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Unformatted text preview: 1 Physics 219  Problem Set 1 Due Date: January 22, 2008 1. Stirling’s formula. The Γ function is defined as: Γ ( z ) = Z ∞ dt t z 1 e t (1) (a) Show that Γ ( z + 1 ) = z ! for integers z ≥ 0. (b) Using the method of steepest descents, derive the Stirling formula ln ( N ! ) ≈ N ln N N (2) (c) What is the next subleading term? 2. Entropy of a Classical Ideal Gas in the Microcanonical Ensemble. (a) What is the surface area of a sphere in d dimensions as a function of its radius, r ? (b) Consider a classical ideal gas of N particles of mass m in 3 dimensions. Their total energy is E = 1 2 m ∑ N i = 1 p 2 i , where p i is the momentum of the i th particle. What is the surface area of the 3 Ndimensional sphere of fixed energy? (c) What is the entropy of the gas, including the volume in position space? (d) What are the Temperature and Pressure of the gas, as a function of N and E ?...
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 Spring '08
 NAYAK
 Physics, mechanics, Statistical Mechanics, Entropy, Classical Ideal Gas, von Neumann entropy

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