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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] ratePop(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 mv2/2.
• Temperature
T in K = E(mv2/3/k)
k = 1.3810-23 joules/ K
• Assume motion in all three dimensions.
kTv m
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 4.510-10 cm (durn rough
estimate).– Mean time between collisions 5.610-10 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 xi(n) the position of particle i at time n.
• The rule is
• So the average is
Computing Ensemble Average
( ) ( 1) ( )i ix n x n rv
1
1 1( ) ( ) [ ( 1) ( )]
N
i ii
X n x n x n rvN N
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, 1713.• Thomas Bayes, Essay towards solving a problem
in the doctrine of chances, 1764.• Pierre-Simon Laplace, Essai philosophique sur les
probabilités, 1812. • 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, 1927.• William Feller, Introduction to Probability Theory
and its Applications (1950-61).• Sir Harold Jefferys, Theory of Probability, 1939. • Samuel Karlin, A First Course in Stochastic
Processes, 1969. A Second Course in Stochastic Processes, 1981.
• Edwin T. James, Probability as Extended Logic, 1995 (bayes.wustl.edu)
