All simulations
Random walk to Brownian motion
Shrink a coin-flip walk until it becomes continuous, and watch uncertainty grow with √t.
Each step moves ±0.500, so more steps means smaller moves.
The red line on the chart.
Spread at t = 41.99theory: √4 = 2.00
Average position-0.03theory: 0
Dark line: one walkDashed: ±1σ = ±√tDotted: ±2σ
Where 3,000 walks are at time t = 4
Bars: simulated positions. Dashed curve: normal distribution with mean 0 and variance 4.
At t = 4, the walks spread out by 1.99, close to √4 = 2.00. Double the time to t = 8 and the spread grows to √8 = 2.83, only √2 ≈ 1.41 times wider, not twice. With so few steps the walk is jagged and lands only on a grid of values.
Try this
- Start at 4 steps per unit and slide up to 1000. Watch the jagged walk turn into a continuous path.
- Set t to 4, then 8. Does the spread double, or grow by about 41%?
- Flip new coins a few times. Each walk is different, but the cloud keeps the same shape.
- At low step counts the histogram is spiky. Raise the steps and it settles onto the bell curve.
The math
- position of the walk at time t
- normal distribution, mean 0, variance t
- spread grows with the square root of time
