Editing Continuity equation

{{Short description|Equation describing the transport of some quantity}}
{{More citations needed|date=December 2023}}
{{continuum mechanics}}

A '''continuity equation''' or '''transport equation''' is an [[equation]] that describes the transport of some quantity. It is particularly simple and powerful when applied to a [[conserved quantity]], but it can be generalized to apply to any [[Intensive and extensive properties|extensive quantity]]. Since [[mass]], [[energy]], [[momentum]], [[electric charge]] and other natural&nbsp;quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described using continuity equations.

Continuity equations are a stronger, local form of [[Conservation law (physics)|conservation law]]s. For example, a weak version of the law of [[conservation of energy]] states that energy can neither be created nor destroyed—i.e., the total amount of energy in the universe is fixed. This statement does not rule out the possibility that a quantity of energy could disappear from one point while simultaneously appearing at another point. A stronger statement is that energy is ''locally'' conserved: energy can neither be created nor destroyed, ''nor'' can it "[[Teleportation|teleport]]" from one place to another—it can only move by a continuous flow. A continuity equation is the mathematical way to express this kind of statement. For example, the continuity equation for [[electric charge]] states that the amount of electric charge in any volume of space can only change by the amount of [[electric current]] flowing into or out of that volume through its boundaries.

Continuity equations more generally can include "source" and "sink" terms, which allow them to describe quantities that are often but not always conserved, such as the density of a molecular species which can be created or destroyed by chemical reactions. In an everyday example, there is a continuity equation for the number of people alive; it has a "source term" to account for people being born, and a "sink term" to account for people dying.

Any continuity equation can be expressed in an "integral form" (in terms of a [[Flux#Flux as a surface integral|flux integral]]), which applies to any finite region, or in a "differential form" (in terms of the [[divergence]] operator) which applies at a point.

Continuity equations underlie more specific transport equations such as the [[convection–diffusion equation]], [[Boltzmann transport equation]], and [[Navier–Stokes equations]].

Flows governed by continuity equations can be visualized using a [[Sankey diagram]].

==General equation==

===Definition of flux===
{{main|Flux}}

A continuity equation is useful when a '''flux''' can be defined. To define flux, first there must be a quantity {{math|''q''}} which can flow or move, such as [[mass]], [[energy]], [[electric charge]], [[momentum]], number of molecules, etc. Let {{math|''ρ''}} be the volume [[density]] of this quantity, that is, the amount of {{math|''q''}} per unit volume.

The way that this quantity {{math|''q''}} is flowing is described by its flux. The flux of {{math|''q''}} is a [[vector field]], which we denote as '''j'''. Here are some examples and properties of flux:
* The dimension of flux is "amount of {{math|''q''}} flowing per unit time, through a unit area". For example, in the mass continuity equation for flowing water, if 1 gram per second of water is flowing through a pipe with cross-sectional area 1&nbsp;cm<sup>2</sup>, then the average mass flux {{math|'''j'''}} inside the pipe is {{nowrap|(1 g/s) / cm<sup>2</sup>}}, and its direction is along the pipe in the direction that the water is flowing. Outside the pipe, where there is no water, the flux is zero.
* If there is a [[velocity field]] {{math|'''u'''}} which describes the relevant flow—in other words, if all of the quantity {{math|''q''}} at a point {{math|'''x'''}} is moving with velocity {{math|'''u'''('''x''')}}—then the flux is by definition equal to the density times the velocity field:
: <math display="block">\mathbf{j} = \rho \mathbf{u}</math>
: For example, if in the mass continuity equation for flowing water, {{math|'''u'''}} is the water's velocity at each point, and {{math|''ρ''}} is the water's density at each point, then {{math|'''j'''}} would be the mass flux, also known as the material [[Discharge (hydrology)|discharge]].
* In a well-known example, the flux of [[electric charge]] is the [[electric current density]].
[[File:Continuity eqn open surface.svg|390px|right|thumb|Illustration of how the fluxes, or flux densities, {{math|'''j'''<sub>1</sub>}} and {{math|'''j'''<sub>2</sub>}} of a quantity {{math|''q''}} pass through open surfaces {{math|''S''<sub>1</sub>}} and {{math|''S''<sub>2</sub>}}. (vectors {{math|'''S'''<sub>1</sub>}} and {{math|'''S'''<sub>2</sub>}} represent [[vector area]]s that can be differentiated into infinitesimal area elements).]]
* If there is an imaginary surface {{math|''S''}}, then the [[surface integral]] of flux over {{math|''S''}} is equal to the amount of {{math|''q''}} that is passing through the surface {{math|''S''}} per unit time:
{{Equation box 1
|indent=:
|equation = <math> (\text{Rate that }q\text{ is flowing through the imaginary surface }S) = \iint_S \mathbf{j} \cdot d\mathbf{S}</math>
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4
}}
: in which <math display="inline">\iint_S d\mathbf{S}</math> is a [[surface integral]].

(Note that the concept that is here called "flux" is alternatively termed '''flux density''' in some literature, in which context "flux" denotes the surface integral of flux density. See the main article on [[Flux]] for details.)

==={{anchor|Integral form|integral form}} Integral form===

The integral form of the continuity equation states that:
* The amount of {{math|''q''}} in a region increases when additional {{math|''q''}} flows inward through the surface of the region, and decreases when it flows outward;
* The amount of {{math|''q''}} in a region increases when new {{math|''q''}} is created inside the region, and decreases when {{math|''q''}} is destroyed;
* Apart from these two processes, there is ''no other way'' for the amount of {{math|''q''}} in a region to change.

Mathematically, the integral form of the continuity equation expressing the rate of increase of {{math|''q''}} within a volume {{math|''V''}} is:
{{Equation box 1
|indent=:
|equation= <math>\frac{\partial q}{\partial t} + \oint_{S}\mathbf{j} \cdot d\mathbf{S} = \Sigma</math>
|cellpadding
|border
|border colour = #0073CF
|background colour=#F5FFFA
}}

[[File:SurfacesWithAndWithoutBoundary.svg|right|thumb|250px|In the integral form of the continuity equation, {{math|''S''}} is any [[closed surface]] that fully encloses a volume {{math|''V''}}, like any of the surfaces on the left. {{math|''S''}} can ''not'' be a surface with boundaries, like those on the right. (Surfaces are blue, boundaries are red.)]]
where
* {{math|''S''}} is any imaginary [[closed surface]], that encloses a volume {{math|''V''}},
* <math>\oint_{S} d\mathbf{S}</math> denotes a [[surface integral]] over that closed surface,
* {{math|''q''}} is the total amount of the quantity in the volume {{math|''V''}},
* {{math|'''j'''}} is the flux of {{math|''q''}},
* {{math|''t''}} is time,
* {{math|Σ}} is the net rate that {{math|''q''}} is being generated inside the volume {{math|''V''}} per unit time. When {{math|''q''}} is being generated (i.e., when   <math>\tfrac{\partial q}{\partial t}>0</math> ), the region is called a ''source'' of {{math|''q''}}, and it makes {{math|Σ}} more positive. When {{math|''q''}} is being destroyed (i.e., when   <math>\tfrac{\partial q}{\partial t}<0</math>), the region is called a ''sink'' of {{math|''q''}}, and it makes {{math|Σ}} more negative. The term {{math|Σ}} is sometimes written as <math>dq/dt|_\text{gen}</math> or the total change of {{math|''q''}} from its generation or destruction inside the control volume.

In a simple example, {{math|''V''}} could be a building, and {{math|''q''}} could be the number of living people in the building. The surface {{math|''S''}} would consist of the walls, doors, roof, and foundation of the building. Then the continuity equation states that the number of living people in the building (1) increases when living people enter the building (i.e., when there is an inward flux through the surface), (2) decreases when living people exit the building (i.e., when there is an outward flux through the surface), (3) increases when someone in the building gives birth to new life (i.e.,  when there is a positive time rate of change within the volume), and (4) decreases when someone in the building no longer lives (i.e.,  when there is a negative time rate of change within the volume). In conclusion, in this example there are four distinct ways that the net rate {{math|Σ}} may be altered.

===Differential form===
{{see also|Conservation law|conservation form}}
By the [[divergence theorem]], a general continuity equation can also be written in a "differential form":
{{Equation box 1
|indent=:
|equation=<math>\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} = \sigma</math>
|cellpadding
|border
|border colour = #0073CF
|background colour=#F5FFFA
}}
where
* {{math|∇⋅}} is [[divergence]],
* {{math|''ρ''}} is the density of the amount {{math|''q''}} (i.e. the quantity {{math|''q''}} per unit volume),
* {{math|'''j'''}} is the flux of {{math|''q''}} (i.e. '''j''' = ρ'''v''', where '''v''' is the vector field describing the movement of the quantity {{math|''q''}}),
* {{math|''t''}} is time,
* {{math|''σ''}} is the generation of {{math|''q''}} per unit volume per unit time. Terms that generate {{math|''q''}} (i.e., {{math|''σ'' > 0}}) or remove {{math|''q''}} (i.e., {{math|''σ'' < 0}}) are referred to as [[sources and sinks]] respectively.

This general equation may be used to derive any continuity equation, ranging from as simple as the volume continuity equation to as complicated as the [[Navier–Stokes equations]]. This equation also generalizes the [[advection equation]]. Other equations in physics, such as [[Gauss's law|Gauss's law of the electric field]] and [[Gauss's law for gravity]], have a similar mathematical form to the continuity equation, but are not usually referred to by the term "continuity equation", because {{math|'''j'''}} in those cases does not represent the flow of a real physical quantity.

In the case that {{math|''q''}} is a [[Conservation law (physics)|conserved quantity]] that cannot be created or destroyed (such as [[energy]]), {{math|1=''σ'' = 0}} and the equations become:
<math display="block">\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} = 0</math>

== Electromagnetism ==

{{Main|Charge conservation}}

In [[electromagnetic theory]], the continuity equation is an empirical law expressing (local) [[charge conservation]]. Mathematically it is an automatic consequence of [[Maxwell's equations]], although charge conservation is more fundamental than Maxwell's equations. It states that the [[divergence]] of the [[current density]] {{math|'''J'''}} (in [[amperes]] per square meter) is equal to the negative rate of change of the [[charge density]] {{math|''ρ''}} (in [[coulomb]]s per cubic meter),
<math display="block"> \nabla \cdot \mathbf{J} = - \frac{\partial \rho}{\partial t} </math>

{{math proof | title = Consistency with Maxwell's equations | proof =
One of [[Maxwell's equations]], [[Ampère's law|Ampère's law (with Maxwell's correction)]], states that
<math display="block"> \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}. </math>

Taking the divergence of both sides (the divergence and partial derivative in time commute) results in
<math display="block"> \nabla \cdot ( \nabla \times \mathbf{H} ) = \nabla \cdot \mathbf{J} + \frac{\partial (\nabla \cdot \mathbf{D})}{\partial t}, </math>
but the divergence of a curl is zero, so that
<math display="block"> \nabla \cdot \mathbf{J} + \frac{\partial (\nabla \cdot \mathbf{D})}{\partial t} = 0. </math>

But [[Gauss's law]] (another Maxwell equation), states that
<math display="block"> \nabla \cdot \mathbf{D} = \rho, </math>
which can be substituted in the previous equation to yield the continuity equation
<math display="block"> \nabla \cdot \mathbf{J} + \frac{\partial \rho}{\partial t} = 0.</math>
}}

Current is the movement of charge. The continuity equation says that if charge is moving out of a differential volume (i.e., divergence of current density is positive) then the amount of charge within that volume is going to decrease, so the rate of change of charge density is negative. Therefore, the continuity equation amounts to a conservation of charge.

If [[magnetic monopole]]s exist, there would be a continuity equation for monopole currents as well, see the monopole article for background and the duality between electric and magnetic currents.

== Fluid dynamics ==

{{see also|Mass flux|Mass flow rate|Vorticity equation}}

In [[fluid dynamics]], the continuity equation states that the rate at which mass enters a system is equal to the rate at which mass leaves the system plus the accumulation of mass within the system.<ref name=Pedlosky>{{Cite book | publisher = [[Springer Science+Business Media|Springer]] | isbn = 978-0-387-96387-7 | last = Pedlosky | first = Joseph | title = Geophysical fluid dynamics | year = 1987 | pages = [https://archive.org/details/geophysicalfluid00jose/page/10 10–13] | url = https://archive.org/details/geophysicalfluid00jose/page/10 }}</ref><ref>Clancy, L.J.(1975), ''Aerodynamics'', Section 3.3, Pitman Publishing Limited, London</ref>
The differential form of the continuity equation is:<ref name=Pedlosky/>
<math display="block"> \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0</math>
where
* {{math|''ρ''}} is fluid [[density]],
* {{math|''t''}} is time,
* {{math|'''u'''}} is the [[flow velocity]] [[vector field]].

The time derivative can be understood as the accumulation (or loss) of mass in the system, while the [[divergence]] term represents the difference in flow in versus flow out. In this context, this equation is also one of the [[Euler equations (fluid dynamics)]]. The [[Navier–Stokes equations]] form a vector continuity equation describing the conservation of [[linear momentum]].

If the fluid is [[Incompressible flow|incompressible]] (volumetric strain rate is zero), the mass continuity equation simplifies to a volume continuity equation:<ref name="Fielding">{{cite web |last1=Fielding |first1=Suzanne |title=The Basics of Fluid Dynamics |url=https://community.dur.ac.uk/suzanne.fielding/teaching/BLT/sec1.pdf |website=Durham University |access-date=22 December 2019}}</ref>
<math display="block">\nabla \cdot \mathbf{u} = 0,</math>
which means that the [[divergence]] of the velocity field is zero everywhere. Physically, this is equivalent to saying that the local volume dilation rate is zero, hence a flow of water through a converging pipe will adjust solely by increasing its velocity as water is largely incompressible.

== Computer vision ==

{{Main|Optical flow}}

In [[computer vision]], optical flow is the pattern of apparent motion of objects in a visual scene. Under the assumption that brightness of the moving object did not change between two image frames, one can derive the optical flow equation as:{{citation needed|date=April 2024}}
<math display="block">\frac{\partial I}{\partial x}V_x + \frac{\partial I}{\partial y}V_y + \frac{\partial I}{\partial t}
= \nabla I\cdot\mathbf{V} + \frac{\partial I}{\partial t}
= 0</math>
where
* {{math|''t''}} is time,
* {{math|''x'', ''y''}} coordinates in the image,
* {{math|''I''}} is the image intensity at image coordinate {{math|(''x'', ''y'')}} and time {{mvar|t}},
* {{math|'''V'''}} is the optical flow velocity vector <math>(V_x, V_y)</math> at image coordinate {{math|(''x'', ''y'')}} and time {{mvar|t}}

== Energy and heat ==

[[Conservation of energy]] says that energy cannot be created or destroyed. (See [[#General relativity|below]] for the nuances associated with general relativity.) Therefore, there is a continuity equation for energy flow:
<math display="block">\frac{ \partial u}{\partial t} + \nabla \cdot \mathbf{q} = 0</math>
where
* {{math|''u''}}, local [[energy density]] (energy per unit volume),
* {{math|'''q'''}}, [[energy flux]] (transfer of energy per unit cross-sectional area per unit time) as a vector,

An important practical example is [[Heat transfer|the flow of heat]]. When heat flows inside a solid, the continuity equation can be combined with [[Thermal conduction#Fourier's law|Fourier's law]] (heat flux is proportional to temperature gradient) to arrive at the [[heat equation]]. The equation of heat flow may also have source terms: Although ''energy'' cannot be created or destroyed, ''heat'' can be created from other types of energy, for example via [[friction]] or [[joule heating]].

== Probability distributions ==

If there is a quantity that moves continuously according to a stochastic (random) process, like the location of a single dissolved molecule with [[Brownian motion]], then there is a continuity equation for its [[probability distribution]]. The flux in this case is the probability per unit area per unit time that the particle passes through a surface. According to the continuity equation, the negative divergence of this flux equals the rate of change of the [[probability density]]. The continuity equation reflects the fact that the molecule is always somewhere—the integral of its probability distribution is always equal to 1—and that it moves by a continuous motion (no [[Teleportation|teleporting]]).

== Quantum mechanics ==
<!-- This section is linked from [[Conservation law (physics)|Conservation law]] -->
{{see also|Madelung equations}}

[[Quantum mechanics]] is another domain where there is a continuity equation related to ''conservation of probability''. The terms in the equation require the following definitions, and are slightly less obvious than the other examples above, so they are outlined here:

* The [[wavefunction]] {{math|Ψ}} for a single [[particle]] in [[position and momentum space|position space]] (rather than [[position and momentum space|momentum space]]), that is, a function of position {{math|'''r'''}} and time {{math|''t''}}, {{math|1=Ψ = Ψ('''r''', ''t'')}}.
* The [[probability density function]] is <math display="block">\rho(\mathbf{r}, t) = \Psi^{*}(\mathbf{r}, t)\Psi(\mathbf{r}, t) = |\Psi(\mathbf{r}, t)|^2. </math>
* The [[probability]] of finding the particle within {{mvar|V}} at {{mvar|t}} is denoted and defined by <math display="block">P = P_{\mathbf{r} \in V}(t) = \int_V \Psi^*\Psi dV = \int_V |\Psi|^2 dV.</math>
* The [[probability current]] (probability flux) is <math display="block">\mathbf{j}(\mathbf{r}, t) = \frac{\hbar}{2mi} \left[ \Psi^{*} \left( \nabla\Psi \right) - \Psi \left( \nabla\Psi^{*} \right) \right].</math>

With these definitions the continuity equation reads:
<math display="block">\nabla \cdot \mathbf{j} + \frac{\partial\rho}{\partial t} = 0 \mathrel{\rightleftharpoons} \nabla \cdot \mathbf{j} + \frac{\partial |\Psi|^2}{\partial t} = 0.</math>

Either form may be quoted. Intuitively, the above quantities indicate this represents the flow of probability. The ''chance'' of finding the particle at some position {{math|'''r'''}} and time {{mvar|t}} flows like a [[fluid]]; hence the term ''probability current'', a [[vector field]]. The particle itself does ''not'' flow [[Deterministic system|deterministically]] in this [[vector field]].

{{math proof|title=Consistency with Schrödinger equation|proof=
The time dependent [[Schrödinger equation]] and its [[complex conjugate]] ({{math|''i'' → −''i''}} throughout) are respectively:<ref>For this derivation see for example {{cite book |title=Quantum Mechanics Demystified |first=D. |last=McMahon |publisher=McGraw Hill |year=2006 |isbn=0-07-145546-9 }}</ref>
<math display="block">\begin{align}
         -\frac{\hbar^2}{2m}\nabla^2\Psi + U\Psi &=  i\hbar\frac{\partial\Psi}{\partial t}, \\
 -\frac{\hbar^2}{2m}\nabla^2\Psi^{*} + U\Psi^{*} &= -i\hbar\frac{\partial\Psi^{*}}{\partial t}, \\
\end{align}</math>
where {{math|''U''}} is the [[Potential|potential function]]. The [[partial derivative]] of {{math|''ρ''}} with respect to {{math|''t''}} is:
<math display="block">
\frac{\partial \rho}{\partial t}
= \frac{\partial | \Psi |^2}{\partial t }
= \frac{\partial}{\partial t} \left( \Psi^{*} \Psi \right)
= \Psi^{*} \frac{\partial\Psi}{\partial t} + \Psi\frac{\partial\Psi^{*}}{\partial t}.
</math>

Multiplying the Schrödinger equation by {{math|Ψ*}} then solving for {{math|Ψ* {{sfrac|∂Ψ|∂''t''}}}}, and similarly multiplying the complex conjugated Schrödinger equation by {{math|Ψ}} then solving for {{math|Ψ {{sfrac|∂Ψ*|∂''t''}}}};
<math display="block">\begin{align}
\Psi^*\frac{\partial\Psi}{\partial t} &=  \frac{1}{i\hbar} \left[ -\frac{\hbar^2\Psi^*}{2m}\nabla^2\Psi + U\Psi^*\Psi \right], \\
\Psi\frac{\partial\Psi^*}{\partial t} &= -\frac{1}{i\hbar} \left[ -\frac{\hbar^2\Psi}{2m}\nabla^2\Psi^* + U\Psi\Psi^* \right], \\
\end{align}</math>

substituting into the time derivative of {{math|''ρ''}}:
<math display="block">\begin{align}
 \frac{\partial \rho}{\partial t}
    &= \frac{1}{i\hbar} \left[ -\frac{\hbar^2\Psi^{*}}{2m}\nabla^2\Psi + U\Psi^{*}\Psi \right] - \frac{1}{i\hbar} \left[ -\frac{\hbar^2\Psi}{2m}\nabla^2\Psi^{*} + U\Psi\Psi^{*} \right] \\
    &= \frac{1}{i\hbar} \left[ -\frac{\hbar^2\Psi^{*}}{2m}\nabla^2 \Psi + U\Psi^{*}\Psi \right] + \frac{1}{i\hbar} \left[ +\frac{\hbar^2\Psi}{2m}\nabla^2\Psi^{*} - U\Psi^{*}\Psi \right] \\[2pt]
    &= -\frac{1}{i\hbar} \frac{\hbar^2\Psi^{*}}{2m}\nabla^2 \Psi + \frac{1}{i\hbar} \frac{\hbar^2\Psi}{2m}\nabla^2 \Psi^{*} \\[2pt]
    &= \frac{\hbar}{2im} \left[ \Psi\nabla^2\Psi^{*} - \Psi^{*}\nabla^2\Psi \right] \\
\end{align} </math>

The [[Laplace operator|Laplacian]] [[Operator (mathematics)|operators]] ({{math|∇<sup>2</sup>}}) in the above result suggest that the right hand side is the divergence of {{math|'''j'''}}, and the reversed order of terms imply this is the negative of {{math|'''j'''}}, altogether:
<math display="block">\begin{align}
\nabla \cdot \mathbf{j}
&=  \nabla \cdot \left[ \frac{\hbar}{2mi} \left( \Psi^{*} \left( \nabla \Psi \right) - \Psi \left( \nabla \Psi^{*} \right) \right) \right] \\
&=  \frac{\hbar}{2mi} \left[ \Psi^{*} \left( \nabla^2 \Psi \right) - \Psi \left( \nabla^2\Psi^{*} \right) \right] \\
&= -\frac{\hbar}{2mi} \left[ \Psi \left( \nabla^2\Psi^{*} \right) - \Psi^{*} \left( \nabla^2 \Psi \right) \right] \\
\end{align}</math>
so the continuity equation is:
<math display="block">\begin{align}
                  &\frac{\partial \rho}{\partial t} = -\nabla \cdot \mathbf{j} \\[3pt]
  {}\Rightarrow{} &\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} = 0 \\
\end{align}</math>

The integral form follows as for the general equation.
}}

==Semiconductor==
The total current flow in the semiconductor consists of [[drift current]] and [[diffusion current]] of both the electrons in the conduction band and holes in the valence band.

General form for electrons in one-dimension:
<math display="block">\frac{\partial n}{\partial t} = n \mu_n \frac{\partial E}{\partial x} + \mu_n E \frac{\partial n}{\partial x} + D_n \frac{\partial^2 n}{\partial x^2} + (G_n - R_n)</math>
where:
* ''n'' is the local concentration of electrons
* <math>\mu_n</math> is [[electron mobility]]
* ''E'' is the electric field across the [[depletion region]]
* ''D<sub>n</sub>'' is the [[diffusion coefficient]] for electrons
* ''G<sub>n</sub>'' is the rate of generation of electrons
* ''R<sub>n</sub>'' is the rate of recombination of electrons

Similarly, for holes:
<math display="block">\frac{\partial p}{\partial t} = -p \mu_p \frac{\partial E}{\partial x} - \mu_p E \frac{\partial p}{\partial x} + D_p \frac{\partial^2 p}{\partial x^2} + (G_p - R_p)</math>
where:
* ''p'' is the local concentration of holes
* <math>\mu_p</math> is hole mobility
* ''E'' is the electric field across the [[depletion region]]
* ''D<sub>p</sub>'' is the [[diffusion coefficient]] for holes
* ''G<sub>p</sub>'' is the rate of generation of holes
* ''R<sub>p</sub>'' is the rate of recombination of holes

===Derivation===

This section presents a derivation of the equation above for electrons. A similar derivation can be found for the equation for holes.

Consider the fact that the number of electrons is conserved across a volume of semiconductor material with cross-sectional area, ''A'', and length, ''dx'', along the ''x''-axis. More precisely, one can say:
<math display="block">\text{Rate of change of electron density} = (\text{Electron flux in} - \text{Electron flux out}) + \text{Net generation inside a volume}</math>

Mathematically, this equality can be written:
<math display="block">\begin{align}
 \frac{dn}{dt} A \, dx &= \left[J(x+dx)-J(x)\right]\frac{A}{e} + (G_n - R_n)A \, dx \\
 &= \left[J(x)+\frac{dJ}{dx}dx-J(x)\right]\frac{A}{e} + (G_n - R_n)A \, dx \\[1.2ex]
 \frac{dn}{dt}         &= \frac{1}{e}\frac{dJ}{dx} + (G_n - R_n)
\end{align}</math>Here ''J'' denotes current density(whose direction is against electron flow by convention) due to electron flow within the considered volume of the semiconductor. It is also called electron current density.

Total electron current density is the sum of drift current and diffusion current densities:
<math display="block">J_n = en\mu_nE + eD_n\frac{dn}{dx}</math>

Therefore, we have
<math display="block">\frac{dn}{dt} = \frac{1}{e}\frac{d}{dx}\left(en\mu_n E + eD_n\frac{dn}{dx}\right) + (G_n - R_n)</math>

Applying the product rule results in the final expression:
<math display="block">\frac{dn}{dt} = \mu_n E\frac{dn}{dx} + \mu_n n\frac{dE}{dx} + D_n\frac{d^2 n}{dx^2} + (G_n - R_n)</math>

===Solution===
The key to solving these equations in real devices is whenever possible to select regions in which most of the mechanisms are negligible so that the equations reduce to a much simpler form.

==Relativistic version==

===Special relativity===
{{see also|4-vector}}

The notation and tools of [[special relativity]], especially [[4-vector]]s and [[4-gradient]]s, offer a convenient way to write any continuity equation.

The density of a quantity {{math|''ρ''}} and its current {{math|'''j'''}} can be combined into a [[4-vector]] called a [[4-current]]:
<math display="block">J = \left(c \rho, j_x, j_y, j_z \right)</math>
where {{math|''c''}} is the [[speed of light]]. The 4-[[divergence]] of this current is:
<math display="block"> \partial_\mu J^\mu = c \frac{ \partial \rho}{\partial ct} + \nabla \cdot \mathbf{j}</math>
where {{math|∂<sub>''μ''</sub>}} is the [[4-gradient]] and {{math|''μ''}} is an [[index notation|index]] labeling the [[spacetime]] [[dimension]]. Then the continuity equation is:
<math display="block">\partial_\mu J^\mu = 0</math>
in the usual case where there are no sources or sinks, that is, for perfectly conserved quantities like energy or charge. This continuity equation is manifestly ("obviously") [[Lorentz invariant]].

Examples of continuity equations often written in this form include electric charge conservation
<math display="block">\partial_\mu J^\mu = 0</math>
where {{math|''J''}} is the electric [[4-current]]; and energy–momentum conservation
<math display="block">\partial_\nu T^{\mu\nu} = 0</math>
where {{math|''T''}} is the [[stress–energy tensor]].

===General relativity===
In [[general relativity]], where spacetime is curved, the continuity equation (in differential form) for energy, charge, or other conserved quantities involves the [[Covariant derivative|''covariant'' divergence]] instead of the ordinary divergence.

For example, the [[stress–energy tensor]] is a second-order [[tensor field]] containing energy–momentum densities, energy–momentum fluxes, and shear stresses, of a mass-energy distribution. The differential form of energy–momentum conservation in general relativity states that the ''covariant'' divergence of the stress-energy tensor is zero:
<math display="block">{T^\mu}_{\nu; \mu} = 0.</math>

This is an important constraint on the form the [[Einstein field equations]] take in [[general relativity]].<ref>{{cite book |title=Relativity DeMystified|author=D. McMahon|publisher=McGraw Hill (USA)|year=2006|isbn=0-07-145545-0}}</ref>

However, the ''ordinary'' [[Tensors in curvilinear coordinates#Second-order tensor field|divergence]] of the stress–energy tensor does ''not'' necessarily vanish:<ref>{{cite book |title=Gravitation |author=C.W. Misner |last2=K.S. Thorne |last3=J.A. Wheeler | publisher=W.H. Freeman & Co |year=1973 |isbn=0-7167-0344-0}}</ref>
<math display="block">\partial_{\mu} T^{\mu\nu} = - \Gamma^{\mu}_{\mu \lambda} T^{\lambda \nu} - \Gamma^{\nu}_{\mu \lambda} T^{\mu \lambda},</math>

The right-hand side strictly vanishes for a flat geometry only.

As a consequence, the ''integral'' form of the continuity equation is difficult to define and not necessarily valid for a region within which spacetime is significantly curved (e.g. around a black hole, or across the whole universe).<ref>{{cite web |url=http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html |title=Is Energy Conserved in General Relativity? |access-date=2014-04-25 |author1=Michael Weiss |author2=John Baez }}</ref>

==Particle physics==

[[Quark]]s and [[gluon]]s have ''[[color charge]]'', which is always conserved like electric charge, and there is a continuity equation for such color charge currents (explicit expressions for currents are given at [[Gluon field strength tensor#Equation of motion|gluon field strength tensor]]).

There are many other quantities in particle physics which are often or always conserved: [[baryon number]] (proportional to the number of quarks minus the number of antiquarks), [[lepton number|electron number, mu number, tau number]], [[isospin]], and others.<ref>{{cite book |title=Gravitation |author=C.W. Misner |last2=K.S. Thorne |last3=J.A. Wheeler | publisher=W.H. Freeman & Co |year=1973 |isbn=0-7167-0344-0|pages=558–559}}</ref> Each of these has a corresponding continuity equation, possibly including source / sink terms.

==Noether's theorem==

{{for|more detailed explanations and derivations|Noether's theorem}}

One reason that conservation equations frequently occur in physics is [[Noether's theorem]]. This states that whenever the laws of physics have a [[continuous symmetry]], there is a continuity equation for some conserved physical quantity. The three most famous examples are:

* The laws of physics are invariant with respect to [[Time translation|time-translation]]—for example, the laws of physics today are the same as they were yesterday. This symmetry leads to the continuity equation for [[conservation of energy]].
* The laws of physics are invariant with respect to space-translation—for example, a rocket in outer space is not subject to different forces or potentials if it is displaced in any given direction (eg. x, y, z), leading to the conservation of the three components of momentum [[conservation of momentum]].
* The laws of physics are invariant with respect to orientation—for example, floating in outer space, there is no measurement you can do to say "which way is up"; the laws of physics are the same regardless of how you are oriented. This symmetry leads to the continuity equation for [[conservation of angular momentum]].

== See also ==

* [[Conservation law (physics)|Conservation law]]
* [[Conservation form]]
* [[Dissipative system]]

== References ==
{{reflist}}

==Further reading==

*{{cite book |title=Hydrodynamics |first=H. |last=Lamb |publisher=Cambridge University Press |year=2006 |orig-year=1932 |edition=6th |isbn=978-0-521-45868-9 }}
*{{cite book |title=Introduction to Electrodynamics |edition=3rd |first=D. J. |last=Griffiths |publisher=Pearson Education Inc |year=1999 |isbn=81-7758-293-3 }}
*{{cite book |title=Electromagnetism |edition=2nd |first1=I. S. |last1=Grant |first2=W. R. |last2=Phillips |series=Manchester Physics Series |year=2008 |isbn=978-0-471-92712-9 }}
*{{cite book |title=[[Gravitation (book)|Gravitation]] |first1=J. A. |last1=Wheeler |first2=C. |last2=Misner |first3=K. S. |last3=Thorne |publisher=W. H. Freeman & Co |year=1973 |isbn=0-7167-0344-0 }}

[[Category:Equations of fluid dynamics]]
[[Category:Conservation equations]]
[[Category:Partial differential equations]]