Eigenstate thermalization hypothesis

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

The eigenstate thermalization hypothesis (or ETH) is a set of ideas which purports to explain when and why an isolated quantum mechanical system can be accurately described using equilibrium statistical mechanics. In particular, it is devoted to understanding how systems which are initially prepared in far-from-equilibrium states can evolve in time to a state which appears to be in thermal equilibrium. The phrase "eigenstate thermalization" was first coined by Mark Srednicki in 1994, after similar ideas had been introduced by Josh Deutsch in 1991. The principal philosophy underlying the eigenstate thermalization hypothesis is that instead of explaining the ergodicity of a thermodynamic system through the mechanism of dynamical chaos, as is done in classical mechanics, one should instead examine the properties of matrix elements of observable quantities in individual energy eigenstates of the system.

Motivation

In statistical mechanics, the microcanonical ensemble is a particular statistical ensemble which is used to make predictions about the outcomes of experiments performed on isolated systems that are believed to be in equilibrium with an exactly known energy. The microcanonical ensemble is based upon the assumption that, when such an equilibrated system is probed, the probability for it to be found in any of the microscopic states with the same total energy have equal probability. With this assumption, the ensemble average of an observable quantity is found by averaging the value of that observable ${\displaystyle A_i}$ over all microstates ${\displaystyle i}$ with the correct total energy:

${\displaystyle {\overline{A}}_{\mathrm{c}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}}=\frac{1}{N}\sum _{i=1}^N{A}_i}$

Importantly, this quantity is independent of everything about the initial state except for its energy.

The assumptions of ergodicity are well-motivated in classical mechanics as a result of dynamical chaos, since a chaotic system will in general spend equal time in equal areas of its phase space. If we prepare an isolated, chaotic, classical system in some region of its phase space, then as the system is allowed to evolve in time, it will sample its entire phase space, subject only to a small number of conservation laws (such as conservation of total energy). If one can justify the claim that a given physical system is ergodic, then this mechanism will provide an explanation for why statistical mechanics is successful in making accurate predictions. For example, the hard sphere gas has been rigorously proven to be ergodic.

This argument cannot be straightforwardly extended to quantum systems, even ones that are analogous to chaotic classical systems, because time evolution of a quantum system does not uniformly sample all vectors in Hilbert space with a given energy. Given the state at time zero in a basis of energy eigenstates

${\displaystyle |\mathrm{\Psi }(0)⟩=\sum _{\alpha }{c}_{\alpha }|E_{\alpha }⟩,}$

the expectation value of any observable ${\displaystyle \hat{A}}$ is

${\displaystyle ⟨\hat{A}{⟩}_t\equiv ⟨\mathrm{\Psi }(t)|\hat{A}|\mathrm{\Psi }(t)⟩=\sum _{\alpha ,\beta }{c}_{\alpha }^{\ast }{c}_{\beta }{A}_{\alpha \beta }{e}^{-i\left(E_{\beta }-E_{\alpha }\right)t/\hslash }.}$

Even if the ${\displaystyle E_{\alpha }}$ are incommensurate, so that this expectation value is given for long times by

${\displaystyle ⟨\hat{A}{⟩}_t\stackrel{t\to \mathrm{\infty }}{\approx }\sum _{\alpha }|c_{\alpha }{|}^2{A}_{\alpha \alpha },}$

the expectation value permanently retains knowledge of the initial state in the form of the coefficients ${\displaystyle c_{\alpha }}$.

In principle it is thus an open question as to whether an isolated quantum mechanical system, prepared in an arbitrary initial state, will approach a state which resembles thermal equilibrium, in which a handful of observables are adequate to make successful predictions about the system. However, a variety of experiments in cold atomic gases have indeed observed thermal relaxation in systems which are, to a very good approximation, completely isolated from their environment, and for a wide class of initial states. The task of explaining this experimentally observed applicability of equilibrium statistical mechanics to isolated quantum systems is the primary goal of the eigenstate thermalization hypothesis.

Statement

Suppose that we are studying an isolated, quantum mechanical many-body system. In this context, "isolated" refers to the fact that the system has no (or at least negligible) interactions with the environment external to it. If the Hamiltonian of the system is denoted ${\displaystyle \hat{H}}$, then a complete set of basis states for the system is given in terms of the eigenstates of the Hamiltonian,

${\displaystyle \hat{H}|E_{\alpha }⟩=E_{\alpha }|E_{\alpha }⟩,}$

where ${\displaystyle |E_{\alpha }⟩}$ is the eigenstate of the Hamiltonian with eigenvalue ${\displaystyle E_{\alpha }}$. We will refer to these states simply as "energy eigenstates." For simplicity, we will assume that the system has no degeneracy in its energy eigenvalues, and that it is finite in extent, so that the energy eigenvalues form a discrete, non-degenerate spectrum (this is not an unreasonable assumption, since any "real" laboratory system will tend to have sufficient disorder and strong enough interactions as to eliminate almost all degeneracy from the system, and of course will be finite in size). This allows us to label the energy eigenstates in order of increasing energy eigenvalue. Additionally, consider some other quantum-mechanical observable ${\displaystyle \hat{A}}$, which we wish to make thermal predictions about. The matrix elements of this operator, as expressed in a basis of energy eigenstates, will be denoted by

${\displaystyle A_{\alpha \beta }\equiv ⟨E_{\alpha }|\hat{A}|E_{\beta }⟩.}$

We now imagine that we prepare our system in an initial state for which the expectation value of ${\displaystyle \hat{A}}$ is far from its value predicted in a microcanonical ensemble appropriate to the energy scale in question (we assume that our initial state is some superposition of energy eigenstates which are all sufficiently "close" in energy). The eigenstate thermalization hypothesis says that for an arbitrary initial state, the expectation value of ${\displaystyle \hat{A}}$ will ultimately evolve in time to its value predicted by a microcanonical ensemble, and thereafter will exhibit only small fluctuations around that value, provided that the following two conditions are met:

1. The diagonal matrix elements ${\displaystyle A_{\alpha \alpha }}$ vary smoothly as a function of energy, with the difference between neighboring values, ${\displaystyle A_{\alpha +1,\alpha +1}-A_{\alpha ,\alpha }}$, becoming exponentially small in the system size.
2. The off-diagonal matrix elements ${\displaystyle A_{\alpha \beta }}$, with ${\displaystyle \alpha \ne \beta }$, are much smaller than the diagonal matrix elements, and in particular are themselves exponentially small in the system size.

These conditions can be written as

${\displaystyle A_{\alpha \beta }\simeq \overline{A}{\delta }_{\alpha \beta }+\sqrt{\frac{\overline{A^2}}{\mathcal{D}}}{R}_{\alpha \beta },}$

where ${\displaystyle \overline{A}=\overline{A}(E_{\alpha })}$ and ${\displaystyle \overline{A^2}=\overline{A^2}(E_{\alpha },E_{\beta })}$ are smooth functions of energy, ${\displaystyle \mathcal{D}=e^{sV}}$ is the many-body Hilbert space dimension, and ${\displaystyle R_{\alpha \beta }}$ is a random variable with zero mean and unit variance. Conversely if a quantum many-body system satisfies the ETH, the matrix representation of any local operator in the energy eigen basis is expected to follow the above ansatz.

Equivalence of the diagonal and microcanonical ensembles

We can define a long-time average of the expectation value of the operator ${\displaystyle \hat{A}}$ according to the expression

${\displaystyle \overline{A}\equiv \underset{\tau \to \mathrm{\infty }}{lim}\frac{1}{\tau }{\int }_0^{\tau }⟨\mathrm{\Psi }(t)|\hat{A}|\mathrm{\Psi }(t)⟩dt.}$

If we use the explicit expression for the time evolution of this expectation value, we can write

${\displaystyle \overline{A}=\underset{\tau \to \mathrm{\infty }}{lim}\frac{1}{\tau }{\int }_0^{\tau }\left[\sum _{\alpha ,\beta =1}^D{c}_{\alpha }^{\ast }{c}_{\beta }{A}_{\alpha \beta }{e}^{-i\left(E_{\beta }-E_{\alpha }\right)t/\hslash }\right]dt.}$

The integration in this expression can be performed explicitly, and the result is

${\displaystyle \overline{A}=\sum _{\alpha =1}^D|c_{\alpha }{|}^2{A}_{\alpha \alpha }+i\hslash \underset{\tau \to \mathrm{\infty }}{lim}\left[\sum _{\alpha \ne \beta }^D\frac{c_{\alpha }^{\ast }{c}_{\beta }{A}_{\alpha \beta }}{E_{\beta }-E_{\alpha }}\left(\frac{e^{-i\left(E_{\beta }-E_{\alpha }\right)\tau /\hslash }-1}{\tau }\right)\right].}$

Each of the terms in the second sum will become smaller as the limit is taken to infinity. Assuming that the phase coherence between the different exponential terms in the second sum does not ever become large enough to rival this decay, the second sum will go to zero, and we find that the long-time average of the expectation value is given by

${\displaystyle \overline{A}=\sum _{\alpha =1}^D|c_{\alpha }{|}^2{A}_{\alpha \alpha }.}$

This prediction for the time-average of the observable ${\displaystyle \hat{A}}$ is referred to as its predicted value in the diagonal ensemble, The most important aspect of the diagonal ensemble is that it depends explicitly on the initial state of the system, and so would appear to retain all of the information regarding the preparation of the system. In contrast, the predicted value in the microcanonical ensemble is given by the equally-weighted average over all energy eigenstates within some energy window centered around the mean energy of the system

${\displaystyle ⟨A{⟩}_{\text{mc}}=\frac{1}{\mathcal{N}}\sum _{{\alpha }^{\prime }=1}^{\mathcal{N}}{A}_{{\alpha }^{\prime }{\alpha }^{\prime }},}$

where ${\displaystyle \mathcal{N}}$ is the number of states in the appropriate energy window, and the prime on the sum indices indicates that the summation is restricted to this appropriate microcanonical window. This prediction makes absolutely no reference to the initial state of the system, unlike the diagonal ensemble. Because of this, it is not clear why the microcanonical ensemble should provide such an accurate description of the long-time averages of observables in such a wide variety of physical systems.

However, suppose that the matrix elements ${\displaystyle A_{\alpha \alpha }}$ are effectively constant over the relevant energy window, with fluctuations that are sufficiently small. If this is true, this one constant value A can be effectively pulled out of the sum, and the prediction of the diagonal ensemble is simply equal to this value,

${\displaystyle \overline{A}=\sum _{\alpha =1}^D|c_{\alpha }{|}^2{A}_{\alpha \alpha }\approx A\sum _{\alpha =1}^D|c_{\alpha }{|}^2=A,}$

where we have assumed that the initial state is normalized appropriately. Likewise, the prediction of the microcanonical ensemble becomes

${\displaystyle ⟨A{⟩}_{\text{mc}}=\frac{1}{\mathcal{N}}\sum _{{\alpha }^{\prime }=1}^{\mathcal{N}}{A}_{{\alpha }^{\prime }{\alpha }^{\prime }}\approx \frac{1}{\mathcal{N}}\sum _{{\alpha }^{\prime }=1}^{\mathcal{N}}A=A.}$

The two ensembles are therefore in agreement.

This constancy of the values of ${\displaystyle A_{\alpha \alpha }}$ over small energy windows is the primary idea underlying the eigenstate thermalization hypothesis. Notice that in particular, it states that the expectation value of ${\displaystyle \hat{A}}$ in a single energy eigenstate is equal to the value predicted by a microcanonical ensemble constructed at that energy scale. This constitutes a foundation for quantum statistical mechanics which is radically different from the one built upon the notions of dynamical ergodicity.

Tests

Several numerical studies of small lattice systems appear to tentatively confirm the predictions of the eigenstate thermalization hypothesis in interacting systems which would be expected to thermalize. Likewise, systems which are integrable tend not to obey the eigenstate thermalization hypothesis.

Some analytical results can also be obtained if one makes certain assumptions about the nature of highly excited energy eigenstates. The original 1994 paper on the ETH by Mark Srednicki studied, in particular, the example of a quantum hard sphere gas in an insulated box. This is a system which is known to exhibit chaos classically. For states of sufficiently high energy, Berry's conjecture states that energy eigenfunctions in this many-body system of hard sphere particles will appear to behave as superpositions of plane waves, with the plane waves entering the superposition with random phases and Gaussian-distributed amplitudes (the precise notion of this random superposition is clarified in the paper). Under this assumption, one can show that, up to corrections which are negligibly small in the thermodynamic limit, the momentum distribution function for each individual, distinguishable particle is equal to the Maxwell–Boltzmann distribution

${\displaystyle f_{\mathrm{M}\mathrm{B}}\left(\mathbf{p},T_{\alpha }\right)={\left(2\pi mkT\right)}^{-3/2}{e}^{-{\mathbf{p}}^2/2mk{T}_{\alpha }},}$

where ${\displaystyle \mathbf{p}}$ is the particle's momentum, m is the mass of the particles, k is the Boltzmann constant, and the "temperature" ${\displaystyle T_{\alpha }}$ is related to the energy of the eigenstate according to the usual equation of state for an ideal gas,

${\displaystyle E_{\alpha }=\frac{3}{2}Nk{T}_{\alpha },}$

where N is the number of particles in the gas. This result is a specific manifestation of the ETH, in that it results in a prediction for the value of an observable in one energy eigenstate which is in agreement with the prediction derived from a microcanonical (or canonical) ensemble. Note that no averaging over initial states whatsoever has been performed, nor has anything resembling the H-theorem been invoked. Additionally, one can also derive the appropriate Bose–Einstein or Fermi–Dirac distributions, if one imposes the appropriate commutation relations for the particles comprising the gas.

Currently, it is not well understood how high the energy of an eigenstate of the hard sphere gas must be in order for it to obey the ETH. A rough criterion is that the average thermal wavelength of each particle be sufficiently smaller than the radius of the hard sphere particles, so that the system can probe the features which result in chaos classically (namely, the fact that the particles have a finite size). However, it is conceivable that this condition may be able to be relaxed, and perhaps in the thermodynamic limit, energy eigenstates of arbitrarily low energies will satisfy the ETH (aside from the ground state itself, which is required to have certain special properties, for example, the lack of any nodes).

Alternatives

Three alternative explanations for the thermalization of isolated quantum systems are often proposed:

1. For initial states of physical interest, the coefficients ${\displaystyle c_{\alpha }}$ exhibit large fluctuations from eigenstate to eigenstate, in a fashion which is completely uncorrelated with the fluctuations of ${\displaystyle A_{\alpha \alpha }}$ from eigenstate to eigenstate. Because the coefficients and matrix elements are uncorrelated, the summation in the diagonal ensemble is effectively performing an unbiased sampling of the values of ${\displaystyle A_{\alpha \alpha }}$ over the appropriate energy window. For a sufficiently large system, this unbiased sampling should result in a value which is close to the true mean of the values of ${\displaystyle A_{\alpha \alpha }}$ over this window, and will effectively reproduce the prediction of the microcanonical ensemble. However, this mechanism may be disfavored for the following heuristic reason. Typically, one is interested in physical situations in which the initial expectation value of ${\displaystyle \hat{A}}$ is far from its equilibrium value. For this to be true, the initial state must contain some sort of specific information about ${\displaystyle \hat{A}}$, and so it becomes suspect whether or not the initial state truly represents an unbiased sampling of the values of ${\displaystyle A_{\alpha \alpha }}$ over the appropriate energy window. Furthermore, whether or not this were to be true, it still does not provide an answer to the question of when arbitrary initial states will come to equilibrium, if they ever do.
2. For initial states of physical interest, the coefficients ${\displaystyle c_{\alpha }}$ are effectively constant, and do not fluctuate at all. In this case, the diagonal ensemble is precisely the same as the microcanonical ensemble, and there is no mystery as to why their predictions are identical. However, this explanation is disfavored for much the same reasons as the first.
3. Integrable quantum systems are proved to thermalize under condition of simple regular time-dependence of parameters, suggesting that cosmological expansion of the Universe and integrability of the most fundamental equations of motion are ultimately responsible for thermalization.

Temporal fluctuations of expectation values

The condition that the ETH imposes on the diagonal elements of an observable is responsible for the equality of the predictions of the diagonal and microcanonical ensembles. However, the equality of these long-time averages does not guarantee that the fluctuations in time around this average will be small. That is, the equality of the long-time averages does not ensure that the expectation value of ${\displaystyle \hat{A}}$ will settle down to this long-time average value, and then stay there for most times.

In order to deduce the conditions necessary for the observable's expectation value to exhibit small temporal fluctuations around its time-average, we study the mean squared amplitude of the temporal fluctuations, defined as

${\displaystyle \overline{{\left(A_t-\overline{A}\right)}^2}\equiv \underset{\tau \to \mathrm{\infty }}{lim}\frac{1}{\tau }{\int }_0^{\tau }{\left(A_t-\overline{A}\right)}^{2}dt,}$

where ${\displaystyle A_t}$ is a shorthand notation for the expectation value of ${\displaystyle \hat{A}}$ at time t. This expression can be computed explicitly, and one finds that

${\displaystyle \overline{{\left(A_t-\overline{A}\right)}^2}=\sum _{\alpha \ne \beta }|c_{\alpha }{|}^2|c_{\beta }{|}^2|A_{\alpha \beta }{|}^2.}$

Temporal fluctuations about the long-time average will be small so long as the off-diagonal elements satisfy the conditions imposed on them by the ETH, namely that they become exponentially small in the system size. Notice that this condition allows for the possibility of isolated resurgence times, in which the phases align coherently in order to produce large fluctuations away from the long-time average. The amount of time the system spends far away from the long-time average is guaranteed to be small so long as the above mean squared amplitude is sufficiently small. If a system poses a dynamical symmetry, however, it will periodically oscillate around the long-time average.

Quantum fluctuations and thermal fluctuations

The expectation value of a quantum mechanical observable represents the average value which would be measured after performing repeated measurements on an ensemble of identically prepared quantum states. Therefore, while we have been examining this expectation value as the principal object of interest, it is not clear to what extent this represents physically relevant quantities. As a result of quantum fluctuations, the expectation value of an observable is not typically what will be measured during one experiment on an isolated system. However, it has been shown that for an observable satisfying the ETH, quantum fluctuations in its expectation value will typically be of the same order of magnitude as the thermal fluctuations which would be predicted in a traditional microcanonical ensemble. This lends further credence to the idea that the ETH is the underlying mechanism responsible for the thermalization of isolated quantum systems.

General validity

Currently, there is no known analytical derivation of the eigenstate thermalization hypothesis for general interacting systems. However, it has been verified to be true for a wide variety of interacting systems using numerical exact diagonalization techniques, to within the uncertainty of these methods. It has also been proven to be true in certain special cases in the semi-classical limit, where the validity of the ETH rests on the validity of Shnirelman's theorem, which states that in a system which is classically chaotic, the expectation value of an operator ${\displaystyle \hat{A}}$ in an energy eigenstate is equal to its classical, microcanonical average at the appropriate energy. Whether or not it can be shown to be true more generally in interacting quantum systems remains an open question. It is also known to explicitly fail in certain integrable systems, in which the presence of a large number of constants of motion prevent thermalization.

It is also important to note that the ETH makes statements about specific observables on a case by case basis - it does not make any claims about whether every observable in a system will obey ETH. In fact, this certainly cannot be true. Given a basis of energy eigenstates, one can always explicitly construct an operator which violates the ETH, simply by writing down the operator as a matrix in this basis whose elements explicitly do not obey the conditions imposed by the ETH. Conversely, it is always trivially possible to find operators which do satisfy ETH, by writing down a matrix whose elements are specifically chosen to obey ETH. In light of this, one may be led to believe that the ETH is somewhat trivial in its usefulness. However, the important consideration to bear in mind is that these operators thus constructed may not have any physical relevance. While one can construct these matrices, it is not clear that they correspond to observables which could be realistically measured in an experiment, or bear any resemblance to physically interesting quantities. An arbitrary Hermitian operator on the Hilbert space of the system need not correspond to something which is a physically measurable observable.

Typically, the ETH is postulated to hold for "few-body operators," observables which involve only a small number of particles. Examples of this would include the occupation of a given momentum in a gas of particles, or the occupation of a particular site in a lattice system of particles. Notice that while the ETH is typically applied to "simple" few-body operators such as these, these observables need not be local in space - the momentum number operator in the above example does not represent a local quantity.

There has also been considerable interest in the case where isolated, non-integrable quantum systems fail to thermalize, despite the predictions of conventional statistical mechanics. Disordered systems which exhibit many-body localization are candidates for this type of behavior, with the possibility of excited energy eigenstates whose thermodynamic properties more closely resemble those of ground states. It remains an open question as to whether a completely isolated, non-integrable system without static disorder can ever fail to thermalize. One intriguing possibility is the realization of "Quantum Disentangled Liquids." It also an open question whether all eigenstates must obey the ETH in a thermalizing system.

The eigenstate thermalization hypothesis is closely connected to the quantum nature of chaos (see quantum chaos). Furthermore, since a classically chaotic system is also ergodic, almost all of its trajectories eventually explore uniformly the entire accessible phase space, which would imply the eigenstates of the quantum chaotic system fill the quantum phase space evenly (up to random fluctuations) in the semiclassical limit ${\displaystyle \hslash \to 0}$. In particular, there is a quantum ergodicity theorem showing that the expectation value of an operator converges to the corresponding microcanonical classical average as ${\displaystyle \hslash \to 0}$. However, the quantum ergodicity theorem leaves open the possibility of non-ergodic states such as quantum scars. In addition to the conventional scarring, there are two other types of quantum scarring, which further illustrate the weak-ergodicity breaking in quantum chaotic systems: perturbation-induced and many-body quantum scars. Since the former arise a combined effect of special nearly-degenerate unperturbed states and the localized nature of the perturbation (potential bums), the scarring can slow down the thermalization process in disordered quantum dots and wells, which is further illustrated by the fact that these quantum scars can be utilized to propagate quantum wave packets in a disordered nanostructure with high fidelity. On the other hand, the latter form of scarring has been speculated to be the culprit behind the unexpectedly slow thermalization of cold atoms observed experimentally.

Biogeographic realm

From Wikipedia, the free encyclopedia

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

  

Map of the world's biogeographic realms in Miklos Udvardy's system.

  Nearctic

  Palearctic

  Afrotropical

  Indomalayan

  Australasian

  Neotropical

  Oceanian

  Antarctic (not shown)

A biogeographic realm or ecozone is the broadest biogeographic division of Earth's land surface, based on distributional patterns of terrestrial organisms. They are subdivided into bioregions, which are further subdivided into ecoregions.

Description

The realms delineate large areas of Earth's surface within which organisms have evolved in relative isolation over long periods of time, separated geographic features, such as oceans, broad deserts, or high mountain ranges, that constitute natural barriers to migration. As such, biogeographic realm designations are used to indicate general groupings of organisms based on their shared biogeography. Biogeographic realms correspond to the floristic kingdoms of botany or zoogeographic regions of zoology.

From 1872, Alfred Russel Wallace developed a system of zoogeographic regions, extending the ornithologist Philip Sclater's system of six regions.

Biogeographic realms are characterized by the evolutionary history of the organisms they contain. They are distinct from biomes, also known as major habitat types, which are divisions of the Earth's surface based on life form, or the adaptation of animals, fungi, micro-organisms and plants to climatic, soil, and other conditions. Biomes are characterized by similar climax vegetation. Each realm may include a number of different biomes. A tropical moist broadleaf forest in Central America, for example, may be similar to one in New Guinea in its vegetation type and structure, climate, soils, etc., but these forests are inhabited by animals, fungi, micro-organisms and plants with very different evolutionary histories.

The patterns of distribution of living organisms in the world's biogeographic realms were shaped by the process of plate tectonics, which has redistributed the world's land masses over geological history.

Concept history

The "biogeographic realms" of Udvardy were defined based on taxonomic composition. The rank corresponds more or less to the floristic kingdoms and zoogeographic regions.

The usage of the term "ecozone" is more variable. It was used originally in stratigraphy (Vella 1962, Hedberg 1971). In Canadian literature, the term was used by Wiken in macro level land classification, with geographic criteria (see Ecozones of Canada). Later, Schültz would use it with ecological and physiognomical criteria, in a way similar to the concept of biome.

In the Global 200/WWF scheme, originally the term "biogeographic realm" in Udvardy sense was used. However, in a scheme of BBC, it was replaced by the term "ecozone".

Terrestrial biogeographic realms

Udvardy biogeographic realms

Main article: List of biogeographic provinces

WWF / Global 200 biogeographic realms

Main article: Global 200

The World Wildlife Fund scheme is broadly similar to Miklos Udvardy's system, the chief difference being the delineation of the Australasian realm relative to the Antarctic, Oceanic, and Indomalayan realms. In the WWF system, the Australasia realm includes Australia, Tasmania, the islands of Wallacea, New Guinea, the East Melanesian Islands, New Caledonia, and New Zealand. Udvardy's Australian realm includes only Australia and Tasmania; he places Wallacea in the Indomalayan Realm, New Guinea, New Caledonia, and East Melanesia in the Oceanian Realm, and New Zealand in the Antarctic Realm.

Biogeographic realm	Area		Lands included
	million square kilometres	million square miles
Palearctic	54.1	20.9	The bulk of Eurasia and North Africa.
Nearctic	22.9	8.8	Greenland and most of North America.
Afrotropic	22.1	8.5	Trans-Saharan Africa, Madagascar and Arabia.
Neotropic	19.0	7.3	South America, Central America, the Caribbean, South Florida and the Falkland Islands.
Australasia	7.6	2.9	Australia, Melanesia, New Zealand, Lesser Sunda Islands, Sulawesi and the neighbouring islands. The northern boundary of this zone is known as the Wallace Line.
Indomalaya	7.5	2.9	The Indian subcontinent, Southeast Asia, southern China and most of the Greater Sunda Islands.
Oceania	1.0	0.39	Polynesia (except New Zealand), Micronesia, and the Fijian Islands.
Antarctic	0.3	0.12	Antarctica, Alexander Island, South Georgia and the South Sandwich Islands.

The Palearctic and Nearctic are sometimes grouped into the Holarctic realm.

Morrone biogeographic kingdoms

Following the nomenclatural conventions set out in the International Code of Area Nomenclature, Morrone defined the next biogeographic kingdoms (or realms) and regions:

• Holarctic kingdom Heilprin (1887)
  • Nearctic region Sclater (1858)
  • Palearctic region Sclater (1858)
• Holotropical kingdom Rapoport (1968)
  • Neotropical region Sclater (1858)
  • Ethiopian region Sclater (1858)
  • Oriental region Wallace (1876)
• Austral kingdom Engler (1899)
  • Cape region Grisebach (1872)
  • Andean region Engler (1882)
  • Australian region Sclater (1858)
  • Antarctic region Grisebach (1872)
• Transition zones:
  • Mexican transition zone (Nearctic–Neotropical transition)
  • Saharo-Arabian transition zone (Palearctic–Ethiopian transition)
  • Chinese transition zone (Palearctic–Oriental transition zone transition)
  • Indo-Malayan, Indonesian or Wallace's transition zone (Oriental–Australian transition)
  • South American transition zone (Neotropical–Austral transition)

Freshwater biogeographic realms

Major continental divides, showing drainage into the major oceans and seas of the world – grey areas are endorheic basins that do not drain to the ocean

The applicability of Udvardy scheme to most freshwater taxa is unresolved.

The drainage basins of the principal oceans and seas of the world are marked by continental divides. The grey areas are endorheic basins that do not drain to the ocean.

Marine biogeographic realms

Longhurst biogeographic provinces

 

See also: Longhurst provinces

According to Briggs and Morrone:

• Indo-West Pacific region
• Eastern Pacific region
• Western Atlantic region
• Eastern Atlantic region
• Southern Australian region
• Northern New Zealand region
• Western South America region
• Eastern South America region
• Southern Africa region
• Mediterranean–Atlantic region
• Carolina region
• California region
• Japan region
• Tasmanian region
• Southern New Zealand region
• Antipodean region
• Subantarctic region
• Magellan region
• Eastern Pacific Boreal region
• Western Atlantic Boreal region
• Eastern Atlantic Boreal region
• Antarctic region
• Arctic region

According to the WWF scheme:

• Arctic realm
• Temperate Northern Atlantic realm
• Temperate Northern Pacific realm
• Tropical Atlantic realm
• Western Indo-Pacific realm
• Central Indo-Pacific realm
• Eastern Indo-Pacific realm
• Tropical Eastern Pacific realm
• Temperate South America realm
• Temperate Southern Africa realm
• Temperate Australasia realm
• Southern Ocean realm

Metabolic theory of ecology

From Wikipedia, the free encyclopedia

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

The metabolic theory of ecology (MTE) is the ecological component of the more general Metabolic Scaling Theory and Kleiber's law. It posits that the metabolic rate of organisms is the fundamental biological rate that governs most observed patterns in ecology. MTE is part of a larger set of theory known as metabolic scaling theory that attempts to provide a unified theory for the importance of metabolism in driving pattern and process in biology from the level of cells all the way to the biosphere.

MTE is based on an interpretation of the relationships between body size, body temperature, and metabolic rate across all organisms. Small-bodied organisms tend to have higher mass-specific metabolic rates than larger-bodied organisms. Furthermore, organisms that operate at warm temperatures through endothermy or by living in warm environments tend towards higher metabolic rates than organisms that operate at colder temperatures. This pattern is consistent from the unicellular level up to the level of the largest animals and plants on the planet.

In MTE, this relationship is considered to be the primary constraint that influences biological processes (via their rates and times) at all levels of organization (from individual up to ecosystem level). MTE is a macroecological theory that aims to be universal in scope and application.

Fundamental concepts in MTE

Metabolism

Metabolic pathways consist of complex networks, which are responsible for processing of both energy and material. The metabolic rate of a heterotroph is defined as the rate of respiration in which energy is obtained by oxidation of carbon compound. The rate of photosynthesis on the other hand, indicates the metabolic rate of an autotroph. According to MTE, both body size and temperature affect the metabolic rate of an organism. Metabolic rate scale as 3/4 power of body size, and its relationship with temperature is described by Van’t Hoff-Arrhenius equation over the range of 0 to 40 °C.

Stoichiometry

From the ecological perspective, stoichiometry is concerned with the proportion of elements in both living organisms and their environment. In order to survive and maintain metabolism, an organism must be able to obtain crucial elements and excrete waste products. As a result, the elemental composition of an organism would be different from the exterior environment. Through metabolism, body size can affect stoichiometry. For example, small organism tend to store most of their phosphorus in rRNA due to their high metabolic rate, whereas large organisms mostly invest this element inside the skeletal structure. Thus, concentration of elements to some extent can limit the rate of biological processes. Inside an ecosystem, the rate of flux and turn over of elements by inhabitants, combined with the influence of abiotic factors, determine the concentration of elements.

Theoretical background

Metabolic rate scales with the mass of an organism of a given species according to Kleiber's law where B is whole organism metabolic rate (in watts or other unit of power), M is organism mass (in kg), and B_o is a mass-independent normalization constant (given in a unit of power divided by a unit of mass. In this case, watts per kilogram):

${\displaystyle B=B_o{M}^{3/4}}$

At increased temperatures, chemical reactions proceed faster. This relationship is described by the Boltzmann factor, where E is activation energy in electronvolts or joules, T is absolute temperature in kelvins, and k is the Boltzmann constant in eV/K or J/K:

${\displaystyle e^{-\frac{E}{kT}}}$

While B_o in the previous equation is mass-independent, it is not explicitly independent of temperature. To explain the relationship between body mass and temperature, building on earlier work showing that the effects of both body mass and temperature could be combined multiplicatively in a single equation, the two equations above can be combined to produce the primary equation of the MTE, where b_o is a normalization constant that is independent of body size or temperature:

${\displaystyle B=b_o{M}^{3/4}{e}^{-\frac{E}{kT}}}$

According to this relationship, metabolic rate is a function of an organism's body mass and body temperature. By this equation, large organisms have higher metabolic rates (in watts) than small organisms, and organisms at high body temperatures have higher metabolic rates than those that exist at low body temperatures. However, specific metabolic rate (SMR, in watts/kg) is given by

${\displaystyle SMR=(B/M)=b_o{M}^{-1/4}{e}^{-\frac{E}{kT}}}$

Hence SMR for large organisms are lower than small organisms.

Past debate over mechanisms and the allometric exponent

Researchers have debated two main aspects of this theory, the pattern and the mechanism. Past debated have focused on the question whether metabolic rate scales to the power of 3⁄4 or 2⁄3w, or whether either of these can even be considered a universal exponent. In addition to debates concerning the exponent, some researchers also disagree about the underlying mechanisms generating the scaling exponent. Various authors have proposed at least eight different types of mechanisms that predict an allometric scaling exponent of either 2⁄3 or 3⁄4. The majority view is that while the 3⁄4 exponent is indeed the mean observed exponent within and across taxa, there is intra- and interspecific variability in the exponent that can include shallower exponents such as2⁄3. Past debates on the exact value of the exponent are settled in part because the observed variability in the metabolic scaling exponent is consistent with a 'relaxed' version of metabolic scaling theory where additional selective pressures lead to a constrained set of variation around the predicted optimal 3⁄4 exponent.

Much of past debate have focused on two particular types of mechanisms. One of these assumes energy or resource transport across the external surface area of three-dimensional organisms is the key factor driving the relationship between metabolic rate and body size. The surface area in question may be skin, lungs, intestines, or, in the case of unicellular organisms, cell membranes. In general, the surface area (SA) of a three dimensional object scales with its volume (V) as SA = cV^2⁄3, where c is a proportionality constant. The Dynamic Energy Budget model predicts exponents that vary between 2⁄3 – 1, depending on the organism's developmental stage, basic body plan and resource density. DEB is an alternative to metabolic scaling theory, developed before the MTE. DEB also provides a basis for population, community and ecosystem level processes to be studied based on energetics of the constituent organisms. In this theory, the biomass of the organism is separated into structure (what is built during growth) and reserve (a pool of polymers generated by assimilation). DEB is based on the first principles dictated by the kinetics and thermodynamics of energy and material fluxes, has a similar number of parameters per process as MTE, and the parameters have been estimated for over 3000 animal species "Add my Pet". Retrieved 23 August 2022. While some of these alternative models make several testable predictions, others are less comprehensive  and of these proposed models only DEB can make as many predictions with a minimal set of assumptions as metabolic scaling theory.

In contrast, the arguments for a 3⁄4 scaling factor are based on resource transport network models, where the limiting resources are distributed via some optimized network to all resource consuming cells or organelles. These models are based on the assumption that metabolism is proportional to the rate at which an organism's distribution networks (such as circulatory systems in animals or xylem and phloem in plants) deliver nutrients and energy to body tissues. Larger organisms are necessarily less efficient because more resource is in transport at any one time than in smaller organisms: size of the organism and length of the network imposes an inefficiency due to size. It therefore takes somewhat longer for large organisms to distribute nutrients throughout the body and thus they have a slower mass-specific metabolic rate. An organism that is twice as large cannot metabolize twice the energy—it simply has to run more slowly because more energy and resources are wasted being in transport, rather than being processed. Nonetheless, natural selection appears to have minimized this inefficiency by favoring resource transport networks that maximize rate of delivery of resources to the end points such as cells and organelles. This selection to maximize metabolic rate and energy dissipation results in the allometric exponent that tends to D/(D+1), where D is the primary dimension of the system. A three dimensional system, such as an individual, tends to scale to the 3/4 power, whereas a two dimensional network, such as a river network in a landscape, tends to scale to the 2/3 power.

Despite past debates over the value of the exponent, the implications of metabolic scaling theory and the extensions of the theory to ecology (metabolic theory of ecology) the theory might remain true regardless of its precise numerical value.

Implications of the theory

The metabolic theory of ecology's main implication is that metabolic rate, and the influence of body size and temperature on metabolic rate, provide the fundamental constraints by which ecological processes are governed. If this holds true from the level of the individual up to ecosystem level processes, then life history attributes, population dynamics, and ecosystem processes could be explained by the relationship between metabolic rate, body size, and body temperature. While different underlying mechanisms make somewhat different predictions, the following provides an example of some of the implications of the metabolism of individuals.

Organism level

Small animals tend to grow fast, breed early, and die young. According to MTE, these patterns in life history traits are constrained by metabolism. An organism's metabolic rate determines its rate of food consumption, which in turn determines its rate of growth. This increased growth rate produces trade-offs that accelerate senescence. For example, metabolic processes produce free radicals as a by-product of energy production. These in turn cause damage at the cellular level, which promotes senescence and ultimately death. Selection favors organisms which best propagate given these constraints. As a result, smaller, shorter lived organisms tend to reproduce earlier in their life histories.

Population and community level

MTE has profound implications for the interpretation of population growth and community diversity. Classically, species are thought of as being either r selected (where populations tend to grow exponentially, and are ultimately limited by extrinsic factors) or K selected (where population size is limited by density-dependence and carrying capacity). MTE explains this diversity of reproductive strategies as a consequence of the metabolic constraints of organisms. Small organisms and organisms that exist at high body temperatures tend to be r selected, which fits with the prediction that r selection is a consequence of metabolic rate. Conversely, larger and cooler bodied animals tend to be K selected. The relationship between body size and rate of population growth has been demonstrated empirically, and in fact has been shown to scale to M^−1/4 across taxonomic groups. The optimal population growth rate for a species is therefore thought to be determined by the allometric constraints outlined by the MTE, rather than strictly as a life history trait that is selected for based on environmental conditions.

Regarding density, MTE predicts carrying capacity of populations to scale as M^-3/4, and to exponentially decrease with increasing temperature. The fact that larger organisms reach carrying capacity sooner than smaller one is intuitive, however, temperature can also decrease carrying capacity due to the fact that in warmer environments, higher metabolic rate of organisms demands a higher rate of supply. Empirical evidence in terrestrial plants, also suggests that density scales as -3/4 power of the body size.

Observed patterns of diversity can be similarly explained by MTE. It has long been observed that there are more small species than large species. In addition, there are more species in the tropics than at higher latitudes. Classically, the latitudinal gradient in species diversity has been explained by factors such as higher productivity or reduced seasonality. In contrast, MTE explains this pattern as being driven by the kinetic constraints imposed by temperature on metabolism. The rate of molecular evolution scales with metabolic rate, such that organisms with higher metabolic rates show a higher rate of change at the molecular level. If a higher rate of molecular evolution causes increased speciation rates, then adaptation and ultimately speciation may occur more quickly in warm environments and in small bodied species, ultimately explaining observed patterns of diversity across body size and latitude.

MTE's ability to explain patterns of diversity remains controversial. For example, researchers analyzed patterns of diversity of New World coral snakes to see whether the geographical distribution of species fit within the predictions of MTE (i.e. more species in warmer areas). They found that the observed pattern of diversity could not be explained by temperature alone, and that other spatial factors such as primary productivity, topographic heterogeneity, and habitat factors better predicted the observed pattern. Extensions of metabolic theory to diversity that include eco-evolutionary theory show that an elaborated metabolic theory can account for differences in diversity gradients by including feedbacks between ecological interactions (size-dependent competition and predation) and evolutionary rates (speciation and extinction) 

Ecosystem processes

At the ecosystem level, MTE explains the relationship between temperature and production of total biomass. The average production to biomass ratio of organisms is higher in small organisms than large ones. This relationship is further regulated by temperature, and the rate of production increases with temperature. As production consistently scales with body mass, MTE provides a framework to assess the relative importance of organismal size, temperature, functional traits, soil and climate on variation in rates of production within and across ecosystems. Metabolic theory shows that variation in ecosystem production is characterized by a common scaling relationship, suggesting that global change models can incorporate the mechanisms governing this relationship to improve predictions of future ecosystem function.