Polymers in Confined Geometry.pdf

46 CHAPTER 4. SIMULATION METHODS
Blocking method There are situations, when a set of samples has been obtained,
but it has not been taken care of measuring the correlation time. When
the whole set of sampling points is available a different approach can be made to
extract an useful error estimate. It is a blocking method and therefore related to
real space renormalization (cf. [13, 35] and the original paper [12]).
The idea is to calculate the error over a successively smaller set of sample
points and observe how this error behaves. The first step is to calculate the error
over the whole set of points. Then the set is reduced to half its size by taking
the mean of two successive point as new points for the smaller sequence. Again
the error is calculated over this new set.
By repeating this until the set is too small, a sequence of errors is obtained,
which is first increasing and then should show a plateau (at the end fluctuations
become very large, so that the last points are unreliable). It can be shown (see
[12], also for further details and plots) that the real uncorrelated error corresponds
to the value of the plateau. If there is no plateau visible, than too few sample
points are available to uncorrelate the error by this method and one can just
get a lower border for the error, which is given by the largest error value in the
sequence.
4.3.4 Recursive averaging and error calculation
In the simulations a lot of averages and errors thereof have to be calculated. If
this is done at the end of the program run all the sample data has to be stored.
This can result in a large memory usage. The smart way is to calculate the
averages and errors recursively from the last value and the just sampled one.
The mean in the Nth step is denoted by (where xi is the sample taken in
MC-step i)
〈x〉 N = 1
N
N
xi. (4.21)
Inserting this in the same formula written down for N +1, results in the recursive
equation for the average
i=1
〈x〉 N+1 = N
N + 1 〈x〉 1
N +
N + 1 xN+1. (4.22)
To equate the relation for the mean-square deviation 〈∆x 2 〉 N = 〈x 2 〉 N − 〈x〉 2
N
the result eq. (4.22) has to be used besides the same for mean-square average
x 2
N+1
N 2
= x
N + 1
1
+ N N + 1 x2N+1. (4.23)
Inserting and after some rearranging of terms the results is
(N + 1) ∆x 2
N+1 = N ∆x 2 N 2. + 〈x〉N − xN+1 (4.24)
N N + 1