probset9

probset9 - Physics 5524 Statistical Mechanics Problem Set 9...

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Physics 5524 Statistical Mechanics Problem Set 9 Due: Wednesday, Mar. 31 9.1 (From the Fall 2006 Qualifying Exam) A system of N bosons in two dimensions has an energy-momentum relationship E = cp 3 / 2 and number density n = N/A ( A is the area). (a) Show that at low temperatures the system will Bose condense (unlike the 2D Bose gas in Problem 8.2), and that the Bose condensation temperature T c n α . Find α . (b) Show that the entropy below T C goes like S T β , and the “pressure” (which in two dimensions has units of force per unit length) goes as P T γ . Find β and γ . 9.2 Consider a D -dimensional ideal Bose gas with single particle energy spectrum E ( ~ k ) = A | ~ k | s where s is some positive number. ( Note: In doing this problem you should not have to evaluate any integrals explicitly.) (a) Determine the conditions on D and s for which this gas exhibits Bose condensation with T c > 0. (b) For the cases when Bose condensation occurs, obtain an expression for the condensate
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probset9 - Physics 5524 Statistical Mechanics Problem Set 9...

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