Wednesday, December 17, 2008

[4] Finite Size PDF of independent points in 1D

The following relates to posts [2] and [3]:



In order to indentify the minimum conditions of a double-peaked universal PDF, the situation can be simplified in several aspects:

A) 1D space:

By restricting the trajectories to one spatial dimension x, rotations are not required any longer.

B) Sets of N independent points:

Trajectories are normally generated by subsequently adding random increments (steps) to the respective last position of a walker. Here, in order to avoid any correlations between successive points, the N positions in each point set are drawn independently from a fixed probability density.

C) Random points equally distributed in [0,1]:

Let this probability density be constant in the intervall [0,1] and zero outside.


Statistical quantities of point sets:

For each point set {x_i}, the center of mass is computed,

\overline{x} = \left\langle x_i \right\rangle_i = \frac{1}{N}\sum_{i=1}^N x_i \;,

as well as the standard deviation

\sigma = \sqrt{\left\langle (x_i-\overline{x})^2\right\rangle_i} \;.

Transformations on point sets:

C-Operation (Center):

x_i \rightarrow (x_i-\overline{x}) \;\;\forall i

S-Operation (Scale):

x_i \rightarrow \overline{x}+(x_i-\overline{x})/\sigma \;\;\forall i

Note that C and S commute with each other.


Effects of transformations:

The effects of C and S on the resulting averaged distributions P(x) are discussed in the following:

* No C, no S:

Direct averaging, without any C or S, yields the expected box-shaped distribution in [0,1], centered around 1/2.

* Only C:

Applying only the C-operation yields PDFs centered around zero. For N=1 one obtains a delta-function, for N=2 a triangular function, etc. For very large N, the box is recovered:

* Only S:

Scaling alone produces already a double-peak structure, centered around 1/2. However, it is a finite size effect that quickly disappears for long trajectories:

* C and S:

The combined effect of C and S yields a double-peak centered around zero:

If instead of the box-shaped distribution, a Gaussian distribution is used for the random points, the results are qualitatively the same. However, the double peak is much weaker pronounced and visible only for very small trajectory lengths.



Summing up, the double-peak is not a very remarkable feature of 1D trajectories.


Origin of double-peak:

Take the extreme case of N=2: After a C-operation the two points lie symmetrically left and right from x=0. An additional S-operation scales the distance between the two points to the norm value, and the average distribution P(x) consequently consists of two delta-functions.

On the other hand, when N becomes very large, the histogram of each individual trajectory will already reflect the ensemble average very closely. The CS-operations therefore do not change the shape of this distribution qualitatively and one expects for P(x) to recover the fundamental distribution (in our case: box-shape).

Then, for intermediate lengths N, one expects a gradual interpolation between the double-delta-peak and a box-distribution.


No comments:

Post a Comment