3. Compartmental Random Particle Movement Model
(35 total points) In a physics experiment a particle is placed in Chamber A of the space
shown below. The scientists observe the length of time for theparticletoﬁrstenterthespecial
chamber, Chamber F,wheretimeismeasuredbythenumberofchamberstheparticlevisits.
For example i f the particle starts in Chamber A and proceeds to Chamber D and then to E
and then to F,thenthelengthoftimetogofrom A to F is 3.
Assume that the particle will move in a random way bouncing o” the walls the chamber that
it is currently in until it randomly exits the current chamberthroughanyavailableopening
out of a chamber with equal likelihood. For example, if there are two openings available
in a particular chamber then the particle will eventually exit this chamber by either of the
openings with equal probability. (In counting the openings,includetheonethattheparticle
used to enter the chamber because the particle could retrace its steps.)
Note: When the particle enters Chamber F (or C)itwillleaveChamber F (or C)viathe
only opening possible and re-enter Chamber E (B).
Figure 2: Compartmental Random Particle Movement Model
In this picture there is only an opening if there is a gap in the lines indicating a wall.
(a) (5 points) Model the particle’s journey through the system to the special chamber,
Chamber F,asaMarkovChainandwritethesingle-steptransitionmatrix (complete
with numbers). You must also write how you specify the initialstate.