Diffusion equation

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The diffusion equation is a partial differential equation. In physics, it describes a behavior of the collective motion of micro particles in a material, resulting from the random movement of each micro particle. In mathematics, it is applicable in common to a subject relevant to the Markov process in various science fields, such as the material science, the information science, the life science, the social science, and so on. These subjects described by the diffusion equation are generally called Brown problems.


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.

More generally, when D is a symmetric positive definite matrix, the equation describes anisotropic diffusion, which is written (for three dimensional diffusion) as:

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

also called the heat equation.

History and Development[edit]

In mathematics, a great many phenomena in various science fields are expressed by using the well-known evolution equations. The diffusion equation is one of them and mathematically corresponds to the Markov process in relation to the normal distribution rule.[1] In physics, the motion of diffusion particles corresponds to the well-known Brown motion satisfying the parabolic law.[2] It is widely accepted that the Brown problem is a general term of investigating subjects in various science fields relevant to the Markov process, such as the material science, the information science, the life science, the social science, and so on. The extended diffusion equations are used for various sciences fields. In that case, they sometimes have a sink and source of their concerned elements, for example, such as a local equilibrium relation between native defects in silicon crystal in the material science or between predation and prey in the life science. We must then solve a system of diffusion equations. In the following, however, we discuss the fundamental diffusion equation of the so-called Fick’s diffusion equation in relation to the material science.

In history, the heat equation proposed by Fourier in 1822 has been applied to investigating a temperature distribution in materials.[3] In 1827, the so-called Brown motion was found, where the self-diffusion of water is visualized by pollen micro particles motion.[4] Nevertheless, the Brown motion had not been recognized as a diffusion problem until the Einstein theory of Brown motion in 1905,[5] although it was a typical diffusion problem. In 1855, Fick applied the heat equation to diffusion phenomena as it had been.[6]

In accordance with the industrial requirement, the solid materials such as alloys, semiconductors, multilayer materials, and so on, have been widely fabricated. The heat treatment is indispensable for their fabrication processes then. The migration of particles in a solid material is caused by the heat treatment. In relation to the migration of their particles, the diffusion problems of various solid materials have been thus widely investigated, although the diffusion equation was mainly applied to problems of liquid material in an early stage after the Fick’s proposition.

The Gauss’s divergence theorem[7] shows that the diffusion equation is valid in the solid, liquid and gas states in every material as a material conservation law, if there is no sink and source in the given diffusion system. It is also shows that the corresponding Fick’s first law to the Fick’s second law is mathematically incomplete without a constant diffusion flux relevant to the Brown motion in the localized space. The constant diffusion flux is indispensable for understanding the self-diffusion mechanism.[8] The self-diffusion mechanism itself was not directly investigated, although it had been indirectly investigated by behavior of impurity diffusion in a pure material as shown in the Einstein’s Brown theory and the Langevin equation.[9]

We found that the diffusivity of diffusion equation depends generally on the concentration of diffusion particles. In that case, the diffusion equation becomes a nonlinear partial differential equation, and the mathematical solution is almost impossible, even if it is a case of the time and one dimension space coordinate . In accordance with the parabolic law , Boltzmann transformed the diffusion equation of , which is a nonlinear partial differential equation, into a nonlinear ordinary differential equation of in 1894. [10] Since then, however, the Boltzmann transformation equation had not been mathematically solved until recently, although Matano empirically used it for analyzing interdiffusion problems in the metallurgy field. [11]

Here, the analytical method of diffusion equation, which is extremely superior in calculation to the existing analytical method such as the integral transformation method of Fourier or Laplace and/or the variable separation method, was thus established in the parabolic space.[12]

In 1947, Kirkendall found that an inert marker set at a point in a binary alloy moves from the initial sate point after the diffusion treatment.[13] The phenomena are so called Kirkendall effect and it was considered that we cannot understand it from the existing theory of binary interdiffusion in those days. As a result, a new concept of intrinsic diffusion was then introduced for understanding the Kirkendall effect in the interdiffusion problems. Based on the intrinsic diffusion concept, Darken derived a relation between an interdiffusion coefficient and intrinsic diffusion coefficients in a binary interdiffusion in 1948.[14] At present, however, it is revealed that the so-called Darken equation itself is mathematically wrong in the derivation process.[8] Although the concentration of diffusion particles is a real quantity in physics, the temperature is a thermodynamic state quantity. As far as the shape of heat conduction material is unchangeable during a thermal treatment, the coordinate system of heat equation set in a material is a fixed one, since the coordinate system is not influenced by variations of the material internal structure. On the other hand, strictly speaking, the coordinate system of diffusion equation set in the diffusion field (solvent) is a moving one, since the origin of coordinate system is generally influenced by such variations.

When Fick proposed the diffusion equation, the Gauss divergence theorem had been already reported in 1840.[15] Nevertheless, the problem of coordinate system of diffusion equation had not been mathematically investigated in accordance with the divergence theorem until recently. In general, however, it is indispensable for understanding the diffusion problems to discuss their coordinate systems, since it is, strictly speaking, considered that the diffusion particles, solvent particles and also the diffusion region space simultaneously move against the experimentation system in the diffusion region outside.

Recently, the diffusion equation was thus mathematically investigated in accordance with the divergence theorem and the coordinate transformation theory.[8] As a result, the diffusion flux should be determined by taking account of the concerned coordinate system of diffusion equation. Using the corresponding diffusion flux to the concerned coordinate system of diffusion equation for interdiffusion, one way diffusion, impurity diffusion and self-diffusion, we found that they are uniformly discussed and the foundation of diffusion problems is included in interdiffusion problems. The interdiffusion theory of an elements system applicable to every material was thus reasonably established. In the analysis of interdiffusion problems, the only difference between a binary system and an N elements system is whether the solvent material is one element or elements.

The coordinate transformation theory reveals that the Kirkendall effect is caused by a shift between the coordinate systems of diffusion equation like the Doppler effect relevant to a wave equation is caused by a shift between the fixed coordinate system and the moving one. Further, it was also found that the concept of intrinsic diffusion is an illusion in the diffusion history.[8] All physical information in the given diffusion system is incorporated into the diffusivity. If we can know a diffusivity behavior in the given diffusion equation, the mathematical solution and/or numerical one at least is possible. In the diffusion problems, it is thus extremely dominant to know the diffusivity behavior. The diffusivity is defined by an interaction between a diffusion particle and the diffusion field near the diffusion particle itself. This indicates that the diffusivity should be essentially investigated in the quantum mechanics, since the behavior of a micro particle should be investigated by analyzing the Schrodinger equation.[16]

From applying the diffusion equation to a problem of diffusion elementary process, the equation was reasonably derived.[8] It was revealed that the diffusivity corresponds to the angular momentum operator in the quantum mechanics. As a result, the universal expression of diffusivity, which is applicable to every material in an arbitrary thermodynamic state, was obtained as one with the proportionality constant composed of the product of Planck constant and Avogadro constant . It was also found that the well-known material wave relation proposed by de Broglie in 1923,[17] which is the most fundamental one in materials science, is obtained from a relation between the given diffusivity expressions.[8] This gives evidence for the theory discussed here.


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:

where j is the flux of the diffusing material. The diffusion equation can be obtained easily from this when combined with the phenomenological Fick's first law, which states that the flux of the diffusing material in any part of the system is proportional to the local density gradient:

If drift must be taken into account, the Smoluchowski equation provides an appropriate generalization.


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.

Discretization (Image)[edit]

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:

where "tr" denotes the trace of the 2nd rank tensor, and superscript "T" denotes transpose, in which in image filtering D(ϕ, r) are symmetric matrices constructed from the eigenvectors of the image structure tensors . The spatial derivatives can then be approximated by two first order and a second order central finite differences. The resulting diffusion algorithm can be written as an image convolution with a varying kernel (stencil) of size 3 × 3 in 2D and 3 × 3 × 3 in 3D.

See also[edit]


  1. ^ AA Markov, The theory of algorithms, American mathematical society translations series [2] 15: 1-14 (1960).
  2. ^ R. Brown, A brief account of microscopical observations made in the months of June, July and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies, Philosophical magazine N. S 4 161-173 (1828).
  3. ^ JBJ Fourier, Theorie analytique de la chaleur, Didot Paris: 499-508 (1822).
  4. ^ R. Brown, A brief account of microscopical observations made in the months of June, July and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies, Philosophical magazine N. S 4 161-173 (1828).
  5. ^ A Einstein, Die von der molekularkinetishen theorie der warme geforderte bewegung von in ruhenden flussiigkeiten sspendierten teilchen, Ann. Phys. 18(8): 599-560 (1905).
  6. ^ A. Fick, Ueber Diffusion, Pogg. Ann. Phys. Chem. 170 (4. Reihe 94), 59-86 (1855).
  7. ^ CF Gauss, Allgemeine lehsatze in beziehung auf die im verkehrten verhaltnisse des quadrats der entfernung wirkenden anziehung-und abstossungs- krafte, Res Beob magn 4 (1): 1 (1840).
  8. ^ a b c d e f Takahisa Okino, Mathematical Physics in Diffusion Problems, Journal of Modern Physics 6: 2109-2144 (2015).
  9. ^ P. Langevin, On the theory of Brownian motion, Compts Rendus Acad Sci (Priss) 146: 530-533 (1908).
  10. ^ L. Boltzmann, Zur integration der diffusionscoefficienten, Ann Rev Phys Chem 53: 959-964 (1894).
  11. ^ C. Matano, On the relation between diffusion-coefficients and concentrations of solid metals, Jpn J Phys 8: 109-113 (1933).
  12. ^ Takahisa Okino, New mathematical solution for analyzing interdiffusion problems, Mater Trans 52: 2220-2227 (2011).
  13. ^ AD Smigelskas, EO Kirkendall, Zinc diffusion in alphabrass, Trans AIME: 171: 130-142 (1947).
  14. ^ LS Darken, Diffusion, mobility and their interrelation through free energy in binary metallic system, Trans AIME: 175: 184-201 (1948).
  15. ^ CF Gauss, Allgemeine lehsatze in beziehung auf die im verkehrten verhaltnisse des quadrats der entfernung wirkenden anziehung-und abstossungs- krafte, Res Beob magn 4 (1): 1 (1840).
  16. ^ E. Schrodinger, Quantisierung als eigenvertproblem, Ann Phys 79: 361-376 (1926).
  17. ^ L de Broglie Wave and quanta, Nture 112: 540 (1923).

Further reading[edit]

  • Carslaw, H. S. and Jager, 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

External links[edit]