Ampère's circuital law

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https://en.wikipedia.org/wiki/Amp%C3%A8re%27s_circuital_law

 

In classical electromagnetism, Ampère's circuital law (not to be confused with Ampère's force law) relates the circulation of a magnetic field around a closed loop to the electric current passing through the loop.

James Clerk Maxwell derived it using hydrodynamics in his 1861 published paper "On Physical Lines of Force". In 1865 he generalized the equation to apply to time-varying currents by adding the displacement current term, resulting in the modern form of the law, sometimes called the Ampère–Maxwell law, which is one of Maxwell's equations which form the basis of classical electromagnetism.

Ampère's original circuital law

In 1820 Danish physicist Hans Christian Ørsted discovered that an electric current creates a magnetic field around it, when he noticed that the needle of a compass next to a wire carrying current turned so that the needle was perpendicular to the wire. He investigated and discovered the rules which govern the field around a straight current-carrying wire:

• The magnetic field lines encircle the current-carrying wire.
• The magnetic field lines lie in a plane perpendicular to the wire.
• If the direction of the current is reversed, the direction of the magnetic field reverses.
• The strength of the field is directly proportional to the magnitude of the current.
• The strength of the field at any point is inversely proportional to the distance of the point from the wire.

This sparked a great deal of research into the relation between electricity and magnetism. André-Marie Ampère investigated the magnetic force between two current-carrying wires, discovering Ampère's force law. In the 1850s Scottish mathematical physicist James Clerk Maxwell generalized these results and others into a single mathematical law. The original form of Maxwell's circuital law, which he derived as early as 1855 in his paper "On Faraday's Lines of Force" based on an analogy to hydrodynamics, relates magnetic fields to electric currents that produce them. It determines the magnetic field associated with a given current, or the current associated with a given magnetic field.

The original circuital law only applies to a magnetostatic situation, to continuous steady currents flowing in a closed circuit. For systems with electric fields that change over time, the original law (as given in this section) must be modified to include a term known as Maxwell's correction (see below).

Equivalent forms

The original circuital law can be written in several different forms, which are all ultimately equivalent:

• An "integral form" and a "differential form". The forms are exactly equivalent, and related by the Kelvin–Stokes theorem (see the "proof" section below).
• Forms using SI units, and those using cgs units. Other units are possible, but rare. This section will use SI units, with cgs units discussed later.
• Forms using either B or H magnetic fields. These two forms use the total current density and free current density, respectively. The B and H fields are related by the constitutive equation: B = μ₀H in non-magnetic materials where μ₀ is the magnetic constant.

Explanation

The integral form of the original circuital law is a line integral of the magnetic field around some closed curve C (arbitrary but must be closed). The curve C in turn bounds both a surface S which the electric current passes through (again arbitrary but not closed—since no three-dimensional volume is enclosed by S), and encloses the current. The mathematical statement of the law is a relation between the circulation of the magnetic field around some path (line integral) due to the current which passes through that enclosed path (surface integral).

In terms of total current, (which is the sum of both free current and bound current) the line integral of the magnetic B-field (in teslas, T) around closed curve C is proportional to the total current I_enc passing through a surface S (enclosed by C). In terms of free current, the line integral of the magnetic H-field (in amperes per metre, A·m⁻¹) around closed curve C equals the free current I_f,enc through a surface S.

Forms of the original circuital law written in SI units

	Integral form	Differential form
Using B-field and total current	${\displaystyle {\oint }_C\mathbf{B}\cdot \mathrm{d}\mathit{l}={\mu }_0{\iint }_S\mathbf{J}\cdot \mathrm{d}\mathbf{S}={\mu }_0{I}_{\mathrm{e}\mathrm{n}\mathrm{c}}}$	${\displaystyle \mathrm{\nabla }\times \mathbf{B}={\mu }_0\mathbf{J}}$
Using H-field and free current	${\displaystyle {\oint }_C\mathbf{H}\cdot \mathrm{d}\mathit{l}={\iint }_S{\mathbf{J}}_{\mathrm{f}}\cdot \mathrm{d}\mathbf{S}=I_{\mathrm{f},\mathrm{e}\mathrm{n}\mathrm{c}}}$	${\displaystyle \mathrm{\nabla }\times \mathbf{H}={\mathbf{J}}_{\mathrm{f}}}$

• J is the total current density (in amperes per square metre, A·m⁻²),
• J_f is the free current density only,
• ∮_C is the closed line integral around the closed curve C,
• ∬_S denotes a 2-D surface integral over S enclosed by C,
• · is the vector dot product,
• dl is an infinitesimal element (a differential) of the 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)
• dS is the vector area of an infinitesimal element of surface S (that is, a vector with magnitude equal to the area of the infinitesimal surface element, and direction normal to surface S. The direction of the normal must correspond with the orientation of C by the right hand rule), see below for further explanation of the curve C and surface S.
• ∇ × is the curl operator.

Ambiguities and sign conventions

There are a number of ambiguities in the above definitions that require clarification and a choice of convention.

1. First, three of these terms are associated with sign ambiguities: the line integral ∮_C could go around the loop in either direction (clockwise or counterclockwise); the vector area dS could point in either of the two directions normal to the surface; and I_enc is the net current passing through the surface S, meaning the current passing through in one direction, minus the current in the other direction—but either direction could be chosen as positive. These ambiguities are resolved by the right-hand rule: With the palm of the right-hand toward the area of integration, and the index-finger pointing along the direction of line-integration, the outstretched thumb points in the direction that must be chosen for the vector area dS. Also the current passing in the same direction as dS must be counted as positive. The right hand grip rule can also be used to determine the signs.
2. Second, there are infinitely many possible surfaces S that have the curve C as their border. (Imagine a soap film on a wire loop, which can be deformed by blowing on the film). Which of those surfaces is to be chosen? If the loop does not lie in a single plane, for example, there is no one obvious choice. The answer is that it does not matter: in the magnetostatic case, the current density is solenoidal (see next section), so the divergence theorem and continuity equation imply that the flux through any surface with boundary C, with the same sign convention, is the same. In practice, one usually chooses the most convenient surface (with the given boundary) to integrate over.

Free current versus bound current

The electric current that arises in the simplest textbook situations would be classified as "free current"—for example, the current that passes through a wire or battery. In contrast, "bound current" arises in the context of bulk materials that can be magnetized and/or polarized. (All materials can to some extent.)

When a material is magnetized (for example, by placing it in an external magnetic field), the electrons remain bound to their respective atoms, but behave as if they were orbiting the nucleus in a particular direction, creating a microscopic current. When the currents from all these atoms are put together, they create the same effect as a macroscopic current, circulating perpetually around the magnetized object. This magnetization current J_M is one contribution to "bound current".

The other source of bound current is bound charge. When an electric field is applied, the positive and negative bound charges can separate over atomic distances in polarizable materials, and when the bound charges move, the polarization changes, creating another contribution to the "bound current", the polarization current J_P.

The total current density J due to free and bound charges is then:

${\displaystyle \mathbf{J}={\mathbf{J}}_{\mathrm{f}}+{\mathbf{J}}_{\mathrm{M}}+{\mathbf{J}}_{\mathrm{P}},}$

with J_f  the "free" or "conduction" current density.

All current is fundamentally the same, microscopically. Nevertheless, there are often practical reasons for wanting to treat bound current differently from free current. For example, the bound current usually originates over atomic dimensions, and one may wish to take advantage of a simpler theory intended for larger dimensions. The result is that the more microscopic Ampère's circuital law, expressed in terms of B and the microscopic current (which includes free, magnetization and polarization currents), is sometimes put into the equivalent form below in terms of H and the free current only. For a detailed definition of free current and bound current, and the proof that the two formulations are equivalent, see the "proof" section below.

Shortcomings of the original formulation of the circuital law

There are two important issues regarding the circuital law that require closer scrutiny. First, there is an issue regarding the continuity equation for electrical charge. In vector calculus, the identity for the divergence of a curl states that the divergence of the curl of a vector field must always be zero. Hence

${\displaystyle \mathrm{\nabla }\cdot (\mathrm{\nabla }\times \mathbf{B})=0,}$

and so the original Ampère's circuital law implies that

${\displaystyle \mathrm{\nabla }\cdot \mathbf{J}=0,}$

i.e. that the current density is solenoidal.

But in general, reality follows the continuity equation for electric charge:

${\displaystyle \mathrm{\nabla }\cdot \mathbf{J}=-\frac{\mathrm{\partial }\rho }{\mathrm{\partial }t},}$

which is nonzero for a time-varying charge density. An example occurs in a capacitor circuit where time-varying charge densities exist on the plates.

Second, there is an issue regarding the propagation of electromagnetic waves. For example, in free space, where

${\displaystyle \mathbf{J}=\mathbf{0},}$

the circuital law implies that

${\displaystyle \mathrm{\nabla }\times \mathbf{B}=\mathbf{0},}$

i.e. that the magnetic field is irrotational, but to maintain consistency with the continuity equation for electric charge, we must have

${\displaystyle \mathrm{\nabla }\times \mathbf{B}=\frac{1}{c^2}\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}.}$

To treat these situations, the contribution of displacement current must be added to the current term in the circuital law.

James Clerk Maxwell conceived of displacement current as a polarization current in the dielectric vortex sea, which he used to model the magnetic field hydrodynamically and mechanically. He added this displacement current to Ampère's circuital law at equation 112 in his 1861 paper "On Physical Lines of Force".

Displacement current

Main article: Displacement current

In free space, the displacement current is related to the time rate of change of electric field.

In a dielectric the above contribution to displacement current is present too, but a major contribution to the displacement current is related to the polarization of the individual molecules of the dielectric material. Even though charges cannot flow freely in a dielectric, the charges in molecules can move a little under the influence of an electric field. The positive and negative charges in molecules separate under the applied field, causing an increase in the state of polarization, expressed as the polarization density P. A changing state of polarization is equivalent to a current.

Both contributions to the displacement current are combined by defining the displacement current as:

${\displaystyle {\mathbf{J}}_{\mathrm{D}}=\frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathbf{D}(\mathbf{r},t),}$

where the electric displacement field is defined as:

${\displaystyle \mathbf{D}={\epsilon }_0\mathbf{E}+\mathbf{P}={\epsilon }_0{\epsilon }_{\mathrm{r}}\mathbf{E},}$

where ε₀ is the electric constant, ε_r the relative static permittivity, and P is the polarization density. Substituting this form for D in the expression for displacement current, it has two components:

${\displaystyle {\mathbf{J}}_{\mathrm{D}}={\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}+\frac{\mathrm{\partial }\mathbf{P}}{\mathrm{\partial }t}.}$

The first term on the right hand side is present everywhere, even in a vacuum. It doesn't involve any actual movement of charge, but it nevertheless has an associated magnetic field, as if it were an actual current. Some authors apply the name displacement current to only this contribution.

The second term on the right hand side is the displacement current as originally conceived by Maxwell, associated with the polarization of the individual molecules of the dielectric material.

Maxwell's original explanation for displacement current focused upon the situation that occurs in dielectric media. In the modern post-aether era, the concept has been extended to apply to situations with no material media present, for example, to the vacuum between the plates of a charging vacuum capacitor. The displacement current is justified today because it serves several requirements of an electromagnetic theory: correct prediction of magnetic fields in regions where no free current flows; prediction of wave propagation of electromagnetic fields; and conservation of electric charge in cases where charge density is time-varying. For greater discussion see Displacement current.

Extending the original law: the Ampère–Maxwell equation

Next, the circuital equation is extended by including the polarization current, thereby remedying the limited applicability of the original circuital law.

Treating free charges separately from bound charges, the equation including Maxwell's correction in terms of the H-field is (the H-field is used because it includes the magnetization currents, so J_M does not appear explicitly, see H-field and also Note):

${\displaystyle {\oint }_C\mathbf{H}\cdot \mathrm{d}\mathit{l}={\iint }_S\left({\mathbf{J}}_{\mathrm{f}}+\frac{\mathrm{\partial }\mathbf{D}}{\mathrm{\partial }t}\right)\cdot \mathrm{d}\mathbf{S}}$

(integral form), where H is the magnetic H field (also called "auxiliary magnetic field", "magnetic field intensity", or just "magnetic field"), D is the electric displacement field, and J_f is the enclosed conduction current or free current density. In differential form,

${\displaystyle \mathrm{\nabla }\times \mathbf{H}={\mathbf{J}}_{\mathrm{f}}+\frac{\mathrm{\partial }\mathbf{D}}{\mathrm{\partial }t}.}$

On the other hand, treating all charges on the same footing (disregarding whether they are bound or free charges), the generalized Ampère's equation, also called the Maxwell–Ampère equation, is in integral form (see the "proof" section below):

${\displaystyle {\oint }_C\mathbf{B}\cdot \mathrm{d}\mathit{l}={\iint }_S\left({\mu }_0\mathbf{J}+{\mu }_0{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}\right)\cdot \mathrm{d}\mathbf{S}}$

In differential form,

${\displaystyle \mathrm{\nabla }\times \mathbf{B}={\mu }_0\mathbf{J}+{\mu }_0{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}}$

In both forms J includes magnetization current density as well as conduction and polarization current densities. That is, the current density on the right side of the Ampère–Maxwell equation is:

${\displaystyle {\mathbf{J}}_{\mathrm{f}}+{\mathbf{J}}_{\mathrm{D}}+{\mathbf{J}}_{\mathrm{M}}={\mathbf{J}}_{\mathrm{f}}+{\mathbf{J}}_{\mathrm{P}}+{\mathbf{J}}_{\mathrm{M}}+{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}=\mathbf{J}+{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t},}$

where current density J_D is the displacement current, and J is the current density contribution actually due to movement of charges, both free and bound. Because ∇ ⋅ D = ρ, the charge continuity issue with Ampère's original formulation is no longer a problem. Because of the term in ε₀∂E/∂t, wave propagation in free space now is possible.

With the addition of the displacement current, Maxwell was able to hypothesize (correctly) that light was a form of electromagnetic wave. See electromagnetic wave equation for a discussion of this important discovery.

Proof of equivalence

Proof that the formulations of the circuital law in terms of free current are equivalent to the formulations involving total current

In this proof, we will show that the equation

${\displaystyle \mathrm{\nabla }\times \mathbf{H}={\mathbf{J}}_{\mathrm{f}}+\frac{\mathrm{\partial }\mathbf{D}}{\mathrm{\partial }t}}$

is equivalent to the equation

${\displaystyle \frac{1}{{\mu }_0}(\mathrm{\nabla }\times \mathbf{B})=\mathbf{J}+{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}.}$

Note that we are only dealing with the differential forms, not the integral forms, but that is sufficient since the differential and integral forms are equivalent in each case, by the Kelvin–Stokes theorem.

We introduce the polarization density P, which has the following relation to E and D:

${\displaystyle \mathbf{D}={\epsilon }_0\mathbf{E}+\mathbf{P}.}$

Next, we introduce the magnetization density M, which has the following relation to B and H:

${\displaystyle \frac{1}{{\mu }_0}\mathbf{B}=\mathbf{H}+\mathbf{M}}$

and the following relation to the bound current:

${\displaystyle \begin{array}{rl}{\mathbf{J}}_{\mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}}& =\mathrm{\nabla }\times \mathbf{M}+\frac{\mathrm{\partial }\mathbf{P}}{\mathrm{\partial }t}\\ & ={\mathbf{J}}_{\mathrm{M}}+{\mathbf{J}}_{\mathrm{P}},\end{array}}$

where

${\displaystyle {\mathbf{J}}_{\mathrm{M}}=\mathrm{\nabla }\times \mathbf{M},}$

is called the magnetization current density, and

${\displaystyle {\mathbf{J}}_{\mathrm{P}}=\frac{\mathrm{\partial }\mathbf{P}}{\mathrm{\partial }t},}$

is the polarization current density. Taking the equation for B:

${\displaystyle \begin{array}{rl}\frac{1}{{\mu }_0}(\mathrm{\nabla }\times \mathbf{B})& =\mathrm{\nabla }\times \left(\mathbf{H}+\mathbf{M}\right)\\ & =\mathrm{\nabla }\times \mathbf{H}+{\mathbf{J}}_{\mathrm{M}}\\ & ={\mathbf{J}}_{\mathrm{f}}+{\mathbf{J}}_{\mathrm{P}}+{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}+{\mathbf{J}}_{\mathrm{M}}.\end{array}}$

Consequently, referring to the definition of the bound current:

${\displaystyle \begin{array}{rl}\frac{1}{{\mu }_0}(\mathrm{\nabla }\times \mathbf{B})& ={\mathbf{J}}_{\mathrm{f}}+{\mathbf{J}}_{\mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}}+{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}\\ & =\mathbf{J}+{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t},\end{array}}$

as was to be shown.

Ampère's circuital law in cgs units

In cgs units, the integral form of the equation, including Maxwell's correction, reads

${\displaystyle {\oint }_C\mathbf{B}\cdot \mathrm{d}\mathit{l}=\frac{1}{c}{\iint }_S\left(4\pi \mathbf{J}+\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}\right)\cdot \mathrm{d}\mathbf{S},}$

where c is the speed of light.

The differential form of the equation (again, including Maxwell's correction) is

${\displaystyle \mathrm{\nabla }\times \mathbf{B}=\frac{1}{c}\left(4\pi \mathbf{J}+\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}\right).}$

Living fossil

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Living_fossil

The coelacanths were thought to have gone extinct 66 million years ago, until a living specimen belonging to the order was discovered in 1938.

A living fossil is an extant taxon that phenotypically resembles related species known only from the fossil record. To be considered a living fossil, the fossil species must be old relative to the time of origin of the extant clade. Living fossils commonly are of species-poor lineages, but they need not be. While the body plan of a living fossil remains superficially similar, it is never the same species as the remote relatives it resembles, because genetic drift would inevitably change its chromosomal structure.

Living fossils exhibit stasis (also called "bradytely") over geologically long time scales. Popular literature may wrongly claim that a "living fossil" has undergone no significant evolution since fossil times, with practically no molecular evolution or morphological changes. Scientific investigations have repeatedly discredited such claims.

The minimal superficial changes to living fossils are mistakenly declared as an absence of evolution, but they are examples of stabilizing selection, which is an evolutionary process—and perhaps the dominant process of morphological evolution.

Characteristics

Fossil and living ginkgos

 

170 million-year-old fossil Ginkgo leaves

 

Living Ginkgo biloba plant

Living fossils have two main characteristics, although some have a third:

1. Living organisms that are members of a taxon that has remained recognisable in the fossil record over an unusually long time span.
2. They show little morphological divergence, whether from early members of the lineage, or among extant species.
3. They tend to have little taxonomic diversity.

The first two are required for recognition as a living fossil; some authors also require the third, others merely note it as a frequent trait.

Such criteria are neither well-defined nor clearly quantifiable, but modern methods for analyzing evolutionary dynamics can document the distinctive tempo of stasis. Lineages that exhibit stasis over very short time scales are not considered living fossils; what is poorly-defined is the time scale over which the morphology must persist for that lineage to be recognized as a living fossil.

The term living fossil is much misunderstood in popular media in particular, in which it often is used meaninglessly. In professional literature the expression seldom appears and must be used with far more caution, although it has been used inconsistently.

One example of a concept that could be confused with "living fossil" is that of a "Lazarus taxon", but the two are not equivalent; a Lazarus taxon (whether a single species or a group of related species) is one that suddenly reappears, either in the fossil record or in nature, as if the fossil had "come to life again". In contrast to "Lazarus taxa", a living fossil in most senses is a species or lineage that has undergone exceptionally little change throughout a long fossil record, giving the impression that the extant taxon had remained identical through the entire fossil and modern period. Because of the mathematical inevitability of genetic drift, though, the DNA of the modern species is necessarily different from that of its distant, similar-looking ancestor. They almost certainly would not be able to cross-reproduce, and are not the same species.

The average species turnover time, meaning the time between when a species first is established and when it finally disappears, varies widely among phyla, but averages about 2–3 million years. A living taxon that had long been thought to be extinct could be called a Lazarus taxon once it was discovered to be still extant. A dramatic example was the order Coelacanthiformes, of which the genus Latimeria was found to be extant in 1938. About that there is little debate – however, whether Latimeria resembles early members of its lineage sufficiently closely to be considered a living fossil as well as a Lazarus taxon has been denied by some authors in recent years.

Coelacanths disappeared from the fossil record some 80 million years ago (in the upper Cretaceous period) and, to the extent that they exhibit low rates of morphological evolution, extant species qualify as living fossils. It must be emphasised that this criterion reflects fossil evidence, and is totally independent of whether the taxa had been subject to selection at all, which all living populations continuously are, whether they remain genetically unchanged or not.

This apparent stasis, in turn, gives rise to a great deal of confusion – for one thing, the fossil record seldom preserves much more than the general morphology of a specimen. To determine much about its physiology is seldom possible; not even the most dramatic examples of living fossils can be expected to be without changes, no matter how persistently constant their fossils and the extant specimens might seem. To determine much about noncoding DNA is hardly ever possible, but even if a species were hypothetically unchanged in its physiology, it is to be expected from the very nature of the reproductive processes, that its non-functional genomic changes would continue at more-or-less standard rates. Hence, a fossil lineage with apparently constant morphology need not imply equally constant physiology, and certainly neither implies any cessation of the basic evolutionary processes such as natural selection, nor reduction in the usual rate of change of the noncoding DNA.

Some living fossils are taxa that were known from palaeontological fossils before living representatives were discovered. The most famous examples of this are:

• Coelacanthiform fishes (2 species)
• Metasequoia, the dawn redwood discovered in a remote Chinese valley (1 species)
• Glypheoid lobsters (2 species)
• Mymarommatid wasps (10 species)
• Eomeropid scorpionflies (1 species)
• Jurodid beetles (1 species)
• Soft sea urchins (59 species)

All the above include taxa that originally were described as fossils but now are known to include still-extant species.

Other examples of living fossils are single living species that have no close living relatives, but are survivors of large and widespread groups in the fossil record. For example:

• Ginkgo biloba
• Syntexis libocedrii, the cedar wood wasp
• Dinoflagellates (typified on coccoid dinocysts: occasionally calcareous cell remnants)

All of these were described from fossils before later being found alive.

The fact that a living fossil is a surviving representative of an archaic lineage does not imply that it must retain all the "primitive" features (plesiomorphies) of its ancestral lineage. Although it is common to say that living fossils exhibit "morphological stasis", stasis, in the scientific literature, does not mean that any species is strictly identical to its ancestor, much less remote ancestors.

Some living fossils are relicts of formerly diverse and morphologically varied lineages, but not all survivors of ancient lineages necessarily are regarded as living fossils. See for example the uniquely and highly autapomorphic oxpeckers, which appear to be the only survivors of an ancient lineage related to starlings and mockingbirds.

Evolution and living fossils

The term living fossil is usually reserved for species or larger clades that are exceptional for their lack of morphological diversity and their exceptional conservatism, and several hypotheses could explain morphological stasis on a geologically long time-scale. Early analyses of evolutionary rates emphasized the persistence of a taxon rather than rates of evolutionary change. Contemporary studies instead analyze rates and modes of phenotypic evolution, but most have focused on clades that are thought to be adaptive radiations rather than on those thought to be living fossils. Thus, very little is presently known about the evolutionary mechanisms that produce living fossils or how common they might be. Some recent studies have documented exceptionally low rates of ecological and phenotypic evolution despite rapid speciation. This has been termed a "non-adaptive radiation" referring to diversification not accompanied by adaptation into various significantly different niches. Such radiations are explanation for groups that are morphologically conservative. Persistent adaptation within an adaptive zone is a common explanation for morphological stasis. The subject of very low evolutionary rates, however, has received much less attention in the recent literature than that of high rates.

Living fossils are not expected to exhibit exceptionally low rates of molecular evolution, and some studies have shown that they do not. For example, on tadpole shrimp (Triops), one article notes, "Our work shows that organisms with conservative body plans are constantly radiating, and presumably, adapting to novel conditions... I would favor retiring the term 'living fossil' altogether, as it is generally misleading." Some scientists instead prefer a new term stabilomorph, being defined as "an effect of a specific formula of adaptative strategy among organisms whose taxonomic status does not exceed genus-level. A high effectiveness of adaptation significantly reduces the need for differentiated phenotypic variants in response to environmental changes and provides for long-term evolutionary success."

The question posed by several recent studies pointed out that the morphological conservatism of coelacanths is not supported by paleontological data. In addition, it was shown recently that studies concluding that a slow rate of molecular evolution is linked to morphological conservatism in coelacanths are biased by the a priori hypothesis that these species are 'living fossils'. Accordingly, the genome stasis hypothesis is challenged by the recent finding that the genome of the two extant coelacanth species L. chalumnae and L. menadoensis contain multiple species-specific insertions, indicating transposable element recent activity and contribution to post-speciation genome divergence. Such studies, however, challenge only a genome stasis hypothesis, not the hypothesis of exceptionally low rates of phenotypic evolution.

History

The term was coined by Charles Darwin in his On the Origin of Species from 1859, when discussing Ornithorhynchus (the platypus) and Lepidosiren (the South American lungfish):

All fresh-water basins, taken together, make a small area compared with that of the sea or of the land; and, consequently, the competition between fresh-water productions will have been less severe than elsewhere; new forms will have been more slowly formed, and old forms more slowly exterminated. And it is in fresh water that we find seven genera of Ganoid fishes, remnants of a once preponderant order: and in fresh water we find some of the most anomalous forms now known in the world, as the Ornithorhynchus and Lepidosiren, which, like fossils, connect to a certain extent orders now widely separated in the natural scale. These anomalous forms may almost be called living fossils; they have endured to the present day, from having inhabited a confined area, and from having thus been exposed to less severe competition.

— On the Origin of Species, 1859

Other definitions

Long-enduring

Elephant shrews resemble the extinct Leptictidium of Eocene Europe.

A living taxon that lived through a large portion of geologic time.

The Australian lungfish (Neoceratodus fosteri), also known as the Queensland lungfish, is an example of an organism that meets this criterion. Fossils identical to modern specimens have been dated at over 100 million years old. Modern Queensland lungfish have existed as a species for almost 30 million years. The contemporary nurse shark has existed for more than 112 million years, making this species one of the oldest, if not actually the oldest extant vertebrate species.

Resembles ancient species

A living taxon morphologically and/or physiologically resembling a fossil taxon through a large portion of geologic time (morphological stasis).

Retains many ancient traits

More primitive trapdoor spiders, such as this female Liphistius sp., have segmented plates on the dorsal surface of the abdomen and cephalothorax, a character shared with scorpions, making it probable that after the spiders diverged from the scorpions, the earliest unique ancestor of trapdoor species was the first to split off from the lineage that contains all other extant spiders.

A living taxon with many characteristics believed to be primitive. This is a more neutral definition. However, it does not make it clear whether the taxon is truly old, or it simply has many plesiomorphies. Note that, as mentioned above, the converse may hold for true living fossil taxa; that is, they may possess a great many derived features (autapomorphies), and not be particularly "primitive" in appearance.

Relict population

Any one of the above three definitions, but also with a relict distribution in refuges.

Some paleontologists believe that living fossils with large distributions (such as Triops cancriformis) are not real living fossils. In the case of Triops cancriformis (living from the Triassic until now), the Triassic specimens lost most of their appendages (mostly only carapaces remain), and they have not been thoroughly examined since 1938.

Low diversity

Any of the first three definitions, but the clade also has a low taxonomic diversity (low diversity lineages).

Oxpeckers are morphologically somewhat similar to starlings due to shared plesiomorphies, but are uniquely adapted to feed on parasites and blood of large land mammals, which has always obscured their relationships. This lineage forms part of a radiation that includes Sturnidae and Mimidae, but appears to be the most ancient of these groups. Biogeography strongly suggests that oxpeckers originated in eastern Asia and only later arrived in Africa, where they now have a relict distribution.

The two living species thus seem to represent an entirely extinct and (as Passerida go) rather ancient lineage, as certainly as this can be said in the absence of actual fossils. The latter is probably due to the fact that the oxpecker lineage never occurred in areas where conditions were good for fossilization of small bird bones, but of course, fossils of ancestral oxpeckers may one day turn up enabling this theory to be tested.

Operational definition

An operational definition was proposed in 2017, where a 'living fossil' lineage has a slow rate of evolution and occurs close to the middle of morphological variation (the centroid of morphospace) among related taxa (i.e. a species is morphologically conservative among relatives). The scientific accuracy of the morphometric analyses used to classify tuatara as a living fossil under this definition have been criticised however, which prompted a rebuttal from the original authors.

Examples

Some of these are informally known as "living fossils".