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Unformatted text preview: Physics 212: Statistical mechanics II, Fall 2006 Problem set 1 : assigned 9/5/06, due 9/14/06, based on lectures I-IV 1. Start from the formula for the entropy of a discrete probability distribution S =- k X i =1 p i log p i . (1) Prove that the entropy for a system with k distinct states is maximized by the uniform distribution i = 1 /k . Hint: one way to do this is by contradiction; given a different probability distribution, can you show that it cannot be the maximum-entropy distribution? 2. Write out an argument that the entropy change in the Boltzmann equation is due only to the collisional term, i.e., that dS dt =- d dt Z f 1 log f 1 d p 1 d r 1 = 0 (2) if f 1 t + p m x f 1 + F p f 1 = 0 , (3) using Liouvilles theorem and the assumption that the one-particle dynamics is Hamiltonian. 3. Consider the global equilibrium distribution (constant density, Maxwellian, zero average velocity) f ( x, p ) = c exp(- p 2 / 2 mk B T ) . (4) (a) Find c if the particle density is...
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