Q.What is the reason for Brownian motion?

Brownian Motion: From Intuition to Precision

Imagine you are in a completely dark room, completely still, and someone throws a hundred ping-pong balls at you from random directions every second. You wouldn't stand still — you'd jerk, stumble, and lurch in a chaotic, unpredictable path. That is the essence of Brownian motion.

In 1827, the botanist Robert Brown looked through his microscope at pollen grains suspended in water. He expected to see them sit still. Instead, they danced — a continuous, jittery, random zig-zag. Brown had no explanation. The real insight came decades later: the water molecules themselves are in constant, violent thermal motion. A pollen grain, though huge compared to a single water molecule, is tiny enough to feel the imbalance of these molecular impacts. At any instant, a few more molecules might hit it from the left than from the right, shoving it sideways. A moment later, the imbalance shifts, and it lurches another way. The result is the random walk we call Brownian motion.

The Intuition: Why "Unbalanced Bombardment"?

A colloidal particle is constantly struck by molecules of the surrounding medium (water, air, etc.). These molecules move at hundreds of metres per second. If the particle were enormous, the billions of trillions of impacts per second would average out perfectly — net force zero, particle stationary. But for a tiny particle, the number of impacts is smaller, and the statistical fluctuations become significant.

Think of a coin toss. Toss a coin 10 times — you might easily get 7 heads and 3 tails (a 4-toss imbalance). Toss it 10,000 times — the imbalance is tiny in percentage terms. The colloidal particle experiences the "7 heads, 3 tails" version of molecular bombardment. That momentary net force shoves it. The next moment, the imbalance is in a different direction. Hence the zig-zag.

The Precise Statement

Mathematically, Brownian motion is formalised as the Wiener process $W(t)$ (or $B(t)$). It is a continuous-time stochastic process satisfying:

1. $W(0) = 0$ almost surely. The motion starts at the origin.

2. Independent increments. For any non-overlapping time intervals $[t_1, t_2]$ and $[t_3, t_4]$, the displacements $W(t_2) - W(t_1)$ and $W(t_4) - W(t_3)$ are independent random variables. What the particle does in one interval has no memory of what it did in another.

3. Gaussian increments. For any $0 \le s < t$, the displacement $W(t) - W(s)$ is normally distributed with mean $0$ and variance $t - s$:

The spread of possible positions grows as the square root of the elapsed time.

4. Continuous paths. The function $t \mapsto W(t)$ is continuous with probability 1. The particle never teleports — its path is an unbroken line, however wild.

Key Properties (Exam-Ready)

• Mean displacement is zero: $\langle W(t) \rangle = 0$. On average, the particle goes nowhere. …