Question

The simplest model of a polymer involves a sequence of bonds of fixed length λ, where each bond can make any angle with the previous bond. This is called the freely-jointed chain model. Here will will study it in 3D. We start at the origin. Then we choose an angle θ randomly between 0 and 2π. We make a step of length λ at an angle of θ top the +x direction. We then repeat, making N steps in total. We can measure the size of the polymer by finding the end to-end-vector R and squaring it. We then take the square root of this to get the root-mean-square size. We will set λ = 1 Note that this model is “off-lattice" or continuum, we are not using any lattice.

You should: (1) Plot an example walk of say 100 steps.

(2) Generate walks between say 10 and 10^6 steps and make a log-log plot of the root mean square size versus total number of steps. You may need to average over several walks for each length to get good data.

Answer 1

i have coded the above problem in MATLAB.

the code is as follows

clc
clear all
x=0;
y=0;
X=zeros(1,101);
Y=zeros(1,101)
for i=1:100
t=rand(1); %a random number is generated in order to create an angle between 0 and 2pi
theta=2*pi*t;
x=x+cos(theta);
y=y+sin(theta);
X(1,i)=x
Y(1,i)=y
end
R=sqrt(x^2+y^2);
size=sqrt(R^2)
plot(X,Y);

the above plot is for 100 steps

this one is for 1000 steps

this one is for 10000 steps.

Now, for finding the RMS length,

clc
clear all
log_size=zeros(1,1000000);
size=zeros(1,1000000);
for j=10:1000000
x=0;
y=0;
X=zeros(1,j+1);
Y=zeros(1,j+1);
log_size(1,j-9)=j;
R=zeros(1,j-9)
for i=1:j
t=rand(1);
theta=2*pi*t;
x=x+cos(theta);
y=y+sin(theta);
X(1,i)=x;
Y(1,i)=y;
R(1,j-9)=sqrt(x^2+y^2)
end
log_size(1,j-9)=log10(R(1,j-9));
end

since this is a lengthy and code, i couldnt complete the run before the ending of the time period for answering this question. so please run this and find the answer for yourself. the answers will vary, as we are using random number to generate the step