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The following graph is the x component of position as a function of time for a particle undergoing simulated Brownian motion. The program that generated the random walk was written in C and was named bsumsci2.c. The code assumes a particle with radius 0.312x10-6 meters, immersed in water with absolute viscosity, 0.0010019 kg/m s, at 293K. The graph depicts five hundred points at which the position of the particle is recorded out of the 80,000 points generated by the bsumsci2.c simulation. Each time step is one thirtieth of a second, chosen because many experiments are done with digital video microscopy, which has a frame rate of 30fps.

Brownian Walk

After recording the positions of the particle in data1.dat and simulation parameters in params1.txt, the program then stores the difference between each position or between positions separated by any fixed time interval (say every third interval). These displacements are written to a file, delt1.dat. data1.dat, params1.txt, and delt1.dat are given a serial number when created. The template file names are, dataXXXX.dat, paramsXXXX.txt, and deltXXXX.dat, where XXXX is an identifying serial number.

These displacements should be normally distributed as described in the Theory section. To check this we use Origin 6.0, a data analysis and graphing program. LabTalk is Origin's built-in programming language, which enabled us to automate some of the analysis. The following code is part of a LabTalk program we wrote called, slopes.ogs. It was written to create histograms of the displacements output by bsumsci2.c. Though this feature is not used immediately, the slopes.ogs program is capable of handling multiple displacement files.

win -t wks ORIGIN Wk$(ii); //Creates a new worksheet template for each ASCII delt file. Assumes delt files have a number on the end of their name.

FDLog.Get(A, ii); //Get the filename to open into %A

open -w %A; //Import the data into the worksheet

#!type "File: %A"; //Report name of imported file

//%H is set to the current active window title.

%B=%H_A; //Adding column A to %H.

win -t wks ORIGIN histdata$(ii); //Creates a workseet for the histogram of the delt files. ii increments histdata wks

ll=1; mm=-0.0000009688;

for(mm=-0.0000009688;mm<=0.0000009688;mm+=0.0000000625){

histdata1_A[ll]=mm; //For loop fills column A of

ll++;}; //histdata(ii) with midpoint bin values for plotting the histogram.

histdata$(ii)_A=histdata1_A; //Fills column A of multiple delt files.

histdata$(ii)_B=histogram(%B,6.25e-8,-9.688e-7,9.688E-7); //Histograms Delt file and puts result in column B of histdata

histdata$(ii)_B=histdata$(ii)_B / 80000; //Converts counts into a frequency.

worksheet -a 1;

window -t graph ORIGIN hist$(ii); //Creates graph histii to plot histogram on

layer -i201 histdata$(ii)_B;

layer -a;

The graph below is the output of the above LabTalk code, applied to a deltXXXX.dat file. Plotted on the vertical axis of the histogram is the number of displacements within a specific bin divided by the total number of steps. The x-axis is divided into 32 bins with width 6.25x10-8 meters, starting at -1x10-6 meters and ending at 1x10-6 meters. The y-axis is thus the probability of a displacement occurring within each bin.

Histogram

The fit to the data points is a non-linear fit to the equation

(see the Normal Distribution section). First we observe that the histogram data is well fit by the Gaussian function. To make a more quantitative comparison we need to note the difference between the histogram and the theoretical probability density. The histogram records the probability, P(x), of a displacement occurring in a range of x from x - Δx/2 to x + Δx/2, where Δx is the bin width. Expressed in terms of the probability density function,

If we have sufficiently narrow bins and a reasonably smooth f(x), this may be approximated as P(x) = f(x) Δx. Origin reports a best fit value of w to be: w = (3.0366 +/- 0.0105) x 10-7m and a best fit value of A to be A = (6.2535 +/- 0.01804) x 10-8. From the Theory page, we expect . Using the values of radius, viscosity, and temperature given above, we expect w = 3.025 x 10-7m. Thus, we see agreement between our simulated Brownian motion and the theoretical expectations, within our uncertainties. The parameter A should just be the bin width and we see that it also agrees with our uncertainties.