Polymers in Confined Geometry.pdf

38 CHAPTER 4. SIMULATION METHODS
polymer with N ≫ 1 monomers. Since in MD simulation the smallest time-scale
of the model must be simulated, it cannot be effectively applied to our problem.
We therefore concentrate on the Monte-Carlo simulation in the following sections.
For a very good introduction also refer to [35].
4.2 The Monte-Carlo method
In statistical physics ensemble averages of fluctuating quantities are of special
interest. To obtain these averages, integrals of the following form have to be
evaluated (schematically for one dimension)
〈A〉 p =
+∞
−∞
dx A(x)p(x) . (4.1)
φ(x)
This example could be an average of an observable A(x) (“score function”)
weighted by the probability distribution p(x), where the integral corresponds—in
the language of statistical mechanics—to a ‘sum’ over all states. But the method
explained in the following is applicable to every integral presentable in this form.
When eq. (4.1) is not analytically solvable then one has to resort to numerical
methods. The direct way would be to divide the integration range—which can
be of any dimension—into n equidistant intervals and evaluate the sum over
{φ(xi)} on the discrete set of points {xi} (Riemann integral). In this way the
error of the approximative result in this deterministic algorithm reduces with
n −1/d depending on the dimensionality d. Therefore this method becomes very
ineffective for higher dimensions d. But in particular high-dimensional integrals
is what we are calculating in statistical mechanical problems, therefore we need
a smart way to increase the efficiency of the numerical integration.
The basic idea of Monte-Carlo 1 evaluation of an integral is to choose the n
discretization points/sample points {xi} randomly out of the probability distribution
p(x) such that the sum has to be performed only over the observable
quantity at the so generated points {A(xi)}. This algorithm has the advantage
of a statistical error proportional to n −1/2 , independent of the dimension which
outperforms the deterministic calculation for n ≥ 3. But both algorithms have
the problem, that they are very inefficient, when the problem under investigation
has a large inherent variance, due to the fact that the computational time grows
proportional to n and the variance is inversely proportional to √ n.
This problem is seen in the following way: If the probability distribution of
the random numbers is such, that the important region of the observable is poorly
sampled and therefore the variance is large, a huge number of samples may be
needed to get a reliable result. It may happen, that the interesting region is
1 The name ought to bring the casinos of Monte-Carlo to mind.