The Random Walker — How Far Do a Hundred Coin-Flip Steps Carry You?

The Random Walker

Teach me

Speed

Random walk — positions and spread

Guided Inquiry

1 · The crowd on one point

Let’s start small. Every walker begins on the same point. Press Step everyone once a few times, and look closely: which sites can be occupied after each tick — and what pattern do you notice in the gaps between them?

2 · Commit: after a hundred steps

Time to commit to a guess. Each walker is about to take 100 coin-flip steps of length s = 1. In the RMS sense, how far from the start do you think a typical walker will stand?

3 · Run to a hundred steps

Let’s run it. Press ▶ Play — the run pauses itself at N = 100. If you like, slow steps per second to a few to watch single hops. As N climbs, what does the histogram do?

4 · What the spread reads

Now compare your choice with the spread readout, and with the predicted s√N panel that’s just been revealed. The sample wobbles a few percent about the prediction — don’t worry, that wobble is finite sampling, not a broken law.

5 · Commit: four times the steps

Let’s push further. Suppose the run continues to N = 400 — four times as many steps as at N = 100. What do you predict the spread will be there?

6 · Run to four hundred

Let’s check. Press ▶ Play again — the run continues and auto-pauses at N = 400. Between N = 100 and N = 400, what did the spread readout do?

7 · Straight against √N

Compare your choice with the readouts. You’ll see two dashed curves have now appeared: the same widths replotted against plain N (read them on their own 100–300 tick row), bending over — and y = N·s, the ballistic path if every step went one way.

8 · Sampling noise, not physics

Now let’s play with the crowd size. Drag walkers W to 50, then to 1000. Each drag restarts and automatically replays the run — raise steps per second if you want it to replay quickly. The shaded band is ±3/√(2W): finite-sample wobble at W = 50, lock-on at W = 1000. It’s noise, not physics.

9 · Why the smell is slow

Here’s one from everyday life. A perfume molecule flies at ~500 m/s but random-walks in steps of ~70 nm. Using the √N law, why do you think the scent takes so long to cross a room?

Listen

walkers W200

102000

step length s1.0

0.52.0

steps per second8

130

Readouts

Steps taken N

Run auto-pauses at N = 400

Spread

0.000

RMS distance from the start, √(Σxᵢ²/W)

Squares add: each step contributes one s² to the mean square displacement, because the cross terms of independent ± steps average to zero.

The standard result — RMS displacement about the start grows as √N, not N. At s = 1, N = 100 it is exactly 10.

The exact N-step distribution is binomial on a lattice: after an even number of steps only even multiples of s are occupied, so occupied sites are 2s apart (parity).

The bell drawn over the count histogram: sites are 2s apart, so the height per site is W·2s·g(x) with g the normal density of variance s²N — the factor 2 is load-bearing.

The shaded band on the width chart: the sample RMS of W walkers scatters about s√N with relative standard deviation ≈ 1/√(2W) (large N), so ±3/√(2W) is the 3σ envelope.

The Physics

The random walker · what this sim shows, the physics behind it, and every control

At a glance

Every walker starts at the same point and, on each tick, steps a distance s left or right on a fair coin flip. The sim runs a crowd of them at once, so you can measure how far random steps carry a walker, and how reliable that measurement is.

What you'll learn

How the typical distance from the start depends on the steps N and the step length s.
How to read the histogram of positions and the curve over it.
How far a spread measured on W walkers can stray from the law.
How the same idea describes molecules moving through a gas.

One tick. From wherever it stands, each walker moves s left or right on a fair coin flip.

What's on screen

Walkers (orange dots, top): where each walker stands. Only the horizontal position matters; the vertical scatter is decoration.
Histogram (grey bars, bottom): walkers per site, on a fixed frame of ±140.
Teal curve over the bars: a reference curve, drawn from N = 10.
Spread chart (right): the measured spread against √N, with reference lines and a shaded band.
Readouts: steps taken N, the Spread, and a Predicted value.

See itRelease 200 walkers from one point. Watch for: the dots spreading out from a single point while the Spread readout climbs.

The controls

Press ⌖ on a row to see where that control sits in the sim.

Control	What it changes	Range
walkers W	How many independent walkers take part. Moving it restarts the run.	10–2000 opens at 200
step length s	The length of every hop, in the units of the ±140 frame. Moving it restarts the run.	0.5–2.0 opens at 1.0
steps per second	How fast ticks arrive. It paces the run and does not change what a tick does.	1–30 opens at 8
Step everyone once	Moves every walker by exactly one tick, so you can go slowly.	button
Readouts	Steps taken N; the Spread, the RMS distance from the start, √(Σxᵢ²/W); and a Predicted value to compare it with.	panel
▶ Play / ⏸ Pause	Runs or freezes the walk. The run stops itself at N = 400; pressing Play again starts a fresh run.	top bar
Speed	Plays the ticks faster or slower on screen, on top of steps per second. The walk itself is unchanged.	0.25×–4×
↻ Reset	Puts W = 200, s = 1.0 and 8 steps per second back, and starts a fresh run.	top bar

Fixed: the coin is fair, every walker starts at 0, and the random sequence is seeded, so the same W and s always replay the same run.

What each tick does

On every tick each walker flips its own fair coin and moves s to the right or to the left. The flips are independent of every other walker and of the walker's own past.

The step rule: ε is +1 or −1.

The average position stays at 0. Left and right are equally likely, so ⟨x⟩ = 0 at every N. That is a statement about signs, not about how far individual walkers have gone.
Positions come in steps of 2s. After N ticks, x/s = (steps right) − (steps left), which has the same parity as N. After an even number of ticks only even multiples of s are occupied, after an odd number only odd ones. The whole set of occupied sites shifts by s on each tick.
One tick is easy to read. Every walker is at +s or −s, so the Spread is exactly s.

See itTake a single tick with your current step length. Watch for: the Spread reading exactly s, whatever the coin flips did.

Squares add, so the spread grows as √N

What accumulates is not the distance but its square. Squaring the position after N ticks:

The cross term averages to 0, because the new coin ignores where the walker already is.

Exact for every N, from N = 1 up. Not a large-N law.

Each tick adds exactly s² to the mean square. The RMS distance, which is what the Spread reads, is s√N.
Four times the steps give twice the spread. Against √N the spread is a straight line of slope s.
The red dashed line y = Ns on the chart is the counterfactual: every step going the same way.

Steps	Spread, s = 1	All steps one way
N = 1	1	1
N = 100	10	100
N = 400	20	400

See itRun 200 walkers with s = 1 out to N = 400, at double speed. Watch for: the Spread at N = 400 coming out about twice its value at N = 100.

A sample wobbles about the law

⟨x²⟩ = Ns² is an average over all possible walks. The Spread is measured on only W walkers, so it scatters about s√N.

Relative scatter of the measured spread (large N), and the band the sim shades.

W s

These sliders move the sim too, and restart its run.

One standard deviation is about 1/√(2W): roughly 10% at W = 50, 5% at W = 200, 2.2% at W = 1000.
A trace that wanders inside the shaded band is sampling scatter, not a failure of the law. More walkers narrow the band; they do not move the law.
Every N is measured on the same W walkers, so the trace drifts rather than re-rolling at each tick.

See itRun W = 50, then W = 1000, each out to N = 400. Watch for: the card reporting the largest miss from s√N for each crowd.

The histogram follows a Gaussian

The exact distribution after N ticks is binomial. It is a bell shape only in the limit of large N.

Probability of ending at x = (2k − N)s: k steps right, N − k steps left.

For large N this approaches a Gaussian of standard deviation s√N. The sim draws it over the bars from N = 10, where the biggest gap is about 2.5% of the curve's peak. By N = 100 it is 0.25%.
Sites are 2s apart, so the curve is scaled by the site spacing to give walkers per site:

Without the 2s the curve would sit at half height.

The Gaussian is an approximation. No walker can be beyond ±Ns, so the real crowd has no tails past that.

See itRelease 2000 walkers with s = 2 and stop at N = 100. Watch for: the grey bars lying along the teal curve.

Where it matters

Same molecule, same speed. Only the direction is re-rolled at each collision.

A scent crossing a room. Take a molecule at about 500 m/s with about 70 nm between collisions (round numbers; real values vary). Crossing 3 m by random steps alone needs N = (3 m / 70 nm)² ≈ 1.8 × 10¹⁵ steps, roughly 3 days. Real rooms mix faster because air currents carry the scent.
Brownian motion. A grain of pollen or a colloidal particle in water is kicked in random directions by molecules, so its mean square displacement grows in proportion to time.
Polymers. An ideal freely jointed chain of N segments of length b has an RMS end-to-end distance b√N.

Go deeper: from steps to a diffusion constant

Give each tick a duration τ. The sim shows no clock, so τ is yours to choose.

With N = t/τ, the law ⟨x²⟩ = Ns² becomes ⟨x²⟩ = 2Dt, with D = s²/2τ. Mean square displacement grows in proportion to time: that is diffusion.

Summary

In one line. Each tick adds s² to the mean square distance, so after N ticks the RMS distance from the start is s√N, and a crowd of W walkers measures it to about 1/√(2W).

Key points

The average position stays at 0 while the spread grows as s√N: four times the steps, twice the spread.
The Spread readout is a sample of W walkers, so it scatters about s√N by roughly 1/√(2W).

Common mix-ups

"Distance from the start grows in proportion to the number of steps." Steps mostly cancel. The square of the distance grows in proportion to N, so the distance grows as √N.
"The mean displacement is zero, so the walkers go nowhere." ⟨x⟩ = 0 and ⟨x²⟩ = Ns² are both true: the crowd stays centred while it spreads.
"The Spread is the average distance from the start." It is the root-mean-square, which weights far walkers more. The mean of |x| is smaller: about 0.80 s√N at large N.

Things to try on your own

Set s = 2, run to N = 400 and compare the Spread with the Predicted value.
Press Step everyone once with s = 0.5 and read the Spread.
Drag W from 50 to 2000 at the same N and note how the shaded band changes.

Predict before reveal	The width chart, its s√N reference and the predicted-width readout are all hidden until after the student commits an answer for N = 100 — nothing on screen contains the answer at commit time.
Name the faulty intuition	The "100" and "about 40" choices encode proportional/ballistic growth; the rebuttal appears only if chosen, and the dashed y = N·s line is left standing while the data abandons it.
Invariant made visible	One s² per step is the invariant: the histogram flattens as it widens on a fixed ±140 frame, while the spread traces a dead-straight line against √N.
Operational numbers	Two readouts only by default (steps N, spread = √(Σxᵢ²/W)); the third, exact s√N, is revealed when a card asks for the comparison.
Signal vs sampling noise	The ±3/√(2W) band is computed from W itself, so the wobble at W = 50 versus the lock-on at W = 1000 is read as finite sampling, not as a failure of the law.
Honest rendering	Dots to W ≤ 300, alpha dots to 1000, a per-site density strip above; the Gaussian overlay is withheld until N = 10, where its deviation from the exact binomial is 2.5% of the peak.