Every click creates a ripple โ in pixels and in sound.
All audio is synthesized in real time from pure mathematics.
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From a drop of ink to the shape of the universe
Drop a stone in still water โ ripples expand, weaken, and vanish. A drop of ink in a glass disperses until it's everywhere. This is diffusion: one of the most fundamental processes in nature, governed by a single, elegant equation.
01 โ TOUCH THE SURFACE
Click anywhere on the surface below. Watch how a concentrated point of energy spreads outward, growing weaker as it travels. This is diffusion in action โ information radiates from where it begins, but it never stays concentrated.
02 โ THE EQUATION
Joseph Fourier discovered this in 1822 while studying heat flow through metal. What he found was shockingly general โ the same equation governs the spread of heat, the diffusion of molecules, the flow of information through networks, even the pricing of financial options.
In words: the rate of change at any point is proportional to how different that point is from its neighbors. Hot spots cool. Concentrations dilute. Peaks flatten. The constant ฮฑ controls the speed.
Click on the bar to add heat. Watch it spread in real time.
What you're seeing โ The simulation solves the heat equation numerically using finite differences. Each pixel's temperature changes based on the difference between it and its neighbors: u(x, t+dt) = u(x,t) + ฮฑยทdtยท(u(x+1,t) โ 2u(x,t) + u(xโ1,t))/dxยฒ. The sharp peak diffuses into a wider, lower Gaussian โ concentrations always spread, and total "heat" is conserved.
03 โ THE FUNDAMENTAL SOLUTION
If you start with all the heat concentrated at a single point, the solution to the heat equation is beautiful and exact:
This is the Gaussian โ the bell curve โ born not from statistics, but from physics. The width grows as โt, and the peak height drops as 1/โt. Energy is conserved: the area under the curve is always exactly 1.
Drag the time slider to watch the fundamental solution evolve. The curve flattens but the total area remains constant.
04 โ THE MICROSCOPIC VIEW
Zoom into the ink drop. Each molecule bounces randomly off its neighbors โ a random walk. In 1905, Einstein showed that this microscopic chaos produces macroscopic diffusion. The key insight:
The average squared displacement grows linearly with time. After N steps, a particle is typically โN steps from where it started โ not N. This is why diffusion is slow: doubling the distance takes four times as long.
Watch 500 particles take independent random walks. The histogram builds a Gaussian โ the Central Limit Theorem in real time.
The deep connection โ The heat equation and random walks are two faces of the same coin. At the macroscopic level, we see smooth diffusion governed by โu/โt = ฮฑโยฒu. At the microscopic level, countless random walkers produce a Gaussian distribution that satisfies the exact same equation. Einstein used this connection to prove that atoms exist โ and won a Nobel Prize for it.
05 โ TWO EQUATIONS, TWO WORLDS
These two equations look almost identical โ but they describe fundamentally different universes:
A wave preserves shape โ it propagates without losing information. Diffusion destroys information โ it smears, smooths, and forgets. One is reversible; the other is not.
Same initial pulse, same spatial setup โ watch how differently they evolve.
Second-order in time. Information travels at speed c without distortion. Reversible โ you can run the clock backward and recover the original pulse. Governs sound, light, and seismic waves.
First-order in time. Information spreads outward and loses its shape. Irreversible โ you cannot "un-diffuse" a dissolved drop of ink. Governs heat, chemical concentration, and probability.
06 โ THE FULL PICTURE
In 2D, the fundamental solution is a circular Gaussian that spreads radially. Click on the canvas to add heat sources and watch them merge, interact, and slowly equilibrate. This is what happens inside a pan of boiling water, a cell membrane, or the atmosphere of a star.
A drop of ink dissolves and never returns.
A hot spot cools and the heat is gone.
Information spreads, weakens, and vanishes.
The second law of thermodynamics says
you cannot un-dissolve the ink.
But what if a machine
could learn to reverse it?
In 2020, researchers discovered that the exact same equation you've been exploring
can teach a machine to generate images, videos, and music โ from pure noise.