Exponential Growth and Decay

Exponential Growth

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

Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. 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).

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

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 rate 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.

Terms like 'exponential growth' are sometimes incorrectly interpreted as 'rapid growth'. Indeed, something that is growing exponentially can in fact be growing slowly.

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. 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.

Physics

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.

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.
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 1=x = 10 requires 10 seconds to complete, and a problem of size 1=x = 11 requires 20 seconds, then a problem of size 1=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 allows you to increase the 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:

{\begin{aligned}a&=3\\b&=2\\r&=5\end{aligned}}

exponential growth:

{\begin{aligned}a&=24\\b&={\frac {1}{2}}\\r&=5\end{aligned}}

A quantity x depends exponentially on time t if

x(t)=a\cdot b^{t/\tau }

where the constant a is the initial value of x,

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:

x(t+\tau )=a\cdot b^{\frac {t+\tau }{\tau }}=a\cdot b^{\frac {t}{\tau }}\cdot b^{\frac {\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.

x(t)=a\cdot b^{t/\tau }=1\cdot 2^{(60{\text{ min}})/(10{\text{ min}})}

x(1{\text{ hr}})=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:

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):

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, $T\simeq 70/r$ .

Reformulation as log-linear growth

If a variable x exhibits exponential growth according to $x(t)=x_{0}(1+r)^{t}$ , 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:

\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 $x(t)=x(0)e^{kt}$ satisfies the linear differential equation:

{\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 $x(0)$ .

The differential equation is solved by direct integration:

{\begin{aligned}{\frac {dx}{dt}}&=kx\\[5pt]{\frac {dx}{x}}&=k\,dt\\[5pt]\int _{x(0)}^{x(t)}{\frac {dx}{x}}&=k\int _{0}^{t}\,dt\\[5pt]\ln {\frac {x(t)}{x(0)}}&=kt.\end{aligned}}

so that

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:

\lim _{t\rightarrow \infty }{t^{\alpha } \over ae^{t}}=0.

There is a whole hierarchy of conceivable growth rates that are slower than exponential and faster than linear (in the long run).

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 $A(n,n)$ , the diagonal of the Ackermann function.

Logistic growth

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

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.

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.

Below are some stories that emphasize this bias.

Rice on a chessboard

According to an old 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, etc. 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 is 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.

Exponential Decay

A quantity undergoing exponential decay. Larger decay constants make the quantity vanish much more rapidly. This plot shows decay for decay constant (λ) of 25, 5, 1, 1/5, and 1/25 for x from 0 to 5.

A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where N is the quantity and λ (lambda) is a positive rate called the exponential decay constant:

{\frac {dN}{dt}}=-\lambda N.

The solution to this equation is:

N(t)=N_{0}e^{-\lambda t},

where N(t) is the quantity at time t, N₀ = N(0) is the initial quantity, that is, the quantity at time t = 0, and the constant λ is called the decay constant, disintegration constant, rate constant, or transformation constant.

Measuring rates of decay

Mean lifetime

If the decaying quantity, N(t), is the number of discrete elements in a certain set, it is possible to compute the average length of time that an element remains in the set. This is called the mean lifetime (or simply the lifetime), where the exponential time constant, $\tau$ , relates to the decay rate, λ, in the following way:

\tau ={\frac {1}{\lambda }}.

The mean lifetime can be looked at as a "scaling time", because the exponential decay equation can be written in terms of the mean lifetime, $\tau$ , instead of the decay constant, λ:

N(t)=N_{0}e^{-t/\tau },

and that $\tau$ is the time at which the population of the assembly is reduced to 1/e ≈ 0.367879441 times its initial value.

For example, if the initial population of the assembly, N(0), is 1000, then the population at time $\tau$ , $N(\tau )$ , is 368.

A very similar equation will be seen below, which arises when the base of the exponential is chosen to be 2, rather than e. In that case the scaling time is the "half-life".

Half-life

A more intuitive characteristic of exponential decay for many people is the time required for the decaying quantity to fall to one half of its initial value. (If N(t) is discrete, then this is the median life-time rather than the mean life-time.) This time is called the half-life, and often denoted by the symbol t_1/2. The half-life can be written in terms of the decay constant, or the mean lifetime, as:

t_{1/2}={\frac {\ln(2)}{\lambda }}=\tau \ln(2).

When this expression is inserted for $\tau$ in the exponential equation above, and ln 2 is absorbed into the base, this equation becomes:

N(t)=N_{0}2^{-t/t_{1/2}}.

Thus, the amount of material left is 2⁻¹ = 1/2 raised to the (whole or fractional) number of half-lives that have passed. Thus, after 3 half-lives there will be 1/2³ = 1/8 of the original material left.

Therefore, the mean lifetime $\tau$ is equal to the half-life divided by the natural log of 2, or:

\tau ={\frac {t_{1/2}}{\ln(2)}}\approx 1.44\cdot t_{1/2}.

For example, polonium-210 has a half-life of 138 days, and a mean lifetime of 200 days.

Solution of the differential equation

The equation that describes exponential decay is

{\frac {dN}{dt}}=-\lambda N

or, by rearranging (applying the technique called separation of variables),

{\frac {dN}{N}}=-\lambda dt.

Integrating, we have

\ln N=-\lambda t+C\,

where C is the constant of integration, and hence

N(t)=e^{C}e^{-\lambda t}=N_{0}e^{-\lambda t}\,

where the final substitution, N₀ = e^C, is obtained by evaluating the equation at t = 0, as N₀ is defined as being the quantity at t = 0.

This is the form of the equation that is most commonly used to describe exponential decay. Any one of decay constant, mean lifetime, or half-life is sufficient to characterize the decay. The notation λ for the decay constant is a remnant of the usual notation for an eigenvalue. In this case, λ is the eigenvalue of the negative of the differential operator with N(t) as the corresponding eigenfunction. The units of the decay constant are s⁻¹.

Derivation of the mean lifetime

Given an assembly of elements, the number of which decreases ultimately to zero, the mean lifetime, $\tau$ , (also called simply the lifetime) is the expected value of the amount of time before an object is removed from the assembly. Specifically, if the individual lifetime of an element of the assembly is the time elapsed between some reference time and the removal of that element from the assembly, the mean lifetime is the arithmetic mean of the individual lifetimes.

Starting from the population formula

N=N_{0}e^{-\lambda t},\,

first let c be the normalizing factor to convert to a probability density function:

1=\int _{0}^{\infty }c\cdot N_{0}e^{-\lambda t}\,dt=c\cdot {\frac {N_{0}}{\lambda }}

or, on rearranging,

c={\frac {\lambda }{N_{0}}}.

Exponential decay is a scalar multiple of the exponential distribution (i.e. the individual lifetime of each object is exponentially distributed), which has a well-known expected value. We can compute it here using integration by parts.

\tau =\langle t\rangle =\int _{0}^{\infty }t\cdot c\cdot N_{0}e^{-\lambda t}\,dt=\int _{0}^{\infty }\lambda te^{-\lambda t}\,dt={\frac {1}{\lambda }}.

Decay by two or more processes

A quantity may decay via two or more different processes simultaneously. In general, these processes (often called "decay modes", "decay channels", "decay routes" etc.) have different probabilities of occurring, and thus occur at different rates with different half-lives, in parallel. The total decay rate of the quantity N is given by the sum of the decay routes; thus, in the case of two processes:

-{\frac {dN(t)}{dt}}=N\lambda _{1}+N\lambda _{2}=(\lambda _{1}+\lambda _{2})N.

The solution to this equation is given in the previous section, where the sum of $\lambda _{1}+\lambda _{2}\,$ is treated as a new total decay constant $\lambda _{c}$ .

N(t)=N_{0}e^{-(\lambda _{1}+\lambda _{2})t}=N_{0}e^{-(\lambda _{c})t}.

Partial mean life associated with individual processes is by definition the multiplicative inverse of corresponding partial decay constant: $\tau =1/\lambda$ . A combined $\tau _{c}$ can be given in terms of $\lambda$ s:

{\frac {1}{\tau _{c}}}=\lambda _{c}=\lambda _{1}+\lambda _{2}={\frac {1}{\tau _{1}}}+{\frac {1}{\tau _{2}}}

\tau _{c}={\frac {\tau _{1}\tau _{2}}{\tau _{1}+\tau _{2}}}.

Since half-lives differ from mean life $\tau$ by a constant factor, the same equation holds in terms of the two corresponding half-lives:

T_{1/2}={\frac {t_{1}t_{2}}{t_{1}+t_{2}}}

where $T_{1/2}$ is the combined or total half-life for the process, $t_{1}$ and $t_{2}$ are so-named partial half-lives of corresponding processes. Terms "partial half-life" and "partial mean life" denote quantities derived from a decay constant as if the given decay mode were the only decay mode for the quantity. The term "partial half-life" is misleading, because it cannot be measured as a time interval for which a certain quantity is halved.

In terms of separate decay constants, the total half-life $T_{1/2}$ can be shown to be

T_{1/2}={\frac {\ln 2}{\lambda _{c}}}={\frac {\ln 2}{\lambda _{1}+\lambda _{2}}}.

For a decay by three simultaneous exponential processes the total half-life can be computed as above:

T_{1/2}={\frac {\ln 2}{\lambda _{c}}}={\frac {\ln 2}{\lambda _{1}+\lambda _{2}+\lambda _{3}}}={\frac {t_{1}t_{2}t_{3}}{(t_{1}t_{2})+(t_{1}t_{3})+(t_{2}t_{3})}}.

Decay series / coupled decay

In nuclear science and pharmacokinetics, the agent of interest might be situated in a decay chain, where the accumulation is governed by exponential decay of a source agent, while the agent of interest itself decays by means of an exponential process.

These systems are solved using the Bateman equation.

In the pharmacology setting, some ingested substances might be absorbed into the body by a process reasonably modeled as exponential decay, or might be deliberately formulated to have such a release profile.

Applications and examples

Exponential decay occurs in a wide variety of situations. Most of these fall into the domain of the natural sciences.

Many decay processes that are often treated as exponential, are really only exponential so long as the sample is large and the law of large numbers holds. For small samples, a more general analysis is necessary, accounting for a Poisson process.

Natural sciences

Chemical reactions: The rates of certain types of chemical reactions depend on the concentration of one or another reactant. Reactions whose rate depends only on the concentration of one reactant (known as first-order reactions) consequently follow exponential decay. For instance, many enzyme-catalyzed reactions behave this way.
Electrostatics: The electric charge (or, equivalently, the potential) contained in a capacitor (capacitance C) changes exponentially, if the capacitor experiences a constant external load (resistance R). The exponential time-constant τ for the process is R C, and the half-life is therefore R C ln2. This applies to both charging and discharging, i.e. a capacitor charges or discharges according to the same law. The same equations can be applied to the current in an inductor. (Furthermore, the particular case of a capacitor or inductor changing through several parallel resistors makes an interesting example of multiple decay processes, with each resistor representing a separate process. In fact, the expression for the equivalent resistance of two resistors in parallel mirrors the equation for the half-life with two decay processes.)
Geophysics: Atmospheric pressure decreases approximately exponentially with increasing height above sea level, at a rate of about 12% per 1000m.
Heat transfer: If an object at one temperature is exposed to a medium of another temperature, the temperature difference between the object and the medium follows exponential decay (in the limit of slow processes; equivalent to "good" heat conduction inside the object, so that its temperature remains relatively uniform through its volume).
Luminescence: After excitation, the emission intensity – which is proportional to the number of excited atoms or molecules – of a luminescent material decays exponentially. Depending on the number of mechanisms involved, the decay can be mono- or multi-exponential.
Pharmacology and toxicology: It is found that many administered substances are distributed and metabolized (see clearance) according to exponential decay patterns. The biological half-lives "alpha half-life" and "beta half-life" of a substance measure how quickly a substance is distributed and eliminated.
Physical optics: The intensity of electromagnetic radiation such as light or X-rays or gamma rays in an absorbent medium, follows an exponential decrease with distance into the absorbing medium. This is known as the Beer-Lambert law.
Radioactivity: In a sample of a radionuclide that undergoes radioactive decay to a different state, the number of atoms in the original state follows exponential decay as long as the remaining number of atoms is large. The decay product is termed a radiogenic nuclide.
Thermoelectricity: The decline in resistance of a Negative Temperature Coefficient Thermistor as temperature is increased.
Vibrations: Some vibrations may decay exponentially; this characteristic is often found in damped mechanical oscillators, and used in creating ADSR envelopes in synthesizers. An overdamped system will simply return to equilibrium via an exponential decay.
Beer froth: Arnd Leike, of the Ludwig Maximilian University of Munich, won an Ig Nobel Prize for demonstrating that beer froth obeys the law of exponential decay.

Social sciences

Finance: a retirement fund will decay exponentially being subject to discrete payout amounts, usually monthly, and an input subject to a continuous interest rate. A differential equation dA/dt = input – output can be written and solved to find the time to reach any amount A, remaining in the fund.
In simple glottochronology, the (debatable) assumption of a constant decay rate in languages allows one to estimate the age of single languages. (To compute the time of split between two languages requires additional assumptions, independent of exponential decay).

Computer science

The core routing protocol on the Internet, BGP, has to maintain a routing table in order to remember the paths a packet can be deviated to. When one of these paths repeatedly changes its state from available to not available (and vice versa), the BGP router controlling that path has to repeatedly add and remove the path record from its routing table (flaps the path), thus spending local resources such as CPU and RAM and, even more, broadcasting useless information to peer routers. To prevent this undesired behavior, an algorithm named route flapping damping assigns each route a weight that gets bigger each time the route changes its state and decays exponentially with time. When the weight reaches a certain limit, no more flapping is done, thus suppressing the route.

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.