In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.

• In the study of transport phenomena (heat transfer, mass transfer and fluid dynamics), flux is defined as flow per unit area, where flow is the movement of some quantity per unit time.^[1] Flux, in this definition, is a vector.

• In the fields of electromagnetism and mathematics, flux is usually the integral of a vector quantity, flux density, over a finite surface. It is an integral operator that acts on a vector field similarly to the gradient, divergence and curl operators found in vector analysis. The result of this integration is a scalar quantity called flux.^[2] The magnetic flux is thus the integral of the magnetic vector field B over a surface, and the electric flux is defined similarly. Using this definition, the flux of the Poynting vector over a specified surface is the rate at which electromagnetic energy flows through that surface. Confusingly, the Poynting vector is sometimes called the power flux, which is an example of the first usage of flux, above.^[3] It has units of watts per square metre (W/m²).

One could argue, based on the work of James Clerk Maxwell,^[4] that the transport definition precedes the more recent way the term is used in electromagnetism. The specific quote from Maxwell is "In the case of fluxes, we have to take the integral, over a surface, of the flux through every element of the surface. The result of this operation is called the surface integral of the flux. It represents the quantity which passes through the surface."

In addition to these few common mathematical definitions, there are many looser, but equally valid, usages to describe observations from other fields such as biology, the arts, history, and humanities.

Transport phenomena [link]

Origin of the term [link]

Look up flux in Wiktionary, the free dictionary.

The word flux comes from Latin: fluxus means "flow", and fluere is "to flow".^[5] As fluxion, this term was introduced into differential calculus by Isaac Newton.

Flux definition and theorems [link]

Flux is surface bombardment rate. There are many fluxes used in the study of transport phenomena. Each type of flux has its own distinct unit of measurement along with distinct physical constants. Seven of the most common forms of flux from the transport literature are defined as:

1. Momentum flux, the rate of transfer of momentum across a unit area (N·s·m⁻²·s⁻¹). (Newton's law of viscosity,)
2. Heat flux, the rate of heat flow across a unit area (J·m⁻²·s⁻¹). (Fourier's law of conduction)^[6] (This definition of heat flux fits Maxwell's original definition.)^[4]
3. Diffusion flux, the rate of movement of molecules across a unit area (mol·m⁻²·s⁻¹). (Fick's law of diffusion)
4. Volumetric flux, the rate of volume flow across a unit area (m³·m⁻²·s⁻¹). (Darcy's law of groundwater flow)
5. Mass flux, the rate of mass flow across a unit area (kg·m⁻²·s⁻¹). (Either an alternate form of Fick's law that includes the molecular mass, or an alternate form of Darcy's law that includes the density.)
6. Radiative flux, the amount of energy moving in the form of photons at a certain distance from the source per steradian per second (J·m⁻²·s⁻¹). Used in astronomy to determine the magnitude and spectral class of a star. Also acts as a generalization of heat flux, which is equal to the radiative flux when restricted to the infrared spectrum.
7. Energy flux, the rate of transfer of energy through a unit area (J·m⁻²·s⁻¹). The radiative flux and heat flux are specific cases of energy flux.
8. Particle flux, the rate of transfer of particles through a unit area ([number of particles] m⁻²·s⁻¹)

These fluxes are vectors at each point in space, and have a definite magnitude and direction. Also, one can take the divergence of any of these fluxes to determine the accumulation rate of the quantity in a control volume around a given point in space. For incompressible flow, the divergence of the volume flux is zero.

Chemical diffusion [link]

Chemical molar flux of a component A in an isothermal, isobaric system is defined in above-mentioned Fick's first law as:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{J}_A = -D_{AB} \nabla c_A

where:

• Failed to parse (Missing texvc executable; please see math/README to configure.): D_{AB}

is the diffusion coefficient (m²/s) of component A diffusing through component B,

• Failed to parse (Missing texvc executable; please see math/README to configure.): c_A

is the concentration (mol/m³) of species A.^[7]

This flux has units of mol·m⁻²·s⁻¹, and fits Maxwell's original definition of flux.^[4]

Note: Failed to parse (Missing texvc executable; please see math/README to configure.): \nabla

("nabla") denotes the del operator.

For dilute gases, kinetic molecular theory relates the diffusion coefficient D to the particle density n = N/V, the molecular mass M, the collision cross section Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma , and the absolute temperature T by

Failed to parse (Missing texvc executable; please see math/README to configure.): D = \frac{2}{3 n\sigma}\sqrt{\frac{kT}{\pi M}}

where the second factor is the mean free path and the square root (with Boltzmann's constant k) is the mean velocity of the particles.

In turbulent flows, the transport by eddy motion can be expressed as a grossly increased diffusion coefficient.

Quantum mechanics [link]

In quantum mechanics, particles of mass m in the state Failed to parse (Missing texvc executable; please see math/README to configure.): \psi(r,t)

have a probability density defined as

Failed to parse (Missing texvc executable; please see math/README to configure.): \rho = \psi^* \psi = |\psi|^2. \,

So the probability of finding a particle in a unit of volume, say Failed to parse (Missing texvc executable; please see math/README to configure.): d^3x , is

Failed to parse (Missing texvc executable; please see math/README to configure.): |\psi|^2 d^3x. \,

Then the number of particles passing through a perpendicular unit of area per unit time is

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{J} = -i \frac{\hbar}{2m} \left(\psi^* \nabla \psi - \psi \nabla \psi^* \right). \,

This is sometimes referred to as the flux density.^[8]

Electromagnetism [link]

Flux definition and theorems [link]

An example of the first definition of flux is the magnitude of a river's current, that is, the amount of water that flows through a cross-section of the river each second. The amount of sunlight that lands on a patch of ground each second is also a kind of flux.

To better understand the concept of flux in Electromagnetism, imagine a butterfly net. The amount of air moving through the net at any given instant in time is the flux. If the wind speed is high, then the flux through the net is large. If the net is made bigger, then the flux would be larger even though the wind speed is the same. For the most air to move through the net, the opening of the net must be facing the direction the wind is blowing. If the net opening is parallel to the wind, then no wind will be moving through the net. Perhaps the best way to think of flux abstractly is "how much stuff goes through your thing", where the stuff is a field and the thing is the virtual surface.

The flux visualized. The rings show the surface boundaries. The red arrows stand for the flow of charges, fluid particles, subatomic particles, photons, etc. The number of arrows that pass through each ring is the flux.

As a mathematical concept, flux is represented by the surface integral of a vector field,

Failed to parse (Missing texvc executable; please see math/README to configure.): \Phi_f = \int_S \mathbf{E} \cdot \mathbf{dA}

where:

• E is a vector field of Electric Force,
• dA is the vector area of the surface S, directed as the surface normal,
• Failed to parse (Missing texvc executable; please see math/README to configure.): \Phi_f

  is the resulting flux.

The surface has to be orientable, i.e. two sides can be distinguished: the surface does not fold back onto itself. Also, the surface has to be actually oriented, i.e. we use a convention as to flowing which way is counted positive; flowing backward is then counted negative.

The surface normal is directed accordingly, usually by the right-hand rule.

Conversely, one can consider the flux the more fundamental quantity and call the vector field the flux density.

Often a vector field is drawn by curves (field lines) following the "flow"; the magnitude of the vector field is then the line density, and the flux through a surface is the number of lines. Lines originate from areas of positive divergence (sources) and end at areas of negative divergence (sinks).

See also the image at right: the number of red arrows passing through a unit area is the flux density, the curve encircling the red arrows denotes the boundary of the surface, and the orientation of the arrows with respect to the surface denotes the sign of the inner product of the vector field with the surface normals.

If the surface encloses a 3D region, usually the surface is oriented such that the influx is counted positive; the opposite is the outflux.

The divergence theorem states that the net outflux through a closed surface, in other words the net outflux from a 3D region, is found by adding the local net outflow from each point in the region (which is expressed by the divergence).

If the surface is not closed, it has an oriented curve as boundary. Stokes' theorem states that the flux of the curl of a vector field is the line integral of the vector field over this boundary. This path integral is also called circulation, especially in fluid dynamics. Thus the curl is the circulation density.

We can apply the flux and these theorems to many disciplines in which we see currents, forces, etc., applied through areas.

Maxwell's equations [link]

The flux of electric and magnetic field lines is frequently discussed in electrostatics. This is because Maxwell's equations in integral form involve integrals like above for electric and magnetic fields.

For instance, Gauss's law states that the flux of the electric field out of a closed surface is proportional to the electric charge enclosed in the surface (regardless of how that charge is distributed). The constant of proportionality is the reciprocal of the permittivity of free space.

Its integral form is:

Failed to parse (Missing texvc executable; please see math/README to configure.): \oint_A \epsilon_0 \mathbf{E} \cdot d\mathbf{A} = Q_A

where:

• Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{E}

is the electric field,

• Failed to parse (Missing texvc executable; please see math/README to configure.): d\mathbf{A}

is the area of a differential square on the surface A with an outward facing surface normal defining its direction,

• Failed to parse (Missing texvc executable; please see math/README to configure.): Q_A \

is the charge enclosed by the surface,

• Failed to parse (Missing texvc executable; please see math/README to configure.): \epsilon_0 \

is the permittivity of free space

• Failed to parse (Missing texvc executable; please see math/README to configure.): \oint_A

is the integral over the surface A.

Either Failed to parse (Missing texvc executable; please see math/README to configure.): \oint_A \epsilon_0 \mathbf{E} \cdot d\mathbf{A}

or Failed to parse (Missing texvc executable; please see math/README to configure.):  \oint_A \mathbf{E} \cdot d\mathbf{A} 
is called the electric flux.

If one considers the flux of the electric field vector, E, for a tube near a point charge in the field the charge but not containing it with sides formed by lines tangent to the field, the flux for the sides is zero and there is an equal and opposite flux at both ends of the tube. This is a consequence of Gauss's Law applied to an inverse square field. The flux for any cross-sectional surface of the tube will be the same. The total flux for any surface surrounding a charge q is q/ε₀.^[9]

In free space the electric displacement vector D = ε₀ E so for any bounding surface the flux of D = q, the charge within it. Here the expression "flux of" indicates a mathematical operation and, as can be seen, the result is not necessarily a "flow".

Faraday's law of induction in integral form is:

Failed to parse (Missing texvc executable; please see math/README to configure.): \oint_C \mathbf{E} \cdot d\mathbf{l} = -\int_{\partial C} \ {d\mathbf{B}\over dt} \cdot d\mathbf{s} = - \frac{d \Phi_D}{ d t}

where:

• Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{d}\mathbf{l}

is an infinitesimal element (differential) of the closed curve C (i.e. a vector with magnitude equal to the length of the infinitesimal line element, and direction given by the tangent to the curve C, with the sign determined by the integration direction).

The magnetic field is denoted by Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{B} . Its flux is called the magnetic flux. The time-rate of change of the magnetic flux through a loop of wire is minus the electromotive force created in that wire. The direction is such that if current is allowed to pass through the wire, the electromotive force will cause a current which "opposes" the change in magnetic field by itself producing a magnetic field opposite to the change. This is the basis for inductors and many electric generators.

Poynting vector [link]

The flux of the Poynting vector through a surface is the electromagnetic power, or energy per unit time, passing through that surface. This is commonly used in analysis of electromagnetic radiation, but has application to other electromagnetic systems as well.

Biology [link]

In general, flux in biology relates to movement of a substance between compartments. There are several cases where the concept of flux is important.

• The movement of molecules across a membrane: in this case, flux is defined by the rate of diffusion or transport of a substance across a permeable membrane. Except in the case of active transport, net flux is directly proportional to the concentration difference across the membrane, the surface area of the membrane, and the membrane permeability constant.
• In ecology, flux is often considered at the ecosystem level - for instance, accurate determination of carbon fluxes using techniques like eddy covariance (at a regional and global level) is essential for modeling the causes and consequences of global warming.
• Metabolic flux refers to the rate of flow of metabolites along a metabolic pathway, or even through a single enzyme. A calculation may also be made of carbon (or other elements, e.g. nitrogen) flux. It is dependent on a number of factors, including: enzyme concentration; the concentration of precursor, product, and intermediate metabolites; post-translational modification of enzymes; and the presence of metabolic activators or repressors. Metabolic control analysis and flux balance analysis provide frameworks for understanding metabolic fluxes and their constraints.

See also [link]

Mathematics portal

Notes [link]

Further reading [link]

• Stauffer, P.H. (2006). "Flux Flummoxed: A Proposal for Consistent Usage". Ground Water 44 (2): 125–128. DOI:10.1111/j.1745-6584.2006.00197.x. PMID 16556188.