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# Selected Topics PHY 300

Syracuse

GPA 3.93

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This 13 page Class Notes was uploaded by Ms. Bryce Wisoky on Wednesday October 21, 2015. The Class Notes belongs to PHY 300 at Syracuse University taught by Staff in Fall. Since its upload, it has received 22 views. For similar materials see /class/225632/phy-300-syracuse-university in Physics 2 at Syracuse University.

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Date Created: 10/21/15

Lec7 o Numerical integration Monte Carlo a Statistical mechanics Motivation Last week studied 2D Ising model Thermodynamic quantities are given by large integrals Used Monte Carlo algorithm to estimate these Step back and explain in more general con text Simple integrals Consider the following integral IzOl lll x2 Options o Evaluate as area using slices calculus def inition 0 Monte Carlo algorithm simple or using im portance sampling Slicing N I N NIHOO 61 1212731 and 61 2 1N Error 06x Pros Can be improved Simpson39s rule error 06x3 Cons For many variables dimensions error decreases slowly If f1 gt fx1xd and N function evalua tions error goes as GOV E Non trivial integration regions Simple Monte Carlo Approx integral by 1 N I N lim N600 N where now points x1 are chosen at random Derive meanL value theorem of calculus Error GOV i independent of number of vari ables Conclusion many variables Monte Carlo meth ods are essential Aside random numbers Most computerslanguages provide a sim ple pseudorandom number generator eg random in python Often use deterministic algorithm to gen erate sequence rn1 2 am b mod m random number in range 0 1 is just rnm Careful choices of abm generate long se quences of seemingly random numbers But such generators will repeat eventually Care must be taken In principle should check all results with at least 2 different random number generators Errors in Monte Carlo Clearly value of integral depends on exact ran dom numbers used Make several eg 10 independent Monte Carlo measurements of the integral and look at how the different estimates of lt f gt differ from one another The error is clearly the standard deviation of these means 1 10 6f 1 0 ltr2gt ltflgt2 i1 Furthermore we can show 1 5f N N j as we increase the number of Monte Carlo runs Compare with error from simple slicing Importance Sampling Simple uniform sampling in interval 01 not optimal Better to try to pick points ml where function is largest Rewrite integral as follows at I fmdm pxpxdx where pl is some arbitrary probability distri bution chosen so that the variance of Then i N ail Metropolis algorithm Want to compute fpxfxdx fpxdx where p1 arbitrary unnormalized probabil ity Metropolis method produces random walk of points x1 whose distribution approaches pCL after a large number of steps Transition probability tzl gt xj satisfies de tailed balance condition ltfgt Pinti gt 133 Pjtj gt iii One solution Mil3 7517139 gt Z min1 Implementation Choose trial position suma mold 6 Calculate w ptrm1Pflfold o If w gt 1 accept suma If w lt 1 generate random number r o If r s w accept change If nOt CL trial CBold Choose 6 so that 50 moves accepted approx 10 Example Consider 1 I 20 xil 12 Take pl il z2 Use Metropolis to esti ma ge lt 1 gt Notice convergence of error like N7 11 Statistical Mechanics Want to understand systems with very many degrees of freedom No needwant to solve all dynamical equa tions Want only time averages of quanti ties summed over all degrees of freedom By ergodic hypothesis can generate these averages by averaging over all microstates of system with a certain probability Probability of finding the system in some state with energy at tem perature T is given by 6 lm 12 Application of Metropolis Monte Carlo Method Here choose pl e HW and point x1 cor responds to value of some degree of freedom eg the spin at a site in case of Ising model tij min1eAH Any thermodynamic quantity can be obtained by simple averaging on the set of configura tions produced by the Monte Carlo simulation 69 lt E gt quiEqeEkT i 1 MC configs ltEgtN El39 NMC i 13

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