Question

I did these so far:

EDIT—————

steps=@ (m) 2*randi([0,1],[1,m])-1;
Walk1D =@ (n) [0,cumsum(steps(n))];
findend=@ (x) x(end);
LastPoint1D=@(n) findend(Walk1D(n));

nsteps=200;
nsq=floor(sqrt(nsteps));
MeanSquareDistance1D= @ (n,m) m.*sum((LastPoint1D(n)).^2)./m;
r2D=MeanSquareDistance1D(100,1000)

data=[ ];
for i=10:20:90
data=[data; i , MeanSquareDistance1D(i,2000)]
end

The only problem now,is that the 2nd column of “data” must give me values around

10
30
50
70
90

but not exactly.Only approximately.For example ,the “data” must be:

10  10.184
30  27.51
50  50.306
70  68.394
90  90.414

Sth is wrong with the sum maybe?

Answer 1

From your code I guess you want to calculate the mean squared distance for a 1D random walk.

The mean squared distance at lag tt is the average squared difference between two positions along the random walk separated by tt steps. I assume that data should be an array where the first column is tt and the second column the corresponding mean squared distance, and where there is an additional parameter that indicates the total number of steps in your random walk.

Here’s how I’d calculate data

%# define parameters
nSteps = 2000;
listOfLags = 10:20:90; %# must be 1-by-n vector

%# create random walk
%# steps can be +1 or -1, add all of them via cumsum
randomWalk = cumsum(randi([0 2],nSteps)-1);

%# calculate msd for the desired lags
%# use a loop for readability
nLags = length(listOfLags);
data = zeros(nLags,2);
data(:,1) = listOfLags;

for lag = listOfLags
    %# lag takes on every lag value, so use logical indexing to find
    %# which lag (in terms of entry into data) we're currently working on

    %# This line corresponds to
    %# 1. get all distances traveled within a duration of `lag`
    %#    vectorOfDistances = randomWalk(lag+1:end) - randomWalk(1:nSteps-lag)
    %#    i.e. the first element is randomWalk(lag+1)-randomWalk(1)
    %# 2. square all: (vectorOfDistances).^2
    %# 3. average all squared distances 
    data(listOfLags==lag,2) = mean( (randomWalk(lag+1:end) - randomWalk(1:end-lag)).^2);
end

%# plot the results
plot(data(:,1),data(:,2),'.')
xlabel('lag'),ylabel('mean squared displacement')