Continuity equation


A continuity equation in physics 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 extensive quantity. Since mass, energy, momentum, electric charge 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 laws. 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 "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", which applies to any finite region, or in a "differential form" 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

A continuity equation is useful when a flux can be defined. To define flux, first there must be a quantity which can flow or move, such as mass, energy, electric charge, momentum, number of molecules, etc. Let be the volume density 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 integral form of the continuity equation states that:
Mathematically, the integral form of the continuity equation expressing the rate of increase of within a volume is:
where
that fully encloses a volume, like any of the surfaces on the left. can not be a surface with boundaries, like those on the right.
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, decreases when people exit the building, increases when someone in the building gives birth, and decreases when someone in the building dies.

Differential form

By the divergence theorem, a general continuity equation can also be written in a "differential form":
where
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, 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, and the equations become:

Electromagnetism

In electromagnetic theory, the continuity equation is an empirical law expressing 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 is equal to the negative rate of change of the charge density ,
One of Maxwell's equations, Ampère's law, states that
Taking the divergence of both sides results in
but the divergence of a curl is zero, so that
But Gauss's law, states that
which can be substituted in the previous equation to yield the continuity equation
Current is the movement of charge. The continuity equation says that if charge is moving out of a differential volume 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, 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
The time derivative can be understood as the accumulation 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. The Navier–Stokes equations form a vector continuity equation describing the conservation of linear momentum.
If the fluid is incompressible, the mass continuity equation simplifies to a volume continuity equation:
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.

Energy and heat

says that energy cannot be created or destroyed. Therefore, there is a continuity equation for energy flow:
where
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 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 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.

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:
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; hence the term probability current, a vector field. The particle itself does not flow deterministically in this vector field.
where is the potential function. The partial derivative of with respect to is:
Multiplying the Schrödinger equation by then solving for, and similarly multiplying the complex conjugated Schrödinger equation by then solving for ;
substituting into the time derivative of :
The Laplacian operators in the above result suggest that the right hand side is the divergence of, and the reversed order of terms imply this is the negative of, altogether:
so the continuity equation is:
The integral form follows as for the general equation.

Relativistic version

Special relativity

The notation and tools of special relativity, 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 4-divergence of this current is:
where is the 4-gradient and is an index labelling the spacetime dimension. 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 Lorentz invariant.
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.

General relativity

In general relativity, where spacetime is curved, the continuity equation for energy, charge, or other conserved quantities involves the 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:
This is an important constraint on the form the Einstein field equations take in general relativity.
However, the ordinary divergence 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.

Particle physics

s and gluons have color charge, which is always conserved like electric charge, and there is a continuity equation for such color charge currents.
There are many other quantities in particle physics which are often or always conserved: baryon number, 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, there is a continuity equation for some conserved physical quantity. The three most famous examples are:
See Noether's theorem for proofs and details.