View source for Physics:Convection–diffusion equation

{{short description|Combination of the diffusion and convection (advection) equations}}
The '''convection–diffusion equation''' is a combination of the [[Diffusion equation|diffusion]] and convection (advection) equations, and describes physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: [[Physics:Diffusion|diffusion]] and [[Astronomy:Convection|convection]]. Depending on context, the same equation can be called the '''[[Advection|advection]]–diffusion equation''', '''[[Physics:Drift velocity|drift]]–diffusion equation''',<ref name=Chandrasekhar/> or '''(generic) scalar transport equation'''.<ref>{{cite book |title=Computational Fluid Dynamics in Industrial Combustion |editor1-last=Baukal |editor2-last=Gershtein |editor3-last=Li |page=67 |publisher=CRC Press |year=2001 |isbn=0-8493-2000-3 |url=https://books.google.com/books?id=YlSbCgjjLrcC&pg=PA67 |via=Google Books }}</ref>

==Equation==

===General===
[[File:ADE contamination plume.gif|thumb|350px|right|A contaminant plume moving in porous media via the Convection-Diffusive Equation(also called Advective-Diffusive Equation) (time stamps of 1 day, 85 days, 462 days and 674 days) with advection as primary transport mechanism.]]
The general equation is<ref>{{cite book |title=Introduction to Climate Modelling |first=Thomas |last=Stocker |location=Berlin |publisher=Springer |year=2011 |isbn=978-3-642-00772-9 |page=57 |url=https://books.google.com/books?id=D4zulgFb5JwC&pg=PA57 |via=Google Books }}</ref><ref name=Socolofsky>{{cite web |title=Advective Diffusion Equation |work=Lecture notes |first1=Scott A. |last1=Socolofsky |first2=Gerhard H. |last2=Jirka |url=https://ceprofs.civil.tamu.edu/ssocolofsky/cven489/downloads/book/ch2.pdf |url-status=dead |archive-date=June 25, 2010 |archive-url=https://web.archive.org/web/20100625232657/https://ceprofs.civil.tamu.edu/ssocolofsky/cven489/downloads/book/ch2.pdf |access-date=April 18, 2012 }}</ref>
<math display="block">\frac{\partial c}{\partial t}  = \mathbf{\nabla} \cdot (D \mathbf{\nabla} c) - \mathbf{\nabla} \cdot (\mathbf{v} c) + R</math>
where
*{{mvar|c}} is the variable of interest (species concentration for [[Physics:Mass transfer|mass transfer]], temperature for [[Physics:Heat transfer|heat transfer]]),
*{{mvar|D}} is the diffusivity (also called diffusion coefficient), such as [[Physics:Mass diffusivity|mass diffusivity]] for particle motion or [[Physics:Thermal diffusivity|thermal diffusivity]] for heat transport,
*{{math|'''v'''}} is the [[Physics:Velocity|velocity]] field that the quantity is moving with. It is a function of time and space. For example, in [[Advection|advection]], {{mvar|c}} might be the concentration of salt in a river, and then {{math|'''v'''}} would be the velocity of the water flow as a function of time and location. Another example, {{mvar|c}} might be the concentration of small bubbles in a calm lake, and then {{math|'''v'''}} would be the velocity of bubbles rising towards the surface by [[Physics:Buoyancy|buoyancy]] (see [[Physics:Convection–diffusion equation#Velocity in response to a force|below]]) depending on time and location of the bubble. For [[Physics:Multiphase flow|multiphase flow]]s and flows in porous media, {{math|'''v'''}} is the (hypothetical) [[Physics:Superficial velocity|superficial velocity]]. 
*{{mvar|R}} describes [[Physics:Current sources and sinks|sources or sinks]] of the quantity {{mvar|c}}. For example, for a chemical species, {{math|''R'' > 0}} means that a [[Chemistry:Chemical reaction|chemical reaction]] is creating more of the species, and {{math|''R'' < 0}} means that a chemical reaction is destroying the species. For heat transport, {{math|''R'' > 0}} might occur if thermal energy is being generated by [[Physics:Friction|friction]].
*{{math|∇}} represents [[Gradient|gradient]] and {{math|∇ ⋅}} represents [[Divergence|divergence]]. In this equation, {{math|∇''c''}} represents concentration gradient.

===Understanding the terms involved===
The right-hand side of the equation is the sum of three contributions.
* The first, {{math|∇ ⋅ (''D''∇''c'')}}, describes [[Diffusion equation|diffusion]]. Imagine that {{mvar|c}} is the concentration of a chemical. When concentration is low somewhere compared to the surrounding areas (e.g. a local minimum of concentration), the substance will diffuse in from the surroundings, so the concentration will increase. Conversely, if concentration is high compared to the surroundings (e.g. a local maximum of concentration), then the substance will diffuse out and the concentration will decrease. The net diffusion is proportional to the Laplacian (or [[Second derivative|second derivative]]) of concentration if the diffusivity {{mvar|D}} is a constant.
* The second contribution, {{math|−∇ ⋅ ('''v'''''c'')}}, describes convection (or advection). Imagine standing on the bank of a river, measuring the water's salinity (amount of salt) each second. Upstream, somebody dumps a bucket of salt into the river. A while later, you would see the salinity suddenly rise, then fall, as the zone of salty water passes by. Thus, the concentration ''at a given location'' can change because of the flow.
* The final contribution, {{mvar|R}}, describes the creation or destruction of the quantity. For example, if {{mvar|c}} is the concentration of a molecule, then {{mvar|R}} describes how the molecule can be created or destroyed by chemical reactions. {{mvar|R}} may be a function of {{mvar|c}} and of other parameters. Often there are several quantities, each with its own convection–diffusion equation, where the destruction of one quantity entails the creation of another. For example, when methane burns, it involves not only the destruction of methane and oxygen but also the creation of carbon dioxide and water vapor. Therefore, while each of these chemicals has its own convection–diffusion equation, they are coupled together and must be solved as a system of ''simultaneous'' differential equations.

===Common simplifications===
In a common situation, the diffusion coefficient is constant, there are no sources or sinks, and the velocity field describes an [[Physics:Incompressible flow|incompressible flow]] (i.e., it has [[Solenoidal vector field|zero divergence]]). Then the formula simplifies to:<ref name=Bejan>{{cite book | author=Bejan A | title=Convection Heat Transfer | year=2004}}</ref><ref name=Bird>{{cite book | author=Bird, Stewart, Lightfoot | title=Transport Phenomena | 
year=1960}}</ref><ref name=Probstein>{{cite book | author=Probstein R | title=Physicochemical Hydrodynamics | 
year=1994}}</ref>
<math display="block">\frac{\partial c}{\partial t}  = D \nabla^2 c - \mathbf{v} \cdot \nabla c. </math>

In this form, the convection–diffusion equation combines both [[Parabolic partial differential equation|parabolic]] and [[Hyperbolic partial differential equation|hyperbolic]] partial differential equations.

In non-interacting material, {{mvar|1=D=0}} (for example, when temperature is close to [[Physics:Absolute zero|absolute zero]], dilute gas has almost zero [[Physics:Mass diffusivity|mass diffusivity]]), hence the transport equation is simply:
<math display="block">\frac{\partial c}{\partial t}  + \mathbf{v} \cdot \nabla c=0. </math>

Using [[Fourier transform]] in both temporal and spatial domain (that is, with integral kernel <math>e^{j\omega t+j\mathbf{k}\cdot\mathbf{x}}</math>), its [[Characteristic equation (calculus)|characteristic equation]] can be obtained:
<math display="block">j\omega \tilde c+\mathbf{v}\cdot j \mathbf{k} \tilde c=0 \rightarrow \omega=-\mathbf{k}\cdot \mathbf{v}, </math>
which gives the general solution:
<math display="block">c = f(\mathbf{x}-\mathbf{v}t), </math>
where <math>f </math> is any [[Differentiable function|differentiable scalar function]]. This is the basis of temperature measurement for ''near'' [[Physics:Bose–Einstein condensate|Bose–Einstein condensate]]<ref>{{cite arXiv|last1=Ketterle|first1=W. |last2=Durfee| first2=D. S.| last3=Stamper-Kurn|first3=D. M. | date=1999-04-01 | title=Making, probing and understanding Bose-Einstein condensates |eprint=cond-mat/9904034}}</ref> via [[Physics:Time of flight|time of flight]] method.<ref>{{Cite journal| last1=Brzozowski|first1=Tomasz M| last2=Maczynska|first2=Maria| last3=Zawada|first3=Michal| last4=Zachorowski|first4=Jerzy| last5=Gawlik|first5=Wojciech| s2cid=67796405| date=2002-01-14| title=Time-of-flight measurement of the temperature of cold atoms for short trap-probe beam distances |journal=Journal of Optics B: Quantum and Semiclassical Optics| language=en| volume=4| issue=1| pages=62–66 | doi=10.1088/1464-4266/4/1/310| issn=1464-4266 | bibcode=2002JOptB...4...62B}}</ref>

===Stationary version===
The '''stationary convection–diffusion equation''' describes the steady-state behavior of a convective-diffusive system. In a steady state, {{math|1={{sfrac|∂''c''|∂''t''}} = 0}}, so the formula is:
<math display="block">0  = \nabla \cdot (D \nabla c) - \nabla \cdot (\mathbf{v} c) + R.</math>

==Derivation ==

The convection–diffusion equation can be derived in a straightforward way<ref name=Socolofsky/> from the [[Continuity equation#Differential form|continuity equation]], which states that the rate of change for a [[Physics:Scalar|scalar quantity]] in a [[Differential (infinitesimal)|differential]] [[Physics:Control volume|control volume]] is given by flow and diffusion into and out of that part of the system along with any generation or consumption inside the control volume:
<math display="block"> \frac{\partial c}{\partial t} + \nabla\cdot\mathbf{j} = R, </math>
where {{math|'''j'''}} is the total [[Physics:Flux|flux]] and {{mvar|R}} is a net volumetric source for {{mvar|c}}. There are two sources of flux in this situation. First, '''diffusive flux''' arises due to [[Physics:Diffusion|diffusion]]. This is typically approximated by Fick's first law:
<math display="block">\mathbf{j}_\text{diff} = -D \nabla c</math>
i.e., the flux of the diffusing material (relative to the bulk motion) in any part of the system is proportional to the local concentration [[Gradient|gradient]]. Second, when there is overall convection or flow, there is an associated flux called '''[[Advection|advective flux]]''':
<math display="block">\mathbf{j}_\text{adv} = \mathbf{v} c</math>
The total flux (in a stationary coordinate system) is given by the sum of these two:
<math display="block">\mathbf{j} = \mathbf{j}_\text{diff} + \mathbf{j}_\text{adv} = -D \nabla c + \mathbf{v} c.</math>
Plugging into the continuity equation:
<math display="block"> \frac{\partial c}{\partial t} + \nabla\cdot \left(-D \nabla c + \mathbf{v} c \right) = R. </math>

==Complex mixing phenomena==
In general, {{mvar|D}}, {{math|'''v'''}}, and {{mvar|R}} may vary with space and time. In cases in which they depend on concentration as well, the equation becomes nonlinear, giving rise to many distinctive mixing phenomena such as [[Physics:Rayleigh–Bénard convection|Rayleigh–Bénard convection]] when {{math|'''v'''}} depends on temperature in the heat transfer formulation and [[Reaction–diffusion system|reaction–diffusion]] pattern formation when {{mvar|R}} depends on concentration in the mass transfer formulation.

==Velocity in response to a force==

In some cases, the average velocity field {{math|'''v'''}} exists because of a force; for example, the equation might describe the flow of ions dissolved in a liquid, with an [[Physics:Electric field|electric field]] pulling the ions in some direction (as in [[Biology:Gel electrophoresis|gel electrophoresis]]). In this situation, it is usually called the '''drift–diffusion equation''' or the '''Smoluchowski equation''',<ref name=Chandrasekhar>{{cite journal |author=Chandrasekhar |year=1943 |title=Stochastic Problems in Physics and Astronomy |journal=Rev. Mod. Phys. |volume=15 |issue=1 |page=1 |doi=10.1103/RevModPhys.15.1 |bibcode = 1943RvMP...15....1C }} See equation (312)</ref> after Marian Smoluchowski who described it in 1915<ref>{{cite journal |first=M. v. |last=Smoluchowski |title=Über Brownsche Molekularbewegung unter Einwirkung äußerer Kräfte und den Zusammenhang mit der verallgemeinerten Diffusionsgleichung |journal=[[Physics:Annalen der Physik|Ann. Phys.]] |volume=353 |series=4. Folge |issue=48 |pages=1103–1112 |year=1915 |doi=10.1002/andp.19163532408 |bibcode=1915AnP...353.1103S |url=http://matwbn.icm.edu.pl/ksiazki/pms/pms2/pms2132.pdf }}</ref> (not to be confused with the [[Physics:Einstein relation (kinetic theory)|Einstein–Smoluchowski relation]] or [[Smoluchowski coagulation equation]]).

Typically, the average velocity is directly proportional to the applied force, giving the equation:<ref>{{cite web |title=Smoluchowski Diffusion Equation |url=https://www.ks.uiuc.edu/~kosztin/PHYCS498NSM/LectureNotes/chp4.pdf }}</ref><ref name=Doi>{{cite book |title=The Theory of Polymer Dynamics |last1=Doi |name-list-style=amp |last2=Edwards |year=1988 |pages=46–52 |isbn=978-0-19-852033-7 |url=https://books.google.com/books?id=dMzGyWs3GKcC&pg=PA46 |via=[[Google Books]] }}</ref>
<math display="block">\frac{\partial c}{\partial t}  = \nabla \cdot (D \nabla c) - \nabla \cdot \left( \zeta^{-1} \mathbf{F} c \right) + R</math>
where {{math|'''F'''}} is the force, and {{mvar|ζ}} characterizes the friction or [[Physics:Drag|viscous drag]]. (The inverse {{math|''ζ''{{isup|−1}}}} is called [[Physics:Einstein relation (kinetic theory)|mobility]].)

===Derivation of Einstein relation===
{{main|Physics:Einstein relation (kinetic theory)}}

When the force is associated with a [[Physics:Potential energy|potential energy]] {{math|'''F''' {{=}} −∇''U''}} (see [[Physics:Conservative force|conservative force]]), a steady-state solution to the above equation (i.e. {{math|0 {{=}} ''R'' {{=}} {{sfrac|∂''c''|∂''t''}}}}) is:
<math display="block">c \propto \exp \left( -D^{-1} \zeta^{-1} U \right)</math>
(assuming {{mvar|D}} and {{mvar|ζ}} are constant). In other words, there are more particles where the energy is lower. This concentration profile is expected to agree with the [[Physics:Boltzmann distribution|Boltzmann distribution]] (more precisely, the [[Gibbs measure]]). From this assumption, the [[Physics:Einstein relation (kinetic theory)|Einstein relation]] can be proven:<ref name=Doi/>
<math display="block">D \zeta = k_\mathrm{B} T.</math>

==Smoluchowski convection-diffusion equation==

The Smoluchowski convective-diffusion equation is a stochastic (Smoluchowski) diffusion equation with an additional convective flow-field,<ref name=Dhont>{{cite book |title=An Introduction to the Dynamics of Colloids |first=J. K. G. |last=Dhont |page=195 |location= |publisher=Elsevier |year=1996 |isbn=0-444-82009-4 |url=https://books.google.com/books?id=mmArTF5SJ9oC&pg=PA195 |via=Google Books }}</ref>
<math display="block">\frac{\partial c}{\partial t}  = \nabla \cdot (D \nabla c) - \mathbf{\nabla} \cdot (\mathbf{v} c) - \nabla \cdot \left( \zeta^{-1} \mathbf{F} c \right)</math>
In this case, the force {{math|'''F'''}} describes the conservative interparticle interaction force between two colloidal particles or the intermolecular interaction force between two molecules in the fluid, and it is unrelated to the externally imposed flow velocity {{math|'''v'''}}. The steady-state version of this equation is the basis to provide a description of the [[Physics:Pair distribution function|pair distribution function]] (which can be identified with {{mvar|c}}) of colloidal suspensions  under shear flows.<ref name=Dhont />

An approximate solution to the steady-state version of this equation has been found using the [[Method of matched asymptotic expansions|method of matched asymptotic expansions]].<ref> {{cite journal | last1 = Zaccone | first1 = A. | last2 = Gentili | first2 = D. | last3 = Wu | first3 = H. | last4 = Morbidelli | first4 = M. | year = 2009| title = Theory of activated-rate processes under shear with application to shear-induced aggregation of colloids. | journal = Physical Review E | volume = 80 | issue = 5| pages = 051404 | doi = 10.1103/PhysRevE.80.051404 | pmid = 20364982 | arxiv = 0906.4879 | bibcode = 2009PhRvE..80e1404Z | hdl = 2434/653702 | s2cid = 22763509 | hdl-access = free }}</ref> This solution provides a theory for the transport-controlled reaction rate of two molecules in a shear flow, and also provides a way to extend the [[Chemistry:DLVO theory#Effect of shear flows|DLVO theory]] of colloidal stability to colloidal systems subject to shear flows (e.g. in [[Physics:Microfluidics|microfluidics]], chemical reactors, environmental flows). 
The full solution to the steady-state equation, obtained using the [[Method of matched asymptotic expansions#Convection-diffusion equation|method of matched asymptotic expansions]], has been developed by Alessio Zaccone and L. Banetta to compute the [[Physics:Pair distribution function|pair distribution function]] of Lennard-Jones interacting particles in [[Physics:Shear flow|shear flow]]<ref> {{cite journal | last1 = Banetta | first1 = L. | last2 = Zaccone | first2 = A. | year = 2019 | title = Radial distribution function of Lennard-Jones fluids in shear flows from intermediate asymptotics. | journal = Physical Review E | volume = 99 | issue = 5| pages = 052606 | doi = 10.1103/PhysRevE.99.052606 | pmid = 31212460 | arxiv = 1901.05175 | bibcode = 2019PhRvE..99e2606B | s2cid = 119011235 }}</ref> and subsequently extended to compute the [[Physics:Pair distribution function|pair distribution function]] of charge-stabilized (Yukawa or [[Chemistry:Debye–Hückel equation|Debye–Hückel]]) colloidal particles in shear flows.<ref>{{cite journal | last1 = Banetta | first1 = L. | last2 = Zaccone | first2 = A. | year = 2020 | title = Pair correlation function of charge-stabilized colloidal systems under sheared conditions. | journal = Colloid and Polymer Science | volume = 298 | issue = 7| pages = 761–771 | doi = 10.1007/s00396-020-04609-4|arxiv=2006.00246| doi-access = free }}</ref>

==As a stochastic differential equation==
The convection–diffusion equation (with no sources or drains, {{math|''R'' {{=}} 0}}) can be viewed as a [[Stochastic differential equation|stochastic differential equation]], describing random motion with diffusivity {{mvar|D}} and bias {{math|'''v'''}}. For example, the equation can describe the [[Brownian motion]] of a single particle, where the variable {{mvar|c}} describes the [[Probability distribution|probability distribution]] for the particle to be in a given position at a given time. The reason the equation can be used that way is because there is no mathematical difference between the probability distribution of a single particle, and the concentration profile of a collection of infinitely many particles (as long as the particles do not interact with each other).

The ''[[Langevin equation]]'' describes advection, diffusion, and other phenomena in an explicitly stochastic way. One of the simplest forms of the Langevin equation is when its "noise term" is [[Physics:Gaussian noise|Gaussian]]; in this case, the Langevin equation is exactly equivalent to the convection–diffusion equation.<ref name=Doi/> However, the Langevin equation is more general.<ref name=Doi/>

==Numerical solution==
{{main|Physics:Numerical solution of the convection–diffusion equation}}
The convection–diffusion equation can only rarely be solved with a pen and paper. More often, computers are used to numerically approximate the solution to the equation, typically using the [[Finite element method|finite element method]]. For more details and algorithms see: [[Physics:Numerical solution of the convection–diffusion equation|Numerical solution of the convection–diffusion equation]].

==Similar equations in other contexts==
The convection–diffusion equation is a relatively simple equation describing flows, or alternatively, describing a stochastically-changing system. Therefore, the same or similar equation arises in many contexts unrelated to flows through space.
*It is formally identical to the [[Fokker–Planck equation]] for the velocity of a particle.
*It is closely related to the [[Finance:Black–Scholes equation|Black–Scholes equation]] and other equations in financial mathematics.<ref>{{Cite journal| last1=Arabas| first1=S.| last2=Farhat| first2=A.| title=Derivative pricing as a transport problem: MPDATA solutions to Black-Scholes-type equations| journal=J. Comput. Appl. Math.| year=2020| language=en| volume=373| page=112275| doi=10.1016/j.cam.2019.05.023| arxiv=1607.01751| s2cid=128273138}}</ref>
*It is closely related to the [[Physics:Navier–Stokes equations|Navier–Stokes equations]], because the flow of [[Physics:Momentum|momentum]] in a fluid is mathematically similar to the flow of mass or energy. The correspondence is clearest in the case of an incompressible Newtonian fluid, in which case the Navier–Stokes equation is: <math display="block">\frac{\partial \mathbf{M}}{\partial t} = \mu \nabla^2 \mathbf{M} -\mathbf{v} \cdot \nabla \mathbf{M} + (\mathbf{f}-\nabla P)</math>
where {{math|'''M'''}} is the momentum of the fluid (per unit volume) at each point (equal to the density {{mvar|ρ}} multiplied by the velocity {{math|'''v'''}}), {{mvar|μ}} is viscosity, {{mvar|P}} is fluid pressure, and {{math|'''f'''}} is any other [[Physics:Body force|body force]] such as [[Physics:Gravity|gravity]]. In this equation, the term on the left-hand side describes the change in momentum at a given point; the first term on the right describes the diffusion of momentum by [[Physics:Viscosity|viscosity]]; the second term on the right describes the advective flow of momentum; and the last two terms on the right describes the external and internal forces which can act as sources or sinks of momentum.

==In semiconductor physics==

[[File:diffusion_center.gif|thumb|350px|right|As carriers are generated (green:electrons and purple:holes) due to light shining at the center of an intrinsic semiconductor, they diffuse towards two ends. Electrons have higher diffusion constant than holes leading to fewer excess electrons at the center as compared to holes.]]

In semiconductor physics, this equation is called the '''drift–diffusion equation'''. The word "drift" is related to [[Physics:Drift current|drift current]] and [[Physics:Drift velocity|drift velocity]]. The equation is normally written:<ref>{{cite journal|last1=Hu|first1=Yue|title=Simulation of a partially depleted absorber (PDA) photodetector|journal=Optics Express|volume=23|issue=16|pages=20402–20417|doi=10.1364/OE.23.020402|pmid=26367895|bibcode=2015OExpr..2320402H|year=2015|hdl=11603/11470|hdl-access=free}}</ref>
<math display="block">\begin{align}
\frac{\mathbf{J}_n}{-q} &= - D_n \nabla n - n \mu_n \mathbf{E} \\
\frac{\mathbf{J}_p}{q} &= - D_p \nabla p + p \mu_p \mathbf{E} \\
\frac{\partial n}{\partial t} &= -\nabla \cdot \frac{\mathbf{J}_n}{-q} + R \\
\frac{\partial p}{\partial t} &= -\nabla \cdot \frac{\mathbf{J}_p}{q} + R
\end{align}</math>
where
*{{mvar|n}} and {{mvar|p}} are the concentrations (densities) of electrons and [[Physics:Electron hole|holes]], respectively,
*{{math|''q'' > 0}} is the [[Physics:Elementary charge|elementary charge]],
*{{mvar|J<sub>n</sub>}} and {{mvar|J<sub>p</sub>}} are the [[Physics:Electric current|electric current]]s due to electrons and holes respectively,
*{{math|{{sfrac|''J<sub>n</sub>''|−''q''}}}} and {{math|{{sfrac|''J<sub>p</sub>''|''q''}}}} are the corresponding "particle currents" of electrons and holes respectively,
*{{mvar|R}} represents [[Physics:Carrier generation and recombination|carrier generation and recombination]] ({{math|''R'' > 0}} for generation of electron-hole pairs, {{math|''R'' < 0}} for recombination.)
*{{math|'''E'''}} is the [[Physics:Electric field|electric field]] vector
*<math>\mu_n</math> and <math>\mu_p</math> are [[Physics:Electron mobility|electron and hole mobility]].

The diffusion coefficient and mobility are related by the [[Physics:Einstein relation (kinetic theory)|Einstein relation]] as above:
<math display="block">\begin{align}
D_n &= \frac{\mu_n k_\mathrm{B} T}{q}, \\
D_p &= \frac{\mu_p k_\mathrm{B} T}{q},
\end{align}</math>
where {{math|''k''<sub>B</sub>}} is the [[Physics:Boltzmann constant|Boltzmann constant]] and {{mvar|T}} is [[Physics:Absolute temperature|absolute temperature]]. The '''[[Physics:Drift current|drift current]]''' and '''[[Physics:Diffusion current|diffusion current]]''' refer separately to the two terms in the expressions for {{math|'''J'''}}, namely:
<math display="block">\begin{align}
\frac{\mathbf{J}_{n,\text{drift}}}{-q} &= - n \mu_n \mathbf{E}, \\
\frac{\mathbf{J}_{p,\text{drift}}}{q} &= p \mu_p \mathbf{E}, \\
\frac{\mathbf{J}_{n,\text{diff}}}{-q} &= - D_n \nabla n, \\
\frac{\mathbf{J}_{p,\text{diff}}}{q} &= - D_p \nabla p.
\end{align}</math>

This equation can be solved together with [[Physics:Poisson's equation|Poisson's equation]] numerically.<ref>{{cite journal| last1=Hu|first1=Yue| title=Modeling sources of nonlinearity in a simple pin photodetector|journal=Journal of Lightwave Technology| volume=32| issue=20| pages=3710–3720|doi=10.1109/JLT.2014.2315740|url=https://www.osapublishing.org/jlt/abstract.cfm?uri=jlt-32-20-3710| bibcode=2014JLwT...32.3710H|year=2014|citeseerx=10.1.1.670.2359|s2cid=9882873}}</ref>

An example of results of solving the drift diffusion equation is shown on the right. When light shines on the center of semiconductor, carriers are generated in the middle and diffuse towards two ends. The drift–diffusion equation is solved in this structure and electron density distribution is displayed in the figure. One can see the gradient of carrier from center towards two ends.

==See also==
{{Div col|colwidth=20em}}
* [[Software:Advanced Simulation Library|Advanced Simulation Library]]
* [[Conservation law|Conservation equations]]
* Incompressible Navier–Stokes equations
* [[Chemistry:Nernst–Planck equation|Nernst–Planck equation]]
* [[Physics:Double diffusive convection|Double diffusive convection]]
* [[Physics:Natural convection|Natural convection]]
* [[Buckley–Leverett equation]]
* [[Burgers' equation]]
{{Div col end}}

==References==
{{reflist}}

==Further reading==
*{{cite book |first=Granville |last=Sewell |title=The Numerical Solution of Ordinary and Partial Differential Equations |publisher=Academic Press |year=1988 |isbn=0-12-637475-9 }}

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