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pset_3 - MIT OpenCourseWare http/ocw.mit.edu 5.72...

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MIT OpenCourseWare http://ocw.mit.edu 5.72 Statistical Mechanics Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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5.72 Problem set #3 ( March 2008) Cao I. 1) Derive the equilibrium value of density fluctuations 2 k 0 1 ˆ S(k) 1 g(k) N = ρ = + ρ 2) 0 B T S(k 0) k T = = ρ χ where T T 1 P ∂ρ χ = ρ is the isothermal compressibility. 3) Show 2 Re ˆ F k,z = − i ω ( ) = S k, ω ( ) where ˆ F k,z ( ) = e zt F k,t ( ) dt 0 is the Laplace transforms 4) To first order in k, we obtain ( ) 2 2 a k t rk t r s 1 1 F k,t S(k) (1 )e cos(kc t)e = + γ γ Use the expression for F(k,t) to derive S(k, ω ) . 5) Make a plot of S(k, ω ) for a typical value of k. ( Ref: Reichl p 544-562 ) II. The transverse current is defined as . ( ) i i N ikz (t) k ix iz J v t e = 1) Show 2 k B T J Nk M = ( note 2 ix jx ij x v v v = δ < > ) 2) Use the Navier-Stokes equation to show 2 k t k k J (t) J (0)e −ν = where ν = η ρ 0 m .
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