An Introduction to Monte Carlo Simulations in Statistical by K. P. N. Murthy

By K. P. N. Murthy

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Additional info for An Introduction to Monte Carlo Simulations in Statistical Physics

Example text

Another way is to calculate the autocorrelation function and from it, the correlation time τ ⋆ in units of MCSS. Discard data from the initial, say 10 τ ⋆ MCSS. Another problem is related to the issue of ergodicity. The system can get trapped in a region of the configuration phase and not come out of it at all. It is also possible that the dynamics is quasi-ergodic, in the sense that the system gets trapped in local minimum and is unable to come out of it in any finite time due to the presence of high energy barriers.

N. Murthy How do we calculate the statistical error associated with the sample mean, m= 1 N N (0) xi ? (92) i=1 ˆ 0 denote the number of data points in the set Ω0 . Note Ω ˆ 0 = N. Calculate the Let Ω sample variance given by, 1 S 2 (Ω0 ) = ˆ0 Ω ˆ0 Ω (0) 2 xi i=1  1 − ˆ0 Ω ˆ0 Ω i=1 2 (0) xi  . (93) ˆ 0 − 1). Then transform the data to another set of data Ω1 = Let ǫ20 = S 2 (Ω0 )/(Ω (1) (1) (1) ˆ1 = Ω ˆ 0 /2 = N/2. e. Ω 1 equations that relate the data set Ωk to Ωk−1 are given by, (k−1) (k) xi = (k−1) x2i−1 + x2i 2 ˆk , i = 1, 2, · · · Ω ˆ ˆ k = Ωk−1 k = 1, 2, · · · .

Discard the initial bins which give averages different from the latter. Another way is to calculate the autocorrelation function and from it, the correlation time τ ⋆ in units of MCSS. Discard data from the initial, say 10 τ ⋆ MCSS. Another problem is related to the issue of ergodicity. The system can get trapped in a region of the configuration phase and not come out of it at all. It is also possible that the dynamics is quasi-ergodic, in the sense that the system gets trapped in local minimum and is unable to come out of it in any finite time due to the presence of high energy barriers.