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Diffusion describes the spread of particles through random motion from regions of higher concentration to regions of lower concentration. The time dependence of the statistical distribution in space is given by the diffusion equation. The concept of diffusion is tied to that of mass transfer driven by a concentration gradient, but diffusion can still occur when there is no concentration gradient (but there will be no net flux).

The concept of diffusion emerged from physical sciences. The paradigmatic examples were heat diffusion, molecular diffusion and Brownian motion. Their mathematical description was elaborated by Joseph Fourier in 1822, Adolf Fick in 1855, and by Albert Einstein and Marian Smoluchowski in 1905/06, respectively.

Applications outside physics were pioneered by Louis Bachelier who in 1900 used a random walk model to describe price fluctuations on financial markets. In a less quantitative way, the concept of diffusion is invoked in the social sciences to describe the spread of ideas (Diffusion of innovations, Lexical diffusion, Trans-cultural diffusion).

Diffusion in physics

In molecular diffusion, the moving entities are small molecules which are self propelled by thermal energy and do not require a concentration gradient to spread out through random motion. They move at random because they frequently collide. Diffusion is this thermal motion of all (liquid and gas) molecules at temperatures above absolute zero. Diffusion rate is a function of only temperature, and is not affected by concentration. Brownian motion is observed in molecules that are so large that they are not driven by their own thermal energy but by collisions with solvent particles.

While Brownian motion of large molecules is observable under a microscope, small-molecule diffusion can only be probed in carefully controlled experimental conditions. Under normal conditions, molecular diffusion is relevant only on length scales between nanometer and millimeter. On larger length scales, transport in liquids and gases is normally due to another transport phenomenon, convection.

Therefore, some often cited examples of diffusion are wrong: If cologne is sprayed in one place, it will soon be smelled in the entire room, but a simple calculation shows that this cannot be due to diffusion; the cause can only be convection. If ink is dropped in water, one usually observes an inhomogeneous evolution of the spatial distribution, which clearly indicates convection; diffusion dominates only in perfect thermal equilibrium.

In contrast, heat conduction through solid media is an everyday occurrence (e.g. a metal spoon partly immersed in a hot liquid). This explains why the diffusion of heat was explained mathematically before the diffusion of mass.

Other types of diffusion

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