The Particle Comes Alive - by CasualPhysicsEnjoyer
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The Particle Comes Alive<br>Modelling a Particle in a Thermal Bath; Langevin Dynamics<br>CasualPhysicsEnjoyer<br>May 23, 2026
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Sometimes small effects don’t matter. Sometimes they do. Maybe you lead a certain French nation but then the whispers of your people become a revolution. Maybe you are a meteorologist that forgot to take into account the wings of a butterfly. But it's hard to know when small things become a big deal. But the thermal physics of systems might give us a hint into when.<br>Brownian motion is an example of thermal effects causing a particle to drift. When you leave a particle alone, you expect it to stay still. But a particle is rarely alone. It may be in a gas, or in water, all of which have an affect on the particle through random effects.<br>And luckily for us, it's observable with a microscope, which makes it possible for independent scientists to study. I'm going to explain what Brownian motion is. Much more than you needed to know. Why it's counterintuitive, the math behind it, and when it breaks (which is when it gets interesting).<br>In 1827 Robert Brown looked at some pollen in water using a microscope. But he saw that the particles drifted, away from the centre, in a motion as if they were alive. This was strange because the water was stagnant, nor the pollen active in any way - and so it couldn't move because of those reasons. Furthermore, it didn't matter which type of pollen he used.<br>[The motions] resemble in a remarkable degree the less rapid motions of some of the simplest animalcules of infusions.
These motions were such as to satisfy me, after frequently repeated observation, that they arose neither from currents in the fluid, nor from its gradual evaporation, but belonged to the particle itself.<br>— Robert Brown, A brief Account of Microscopical Observations… (1828)
It was only later on that Jean Perrin drew these paths of such particles in water. He got something that looked like this.
Jean Perrin's tracings of three colloidal particles, recorded at 30-second intervals. From Les Atomes (Paris: Félix Alcan, 1913). SVG rendering by ElioPrrl via [Wikimedia Commons](https://commons.wikimedia.org/wiki/File:PerrinPlot1.svg), CC0 / public domain.Wikimedia Commons, CC0 / public domain.*<br>But to the peoples of old, this drift was weird.<br>Water molecules are small and collisions are random. And because it’s random, the collisions against the pollen particle shouldn’t prefer any direction. Each collision placed a force on the particle. And since these forces averaged to zero, by symmetry, then the net force should be zero. Which means that the particle should not accelerate at all due to Newton's law of motion.<br>This was the classical view
Water molecules hit the pollen particle from all sides. With collisions equally likely in every direction, the forces should cancel out and the particle should sit still.<br>By Newton's second law, we had<br>\(F = ma\)
And if we take the average of the forces, then<br>\(\langle F \rangle = 0 = m \langle a \rangle\)
which means that the acceleration should also be zero. Which means that if the particle was stationary at the beginning, then it shouldn't move. But this didn't happen according to Brown's experiments!<br>The problem with the logic above is that the averaging is done too soon. What we are interested in is the distance from the start point, which is the magnitude of the displacement, given by<br>\(\text{distance} = \sqrt{x^2}\)
For now, forget about the square root - we can add that later. We want to compute the average squared displacement from the starting point.<br>In other words, if the displacement from a start point is x, then we want to compute<br>\(\langle x^2 \rangle\)
The reason why this is not zero is because the average of a squared random variable is not the same as the average of that variable, squared. ⟨a⟩ = 0 doesn't imply ⟨x⟩ = 0, nor does ⟨x⟩ = 0 imply ⟨x²⟩ = 0.<br>So how do we compute the average displacement ⟨x²⟩ in this scenario? Einstein originally came up with the derivation of what happens in 1905, but I find Langevin's approach made in 1908 more constructive. I will explain it here.<br>As usual, start with the forces.<br>If a pollen is in water the first force is the friction that the pollen has by travelling through water. This friction is given by a law discovered by Stokes, which says that the friction force is proportional to its speed. And the second term is this random kick felt by the molecules colliding with it. The force diagram is like this:
The pollen particle has a friction force -γ v against its motion and a random kick η(t) from the surrounding water molecules. The net force is F_net. In motion, it causes the particle to move like this:
If we try to simulate the particles, it looks like they move away from the centre as a function of time. What is this function?<br>Since force is mass times acceleration, the equation of motion is the...