Computational Physics - Department of Physics

11.4 Improved Monte Carlo Integration 369 and y= √ −2 ln( 1 −x′)sin(θ), withx′∈[ 0 , 1 ]andθ∈ 2 π[ 0 , 1 ]. A function which yie ...

370 11 Outline of the Monte Carlo Strategy we can sample over relevant values for the integrand. It is however not trivial to fi ...

11.5 Monte Carlo Integration of Multidimensional Integrals 371 we accept the new value ofx, else we generate again two new rando ...

372 11 Outline of the Monte Carlo Strategy where g(x,y) =exp(−x^2 −y^2 )(x−y)^2 withd= 6. We can solve this integral by employin ...

11.5 Monte Carlo Integration of Multidimensional Integrals 373 fx=brute_force_MC(x); int_mc += fx; sum_sigma += fxfx; } int_mc = ...

374 11 Outline of the Monte Carlo Strategy } fx=gaussian_MC(x); int_mc += fx; sum_sigma += fxfx; } int_mc = int_mc/((double) n ) ...

11.6 Classes for Random Number Generators 375 Table 11.4Results as function of number of Monte Carlo samplesN. The exact answer ...

376 11 Outline of the Monte Carlo Strategy /** *@return A random number from a particular Random Number Generator *implemented i ...

11.7 Exercises 377 and dNY(t) dt =−ωYNY(t)+ωXNX(t). We assume that att= 0 we haveNY( 0 ) = 0. In the beginning we will have an i ...

378 11 Outline of the Monte Carlo Strategy doubleint_mc = 0.;doublevariance = 0.; doublesum_sigma= 0. ;longidum=-1 ; doublelengt ...

11.7 Exercises 379 The importance sampling improves considerably the results, as we noted in the example with the normal distrib ...

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Chapter 12 Random walks and the Metropolis algorithm The way that can be spoken of is not the constant way. (Tao Te Ching, Book ...

382 12 Random walks and the Metropolis algorithm To reach this distribution, the Markov process needs to obeytwo important condi ...

12.2 Diffusion Equation and Random Walks 383 whereDis the so-called diffusion constant, with dimensionality length^2 per time. I ...

384 12 Random walks and the Metropolis algorithm where we have performed an integration by parts as we did for∂∂〈xt〉. A further ...

12.2 Diffusion Equation and Random Walks 385 12.2.2Random Walks Consider now a random walker in one dimension, with probabilityR ...

386 12 Random walks and the Metropolis algorithm The program below demonstrates the simplicity of the one-dimensional random wal ...

12.3 Microscopic Derivation of the Diffusion Equation 387 cin >> max_trials; cout <<"Number of attempted walks="; ci ...

388 12 Random walks and the Metropolis algorithm 0 20 40 60 80 100 0 20 40 60 80 100 σ^2 Time stepst Fig. 12.3Time development o ...