Diffusion

Mathematical Modeling · Axiom Academy

LESSON Diffusion - Mathematical Modeling Unit 6: Spatial Models - Mathematical Modeling Diffusion is the net movement of particles, molecules, or other quantities from regions of higher concentration to regions of lower concentration, driven by random molecular motion. It is one of the most fundamental transport phenomena in nature. "From high concentration to low, the random walk flows." Chemical reactions and mass transport Heat conduction (thermal diffusion) Biological processes (oxygen transport, neural signaling) Population dispersal and spatial ecology Financial modeling (option pricing via diffusion processes) Understanding diffusion starts with the random motion of individual particles. At the molecular level, particles undergo random walks due to thermal motion and collisions. Each particle moves randomly, but when there are more particles in one region than another, more particles randomly move out of the crowded region than into it. This creates a net flow from high to low concentration. Although individual particles move randomly, the collective behavior is predictable. The concentration spreads out smoothly over time, eventually reaching a uniform distribution (equilibrium). A classic demonstration of diffusion: t = 0: Concentrated drop of ink at one location t > 0: Ink molecules spread outward in all directions Later: Distribution becomes more uniform Equilibrium: Ink is evenly distributed throughout

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