FR21 · Reaction and diffusion
Why can diffusion help a uniform mixture grow a pattern? Two reacting substances can amplify small spatial differences when they spread at different rates. The growing wavelengths come from the reaction rates, diffusion and the size of the domain.
Watch small concentration differences become spots. Change diffusion, paint a disturbance and compare measured growth with a predicted spectrum.
SIMULATION RECORD AND ANALYTIC REFERENCE
Cyan: continuum growth rate. Dashed gold: the 32-cell finite-difference prediction. White points: slopes fitted to nine saved Fourier-amplitude samples from separate tiny-perturbation simulations. Those runs use the same current reaction parameters as the field.
The field uses one fixed color scale: dark blue at zero, cyan near one and amber at 2.5 and above. Opposite edges touch.
Physics tutorial
BackgroundA mixture can return to equilibrium after a uniform disturbance and still amplify a striped disturbance. Different diffusion rates change the response to each spatial scale.
Why it mattersThis gives a precise way to ask how a pattern starts: calculate which small ripples grow, then measure their growth before the peaks become large.
Start with the essentials
The activator consumes substrate to make more activator. Both species diffuse, and the surrounding reservoir supplies each at a fixed rate. Every quantity here is dimensionless.
The matrix tells how a sinusoidal perturbation changes. The largest real part of its eigenvalues is the exponential growth rate. Positive values predict early growth.
Typical misconceptionDiffusion always smooths things out, so it cannot make a pattern.
Better mental modelEach species does smooth out. But when substrate moves much faster, its redistribution changes the local reaction feedback. The coupled system can amplify some disturbances.
Typical misconceptionThe predicted growth rate should describe every frame.
Better mental modelThe spectrum applies close to the homogeneous equilibrium. Large spots deplete their surroundings and interact nonlinearly. Their late growth is governed by the full reaction equations.
Start with Spots emerge. Find the white points above the zero line in the spectrum.
What to observe: Those modes grow from tiny perturbations. Their finite-grid reference differs slightly from the smooth continuum prediction.Select Diffusion smooths, then drag across the field to add activator.
What to observe: The disturbance spreads and fades. Equal diffusion subtracts the same damping from both coupled modes.Select Mixture oscillates and watch the field change in time.
What to observe: The uniform mode is now unstable too. A changing color across the mixture is a temporal oscillation; a stationary pattern grows at nonzero spatial wave number.Morphogenesis and reaction–diffusion · Reading path
Articles on this path that use this experiment, followed by the other experiments they suggest.