Diffusion 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). In mathematics, it is related to Markov processes, such as random walks, and applied in many other fields, such as materials science, information theory, and biophysics. The diffusion equation is a special case of the convection–diffusion equation when bulk velocity is zero. It is equivalent to the heat equation under some circumstances.

The equation is usually written as:

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 equation above applies when the diffusion coefficient is isotropic; in the case of anisotropic diffusion, D is a symmetric positive definite matrix, and the equation is written (for three dimensional diffusion) as:

If D is constant, then the equation reduces to the following linear differential equation:

${\displaystyle {\frac {\partial \phi (\mathbf {r} ,t)}{\partial t}}=D\nabla ^{2}\phi (\mathbf {r} ,t),}$ 

which is identical to the heat equation.

The particle diffusion equation was originally derived by Adolf Fick in 1855.

The diffusion equation can be trivially derived from the continuity equation, which states that a change in density in any part of the system is due to inflow and outflow of material into and out of that part of the system. Effectively, no material is created or destroyed:

$${\displaystyle {\frac {\partial \phi }{\partial t}}+\nabla \cdot \mathbf {j} =0,}$$

 

$${\displaystyle \mathbf {j} =-D(\phi ,\mathbf {r} )\,\nabla \phi (\mathbf {r} ,t).}$$

 

If drift must be taken into account, the Fokker–Planck equation provides an appropriate generalization.

The thermodynamic view

The chemical potential of a solute in solution is given by:

$${\displaystyle \mathbf {\mu } ={\mu _{0}}-RT\ln(c)}$$

 

µ: chemical potential
µ°: chemical potential in the standard state
R: universal gas constant
T: absolute temperature
c: concentration of the solute

The change in chemical potential dµ due to a concentration gradient dc is then

$${\displaystyle {d\mu \over dc}={d(\mu _{0}-RT\ln(c)) \over dc}}$$

 

$${\displaystyle {d\mu }=RT({\frac {dc}{c}})}$$

 

$${\displaystyle F=-{d\mu \over dx}=-{\frac {RT}{c}}({dc \over dx})}$$

 

The negative sign arises because the concentrations (c) and distances (x) increase in opposite directions.

When a molecule (or particle) experiences a driving force, its velocity increases until the frictional force acting on it balances the driving force. This frictional force (Ff) is directly proportional to the molecule's velocity, with the constant of proportionality (f) termed the frictional coefficient, denoted as:

$${\displaystyle F_{f}=N_{\text{A}}vf}$$

 

$${\displaystyle Nvf=-{\frac {RT}{c}}({dc \over dx})}$$

 

$${\displaystyle cv=-{\frac {RT}{Nf}}({dc \over dx})=-{\frac {kT}{f}}({dc \over dx})}$$

 

$${\displaystyle J=-{\frac {kT}{f}}({dc \over dx})}$$

 

When the temperature is maintained constant, the proportionality (kT/f) constant relies on the molecular quantity . This constant equates to the diffusion coefficient (D), which is a macroscopic quantity measurable through experimentation.

$${\displaystyle J=-D{dc \over dx}}$$

 

This is the one-dimensional form of Fick's first law.

The diffusion equation is continuous in both space and time. One may discretize space, time, or both space and time, which arise in application. Discretizing time alone just corresponds to taking time slices of the continuous system, and no new phenomena arise. In discretizing space alone, the Green's function becomes the discrete Gaussian kernel, rather than the continuous Gaussian kernel. In discretizing both time and space, one obtains the random walk.

The product rule is used to rewrite the anisotropic tensor diffusion equation, in standard discretization schemes, because direct discretization of the diffusion equation with only first order spatial central differences leads to checkerboard artifacts. The rewritten diffusion equation used in image filtering:

$${\displaystyle {\frac {\partial \phi (\mathbf {r} ,t)}{\partial t}}=\nabla \cdot \left[D(\phi ,\mathbf {r} )\right]\nabla \phi (\mathbf {r} ,t)+{\rm {tr}}{\Big [}D(\phi ,\mathbf {r} ){\big (}\nabla \nabla ^{\text{T}}\phi (\mathbf {r} ,t){\big )}{\Big ]}}$$

 

• Continuity equation
• Heat equation
• Fokker–Planck equation
• Fick's laws of diffusion
• Maxwell–Stefan equation
• Radiative transfer equation and diffusion theory for photon transport in biological tissue
• Streamline diffusion
• Numerical solution of the convection–diffusion equation