. Here σ is the Stefan-Boltzmann constant σ = π2k4/60h̄3c2. Use this result to estimate how massive a black hole formed in the big bang must be if it is to have survived ≈ 13.7 × 109 yr to the present. LD 5: Increase of entropy and heat flow. Two large many-particle systems A and B with initial energies E0A and E 0 B are brought into contact. Assume that there is no mixing of the constituents (for example, A and B could be solids), but that energy can be transferred between A and B. State your physical assumptions, and use the Boltzmann definition of entropy in terms of ΓAB to show (i) that entropy increases when the two systems are combined, i.e., that SAB is greater than or equal to the total entropy of the separate systems once AB reaches equi- librium, SAB(E) ≥ SA(E 0 A) + SB(E 0 B), E = EA + EB; (ii) that energy flows from the hotter to the colder system. [Hint: use ∆S ≥ 0, the extensive property of the entropy, and appropriate Taylor series expansions to show that ( 1 T 0A − 1 T 0B ) (ĒA − E 0 A) ≥ 0, for ĒA − E 0 A small and discuss the consequences.] 1