I set out to study relative diffusion in the two experiments I have presented in chapter 2 and 3. The quantity I extract from my measurements is the time derivative of the squared separation of pairs of particles as a function of their separation (or rather an approximation thereof).
The data I collect from the experiments are tracks of particles. Each particle track consists of positions of the particle sampled at a fixed frequency. For the Faraday experiment, the sampling frequency was , and several tracks consisted of more than 1000 positions. In the soap film experiment, I worked with a sampling frequency of , and all tracks consisted of two positions (the two ends of an image of a particle track).
Figure 5.1: Particle separation. The sketch shows the tracks of two particles P and Q, and the positions of the particles at two times. The time between successive known positions is in the data from the Faraday experiment and in the data from the soap film experiment.
Since I do not have continous data
for the locations, I have to approximate the time derivative of the squared
particle separation with a difference
Richardson's collection of experimental data for the atmosphere and
Kolmogorov's derivations gives a scaling relation
Instead of just splitting the R-axis in small intervals, and calculating
the average of in each interval, as we did for our articles
on the Faraday experiment, I sorted all the by the particle
separation and calculated an approximation to the integral of
with respect to R:
In the following, an integral with respect to R of some quantity measured on the experiments, denotes such an approximation.
The integral with respect to R of a quantity which scales as a power of R
is proportional to the average of the quantity times R.