Foundation lesson

From steps to spread

Random steps partly cancel, but their squares add up. After n steps of one metre the average squared distance from the start is n, so the typical distance grows only as the square root of n.

Written for this edition, not translated from Einstein.

Why does typical displacement grow as the square root of the step count?

A walker flips a coin before every step: heads, one metre to the right; tails, one metre to the left. After four steps, how far from the start is the walker likely to be? Not four metres: the steps partly cancel. Count the walks, and square each final distance so that left and right count alike:

After one step the walker is 1 m away, every time. The squared distance is 1.
After two steps there are four equally likely walks, ending 2 m right, at the start, at the start, or 2 m left. The squared distances 4, 0, 0 and 4 average 2.
After four steps there are sixteen equally likely walks. Two end 4 m out, eight end 2 m out and six end at the start. The squared distances average (2 × 16 + 8 × 4) ÷ 16 = 4.

The average squared distance grows by one square metre per step: after n steps it is n. The typical distance, its square root, grows only as the square root of n: 1 m after one step, about 1.4 m after two, 2 m after four, 10 m after a hundred. To get twice as far takes four times as many steps.

In symbols, call the steps Δ₁, Δ₂ and so on, and the position after n steps X, their sum:

X_n=\Delta_1+\ldots+\Delta_n

Position after n steps is the sum of the n signed increments.

Suppose each step averages zero and has the same mean square l², where l is the typical size of one step (1 m in the coin walk). Squaring the sum gives the n squared steps and many products of two different steps. When the steps are independent and each averages zero, every such product averages zero, as the lesson on independence shows for two coin steps. Only the squares are left to add up:

\left\langle X_{n}^{2}\right\rangle = n\,l^{2}

\sqrt{\left\langle X_{n}^{2}\right\rangle} = l\,\sqrt{n}

The mean square is n times ell squared, and RMS displacement is ell times the square root of n.

Worked example: A hundred steps, then four hundred

A hundred steps of typical size 1 m leave a typical distance of 10 m from the start, because the square root of 100 is 10. Four hundred steps leave 20 m: four times the steps, twice the distance. No single walker has to end exactly 10 or 20 m away; these are typical sizes over many walkers. Nor is the distance from the start the length of the path: to end 10 m from home, the walker has walked 100 m.

Four steps is not four metres

The four two-step walks the lesson lists, each drawn from the start. Two arrows a walk, and a dot where it ends.

The four ends add to zero, so the average end is the start. Their squares are 4, 0, 0, 4, which add to 8 and average 2. Squares do not cancel, and that is the quantity that keeps count.

So the typical distance climbs, but slowly. The straight line is what a reader expects before reading the lesson: every step in the same direction.

What it shows, in words

The first drawing is the four equally likely two-step walks. Right then right ends 2 metres right; right then left and left then right both end back at the start; left then left ends 2 metres left. The four ends add to zero, so the average end is the start, and nobody should read that as the walker not having moved. The squares of those ends are 4, 0, 0, 4, and their average is 2 square metres, which is two step-squares for two steps.

The second drawing is the typical distance against the number of steps. It passes through 1 metre at 1 step, 2 at 4, 3 at 9 and 4 at 16: four times the steps for twice the distance, every time. The broken straight line is a walker whose every step went the same way, one metre per step, and it leaves the top of the graph after 5 steps. At a hundred steps the curve is at 10 metres where that line would be at a hundred.

Where this lesson stops

The coin walk is a picture, not a claim that molecules push a particle in fixed jumps. The Brownian paper needs only that the particle's moves in successive intervals are independent of one another.

If you came here from a passage, Back returns you to the exact place you left.

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