Probability in Modeling D. E. Stevenson Shodor Education Foundation

An Aside on Matlab

Population in Stella Revisited

Stella Model with Random Population Change Pop(t+1) = Pop(t)  average rate of change + random deviation [-6,6]  rate  Pop(t) 2.

Diffusion

Diffusion Processes “Diffusion refers to the process by which molecules intermingle as a result of their kinetic energy of random motion. Molecules are in constant motion and make numerous collisions.” (edited version from hyperphysics.phy- astr.gsu.edu)

Modeling the Physics Kinetic Energy is mv 2 /2. Temperature T in  K = E(mv 2 /3/k) k = 1.38  joules/  K Assume motion in all three dimensions.

Some Real Stuff What does this all mean for a pile of sugar? –Mass of sucrose is 342 daltons. –Velocity in sucrose 81 m/sec. –Mean free path about 4.5  cm (durn rough estimate). –Mean time between collisions 5.6  sec.

Random Walks

Model=Random Walk Let x(n) be the location of a particle at time t. x(0)=0 The particle moves a fixed (unit) distance every time interval  at a speed of u for an effective length of  u. The probability that particle moves to the right is p and to the left q. Time step directions are independent.

Question 1: Where do the particles end up?

Assume that the particles don’t transfer momentum. Consider the trajectory of a single particle. Assume p=q=1/2. Where does the particle end up? Matlab d1drwalk1.m d1drwalk2.m Final Location

Let x i (n) the position of particle i at time n. The rule is So the average is Computing Ensemble Average

Finalizing The average of the steps is zero if p=q. Then the average location at time n is the same as that of n-1. Recursively, then the average location is the same as the starting location…zero.

Computing Ensembles Now let’s consider many particles, all starting at X(0)=0. Assume these do not collide with one another. All these particles together form an ensemble. What can we say about the ensemble? d1drwalk4.m But isn’t it zero? d1drwalk4bin.m

Ensemble Average Here’s the uncertainty. We ran a small number of trials (M=100) for a short period of time (N=500 steps). I need to consider –Is M big enough? –Is N big enough? –Ah, is the random sequence good enough? d1drwalk5.m

So What?

Summary We have considered some of the history of probability in science as opposed to its use as a mathematical subject. We considered very briefly the diffusion process and random walks as a implementation. We saw that ensembles may or may not be well constructed by Matlab.

A Little Background

A little history Jakob (Jacques) Bernoulli, Ars Conjectandi, Thomas Bayes, Essay towards solving a problem in the doctrine of chances, Pierre-Simon Laplace, Essai philosophique sur les probabilités, George Boole, An investigation into the Laws of Thought, on Which are founded the Mathematical Theories of Logic and Probabilities, 1854.

More Modern… Copenhagen Meeting, William Feller, Introduction to Probability Theory and its Applications ( ). Sir Harold Jefferys, Theory of Probability, Samuel Karlin, A First Course in Stochastic Processes, A Second Course in Stochastic Processes, Edwin T. James, Probability as Extended Logic, 1995 (bayes.wustl.edu)