Virtual particle

From Wikipedia, the free encyclopedia

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

A virtual particle is a theoretical transient particle that exhibits some of the characteristics of an ordinary particle, while having its existence limited by the uncertainty principle, which allows the virtual particles to spontaneously emerge from vacuum at short time and space ranges. The concept of virtual particles arises in the perturbation theory of quantum field theory (QFT) where interactions between ordinary particles are described in terms of exchanges of virtual particles. A process involving virtual particles can be described by a schematic representation known as a Feynman diagram, in which virtual particles are represented by internal lines.

Virtual particles do not necessarily carry the same mass as the corresponding ordinary particle, although they always conserve energy and momentum. The closer its characteristics come to those of ordinary particles, the longer the virtual particle exists. They are important in the physics of many processes, including particle scattering and Casimir forces. In quantum field theory, forces—such as the electromagnetic repulsion or attraction between two charges—can be thought of as resulting from the exchange of virtual photons between the charges. Virtual photons are the exchange particles for the electromagnetic interaction.

The term is somewhat loose and vaguely defined, in that it refers to the view that the world is made up of "real particles". "Real particles" are better understood to be excitations of the underlying quantum fields. Virtual particles are also excitations of the underlying fields, but are "temporary" in the sense that they appear in calculations of interactions, but never as asymptotic states or indices to the scattering matrix. The accuracy and use of virtual particles in calculations is firmly established, but as they cannot be detected in experiments, deciding how to precisely describe them is a topic of debate. Although widely used, they are by no means a necessary feature of QFT, but rather are mathematical conveniences — as demonstrated by lattice field theory, which avoids using the concept altogether.

Properties

The concept of virtual particles arises in the perturbation theory of quantum field theory, an approximation scheme in which interactions (in essence, forces) between actual particles are calculated in terms of exchanges of virtual particles. Such calculations are often performed using schematic representations known as Feynman diagrams, in which virtual particles appear as internal lines. By expressing the interaction in terms of the exchange of a virtual particle with four-momentum q, where q is given by the difference between the four-momenta of the particles entering and leaving the interaction vertex, both momentum and energy are conserved at the interaction vertices of the Feynman diagram.

A virtual particle does not precisely obey the energy–momentum relation m²c⁴ = E² − p²c². Its kinetic energy may not have the usual relationship to velocity. It can be negative. This is expressed by the phrase off mass shell. The probability amplitude for a virtual particle to exist tends to be canceled out by destructive interference over longer distances and times. As a consequence, a real photon is massless and thus has only two polarization states, whereas a virtual one, being effectively massive, has three polarization states.

Quantum tunnelling may be considered a manifestation of virtual particle exchanges. The range of forces carried by virtual particles is limited by the uncertainty principle, which regards energy and time as conjugate variables; thus, virtual particles of larger mass have more limited range.

Written in the usual mathematical notations, in the equations of physics, there is no mark of the distinction between virtual and actual particles. The amplitudes of processes with a virtual particle interfere with the amplitudes of processes without it, whereas for an actual particle the cases of existence and non-existence cease to be coherent with each other and do not interfere any more. In the quantum field theory view, actual particles are viewed as being detectable excitations of underlying quantum fields. Virtual particles are also viewed as excitations of the underlying fields, but appear only as forces, not as detectable particles. They are "temporary" in the sense that they appear in some calculations, but are not detected as single particles. Thus, in mathematical terms, they never appear as indices to the scattering matrix, which is to say, they never appear as the observable inputs and outputs of the physical process being modelled.

There are two principal ways in which the notion of virtual particles appears in modern physics. They appear as intermediate terms in Feynman diagrams; that is, as terms in a perturbative calculation. They also appear as an infinite set of states to be summed or integrated over in the calculation of a semi-non-perturbative effect. In the latter case, it is sometimes said that virtual particles contribute to a mechanism that mediates the effect, or that the effect occurs through the virtual particles.

Manifestations

There are many observable physical phenomena that arise in interactions involving virtual particles. For bosonic particles that exhibit rest mass when they are free and actual, virtual interactions are characterized by the relatively short range of the force interaction produced by particle exchange. Confinement can lead to a short range, too. Examples of such short-range interactions are the strong and weak forces, and their associated field bosons.

For the gravitational and electromagnetic forces, the zero rest-mass of the associated boson particle permits long-range forces to be mediated by virtual particles. However, in the case of photons, power and information transfer by virtual particles is a relatively short-range phenomenon (existing only within a few wavelengths of the field-disturbance, which carries information or transferred power), as for example seen in the characteristically short range of inductive and capacitative effects in the near field zone of coils and antennas.

Some field interactions which may be seen in terms of virtual particles are:

• The Coulomb force (static electric force) between electric charges. It is caused by the exchange of virtual photons. In symmetric 3-dimensional space this exchange results in the inverse square law for electric force. Since the photon has no mass, the coulomb potential has an infinite range.
• The magnetic field between magnetic dipoles. It is caused by the exchange of virtual photons. In symmetric 3-dimensional space, this exchange results in the inverse cube law for magnetic force. Since the photon has no mass, the magnetic potential has an infinite range. Even though the range is infinite, the time lapse allowed for a virtual photon existence is not infinite.
• Electromagnetic induction. This phenomenon transfers energy to and from a magnetic coil via a changing (electro)magnetic field.
• The strong nuclear force between quarks is the result of interaction of virtual gluons. The residual of this force outside of quark triplets (neutron and proton) holds neutrons and protons together in nuclei, and is due to virtual mesons such as the pi meson and rho meson.
• The weak nuclear force is the result of exchange by virtual W and Z bosons.
• The spontaneous emission of a photon during the decay of an excited atom or excited nucleus; such a decay is prohibited by ordinary quantum mechanics and requires the quantization of the electromagnetic field for its explanation.
• The Casimir effect, where the ground state of the quantized electromagnetic field causes attraction between a pair of electrically neutral metal plates.
• The van der Waals force, which is partly due to the Casimir effect between two atoms.
• Vacuum polarization, which involves pair production or the decay of the vacuum, which is the spontaneous production of particle-antiparticle pairs (such as electron-positron).
• Lamb shift of positions of atomic levels.
• The impedance of free space, which defines the ratio between the electric field strength |E| and the magnetic field strength |H|: Z₀ = |E| / |H|.
• Much of the so-called near-field of radio antennas, where the magnetic and electric effects of the changing current in the antenna wire and the charge effects of the wire's capacitive charge may be (and usually are) important contributors to the total EM field close to the source, but both of which effects are dipole effects that decay with increasing distance from the antenna much more quickly than do the influence of "conventional" electromagnetic waves that are "far" from the source. These far-field waves, for which E is (in the limit of long distance) equal to cB, are composed of actual photons. Actual and virtual photons are mixed near an antenna, with the virtual photons responsible only for the "extra" magnetic-inductive and transient electric-dipole effects, which cause any imbalance between E and cB. As distance from the antenna grows, the near-field effects (as dipole fields) die out more quickly, and only the "radiative" effects that are due to actual photons remain as important effects. Although virtual effects extend to infinity, they drop off in field strength as 1/r² rather than the field of EM waves composed of actual photons, which drop as 1/r.

Most of these have analogous effects in solid-state physics; indeed, one can often gain a better intuitive understanding by examining these cases. In semiconductors, the roles of electrons, positrons and photons in field theory are replaced by electrons in the conduction band, holes in the valence band, and phonons or vibrations of the crystal lattice. A virtual particle is in a virtual state where the probability amplitude is not conserved. Examples of macroscopic virtual phonons, photons, and electrons in the case of the tunneling process were presented by Günter Nimtz and Alfons A. Stahlhofen.

Feynman diagrams

One particle exchange scattering diagram

The calculation of scattering amplitudes in theoretical particle physics requires the use of some rather large and complicated integrals over a large number of variables. These integrals do, however, have a regular structure, and may be represented as Feynman diagrams. The appeal of the Feynman diagrams is strong, as it allows for a simple visual presentation of what would otherwise be a rather arcane and abstract formula. In particular, part of the appeal is that the outgoing legs of a Feynman diagram can be associated with actual, on-shell particles. Thus, it is natural to associate the other lines in the diagram with particles as well, called the "virtual particles". In mathematical terms, they correspond to the propagators appearing in the diagram.

In the adjacent image, the solid lines correspond to actual particles (of momentum p₁ and so on), while the dotted line corresponds to a virtual particle carrying momentum k. For example, if the solid lines were to correspond to electrons interacting by means of the electromagnetic interaction, the dotted line would correspond to the exchange of a virtual photon. In the case of interacting nucleons, the dotted line would be a virtual pion. In the case of quarks interacting by means of the strong force, the dotted line would be a virtual gluon, and so on.

One-loop diagram with fermion propagator

Virtual particles may be mesons or vector bosons, as in the example above; they may also be fermions. However, in order to preserve quantum numbers, most simple diagrams involving fermion exchange are prohibited. The image to the right shows an allowed diagram, a one-loop diagram. The solid lines correspond to a fermion propagator, the wavy lines to bosons.

Vacuums

Main articles: Quantum fluctuation, QED vacuum, QCD vacuum, and Vacuum state

In formal terms, a particle is considered to be an eigenstate of the particle number operator a^†a, where a is the particle annihilation operator and a^† the particle creation operator (sometimes collectively called ladder operators). In many cases, the particle number operator does not commute with the Hamiltonian for the system. This implies the number of particles in an area of space is not a well-defined quantity but, like other quantum observables, is represented by a probability distribution. Since these particles are not certain to exist, they are called virtual particles or vacuum fluctuations of vacuum energy. In a certain sense, they can be understood to be a manifestation of the time-energy uncertainty principle in a vacuum.

An important example of the "presence" of virtual particles in a vacuum is the Casimir effect. Here, the explanation of the effect requires that the total energy of all of the virtual particles in a vacuum can be added together. Thus, although the virtual particles themselves are not directly observable in the laboratory, they do leave an observable effect: Their zero-point energy results in forces acting on suitably arranged metal plates or dielectrics. On the other hand, the Casimir effect can be interpreted as the relativistic van der Waals force.

Pair production

Main article: Pair production

Virtual particles are often popularly described as coming in pairs, a particle and antiparticle, which can be of any kind. These pairs exist for an extremely short time, and then mutually annihilate, or in some cases, the pair may be boosted apart using external energy so that they avoid annihilation and become actual particles, as described below.

This may occur in one of two ways. In an accelerating frame of reference, the virtual particles may appear to be actual to the accelerating observer; this is known as the Unruh effect. In short, the vacuum of a stationary frame appears, to the accelerated observer, to be a warm gas of actual particles in thermodynamic equilibrium.

Another example is pair production in very strong electric fields, sometimes called vacuum decay. If, for example, a pair of atomic nuclei are merged to very briefly form a nucleus with a charge greater than about 140, (that is, larger than about the inverse of the fine-structure constant, which is a dimensionless quantity), the strength of the electric field will be such that it will be energetically favorable to create positron–electron pairs out of the vacuum or Dirac sea, with the electron attracted to the nucleus to annihilate the positive charge. This pair-creation amplitude was first calculated by Julian Schwinger in 1951.

Compared to actual particles

As a consequence of quantum mechanical uncertainty, any object or process that exists for a limited time or in a limited volume cannot have a precisely defined energy or momentum. For this reason, virtual particles – which exist only temporarily as they are exchanged between ordinary particles – do not typically obey the mass-shell relation; the longer a virtual particle exists, the more the energy and momentum approach the mass-shell relation.

The lifetime of real particles is typically vastly longer than the lifetime of the virtual particles. Electromagnetic radiation consists of real photons which may travel light years between the emitter and absorber, but (Coulombic) electrostatic attraction and repulsion is a relatively short-range force that is a consequence of the exchange of virtual photons.

Exponential growth

From Wikipedia, the free encyclopedia

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

The graph illustrates how exponential growth (green) eventually surpasses both linear (red) and cubic (blue) growth.

  Linear growth

  Cubic growth

  Exponential growth

Exponential growth occurs when a quantity grows as an exponential function of time. The quantity grows at a rate directly proportional to its present size. For example, when it is 3 times as big as it is now, it will be growing 3 times as fast as it is now.

In more technical language, its instantaneous rate of change (that is, the derivative) of a quantity with respect to an independent variable is proportional to the quantity itself. Often the independent variable is time. Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as quadratic growth). Exponential growth is the inverse of logarithmic growth.

Terms like "exponential growth" are sometimes incorrectly interpreted as "rapid growth." Indeed, something that grows exponentially can in fact be growing slowly at first. Also, not all cases of growth at an always increasing rate are instances of exponential growth. For example the function $f(x)=x^3$ grows at an ever increasing rate, but is much slower than growing exponentially. For example, when $x=1,$ it grows at 3 times its size, but when $x=10$ it grows at 30% of its size. If an exponentially growing function grows at a rate that is 3 times its present size, then it always grows at a rate that is 3 times its present size: when it is 10 times as big as it is now, it will grow 10 times as fast. On the other hand, functions may also grow faster than exponentially, for instance, the factorial or double exponential function.

If the constant of proportionality is negative, then the quantity decreases over time, and is said to be undergoing exponential decay instead. In the case of a discrete domain of definition with equal intervals, it is also called geometric growth or geometric decay since the function values form a geometric progression.

The formula for exponential growth of a variable x at the growth rate r, as time t goes on in discrete intervals (that is, at integer times 0, 1, 2, 3, ...), is

$${\displaystyle x_t=x_0(1+r{)}^t}$$

where x₀ is the value of x at time 0. The growth of a bacterial colony is often used to illustrate it. One bacterium splits itself into two, each of which splits itself resulting in four, then eight, 16, 32, and so on. The amount of increase keeps increasing because it is proportional to the ever-increasing number of bacteria. Growth like this is observed in real-life activity or phenomena, such as the spread of virus infection, the growth of debt due to compound interest, and the spread of viral videos. In real cases, initial exponential growth often does not last forever, instead slowing down eventually due to upper limits caused by external factors and turning into logistic growth.

Examples

Bacteria exhibit exponential growth under optimal conditions.

Biology

• The number of microorganisms in a culture will increase exponentially until an essential nutrient is exhausted, so there is no more of that nutrient for more organisms to grow. Typically the first organism splits into two daughter organisms, who then each split to form four, who split to form eight, and so on. Because exponential growth indicates constant growth rate, it is frequently assumed that exponentially growing cells are at a steady-state. However, cells can grow exponentially at a constant rate while remodeling their metabolism and gene expression.
• A virus (for example COVID-19, or smallpox) typically will spread exponentially at first, if no artificial immunization is available. Each infected person can infect multiple new people.

Physical sciences

Wind damage in cyclones/hurricanes varies exponentially with wind speed, so that small increases in wind strength can dramatically increase damage.

• Avalanche breakdown within a dielectric material. A free electron becomes sufficiently accelerated by an externally applied electrical field that it frees up additional electrons as it collides with atoms or molecules of the dielectric media. These secondary electrons also are accelerated, creating larger numbers of free electrons. The resulting exponential growth of electrons and ions may rapidly lead to complete dielectric breakdown of the material.
• Nuclear chain reaction (the concept behind nuclear reactors and nuclear weapons). Each uranium nucleus that undergoes fission produces multiple neutrons, each of which can be absorbed by adjacent uranium atoms, causing them to fission in turn. If the probability of neutron absorption exceeds the probability of neutron escape (a function of the shape and mass of the uranium), the production rate of neutrons and induced uranium fissions increases exponentially, in an uncontrolled reaction. "Due to the exponential rate of increase, at any point in the chain reaction 99% of the energy will have been released in the last 4.6 generations. It is a reasonable approximation to think of the first 53 generations as a latency period leading up to the actual explosion, which only takes 3–4 generations."
• Positive feedback within the linear range of electrical or electroacoustic amplification can result in the exponential growth of the amplified signal, although resonance effects may favor some component frequencies of the signal over others. This is the mechanism behing the super-regenerative receiver.

Economics

• Economic growth is expressed in percentage terms, implying exponential growth.

Finance

• Compound interest at a constant interest rate provides exponential growth of the capital. See also rule of 72.
• Pyramid schemes or Ponzi schemes also show this type of growth resulting in high profits for a few initial investors and losses among great numbers of investors.

Computer science

• Processing power of computers. See also Moore's law and technological singularity. (Under exponential growth, there are no singularities. The singularity here is a metaphor, meant to convey an unimaginable future. The link of this hypothetical concept with exponential growth is most vocally made by futurist Ray Kurzweil.)
• In computational complexity theory, computer algorithms of exponential complexity require an exponentially increasing amount of resources (e.g. time, computer memory) for only a constant increase in problem size. So for an algorithm of time complexity 2^x, if a problem of size x = 10 requires 10 seconds to complete, and a problem of size x = 11 requires 20 seconds, then a problem of size x = 12 will require 40 seconds. This kind of algorithm typically becomes unusable at very small problem sizes, often between 30 and 100 items (most computer algorithms need to be able to solve much larger problems, up to tens of thousands or even millions of items in reasonable times, something that would be physically impossible with an exponential algorithm). Also, the effects of Moore's Law do not help the situation much because doubling processor speed merely increases the feasible problem size by a constant. E.g. if a slow processor can solve problems of size x in time t, then a processor twice as fast could only solve problems of size x + constant in the same time t. So exponentially complex algorithms are most often impractical, and the search for more efficient algorithms is one of the central goals of computer science today.

Internet phenomena

• Internet contents, such as internet memes or videos, can spread in an exponential manner, often said to "go viral" as an analogy to the spread of viruses. With media such as social networks, one person can forward the same content to many people simultaneously, who then spread it to even more people, and so on, causing rapid spread. For example, the video Gangnam Style was uploaded to YouTube on 15 July 2012, reaching hundreds of thousands of viewers on the first day, millions on the twentieth day, and was cumulatively viewed by hundreds of millions in less than two months.

Basic formula

Exponential growth:
${\displaystyle \begin{array}{rl}a& =3\\ b& =2\\ r& =5\end{array}}$

Exponential decay:
${\displaystyle \begin{array}{rl}a& =24\\ b& =\frac{1}{2}\\ r& =5\end{array}}$

A quantity x depends exponentially on time t if $${\displaystyle x(t)=a\cdot b^{t/\tau }}$$ where the constant a is the initial value of x, $${\displaystyle x(0)=a,}$$ the constant b is a positive growth factor, and τ is the time constant—the time required for x to increase by one factor of b: $${\displaystyle x(t+\tau )=a\cdot b^{(t+\tau )/\tau }=a\cdot b^{t/\tau }\cdot b^{\tau /\tau }=x(t)\cdot b.}$$

If τ > 0 and b > 1, then x has exponential growth. If τ < 0 and b > 1, or τ > 0 and 0 < b < 1, then x has exponential decay.

Example: If a species of bacteria doubles every ten minutes, starting out with only one bacterium, how many bacteria would be present after one hour? The question implies a = 1, b = 2 and τ = 10 min.

$${\displaystyle x(t)=a\cdot b^{t/\tau }=1\cdot 2^{t/(10\text{ min})}}$$ $${\displaystyle x(1\text{ hr})=1\cdot 2^{(60\text{ min})/(10\text{ min})}=1\cdot 2^6=64.}$$

After one hour, or six ten-minute intervals, there would be sixty-four bacteria.

Many pairs (b, τ) of a dimensionless non-negative number b and an amount of time τ (a physical quantity which can be expressed as the product of a number of units and a unit of time) represent the same growth rate, with τ proportional to log b. For any fixed b not equal to 1 (e.g. e or 2), the growth rate is given by the non-zero time τ. For any non-zero time τ the growth rate is given by the dimensionless positive number b.

Thus the law of exponential growth can be written in different but mathematically equivalent forms, by using a different base. The most common forms are the following: $${\displaystyle x(t)=x_0\cdot e^{kt}=x_0\cdot e^{t/\tau }=x_0\cdot 2^{t/T}=x_0\cdot {\left(1+\frac{r}{100}\right)}^{t/p},}$$ where x₀ expresses the initial quantity x(0).

Parameters (negative in the case of exponential decay):

• The growth constant k is the frequency (number of times per unit time) of growing by a factor e; in finance it is also called the logarithmic return, continuously compounded return, or force of interest.
• The e-folding time τ is the time it takes to grow by a factor e.
• The doubling time T is the time it takes to double.
• The percent increase r (a dimensionless number) in a period p.

The quantities k, τ, and T, and for a given p also r, have a one-to-one connection given by the following equation (which can be derived by taking the natural logarithm of the above): $${\displaystyle k=\frac{1}{\tau }=\frac{\ln 2}{T}=\frac{\ln \left(1+\frac{r}{100}\right)}{p}}$$ where k = 0 corresponds to r = 0 and to τ and T being infinite.

If p is the unit of time the quotient t/p is simply the number of units of time. Using the notation t for the (dimensionless) number of units of time rather than the time itself, t/p can be replaced by t, but for uniformity this has been avoided here. In this case the division by p in the last formula is not a numerical division either, but converts a dimensionless number to the correct quantity including unit.

A popular approximated method for calculating the doubling time from the growth rate is the rule of 70, that is, ${\displaystyle T\simeq 70/r}$.

Graphs comparing doubling times and half lives of exponential growths (bold lines) and decay (faint lines), and their 70/t and 72/t approximations. In the SVG version, hover over a graph to highlight it and its complement.

Reformulation as log-linear growth

If a variable x exhibits exponential growth according to ${\displaystyle x(t)=x_0(1+r{)}^t}$, then the log (to any base) of x grows linearly over time, as can be seen by taking logarithms of both sides of the exponential growth equation: $${\displaystyle \log x(t)=\log x_0+t\cdot \log (1+r).}$$

This allows an exponentially growing variable to be modeled with a log-linear model. For example, if one wishes to empirically estimate the growth rate from intertemporal data on x, one can linearly regress log x on t.

Differential equation

The exponential function ${\displaystyle x(t)=x_0{e}^{kt}}$ satisfies the linear differential equation: $${\displaystyle \frac{dx}{dt}=kx}$$ saying that the change per instant of time of x at time t is proportional to the value of x(t), and x(t) has the initial value ${\displaystyle x(0)=x_0}$.

The differential equation is solved by direct integration: $${\displaystyle \begin{array}{rl}\frac{dx}{dt}& =kx\\ \frac{dx}{x}& =kdt\\ {\int }_{x_0}^{x(t)}\frac{dx}{x}& =k{\int }_0^{t}dt\\ \ln \frac{x(t)}{x_0}& =kt.\end{array}}$$ so that $${\displaystyle x(t)=x_0{e}^{kt}.}$$

In the above differential equation, if k < 0, then the quantity experiences exponential decay.

For a nonlinear variation of this growth model see logistic function.

Other growth rates

In the long run, exponential growth of any kind will overtake linear growth of any kind (that is the basis of the Malthusian catastrophe) as well as any polynomial growth, that is, for all α: $${\displaystyle \underset{t\to \mathrm{\infty }}{lim}\frac{t^{\alpha }}{a{e}^t}=0.}$$

There is a whole hierarchy of conceivable growth rates that are slower than exponential and faster than linear (in the long run). See Degree of a polynomial § Computed from the function values.

Growth rates may also be faster than exponential. In the most extreme case, when growth increases without bound in finite time, it is called hyperbolic growth. In between exponential and hyperbolic growth lie more classes of growth behavior, like the hyperoperations beginning at tetration, and ${\displaystyle A(n,n)}$, the diagonal of the Ackermann function.

Logistic growth

The J-shaped exponential growth (left, blue) and the S-shaped logistic growth (right, red)

Main article: Logistic curve

In reality, initial exponential growth is often not sustained forever. After some period, it will be slowed by external or environmental factors. For example, population growth may reach an upper limit due to resource limitations. In 1845, the Belgian mathematician Pierre François Verhulst first proposed a mathematical model of growth like this, called the "logistic growth".

Limitations of models

Exponential growth models of physical phenomena only apply within limited regions, as unbounded growth is not physically realistic. Although growth may initially be exponential, the modelled phenomena will eventually enter a region in which previously ignored negative feedback factors become significant (leading to a logistic growth model) or other underlying assumptions of the exponential growth model, such as continuity or instantaneous feedback, break down.

Further information: Limits to Growth, Malthusian catastrophe, and Apparent infection rate

Exponential growth bias

Studies show that human beings have difficulty understanding exponential growth. Exponential growth bias is the tendency to underestimate compound growth processes. This bias can have financial implications as well.

Rice on a chessboard

See also: Wheat and chessboard problem

According to legend, vizier Sissa Ben Dahir presented an Indian King Sharim with a beautiful handmade chessboard. The king asked what he would like in return for his gift and the courtier surprised the king by asking for one grain of rice on the first square, two grains on the second, four grains on the third, and so on. The king readily agreed and asked for the rice to be brought. All went well at first, but the requirement for 2ⁿ⁻¹ grains on the nth square demanded over a million grains on the 21st square, more than a million million (a.k.a. trillion) on the 41st and there simply was not enough rice in the whole world for the final squares. (From Swirski, 2006)

The "second half of the chessboard" refers to the time when an exponentially growing influence is having a significant economic impact on an organization's overall business strategy.

Water lily

French children are offered a riddle, which appears to be an aspect of exponential growth: "the apparent suddenness with which an exponentially growing quantity approaches a fixed limit". The riddle imagines a water lily plant growing in a pond. The plant doubles in size every day and, if left alone, it would smother the pond in 30 days killing all the other living things in the water. Day after day, the plant's growth is small, so it is decided that it won't be a concern until it covers half of the pond. Which day will that be? The 29th day, leaving only one day to save the pond.