pset_1 - 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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Problem set #1 for 5.72 (Feb. 2008) Cao I. Consider a random walk on a linear three-site model. p and q are the probabilities of moving right and left, respectively. (p+q=1) p p 1) Write the transition matrix Q. 2) Find the stationary distribution and show that it satisfies detailed balance. 3) For the special case of p = q = 1 2 , compute the probability at n=3 after s steps. , given p n (s) p n 0 ()=δ n1 . 4)* Repeat the calculation in 3) for p q. II. One-dimensional random walk on an infinite lattice is described by a master equation dp n dt = ap n 1 p n () + bp n + 1 p n where a and b are the forward and backward rate constants. 1) Calculate nt = n n p n t ( ) , n 2 t ( ) = n 2 p n t ( ) n and δ n 2 = n 2 t ( ) ( ) 2 2)* Given the initial state at n 0 = 0, find p(n,t). III. In the three-site model, the forward rate is a and the backward rate is b. The walk is initially at site A 1 .
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pset_1 - MIT OpenCourseWare http/ocw.mit.edu 5.72...

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