__HW 1__

__HW2__**Question 1:**I have wrote the program

**HW2.f**using Fortran77 the output file**hw2.out**presents the enumeration of total number

of random walks of N steps that reach the point x, for all x for N=4,5,6,7 and 8.The following conclusions can be made from the output file:

- The even positions can be reached only by even number of steps.
- The odd positions can be reached only with the odd number of steps.
- The general results form a triangular structure. The results which are presented in the output file form a trapezoid structure.
- For an even N, the highest number of random walks is to return to position zero.
- For an odd N the highest number of random walks is to gett to position +/-1.

Like it was asked in the question 2, I have compiled the program rwalk1.f**Question 2:**

and cplot.f in order to draw graphs of P(x) (the probability that after N steps a walker has nett displacement X).

I checked it for N = 8,16,32 and 64. I wanted to check the statment that the better seed

is big and odd, for this purpose I have took diferent seeds, for example

1) small but odd - 123

2) smal and even - 242

3) big and odd - 76539

4) big but even - 34326

All the seeds were tested for every one of 4 numbers of steps (N). Unfortunatly, it was hard to distinct clearly a specific pattern

of influencing the change of the seed numbers on the mean displacement

of the graph. However it is possible to make the following conclusion: The range of the probability that a walker has nett displacement X

after N steps (N = 8,16,32, 64) was between 0 to ~0.3, for the checked seed

numbers. With increasing the steps number, the mean of x approaches zero

that strengthens the observation in question 1: X0 is the position with

the highest probability that a walker will end in.