Continuity equation

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.

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.

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 operator) which applies at a point.

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

To define flux, first there must be a quantity q which can flow or move, such as mass, energy, electric charge, momentum, number of molecules, etc.

The surface S would consist of the walls, doors, roof, and foundation of the building.

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

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 j in those cases does not represent the flow of a real physical quantity.

In the case that q is a conserved quantity that cannot be created or destroyed (such as energy), σ = 0 and the equations become:

In electromagnetic theory, the continuity equation is an empirical law expressing (local) charge conservation.

It states that the divergence of the current density J (in amperes per square meter) is equal to the negative rate of change of the charge density ρ (in coulombs per cubic meter),

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.

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]

According to the continuity equation, the negative divergence of this flux equals the rate of change of the probability density.

Quantum mechanics is another domain where there is a continuity equation related to conservation of probability.

The chance of finding the particle at some position r and time t flows like a fluid; hence the term probability current, a vector field.

The time dependent Schrödinger equation and its complex conjugate (i → −i throughout) are respectively:[4]

The Laplacian operators (∇2) in the above result suggest that the right hand side is the divergence of j, and the reversed order of terms imply this is the negative of j, altogether:

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.

The notation and tools of special relativity, especially 4-vectors and 4-gradients, offer a convenient way to write any continuity equation.

in the usual case where there are no sources or sinks, that is, for perfectly conserved quantities like energy or charge.

Examples of continuity equations often written in this form include electric charge conservation

In general relativity, 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.

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.

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).

[8] Each of these has a corresponding continuity equation, possibly including source / sink terms.

One reason that conservation equations frequently occur in physics is Noether's theorem.