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The following is a problem I ran into during my internship. The background of my problem is biological/chemical in nature, so I’ll try to bother you as little as possible with the details.

Thanks in advance for any help given.

**1. Homework Statement**

I have a point p0 on a 2-dimensional surface. The next point (p1) is at distance L1 from p0. The next point p2 is at distance L2 from p1. This continues, up to p28.

I would like to find an equation to calculate the expected distance between the origin and each subsequent point (d1, d2, etc), as a direct function of the number of steps (s) between the origin and that point.

The step length L between each two subsequent points is assumed to be variable, but with a limited maximum value Lmax. L = x*Lmax, where x is a random (or at least poorly understood) variable between 0 and 1.

**2. Homework Equations**

My dear friend Pythagoras:

[tex]A^2+B^2=C^2[/tex]

Also:

[tex]\sin\left(\alpha\right)^2+\cos\left(\alpha\right)^2=1[/tex]

**3. The Attempt at a Solution**

The distance d1 (between p0 and p1) is obviously trivial: It is L1.

The angle between d1 and the line p1-p2 is called a2.

d2 is then calculated as:

[tex]d2^2=(L1+L2*\cos\left(a2\right))^2+(L2+\sin\left(a2\right))^2

d2^2=L1^2+L2^2+2*L1*L2*\cos\left(a2\right)[/tex]

d3 is calculated in a very similar manner, with a3 the angle between d2 and the line p2-p3:

[tex]d3^2=(d2+L3*\cos\left(a3\right))^2+(L3+\sin\left(a3\right))^2

d3^2=d2^2+L3^2+2*d2*L3*\cos\left(a3\right)[/tex]

In general, it can be said that:

[tex]ds^2=d\left(s-1\right)^2+Ls^2+2*d\left(s-1\right)*\cos\left(as\right)[/tex]

Now, this equation requires that I always calculate d(s-1) before I can calculate ds. I would like to find a direct equation, if that is at all possible. I am aware that that for any single iteration of this process, one

**needs**to know all previous steps, but I would say that there is some expected value if the process is repeated often enough.

To establish this, I used Excel to simulate this process 12,000 times, and noted the average distance between the origin and ps after s steps. What resulted was a clear correlation, with very little variability (due to the large number of iterations). This to me seems like a clear indicator that there is indeed some expected value for ds, but I can’t seem to nail down exactly how I could calculate it.

Again, thanks in advance for any help provided. You would really help me out if you could point me in the right direction.

Regards,

DaanV