Wandering Molecules and Bell Curves: A Dive into Feynman’s Probability Distributions

Richard Feynman’s Lectures on Physics are celebrated not only for their intellectual clarity but for their rare ability to infuse deep physical principles with vivid, relatable imagery. In Chapter 6-4, “A Probability Distribution,” Feynman explores the nature of randomness, step-by-step motion, and the concept of a probability density—all through the lens of a slightly modified “random walk.” The section offers a gently meandering journey through one of the most foundational ideas in statistical physics: the Gaussian distribution.

Feynman begins by returning to the concept of a random walk. Traditionally, this involves a particle taking steps either forwards or backwards, with each step being the same length. But now, he introduces a twist: the step lengths themselves are allowed to vary unpredictably, although their average length remains one unit. This modification makes the model far more realistic, more akin to the thermal motion of molecules in a gas, where step directions and distances are random but governed by statistical constraints.

What makes Feynman’s writing so engaging is his ability to shift from pure mathematics to visual intuition. He notes that under these new conditions, the probability of a molecule returning to exactly zero after thirty such variable steps is zero—because it’s almost impossible for a set of backwards and forwards steps of arbitrary length to cancel out precisely. You can’t now plot a histogram of discrete outcomes like “step = 1 or 2 or 3,” because the results are spread out across a continuum.

This is where Feynman introduces the probability density function. He defines a new measure, P(x, Δx), as the probability that a result lies within an interval near a given value x. For small Δx, the probability is proportional to the width of the interval, which leads us to the core concept:

P(x, Δx) = p(x)Δx,

where p(x) is the probability density, the smooth function describing how likely the variable D is to take a value near x.

Feynman uses simple but effective imagery to explain this: instead of counting exact outcomes, we now shade areas under a curve to determine the chance of something happening between two values. The curve he describes is the normal distribution, the familiar bell curve, which widens with the square root of N (the number of steps) and flattens proportionally—preserving the total area under the curve, which must be 1. It’s a beautiful visual: as time passes and more steps accumulate, the “cloud” of probabilities spreads out, but the total likelihood remains constant, merely redistributed.

He then links this mathematical idea to a physical scenario that is both relatable and atmospheric: the diffusion of a smell. Imagine opening a bottle of an organic compound in still air. Over time, the odour spreads throughout the room—not because molecules are seeking out your nose, but due to countless random motions. The random walk becomes real. If we know the average step size and how frequently steps occur, we can predict how the vapour disperses over time, just like the probability curves widen in the earlier graph (Fig. 6-7).

Finally, Feynman leads us to the velocity distribution in a gas—a foretaste of Maxwell’s work. Again, we’re in a world of probabilities: molecules don’t have one set speed but rather a spread of likely speeds. The chance of finding a molecule with a speed in the range v to v + Δv is described by a probability density p(v), and when scaled up for N molecules, the product Np(v) gives the expected number of molecules with those speeds. It’s all an extrapolation of the earlier random walk logic—just applied to velocity instead of position.

What Feynman does here is more than teach physics; he animates probability. The random walk becomes the jittery journey of a molecule. The bell curve becomes a landscape of possibilities. The spreading of scent in a room becomes a tangible metaphor for diffusion. His genius lies in making the abstract physical and making the physical beautiful.

In conclusion, Chapter 6-4 is not merely a lesson in mathematical distributions; it is a poetic reflection on randomness, structure, and the invisible order that arises from chaos. With elegant mathematics and vivid metaphor, Feynman shows us that even in uncertainty, there are patterns—and in those patterns, a deeper truth about the nature of motion, matter, and probability itself.