pset_4 - 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 #4 2008 J. Cao (The problems can be solved by following the lecture notes.) ik z(t ) z(0) ] [ I. The self-intermediate scattering function F s (k, t) = e can be expressed in even moments C 2n = ( 1) n F s (2n) (k,0). 1) Show C 0 = 1, C 2 0 2 , and C 4 = 3 ω 0 4 0 2 Ω 0 2 where ω o 2 = k 2 K B T and Ω 0 is the Einstein frequency. M 1 2 t 2 ω 0 2) For free particles, show F s (k, t) = e 2 . 3) Under what conditions for k, can we ignore molecular collisions? (see Balucani: Dynamics of the liquid state p11-20) II. Calculate the velocity auto correlation function C(t) = V(t)V(0) from the generalized Langevin equation. 1) Show C satisfies C + 0 t γ ( t τ τ = 0 ) C ( ) d . 2) Show γ (0)
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pset_4 - MIT OpenCourseWare http/ocw.mit.edu 5.72...

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