1549380323-Statistical Mechanics Theory and Molecular Simulation

312 Monte Carlo

methods can become trapped in local regions of configuration space, so transition path
sampling can become trapped in local regions of path space where substantial barriers
keep the system from accessing regions of the space containing paths of higher proba-
bility. The design of path-generation algorithms capable of enhancing sampling of the
transition path ensemble is still an open and interesting question withroom for novel
improvements.

7.8 Problems

7.1. Write a Monte Carlo program to calculate the integral

I=

∫ 1

0

e−x

2

dx

using

a. uniform sampling ofxon the interval [0,1], and

b. an importance functionh(x), where

h(x) =

3

2

(

1 −x^2

)

constitutes the first two terms in the Taylor series expansion of exp(−x^2 ).

In both cases, compare the converged result you obtain to the value ofI

generated using a simple numerical integration algorithm such as Simpson’s

rule.

7.2. Devise an importance function for performing the integral

I=

∫ 1

0

cos

(πx

2

)

dx,

and show using a Monte Carlo program that your importance function leads

to a smaller variance than uniform sampling for the same number of Monte

Carlo moves. How many steps are required with your importance function to

converge the Monte Carlo estimator to within 10−^6 of the analytical value of

I?

7.3. The following example (Kalos and Whitlock, 1986) illustrates the recursion

associated with the M(RT)^2 algorithm. Consider the M(RT)^2 algorithm for

sampling the one-dimensional probability distribution f(x) = 2xforx∈

(0,1). Let the probability for trial moves fromytoxbe

T(x|y) =

{

1 x∈(0,1)

0 otherwise

}

.

In this case,r(x|y) =x/yfor iny∈(0,1).