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PHYS 420 Thermal Physics Spring 2016, Problem Set 6

‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ PHYS 420 Thermal Physics Spring 2016, Problem Set 6 ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ Reading material: Gould and Tobochnik, Chapter 4.1 4.2 Problem 1 (5 points) Problem 4.2, Counting microstates (page 182). Problem 2 (5 points) Consider a one D chain consisting of 1  N segments as illustrated in the figure. Let the length of each segment be a when the long dimension of the segment is parallel to the chain and zero when the segment is vertical. Each segment has just two states, a horizontal orientation and a vertical orientation, and each of these states is not degenerate. The distance between the chain ends is Nx ( a x 0 ). Calculate the entropy of the chain as a function of x . Problem 3 (7 points) The damped pendulum has a force proportional to the momentum slowing down the pendulum. It satisfies the equations: ݔሶ ൌ ݌ ݉ ሶൌെߛ݌െܭݏ݅݊ሺݔሻ At long times, the pendulum will tend to an equilibrium stationary state, with zero velocity at x=0. (p, x) = (0,0) is an attractor for the damped pendulum. An ensemble of damped pendulums is started with initial conditions distributed with probability ρ (p0,x0). At late times, these initial conditions are gathered together near the equilibrium stationary state; Liouville’s theorem clearly is not satisfied. (a) Why does Liouville’s theorem not apply to the damped pendulum? (b) Find an expression for the total derivative d ρ /dt in terms of ρ for the damped pendulum. If we evolve a region of phase space of initial volume A= Δ p Δ x, how will its volume depend upon time? Problem 4 (8 points): N particles are separated into two groups, 2 1 n n N . All the particles in the former group has energy 1 E , and the latter has energy 2 E ( 1 2 E E ). If a single quantum emission happens, 1 2 2 n n , 1 1 1 n n . a) Assume 1 1  n and 1 2  n , calculate the change of entropy during the quantum emission process. b) Say the energy of the quantum emission is released into a heat reservoir with energy T. Calculate the change of entropy for the reservoir. c) From a and b calculate the ratio 1 2 / n n . (remember microscopic process is always reversible).

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