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A continuity equation or transport equation is an
equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in F ...
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 extensive quantity. Since
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
,
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
,
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
,
electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons res ...
and other natural 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 In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, c ...
s. For example, a weak version of the law of
conservation of energy In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means tha ...
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 " 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 Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons res ...
states that the amount of electric charge in any volume of space can only change by the amount of
electric current An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The movi ...
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 integral), which applies to any finite region, or in a "differential form" (in terms of the
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...
operator) which applies at a point. Continuity equations underlie more specific transport equations such as the
convection–diffusion equation The convection–diffusion equation is a combination of the 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 ...
, Boltzmann transport equation, and Navier–Stokes equations. Flows governed by continuity equations can be visualized using a Sankey diagram.


General equation


Definition of flux

A continuity equation is useful when a ''
flux Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ...
'' can be defined. To define flux, first there must be a quantity which can flow or move, such as
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
,
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
,
electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons res ...
,
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
, number of molecules, etc. Let be the volume
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
of this quantity, that is, the amount of per unit volume. The way that this quantity is flowing is described by its flux. The flux of is a vector field, which we denote as j. Here are some examples and properties of flux: (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 Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ...
for details.)


Integral form

The integral form of the continuity equation states that: * The amount of in a region increases when additional flows inward through the surface of the region, and decreases when it flows outward; * The amount of in a region increases when new is created inside the region, and decreases when is destroyed; * Apart from these two processes, there is ''no other way'' for the amount of in a region to change. Mathematically, the integral form of the continuity equation expressing the rate of increase of within a volume is: where * is any imaginary closed surface, that encloses a volume , * denotes a
surface integral In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one ...
over that closed surface, * is the total amount of the quantity in the volume , * is the flux of , * is time, * is the net rate that is being generated inside the volume per unit time. When is being generated, it is called a ''source'' of , and it makes more positive. When is being destroyed, it is called a ''sink'' of , and it makes more negative. This term is sometimes written as dq/dt, _\text or the total change of q from its generation or destruction inside the control volume. In a simple example, could be a building, and could be the number of people in the building. The surface would consist of the walls, doors, roof, and foundation of the building. Then the continuity equation states that the number of people in the building increases when people enter the building (an inward flux through the surface), decreases when people exit the building (an outward flux through the surface), increases when someone in the building gives birth (a source, ), and decreases when someone in the building dies (a sink, ).


Differential form

By the
divergence theorem In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the '' flux'' of a vector field through a closed surface to the ''divergence'' of the field in the ...
, a general continuity equation can also be written in a "differential form": where * is
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...
, * is the amount of the quantity per unit volume, * is the flux of , * is time, * is the generation of per unit volume per unit time. Terms that generate (i.e., ) or remove (i.e., ) are referred to as a "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 of the electric field and
Gauss's law for gravity In physics, Gauss's law for gravity, also known as Gauss's flux theorem for gravity, is a law of physics that is equivalent to Newton's law of universal gravitation. It is named after Carl Friedrich Gauss. It states that the flux (surface int ...
, have a similar mathematical form to the continuity equation, but are not usually referred to by the term "continuity equation", because in those cases does not represent the flow of a real physical quantity. In the case that is a conserved quantity that cannot be created or destroyed (such as
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
), and the equations become: \frac + \nabla \cdot \mathbf = 0


Electromagnetism

In electromagnetic theory, the continuity equation is an empirical law expressing (local)
charge conservation In physics, charge conservation is the principle that the total electric charge in an isolated system never changes. The net quantity of electric charge, the amount of positive charge minus the amount of negative charge in the universe, is al ...
. Mathematically it is an automatic consequence of
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits ...
, although charge conservation is more fundamental than Maxwell's equations. It states that the
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...
of the
current density In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional a ...
(in
amperes The ampere (, ; symbol: A), often shortened to amp,SI supports only the use of symbols and deprecates the use of abbreviations for units. is the unit of electric current in the International System of Units (SI). One ampere is equal to elect ...
per square metre) is equal to the negative rate of change of the
charge density In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system i ...
(in
coulomb The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI). In the present version of the SI it is equal to the electric charge delivered by a 1 ampere constant current in 1 second and to elementary char ...
s per cubic metre), \nabla \cdot \mathbf = - \frac 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 monopoles 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

In
fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) a ...
, 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. The differential form of the continuity equation is: \frac + \nabla \cdot (\rho \mathbf) = 0 where * is fluid
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
, * is time, * 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 In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...
term represents the difference in flow in versus flow out. In this context, this equation is also one of the
Euler equations (fluid dynamics) In fluid dynamics, the Euler equations are a set of quasilinear partial differential equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. In particular, they correspond to the Navier–Stokes equations wit ...
. The Navier–Stokes equations form a vector continuity equation describing the conservation of linear momentum. If the fluid is incompressible (volumetric strain rate is zero), the mass continuity equation simplifies to a volume continuity equation: \nabla \cdot \mathbf = 0, which means that the
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...
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

In
computer vision Computer vision is an interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate tasks that the human ...
, 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: \fracV_x + \fracV_y + \frac = \nabla I\cdot\mathbf + \frac = 0 where * is time, * coordinates in the image, * is the image intensity at image coordinate and time , * is the optical flow velocity vector (V_x, V_y) at image coordinate and time


Energy and heat

Conservation of energy In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means tha ...
says that energy cannot be created or destroyed. (See
below Below may refer to: *Earth * Ground (disambiguation) *Soil *Floor * Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Ernst von Below (1863–1955), German World War I general *Fred Below ...
for the nuances associated with general relativity.) Therefore, there is a continuity equation for energy flow: \frac + \nabla \cdot \mathbf = 0 where * , local
energy density In physics, energy density is the amount of energy stored in a given system or region of space per unit volume. It is sometimes confused with energy per unit mass which is properly called specific energy or . Often only the ''useful'' or extrac ...
(energy per unit volume), * ,
energy flux Energy flux is the rate of transfer of energy through a surface. The quantity is defined in two different ways, depending on the context: # Total rate of energy transfer (not per unit area); SI units: W = J⋅s−1. # Specific rate of energy transf ...
(transfer of energy per unit cross-sectional area per unit time) as a vector, An important practical example is the flow of heat. When heat flows inside a solid, the continuity equation can be combined with
Fourier's law Conduction is the process by which heat is transferred from the hotter end to the colder end of an object. The ability of the object to conduct heat is known as its '' thermal conductivity'', and is denoted . Heat spontaneously flows along a t ...
(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 Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of ...
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 In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
. 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 teleporting).


Quantum mechanics

Quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
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 A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
for a single
particle In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from ...
in position space (rather than momentum space), that is, a function of position and time , . * The
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
is \rho(\mathbf, t) = \Psi^(\mathbf, t)\Psi(\mathbf, t) = , \Psi(\mathbf, t), ^2. * The
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
of finding the particle within at is denoted and defined by P = P_(t) = \int_V \Psi^*\Psi dV = \int_V , \Psi, ^2 dV. * The probability current (aka probability flux) is \mathbf(\mathbf, t) = \frac \left \Psi^ \left( \nabla\Psi \right) - \Psi \left( \nabla\Psi^ \right) \right With these definitions the continuity equation reads: \nabla \cdot \mathbf + \frac = 0 \mathrel \nabla \cdot \mathbf + \frac = 0. 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 and time flows like a fluid; hence the term ''probability current'', a vector field. The particle itself does ''not'' flow deterministically in this vector field.


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: \frac = n \mu_n \frac + \mu_n E \frac + D_n \frac + (G_n - R_n) where: * ''n'' is the local concentration of electrons * \mu_n is electron mobility * ''E'' is the electric field across the depletion region * ''Dn'' is the
diffusion coefficient Diffusivity, mass diffusivity or diffusion coefficient is a proportionality constant between the molar flux due to molecular diffusion and the gradient in the concentration of the species (or the driving force for diffusion). Diffusivity is enc ...
for electrons * ''Gn'' is the rate of generation of electrons * ''Rn'' is the rate of recombination of electrons Similarly, for holes: \frac = -p \mu_p \frac - \mu_p E \frac + D_p \frac + (G_p - R_p) where: * ''p'' is the local concentration of holes * \mu_p is hole mobility * ''E'' is the electric field across the depletion region * ''Dp'' is the
diffusion coefficient Diffusivity, mass diffusivity or diffusion coefficient is a proportionality constant between the molar flux due to molecular diffusion and the gradient in the concentration of the species (or the driving force for diffusion). Diffusivity is enc ...
for holes * ''Gp'' is the rate of generation of holes * ''Rp'' 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: \text = (\text - \text) + \text Mathematically, this equality can be written: \begin \frac A \, dx &= (x+dx)-J(x)frac + (G_n - R_n)A \, dx \\ pt \frac A \, dx &= (x)+\fracdx-J(x)frac + (G_n - R_n)A \, dx \\ pt \frac &= \frac\frac + (G_n - R_n) \endHere 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: J_n = en\mu_nE + eD_n\frac Therefore, we have \frac = \frac\frac\left(en\mu_n E + eD_n\frac\right) + (G_n - R_n) Applying the product rule results in the final expression: \frac = \mu_n E\frac + \mu_n n\frac + D_n\frac + (G_n - R_n)


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

The notation and tools of
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
, especially 4-vectors and 4-gradients, offer a convenient way to write any continuity equation. The density of a quantity and its current can be combined into a 4-vector called a
4-current In special and general relativity, the four-current (technically the four-current density) is the four-dimensional analogue of the electric current density. Also known as vector current, it is used in the geometric context of ''four-dimensional spa ...
: J = \left(c \rho, j_x, j_y, j_z \right) where is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
. The 4-
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...
of this current is: \partial_\mu J^\mu = c \frac + \nabla \cdot \mathbf where is the 4-gradient and is an index labeling the
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
. Then the continuity equation is: \partial_\mu J^\mu = 0 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 \partial_\mu J^\mu = 0 where is the electric
4-current In special and general relativity, the four-current (technically the four-current density) is the four-dimensional analogue of the electric current density. Also known as vector current, it is used in the geometric context of ''four-dimensional spa ...
; and energy–momentum conservation \partial_\nu T^ = 0 where is the
stress–energy tensor The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the str ...
.


General relativity

In
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, where spacetime is curved, the continuity equation (in differential form) for energy, charge, or other conserved quantities involves the ''covariant'' divergence instead of the ordinary divergence. For example, the
stress–energy tensor The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the str ...
is a second-order
tensor field In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
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: _ = 0. This is an important constraint on the form the Einstein field equations take in
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. However, the ''ordinary''
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...
of the stress–energy tensor does ''not'' necessarily vanish: \partial_ T^ = - \Gamma^_ T^ - \Gamma^_ T^, 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).


Particle physics

Quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly ...
s and
gluon A gluon ( ) is an elementary particle that acts as the exchange particle (or gauge boson) for the strong force between quarks. It is analogous to the exchange of photons in the electromagnetic force between two charged particles. Gluons bind ...
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 In theoretical particle physics, the gluon field strength tensor is a second order tensor field characterizing the gluon interaction between quarks. The strong interaction is one of the fundamental interactions of nature, and the quantum fiel ...
). 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), electron number, mu number, tau number, isospin, and others. Each of these has a corresponding continuity equation, possibly including source / sink terms.


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 In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some symmetries as motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invariant under a kind of flip from one state to ano ...
, 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—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 In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means tha ...
. * The laws of physics are invariant with respect to space-translation—for example, the laws of physics in Brazil are the same as the laws of physics in Argentina. This symmetry leads to the continuity equation for 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 In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed sy ...
.


See also

* One-Way Wave Equation *
Conservation law In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, c ...
* Conservation form * Dissipative system


References


Further reading

*''Hydrodynamics, H. Lamb'', Cambridge University Press, (2006 digitalization of 1932 6th edition) *''Introduction to Electrodynamics (3rd Edition), D.J. Griffiths'', Pearson Education Inc, 1999, *''Electromagnetism (2nd edition), I.S. Grant, W.R. Phillips'', Manchester Physics Series, 2008 *''Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne'', W.H. Freeman & Co, 1973, {{ISBN, 0-7167-0344-0 Equations of fluid dynamics Conservation equations Partial differential equations