By K. P. N. Murthy
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Let S = x(i) + x(j). Split S randomly into two parts. Set x(i) to one part and x(j) to the other part. Repeat the above for a warm up time of say 4 × N iterations. Then every subsequent time you select two particles (k and l), the corresponding x(k) and x(l) are two independent random numbers with exponential distribution: λ exp(−λx) for 0 ≤ x ≤ ∞. (a) Implement the above algorithm and generate a large number of random numbers. Plot their frequency distribution and check if they follow exponential distribution.
We call W the trial matrix. The sum of the elements in each row as well as each column of the trial matrix W ⋆ is unity. As we shall see below, we need W ⋆ matrix to select a trial state from the current state. Hence W ⋆ is chosen conveniently to make the selection of the trial state simple. Assignment 18 Verify that the transition matrix W whose elements are calculated as per the prescriptions in Eqns. (112) obeys the detailed balance given by Eq. (110 ). The implementation of the Metropolis sampling procedure proceeds as follows.
We get, ∞ 1 e−x xN −1 . (91) dx eikx = (N − 1)! (1 − ik)N 0 We immediately identify the right hand side of the above, as the characteristic function of the sum of N independent and identically distributed exponential random variables with mean unity, see Eq. (67). In Eq. (91) above, we replace, on both sides, k by k/N , x by N x, and dx by N dx, to get, ∞ dx eikx 0 e−N x N N xN −1 1 = (N − 1)! 1 − i Nk N . (92) The right hand side of the above is the characteristic function of Y¯N see Eq. (68).