homework4 - Homework #4 PHYS 603 Spring 2007 Deadline:...

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Homework #4 — PHYS 603 — Spring 2007 Deadline: Tuesday, April 17, 2007, in class Professor Victor Yakovenko Office: 2115 Physics Web page: http://www2.physics.umd.edu/˜yakovenk/teaching/ Textbook: Gregory H. Wannier, Statistical Physics Dover 1987 reprint of the 1966 edition, ISBN 0-486-65401-X Do not forget to write your name and the homework number! Each problem is worth 10 points. Ch. 4 The Gibbs-Boltzmann Distribution 1. Problem 4.2, Derive the Maxwell distribution p ( v ) for the speed v . 2. Problem 4.3, Compare h v i with q h v 2 i 3. Problem 4.6, Calculate the most probable speed v * . 4. Energy distribution p ( E ) for the 3D Maxwell gas. (a) Derive the probability density p ( E ) for the distribution of energy E of an atom in a three-dimensional (3D) Maxwell gas. The definition of the probability density is dP = p ( E ) dE , where dP is the probability to have energy in the interval ( E,E + dE ). Hint: Follow the discussion around Eq. (4.35) for one particle n = 1. (b) From the probability density p ( E ) derived above, calculate the most probable energy E * which maximizes p ( E ). Compare E * with mv 2 * / 2, where v * is the most probable speed maximizing p ( v ), as derived in Problem 3. Are E * and mv 2 * / 2 equal? If not, can you explain why? 5. Problem 4.7, Mean kinetic energy in a leak. Hint: To calculate the mean kinetic energy for a leak in the x direction, calculate the energy flux h Ev x ρ ( E ) i and divide it by the particle flux h v x ρ ( E ) i . Here E = m ( v 2 x + v 2 y + v 2 x ) / 2 is the energy of an atom, ρ [ E ( v )] is the Gibbs-Boltzmann distribution
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homework4 - Homework #4 PHYS 603 Spring 2007 Deadline:...

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