hw5 - Department of Chemical Engineering University of...

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Unformatted text preview: Department of Chemical Engineering University of California, Santa Barbara ChE 210B Winter 2010 Instructor: Glenn Fredrickson Homework #5 Homework Due: Friday, March 5, 2010 1. Prepare solutions to the following problems in Chandler: 5.15-5.18. 2. The surface area of a unit sphere in four dimensions is needed in ² expansion calculations. Find the surface area of a unit sphere in d dimensions, S d , by evaluating the integral R d d k exp(- k 2 ) ≡ S d R ∞ dk k d- 1 exp(- k 2 ). The second expression can be viewed as a defining relationship for S d , while the first is the target of your calculation. Note the following definition of the Gamma function in expressing your general result: Γ( z ) ≡ Z ∞ dt t z- 1 exp(- t ) Specialize to four dimensions. 3. (a). Consider the RG flow equations in differential form for the “ ψ 4 field theory” described in class. Linearize the flow equations about the Wilson-Fisher fixed point and find the eigenvalues and eigenvectors to confirm the results quoted in the lecture for d < 4 ( ² > 0). Identify the eigenvector and eigenvalue that are0)....
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hw5 - Department of Chemical Engineering University of...

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