Figure: An illustration of 3 particles doing a random walk diffusion process

Caption: "Isotropic random walk on the Euclidean lattice. This picture shows three different random walks (red, blue, green paths) after 10,000 unit steps, all three starting from the origin." (Slightly edited.)

"Isotropic" means the traveling particles scattering all directions with equal probability.

The particles do slowly diffuse from the origin though their average position remains the origin.

What changes is the width of their distribution. It increases with number of steps.

Below, we give some of the mathematical detials of the process.

  r_n = r_(n-1) + s*cos(θ) , 

where r_n is the radial distance from the origin after n steps, r_(n-1) is the radial distance from the origin after (n-1) steps, s is the step size, θ is the angle from the radial direction, and the formula is in the limit of r_n >> s.

If we take the average, we get

  < r_n> = < r_(n-1)> + s*< cos(θ)> = < r_(n-1)>  + s*0
  < r_n> = < r_(n-1)> 
  < r_n> = 0  .  

So the average position remains the origin.

But if we take the root mean square (RMS), we get

  < r_n**2> = < r_(n-1)**2> + (s**2)*< cos(θ)**2> 
            = < r_(n-1)>  + (s**2)*(1/3)
  < r_n**2> = n*(s**2)*(1/3)
  σ = sqrt(< r_n>) = sqrt(n/3)*s  .  

The RMS or σ is the width of the distribution of particles.

As you can see, the RMS increases linearly as the square root of the number of steps.

So for 10,000 steps you expect the particles to of order 100 step lengths away from the origin which is what the image shows.

In many physical cases, the number of steps is proportional to time since the particle started moving. In those cases, the distribution of particles increases linearly as the square root of time.

In many cases, photons in gases approximately random walk like the case in the image.

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