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In the ultrashort time limit, in the order of the diffusion time a 2 /D, where a is the particle radius, the diffusion is described by the Langevin equation. At a longer time, the Langevin equation merges into the Stokes–Einstein equation. The latter is appropriate for the condition of the diluted solution, where long-range diffusion is ...
There are several interesting features associated with the diffusion distance, based on our previous discussion that also serves as a scale parameter: Points are closer at a given scale (as specified by D t ( x i , x j ) {\displaystyle D_{t}(x_{i},x_{j})} ) if they are highly connected in the graph, therefore emphasizing the concept of a cluster.
Another method to describe the motion of a Brownian particle was described by Langevin, now known for its namesake as the Langevin equation.) (,) = (,), given the initial condition (, =) = (); where () is the position of the particle at some given time, is the tagged particle's initial position, and is the diffusion constant with the S.I. units ...
The diffusion equation is a parabolic partial differential equation. ... where ϕ(r, t) is the density of the diffusing material at location r and time t and D ...
Reaction–diffusion systems are mathematical models that 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 ...
For the diffusion happens in a much longer time compared to the advection, and therefore the latter of the two phenomena predominates in the mass transport. Plan view: For P e L → ∞ {\displaystyle Pe_{L}\rightarrow \infty } , diffusion is negligible, and advection dominates mass transport.
The Fourier number can be derived by nondimensionalizing the time-dependent diffusion equation.As an example, consider a rod of length that is being heated from an initial temperature by imposing a heat source of temperature > at time = and position = (with along the axis of the rod).
The diffusion coefficient is the coefficient in the Fick's first law = /, where J is the diffusion flux (amount of substance) per unit area per unit time, n (for ideal mixtures) is the concentration, x is the position [length].