We will now study the general properties of random walks.
Consider a point particle in the real axis starting form the origin and moving by steps (which can be thought to be done at times ) independent from each other. Supposing that (namely the particle can make one step to the right or left each time) the position of the particle after steps (note that the number of steps done is proportional to the time passed) will be:
What we would like to understand is the following: if we make the same particle start from the origin for
times and move freely, what is its average behaviour? Equivalently, if we make
different independent particles start at the same time from the origin, how will the system evolve and which will be its properties?
Since the particles at every step have the same probability to go right or left, namely
so the mean position is not really relevant. We can however compute the variance from this mean position to understand how the particles are distributed:
However, since the steps are independent we have
. In other words:
steps the particles are distributed in a region of width
This allows us to highlight a great difference with the more familiar uniform linear motion: for random walks the mean displacement does not
grow linearly with time, but depends on its square root
If we now move to three dimensions, calling the random steps the position of one particle after steps will be:
-s are uniformly distributed on a sphere.
We will also have
and therefore this time:
so that the variance after
Therefore, this time the particles will be distributed in a region of linear dimension
; as in the previous case, the variance increases with the square root of the number of steps (namely, the square root of time).