Brownian motion is the observed movement of small particles as they are randomly bombarded by the molecules of the surrounding medium. This was first observed by the biologist Robert Brown and was eventually explained by Albert Einstein, for which work he received the Nobel prize.
We can simulate Brownian motion in one dimension by tossing coins. If the sequence of tosses is
1 1 0 1 1 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1then we can move are particle one unit to the right each time we see a 1, and on unit to the left each time we see a 0. This results in the new sequence
1 2 1 2 3 4 3 2 3 2 1 0 1 0 1 0 -1 0 -1 0We haven't dispalyed the starting position (0). We can also display this graphically:
* * * * * * * * * * * * * -------------*-*-*-*-*--- x = 0 * *
We can simulate Brownian motion using the computer if we have a good generator of uniformly distributed random numbers. This is what the program uni.c does. To copy uni.c to your directory, do this
% cp /u/cl/doc/ma217/uni.c . % ls uni.c % uni.cWe copied uni.c from the directory /u/cl/doc/ma217 . Note the period (.) in the copy (cp) command. It stands for your current directory. Now type the program r1.c . Finally, compile it like this:
% gcc r1.c uni.cThis compiles both your program and uni.c. The two compiled files are then linked together, and you can run them using
% a.outRecall that a.out is the default name of a compiled C program. You should get this output:
1: 0.775535 2: 0.847905 3: 0.297409 4: 0.017542 5: 0.121674 6: 0.527103 7: 0.889287 8: 0.182352 9: 0.387452 10: 0.564470Once you have a good way of producing uniformly distributed random numbers in the range [0,1], you can compute other sequences of random numbers. For example, to get a uniformly distributed sequence of zeros and ones, use the rule:
if u < 1/2 then v = 0 else v = 1where u is a random variable uniformly distributed in the range [0,1]. To get a sequence of uniformly distributed numbers in the range [-1,+1], set
v = 2u - 1
Redo the previous problem for 100-step random walks. Study the relative shape of the frequency histograms.
Problems 4 and 5 of the last problem set dealt with random variables that are themselves sums of random variables:
Y = X_1 + X_2 + ... + X_NHere the X_i are independent and have identical probability distributions. The central limit theorem, which we have mentioned before, says that as N gets larger and larger, the probability distribution of Y becomes closer and closer to a fixed distribution, the normal distribution. Its general shape is that of a bell curve, with the center of the bell at
Y = E(Y) = N E(X)This bell can be a narrow one or a broad one. About 68% of the area under the bell lies above the interval
[ E(Y) - SD(Y), E(Y) + SD(Y) ]where
SD(Y) = sqrt(N) SD(X)is the standard deviation. Let us call this the core of the bell. Note that about 95% of the area under the bell lies above the two standard deviation interval.
Suppose that after one second the core is one millimeter wide. How wide will it be afer 100 seconds? How wide will it be after 10,000 seconds (almost three hours later). If the concentration is 10 units at one second, what will it be at 100 seconds and at 10,000 seconds?
Make a sketch of the width of the core as a function of time.
Conclude by make a brief, labeled sketch of the diffusion process we have studied. Feature bell-shaped blobs getting wider and shorter, and explain what is going on.
The authors strive to develop statistical intution with as little mathematics as possible.