Question

Compute and plot the path(s) of a set of random walkers which are confined by

a pair of barriers at +B units and -B units from the origin (where the walkers

all start from).

A random walk is computed by repeatedly performing the calculation

x

j+1

= x

j

+ s

where s is a number drawn from the standard normal distribution (‘randn

’

in

MATLAB). For example, a walk of N steps would be handled by the code fragment

x(1) = 0;

for j = 1:N

x(j+1) = x(j) + randn(1,1);

end

There are three possible ways that the walls can "act":

a. Reflecting - In this case, when the new position is outside the walls, the

walker is "bounced" back by the amount that it exceeded the barrier.

That is,

when x

j+1

> B,

x

j+1

= B - |B - x

j+1

|

when x

j+1

< (-B),

x

j+1

= (-B) + |(-B) - x

j+1

|

If you plot the paths, you should not see any positions that are beyond |B|

units from the origin.

b. Absorbing - In this case, if a walker hits or exceeds the wall positions, it

"dies" or is absorbed and the walk ends. For this case, it is of interest

to determine the mean lifetime of a walker (i.e., the mean and distribution

of the number of steps the "average" walker will take before being absorbed).

c. Partially absorbing - This case is a combination of the previous two cases.

When a walker encounters a wall, "a coin is flipped" to see if the walker

reflects or is absorbed. Assuming a probability p (0 < p < 1) for

reflection, the pseudo-code fragment that follows uses the MATLAB uniform

random-number generator to make the reflect/absorb decision:

if rand < p

reflect

else

absorb

end

What do you do with all the walks that you generate? Compute statistics, of course.

You should answer questions like

What is the average position of the walkers as a function of time?

What is the standard deviation of the position of the walkers as a function

of time?

Does the absorbing or reflecting character influence these summaries?

For the absorbing/partial-reflection case, a plot of the number of surviving

walkers as a function of step numbers is a very interesting thing. Useful, informative and

interesting, particularly if graphically displayed.

Report Requirement

In your project report, you need to

Describe the process of getting the answers and explain your graph or table. You have

to make it understandable to students with basic knowledge of the pre-requisites and

the course materials covered so far. In general, you should type it up in a word

document and copy and paste any relevant results from MATLAB.

Append your MATLAB programs at the end of your document or submit them as

separate files.

Organize your report according to the questions asked.

Submitting only MATLAB files will result in very few points.

Answer 1

• matlab code:

  %% Program Starts here
  clc;
  close all;
  clear all;
  walkers=1000;
  %walkers=input('how many walkers do you want to simulate?')
  Nsteps = 1000;
  B=3;
  %N = input('how many steps do you want?')
  %menu for how barrier acts
  %case 1 reflect/ bounce back in
  %case 2 end walk
  %case 3 flip coin to decide bounce or end walk
  prompt = 'Please Enter your Choice for how barrier acts? ';
  option = input(prompt);
  if option==3
  prompt = 'Enter the Probability of reflection(0-1) ';
  p_val = input(prompt);
  end
  x = zeros(walkers,Nsteps);
  for w= 1:walkers
  for k = 1:Nsteps %for each walker
  x(w,k+1)= x(k)+ randn(1,1);
  switch option
  case 1
    
  if x(w,k+1)> B
  x(w,k+1)=B-abs(B-x(w,k+1));
  else
  x(w,k+1)=-B+abs(-B-x(w,k+1));
  end
    
  case 2
  if (x(w,k+1)>= B) || (x(w,k+1)<= -B)
  break;
  end
    
  case 3
  p=randn(1,1);
  if p<p_val
  if x(w,k+1)> B
  x(w,k+1)=B-abs(B-x(w,k+1));
  else
  x(w,k+1)=-B+abs(-B-x(w,k+1));
  end
  else
  if x(w,k+1)>= B || x(w,k+1)<= -B
  break;
  end
  end
  end
  end
  hold on;
  grid on;

  plot(x(w,:))
  end

  output: