Diffusion equation
The diffusion equation is a
Statement
The equation is usually written as:
where ϕ(r, t) is the
The equation above applies when the diffusion coefficient is
If D is constant, then the equation reduces to the following linear differential equation:
which is identical to the heat equation.
Historical origin
The
Derivation
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:
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:
- 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
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:
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.
This is the one-dimensional form of Fick's first law.[2]
Discretization
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
Discretization (image)
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:
See also
- 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
References
- ISSN 0003-3804.
- ^ Mohammad, Alsalamat (2024-03-12). "Diffusion — Fick's Laws of Diffusion and Steady State". Retrieved 2024-04-10.
Further reading
- Carslaw, H. S. and Jaeger, J. C. (1959). Conduction of Heat in Solids. Oxford: Clarendon Press
- Crank, J. (1956). The Mathematics of Diffusion. Oxford: Clarendon Press
- Mathews, Jon; Walker, Robert L. (1970). Mathematical methods of physics (2nd ed.), New York: W. A. Benjamin, ISBN 0-8053-7002-1
- Thambynayagam, R. K. M (2011). The Diffusion Handbook: Applied Solutions for Engineers. McGraw-Hill