Answer:
This problem involves the concept of a random walk, which is a mathematical model of a path consisting of a succession of random steps.
The question asks for the distance, R(n), from the center of a star after n steps of a photon, assuming a 2D random walk.
The random walk in two dimensions has a step length of A_i and the direction of the steps is uniformly distributed in [0, 2π). The change in position after each step can be written in Cartesian coordinates (Δx, Δy), where Δx = A_i cos(θ_i) and Δy = A_i sin(θ_i).
The displacement from the center after n steps is given by the vector sum of all the individual steps. This vector sum can be written in terms of its Cartesian coordinates, (X, Y), where X = Σ Δx and Y = Σ Δy. This sum over n random vectors is itself a random variable. The net displacement R(n) from the center of the star after n steps is given by the magnitude of the net displacement vector:
R(n) = √(X² + Y²)
Because each step is independent and has a random direction, the expected value of the cosine and sine for any step is zero. This means that the expected values of X and Y are both zero.
However, the mean square displacement is not zero. Because the steps are independent, the mean square displacement in each direction is additive. For a 2D random walk:
<X²> = Σ <(Δx)²> = n <(A cos θ)²> = n A²/2
<Y²> = Σ <(Δy)²> = n <(A sin θ)²> = n A²/2
Because <X²> = <Y²>, we can write:
<R²> = <X²> + <Y²> = n A²
So, the root mean square distance (the square root of the mean square displacement) after n steps is:
R(n) = √(<R²>) = √(n) * A
Therefore, the distance R(n) that the photon is expected to be from the center of the star after n steps grows as the square root of the number of steps, with each step having a length A. Please note that this result holds for a 2D random walk. A real photon in a star would be performing a 3D random walk, which would have slightly different characteristics.