What is a Brownian motion?

Imagine a delicious… poisoned brownie, served at a rather ill-intentioned dinner party. Each guest, polite but keen to stay alive, carefully avoids eating it by gently pushing it away. Everyone does so at random moments and in random directions.

As a result, the brownie gets jostled around the table, bumping into glasses, plates, and cutlery, following a chaotic and unpredictable path. If we measured its position over time, we would see it moving randomly, with no preferred direction. And since the guests are evenly spaced around the table, there’s no reason for it to drift more one way than another.

Over time, its average position would stay close to where it started, with random fluctuations around it — a perfect illustration of Brownian motion, dinner-party edition.

"Observing Brownian motion of micro beads", by Forrest Charnock on Youtube.

Of course, Brownian motion isn't really about a brownie! In reality, it's the random movement of microscopic or nanoscopic particles — known as colloids — suspended in a fluid. These colloids are constantly jostled by the surrounding fluid molecules. Why? Because molecules are always in motion. The faster they move, the more energy they have — this is what we call thermal energy, which is directly related to the temperature we can measure.

In our dinner analogy, the guests represent the molecules of the fluid. Temperature plays a key role: the higher it is, the more agitated the molecules — or here, the guests’ patience — become, and the more frequently and forcefully they push the colloid — or in this case, the brownie. These random pushes, accumulated over time, cause the colloid to move in a chaotic and unpredictable way through the fluid.

This disordered motion, driven by countless small and random interactions, is precisely what we observe as Brownian motion. The effect becomes negligible when the colloids are too large to be significantly influenced by the surrounding molecular impacts. The video above shows this phenomenon in action, by tracking the Brownian motion of micrometer-sized pollen grains.

“Robert Brown” by Skara kommun (licensed under CC BY 2.0).

To understand the origins of this phenomenon, we need to travel back in time. In 1827, botanist and explorer Robert Brown was using one of the first microscopes to study pollen grains. To his surprise, he noticed that once placed in water, the grains moved erratically, without any clear pattern.

At the time, many scientists were skeptical — they thought nature had already revealed all her secrets — and dismissed Brown’s observations, blaming poor-quality microscope lenses. But later, it became clear that this motion was real. It was eventually named Brownian motion, not in honor of brownies, of course, but of Mr. Brown.

Years later, in 1901, Louis Bachelier developed the first mathematical model of this motion, laying the groundwork for the theory of stochastic processes. Interestingly, his work was not initially applied to physics, but to finance — another domain where random fluctuations abound.

“Albert Einstein” by ThomasThomas (License CC BY-NC 2.0).

Then came Albert Einstein. In 1905, he derived a fundamental equation describing how colloids diffuse through a fluid. He showed that the diffusion coefficient — which measures how fast a particle spreads — is determined by the ratio between the thermal energy of the environment and the mobility of the colloid (which depends on its size and the fluid’s viscosity). This diffusion coefficient captures the balance between random thermal kicks and frictional resistance.

Remarkably, Einstein demonstrated that by observing this motion, one could estimate the size of molecules — a major step forward in understanding matter at the microscopic scale.

In 1908, Paul Langevin proposed a dynamic description of Brownian motion. Building on Einstein’s work, he wrote an equation of motion using Newton’s laws, but added a random force to model thermal agitation. A year later, in 1909, Jean Perrin used Einstein’s theory to experimentally measure Avogadro’s number — the number of atoms in 12 grams of carbon-12. His work provided crucial evidence for the atomic theory of matter and the reality of Brownian motion.

What about now?

Today, we have a solid understanding of how microscopic particles diffuse in an ideal, uniform fluid far from any obstacles. But here’s the catch: nature is rarely that simple. In reality, nothing is truly “free” or isolated.

“Red Blood cells” by SciTechTrend (license CC PDM 1.0).

Take red blood cells, for instance. They move through narrow blood vessels, often surrounded by many other cells. Can we still apply classical Brownian motion laws in such conditions?

Not directly. In these more realistic environments, many additional physical effects come into play. For example, the nearby vessel wall affects the motion of the particles. Their diffusion depends on how close they are to the wall — so the diffusion coefficient is no longer a constant, but varies with position. Furthermore, red blood cells don’t travel alone: interactions between particles also influence the dynamics.

These are exactly the kinds of questions that drive my research. During my PhD, I studied how confinement — by hard or soft walls — affects Brownian motion, and how interactions between particles modify their behavior. Even today, nearly 200 years after Robert Brown’s discovery, Brownian motion remains a rich playground for scientific exploration.