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The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion , resulting from the random movements and collisions of the particles (see Fick's laws of diffusion ).
The accumulated number of molecules adsorbed on the surface is expressed by the Langmuir-Schaefer equation by integrating the diffusion flux equation over time as shown in the simulated molecular diffusion in the first section of this page: [13] =. A is the surface area (m 2).
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]
Assuming that N particles start from the origin at the initial time t = 0, the diffusion equation has the solution (,) = (). This expression (which is a normal distribution with the mean μ = 0 {\displaystyle \mu =0} and variance σ 2 = 2 D t {\displaystyle \sigma ^{2}=2Dt} usually called Brownian motion B t {\displaystyle B_{t}} ) allowed ...
The convection–diffusion equation can be derived in a straightforward way [4] from the continuity equation, which states that the rate of change for a scalar quantity in a differential control volume is given by flow and diffusion into and out of that part of the system along with any generation or consumption inside the control volume: + =, where j is the total flux and R is a net ...
The self-diffusion coefficient of neat water is: 2.299·10 −9 m 2 ·s −1 at 25 °C and 1.261·10 −9 m 2 ·s −1 at 4 °C. [2] Chemical diffusion occurs in a presence of concentration (or chemical potential) gradient and it results in net transport of mass. This is the process described by the diffusion equation.
The Fokker–Planck equation for this particle is the Smoluchowski diffusion equation: (, |,) = [(()) (, |,)] Where is the diffusion constant and =. The importance of this equation is it allows for both the inclusion of the effect of temperature on the system of particles and a spatially dependent diffusion constant.
Diffusion models may also be used to solve inverse boundary value problems in which some information about the depositional environment is known from paleoenvironmental reconstruction and the diffusion equation is used to figure out the sediment influx and time series of landform changes.