Home / Topics / Kinetic Theory & Statistical Physics
Random walks
Advanced
Steps in random directions give diffusion.
⟨x²⟩ = 2DtA particle taking random steps of length l moves a typical distance l√N after N steps. This links microscopic randomness to diffusion.
After 10⁶ steps, only 1000 step lengths from the origin.
Net distance grows like √t, not t.
In more detail
A random walk is a path made of random steps. Its displacement grows with the square root of the number of steps, which is why diffusion is slow over long distances.
Real-world uses
- Modelling molecular diffusion and stock price fluctuations.
- Animal foraging paths are often modelled as random walks.
- Physicists model the path of a pollen grain or a drunk walker with a random walk, which gives diffusion.
Open the interactive lesson, with simulation and quiz →
Frequently asked questions
How far does a random walker get?
Typically about the square root of the number of steps times the step length.
Is a random walk always random?
Yes, the steps are chosen randomly, though biased walks have drift.
Why is the spread proportional to the square root of time?
Random steps partially cancel, so the typical distance grows with the square root of the number of steps.
Related topics in Kinetic Theory & Statistical Physics
Debye model · Photon gas · Planck distribution · Equation of state of ideal gas in statistical terms · Heat conduction at microscopic level · Transport phenomena