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 ...
that describes the transport of some quantity. It is particularly simple and powerful when applied to a
conserved quantity
In mathematics, a conserved quantity of a dynamical system is a function of the dependent variables, the value of which remains constant along each trajectory of the system.
Not all systems have conserved quantities, and conserved quantities are ...
, but it can be generalized to apply to any
extensive quantity
Physical properties of materials and systems can often be categorized as being either intensive or extensive, according to how the property changes when the size (or extent) of the system changes. According to IUPAC, an intensive quantity is on ...
. 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 elementar ...
,
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 heat a ...
,
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 an ...
,
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 respe ...
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 th ...
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 respe ...
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 moving pa ...
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 the ...
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 equation, diffusion and convection (advection equation, advection) equations, and describes physical phenomena where particles, energy, or other physical quantities are transferr ...
,
Boltzmann transport equation
The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of Thermodynamic equilibrium, equilibrium, devised by Ludwig Boltzmann in 1872.Encyclopaedia of Physics ( ...
, and
Navier–Stokes equations
In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
.
Flows governed by continuity equations can be visualized using a
Sankey diagram
Sankey diagrams are a type of flow diagram in which the width of the arrows is proportional to the flow rate.
Sankey diagrams can also visualize the energy accounts, material flow accounts on a regional or national level, and cost breakdowns. ...
.
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 ph ...
'' 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 elementar ...
,
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 heat a ...
,
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 respe ...
,
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 an ...
, 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. Mathematical ...
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:
- The dimension of flux is "amount of 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 cm2, then the average mass flux inside the pipe is , 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
In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
which describes the relevant flow—in other words, if all of the quantity at a point is moving with velocity —then the flux is by definition equal to the density times the velocity field:
For example, if in the mass continuity equation for flowing water, is the water's velocity at each point, and is the water's density at each point, then would be the mass flux.
- In a well-known example, the flux 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 respe ...
is the electric 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 are ...
.
- If there is an imaginary surface , then the surface integral of flux over is equal to the amount of that is passing through the surface per unit time:
in which 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
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 ph ...
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:
![SurfacesWithAndWithoutBoundary](https://upload.wikimedia.org/wikipedia/commons/1/1e/SurfacesWithAndWithoutBoundary.svg)
where
* is any imaginary
closed surface
In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as g ...
, that encloses a volume ,
* denotes a
surface integral 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
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, 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 the ...
,
* 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
In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
. This equation also generalizes the
advection equation
In the field of physics, engineering, and earth sciences, advection is the transport of a substance or quantity by bulk motion of a fluid. The properties of that substance are carried with it. Generally the majority of the advected substance is al ...
. 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 inte ...
, 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
In mathematics, a conserved quantity of a dynamical system is a function of the dependent variables, the value of which remains constant along each trajectory of the system.
Not all systems have conserved quantities, and conserved quantities are ...
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 heat a ...
), and the equations become:
Electromagnetism
In
electromagnetic theory
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
, 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 alwa ...
. 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 the ...
of the
current density (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 in ...
(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),
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
In particle physics, a magnetic monopole is a hypothetical elementary particle that is an isolated magnet with only one magnetic pole (a north pole without a south pole or vice versa). A magnetic monopole would have a net north or south "magneti ...
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
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) an ...
, 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:
[
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. Mathematical ...
,
* is time,
* is the flow velocity
In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
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 the ...
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
In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
form a vector continuity equation describing the conservation of linear 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 a ...
.
If the fluid is incompressible (volumetric strain rate is zero), the mass continuity equation simplifies to a volume continuity equation:
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 the ...
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 hum ...
, 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:
where
* is time,
* coordinates in the image,
* is the image intensity at image coordinate and time ,
* is the optical flow velocity vector 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 th ...
says that energy cannot be created or destroyed. (See below for the nuances associated with general relativity.) Therefore, there is a continuity equation for energy flow:
where
* , local energy density (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 transfe ...
(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 te ...
(heat flux is proportional to temperature gradient) to arrive at the heat equation
In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for t ...
. 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 t ...
or joule heating
Joule heating, also known as resistive, resistance, or Ohmic heating, is the process by which the passage of an electric current through a conductor (material), conductor produces heat.
Joule's first law (also just Joule's law), also known in c ...
.
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
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
, 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
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) can ...
. 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
In physics and geometry, there are two closely related vector spaces, usually three-dimensional but in general of any finite dimension.
Position space (also real space or coordinate space) is the set of all ''position vectors'' r in space, and h ...
(rather than momentum space
In physics and geometry, there are two closely related vector spaces, usually three-dimensional but in general of any finite dimension.
Position space (also real space or coordinate space) is the set of all ''position vectors'' r in space, and h ...
), 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) can ...
is
* The probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), 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 ...
of finding the particle within at is denoted and defined by
* The probability current
In quantum mechanics, the probability current (sometimes called probability flux) is a mathematical quantity describing the flow of probability. Specifically, if one thinks of probability as a heterogeneous fluid, then the probability current is th ...
(aka probability flux) is
With these definitions the continuity equation reads:
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
In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
; 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
In condensed matter physics and electrochemistry, drift current is the electric current, or movement of charge carriers, which is due to the applied electric field, often stated as the electromotive force over a given distance. When an electric fi ...
and diffusion current Diffusion current Density is a current in a semiconductor caused by the diffusion of charge carriers (electrons and/or electron holes). This is the current which is due to the transport of charges occurring because of non-uniform concentration of ch ...
of both the electrons in the conduction band and holes in the valence band.
General form for electrons in one-dimension:
where:
* ''n'' is the local concentration of electrons
* is electron mobility
* ''E'' is the electric field across the depletion region
In semiconductor physics, the depletion region, also called depletion layer, depletion zone, junction region, space charge region or space charge layer, is an insulating region within a conductive, doped semiconductor material where the mobile ...
* ''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 enco ...
for electrons
* ''Gn'' is the rate of generation of electrons
* ''Rn'' is the rate of recombination of electrons
Similarly, for holes:
where:
* ''p'' is the local concentration of holes
* is hole mobility
* ''E'' is the electric field across the depletion region
In semiconductor physics, the depletion region, also called depletion layer, depletion zone, junction region, space charge region or space charge layer, is an insulating region within a conductive, doped semiconductor material where the mobile ...
* ''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 enco ...
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:
Mathematically, this equality can be written:
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:
Therefore, we have
Applying the product rule results in the final expression:
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 laws o ...
, 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:
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 ...
. 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 the ...
of this current is:
where is the 4-gradient and is an index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
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 differen ...
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
. Then the continuity equation is:
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
In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of v ...
.
Examples of continuity equations often written in this form include electric charge conservation
where is the electric 4-current; and energy–momentum conservation
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 stress ...
.
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 stress ...
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:
This is an important constraint on the form the Einstein field equations
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
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 the ...
of the stress–energy tensor does ''not'' necessarily vanish:
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 o ...
s and gluons have ''color charge
Color charge is a property of quarks and gluons that is related to the particles' strong interactions in the theory of quantum chromodynamics (QCD).
The "color charge" of quarks and gluons is completely unrelated to the everyday meanings of colo ...
'', 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 fie ...
).
There are many other quantities in particle physics which are often or always conserved: baryon number
In particle physics, the baryon number is a strictly conserved additive quantum number of a system. It is defined as
::B = \frac\left(n_\text - n_\bar\right),
where ''n''q is the number of quarks, and ''n'' is the number of antiquarks. Bary ...
(proportional to the number of quarks minus the number of antiquarks), electron number, mu number, tau number, isospin
In nuclear physics and particle physics, isospin (''I'') is a quantum number related to the up- and down quark content of the particle. More specifically, isospin symmetry is a subset of the flavour symmetry seen more broadly in the interactions ...
, 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
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in ...
. 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 th ...
.
* 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
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 an ...
.
* 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 syste ...
.
See also
* One-Way Wave Equation
A one-way wave equation is a first-order partial differential equation describing one wave traveling in a direction defined by the vector wave velocity. It contrasts with the second-order two-way wave equation describing a standing wavefield resu ...
* 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 Conservation form or ''Eulerian form'' refers to an arrangement of an equation or system of equations, usually representing a hyperbolic system, that emphasizes that a property represented is conserved, i.e. a type of continuity equation. The term i ...
* Dissipative system
A dissipative system is a thermodynamically open system which is operating out of, and often far from, thermodynamic equilibrium in an environment with which it exchanges energy and matter. A tornado may be thought of as a dissipative system. Dis ...
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