Question

Explain modelling diffusion without reaction.

Answer 1

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Answer 2

A simulation of two virtual chemicals reacting and diffusing on a Torus using the Gray–Scott model

Reaction–diffusion systems are mathematical models which correspond to several physical phenomena: the most common is the change in space and time of the concentration of one or more chemical substances: local chemical reactions in which the substances are transformed into each other, and diffusion which causes the substances to spread out over a surface in space.

Reaction–diffusion systems are naturally applied in chemistry. However, the system can also describe dynamical processes of non-chemical nature. Examples are found in biology, geology and physics (neutron diffusion theory) and ecology. Mathematically, reaction–diffusion systems take the form of semi-linear parabolic partial differential equations. They can be represented in the general form

{\displaystyle \partial _{t}{\boldsymbol {q}}={\underline {\underline {\boldsymbol {D}}}}\,\nabla ^{2}{\boldsymbol {q}}+{\boldsymbol {R}}({\boldsymbol {q}}),}

where q(x, t) represents the unknown vector function, D is a diagonal matrix of diffusion coefficients, and R accounts for all local reactions. The solutions of reaction–diffusion equations display a wide range of behaviours, including the formation of travelling waves and wave-like phenomena as well as other self-organized patterns like stripes, hexagons or more intricate structure like dissipative solitons. Such patterns have been dubbed "Turing patterns".[1] Each function, for which a reaction diffusion differential equation holds, represents in fact a concentration variable