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 $s_{i}$ (which can be thought to be done at times $i\Delta t$) independent from each other. Supposing that $s_{i}=\pm 1$ (namely the particle can make one step to the right or left each time) the position of the particle after $N$ steps (note that the number of steps done is proportional to the time passed) will be:

$x_{N}=s_{1}+s_{2}+\cdots +s_{N}$

What we would like to understand is the following: if we make the same particle start from the origin for

$N$ times and move freely, what is its average behaviour? Equivalently, if we make

$N$ 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

$p(s_{i}=1)=p(s_{i}=-1)=1/2$, then:

$\left\langle s_{i}\right\rangle ={\frac {1}{2}}\cdot 1+{\frac {1}{2}}\cdot (-1)=0\quad \Rightarrow \quad \left\langle x_{N}\right\rangle =\sum _{i=1}^{N}\left\langle s_{i}\right\rangle =0$

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:

$\sigma _{N}^{2}=\left\langle x_{N}^{2}\right\rangle -\left\langle x_{N}\right\rangle ^{2}=\left\langle x_{N}^{2}\right\rangle =\left\langle \left(\sum _{i=1}^{N}s_{i}^{2}\right)^{2}\right\rangle =\left\langle \sum _{i,j=1}^{N}s_{i}s_{j}\right\rangle =\sum _{i,j=1}^{N}\left\langle s_{i}s_{j}\right\rangle$

However, since the steps are independent we have

$\left\langle s_{i}s_{j}\right\rangle =\left\langle s_{i}\right\rangle \left\langle s_{j}\right\rangle$ if

$i\neq j$. In other words:

$\left\langle s_{i}s_{j}\right\rangle ={\begin{cases}\left\langle s_{i}\right\rangle \left\langle s_{j}\right\rangle =0&i\neq j\\\left\langle s_{i}^{2}\right\rangle =1&i=j\end{cases}}\quad \Rightarrow \quad \left\langle s_{i}s_{j}\right\rangle =\delta _{ij}$

so that:

$\sigma _{N}^{2}=\sum _{i,j=1}^{N}\left\langle s_{i}s_{j}\right\rangle =\sum _{i,j=1}^{N}\delta _{ij}=N$

Therefore, after

$N$ steps the particles are distributed in a region of width

$\sigma _{N}={\sqrt {N}}$.
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 ${\vec {s}}_{i}$ the random steps the position of one particle after $N$ steps will be:

${\vec {r}}_{N}=\sum _{i=1}^{N}{\vec {s}}_{i}$

and supposing

$|{\vec {s}}_{i}|=\ell \quad \forall i$, then:

$\left\langle {\vec {r}}_{N}\right\rangle =\sum _{i=1}^{N}\left\langle {\vec {s}}_{i}\right\rangle =0$

since the

${\vec {s}}_{i}$-s are uniformly distributed on a sphere.
We will also have

$\left\langle \right\rangle =\ell ^{2}$ and therefore this time:

$\left\langle {\vec {s}}_{i}\cdot {\vec {s}}_{j}\right\rangle ={\begin{cases}\left\langle {\vec {s}}_{i}\right\rangle \left\langle {\vec {s}}_{j}\right\rangle =0&i\neq j\\\left\langle {\vec {s}}_{i}^{2}\right\rangle =\ell ^{2}&i=j\end{cases}}\quad \Rightarrow \quad \left\langle {\vec {s}}_{i}\cdot {\vec {s}}_{j}\right\rangle =\ell ^{2}\delta _{ij}$

so that the variance after

$N$ steps is:

$\sigma _{N}^{2}=\left\langle {\vec {r}}_{N}^{2}\right\rangle =\left\langle \sum _{i=1}^{N}{\vec {s}}_{i}\cdot \sum _{j=1}^{N}{\vec {s}}_{j}\right\rangle =\sum _{i,j=1}^{N}\left\langle {\vec {s}}_{i}\cdot {\vec {s}}_{j}\right\rangle =N\ell ^{2}$

Therefore, this time the particles will be distributed in a region of linear dimension

$\sigma _{N}=\ell {\sqrt {N}}$; as in the previous case, the variance increases with the square root of the number of steps (namely, the square root of time).