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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 ...
The diffusion distance at time between two points can be measured as the similarity of two points in the observation space with the connectivity between them. It is given by It is given by D t ( x i , x j ) 2 = ∑ y ( p ( y , t | x i ) − p ( y , t | x j ) ) 2 ϕ 0 ( y ) {\displaystyle D_{t}(x_{i},x_{j})^{2}=\sum _{y}{\frac {(p(y,t|x_{i})-p(y ...
where ϕ(r, t) is the density of the diffusing material at location r and time t and D(ϕ, r) is the collective diffusion coefficient for density ϕ at location r; and ∇ represents the vector differential operator del. If the diffusion coefficient depends on the density then the equation is nonlinear, otherwise it is linear.
The concentration profile c at time t can then be extracted as a function of the x coordinate. In Matano's notation, the two concentrations are indicated as c L and c R (L and R for left and right, as shown in most diagrams), with the implicit assumption that c L > c R; this is however not strictly necessary as the formulas hold also if c R is ...
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 ...
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].
Darken’s equations can be applied to almost any scenario involving the diffusion of two different components that have different diffusion coefficients. This holds true except in situations where there is an accompanying volume change in the material because this violates one of Darken’s critical assumptions that atomic volume is constant.
This diffusion proceeds in an ordered fashion, analytically determinable from the diffusion equation. Assuming a Gaussian profile for the bleaching beam, the diffusion constant D can be simply calculated from: = where w is the radius of the beam and t D is the "Characteristic" diffusion time. [1] [2]