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The MSD is symmetrically distributed over the 1D, 2D, and 3D space. Thus, the probability distribution of the magnitude of MSD in 1D is Gaussian and 3D is Maxwell-Boltzmann distribution.

The Chapman—Enskog formulae for diffusion in gases include exactly the same terms. Earlier, such terms were introduced in the Maxwell—Stefan diffusion equation.

Equations based on Fick's law have been commonly used to model transport processes in foods, neurons , biopolymers , pharmaceuticals , porous soils , population dynamics , nuclear materials, plasma physics , and semiconductor doping processes.

Theory of all voltammetric methods is based on solutions of Fick's equation. Much experimental research in polymer science and food science has shown that a more general approach is required to describe transport of components in materials undergoing glass transition.

In the vicinity of glass transition the flow behavior becomes "non-Fickian". The Fick's law is limiting case of the Maxwell—Stefan equations, when the mixture is extremely dilute and every chemical species is interacting only with the bulk mixture and not with other species.

To account for the presence of multiple species in a non-dilute mixture, several variations of the Maxwell—Stefan equations are used.

See also non-diagonal coupled transport processes Onsager relationship. When two miscible liquids are brought into contact, and diffusion takes place, the macroscopic or average concentration evolves following Fick's law.

On a mesoscopic scale, that is, between the macroscopic scale described by Fick's law and molecular scale, where molecular random walks take place, fluctuations cannot be neglected.

Such situations can be successfully modeled with Landau-Lifshitz fluctuating hydrodynamics. In this theoretical framework, diffusion is due to fluctuations whose dimensions range from the molecular scale to the macroscopic scale.

In particular, fluctuating hydrodynamic equations include a Fick's flow term, with a given diffusion coefficient, along with hydrodynamics equations and stochastic terms describing fluctuations.

When calculating the fluctuations with a perturbative approach, the zero order approximation is Fick's law.

The first order gives the fluctuations, and it comes out that fluctuations contribute to diffusion.

This represents somehow a tautology , since the phenomena described by a lower order approximation is the result of a higher approximation: this problem is solved only by renormalizing the fluctuating hydrodynamics equations.

The adsorption or absorption rate of a dilute solute to a surface or interface in a gas or liquid solution can be calculated using Fick's laws of diffusion.

The accumulated number of molecules adsorbed on the surface is expressed by the Langmuir-Schaefer equation at the short-time limit by integrating the diffusion equation over time: [9].

The Langmuir-Schaefer equation is extended to the Ward-Tordai Equation to account for the back-diffusion of the rejected molecules from the surface in the later time of the adsorption: [10].

The square root t dependent on the adsorption is because when the molecules are adsorbed, the concentration in the sub-surface drops and creates a concentration gradient near the surface which slows down the absorption over time.

This assumption can be confirmed with a Monte Carlo simulation. The langmuir-Schaefer equation can also be obtained from analysing the single-molecule diffusion.

Considering one dimension that is perpendicular to the surface, the probability of any given solute molecule in the solution hit the surface is the error function of its diffusive broadening over the time of interest.

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 considered.

According to the fluctuation-dissipation theorem based on the Langevin equation in the long-time limit and when the particle is significantly denser than the surrounding fluid, the time-dependent diffusion constant is: [13].

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