hw4 - 3 Explicitly ll in the details of Exercise 3-3(from...

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Department of Chemical Engineering University of California, Santa Barbara ChE 210B Winter 2010 Instructor: Glenn Fredrickson Homework #4 Homework Due: Friday, February 19, 2010 1. For the vibrating elastic string discussed in class, compute the fourth moment of the shape fluctuations at some point x , i.e. h [ u ( x )] 4 i , and plot this quantity over 0 x L . Is your expression simply related to the second moment derived in class? Research a result from probability theory called Wick’s theorem to explain why. 2. Relate ∂F/∂ψ ) t to the chemical potential μ of the VDW fluid. Explain why the O( ψ 0 ) and O( ψ 1 ) terms in the Landau expansion of F ( ψ,t ) are irrelevant to phase behavior and critical phenomena.
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Unformatted text preview: 3. Explicitly ll in the details of Exercise 3-3 (from the pages in your handout copied from the text by Goldenfeld), which sketches the derivation of the free energy of the nearest-neighbor Ising model in the mean eld approximation. 4. Develop a Landau expansion of the mean-eld free energy per spin for the Ising model (with coordination number z ) in powers of the average magnetization to O( 4 ). Show that this leads to the same classical or mean-eld critical exponents as the VDW uid produced. 5. Prepare solutions to the following problems in Chandler: 5.1, 5.2, 5.21, 6.3....
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