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ps7_molecular-mechanicsF07

ps7_molecular-mechanicsF07 - of sample points How does the...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.370/2.37 Molecular Mechanics Fall 2006 PROBLEM SET 7 DUE: Friday, November 16 1. In this problem we will use various methods to integrate Z π 0 sin( x ) dx Use the trapezoidal method. Plot the diﬀerence between your result and the exact answer as a function of the number of integration points. Use hit-and-miss integration. Plot the diﬀerence between your result and the exact answer as a function of the number of sample points. Use sample mean integration. Plot the diﬀerence between your result and the exact answer as a function of the number of sample points. For students taking 2.37 Use sample mean integration with points dis- tributed non-uniformly. What is the optimal distribution? Plot the diﬀer- ence between your result and the exact answer as a function of the number
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Unformatted text preview: of sample points. How does the convergence rate compare to the results from part 3? What about the error level? Use log-log axes for all plots. Which method converges the fastest? Slowest? How do the convergence rates compare amoung the various Monte Carlo schemes? Do the convergence rates match the predictions from class? 2. Use Monte Carlo simulation to ﬁnd the average energy of the DNA described in problem 5 (for 2.370)/ problem 6 (for 2.37) of Problem Set 6 as a function of kT . Compare your results to the theoretical prediction for N = 10 and N = 100. 3. Find the average energy of the DNA described in problem 5 (for 2.370)/ problem 6 (for 2.37) of Problem Set 6 as a function of kT and for N = 10 and N = 100, if the energy associated an open s th link is ε/s ....
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