A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. Symbolically, this can be expressed as the following differential equation, where N is the quantity and λ is a positive number called the decay constant: An illustration of a differential equation. ... A negative number is a number that is less than zero, such as −3. ...

The solution to this equation is [1]:

Here N(t) is the quantity at time t, and N₀ = N(0) is the (initial) quantity, at time t=0.

This is the form of the equation that is most commonly used to describe exponential decay. The constant of integration N₀ denotes the original quantity at t = 0. (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 opposite of the differentiation operator with N(t) as the corresponding eigenfunction). In calculus, the indefinite integral of a given function (i. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... The additive inverse, or opposite, of a number n is the number which, when added to n, yields zero. ... In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ... In this transformation of the Mona Lisa, the blue vector has been rotated, but the red one has not. ...

Large decay constants make the quantity vanish almost immediately; smaller decay constants lead to almost-imperceptible decrease. This plot shows decay for decay constants of 25, 5, 1, 1/5, and 1/25. Plot generated using MAPLE.

Image File history File links Download high-resolution version (861x577, 8 KB) This is a plot of exponential decay for several different decay constants File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Exponential decay ... Image File history File links Download high-resolution version (861x577, 8 KB) This is a plot of exponential decay for several different decay constants File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Exponential decay ... Maple 9. ...

Measuring rates of decay

Mean lifetime

If the decaying quantity is the number of discrete elements of a set, it is possible to compute the average length of time for which an element remains in the set. This is called the mean lifetime, and it can be shown that it relates to the decay rate, In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... Given an assembly of elements, the number of which decreases ultimately to zero, the lifetime (also called the mean lifetime) is a certain number that characterizes the rate of reduction (decay) of the assembly. ...

The mean lifetime (also called the exponential time constant) is thus seen to be a simple "scaling time": In physics and engineering, the time constant, usually denoted by the Greek letter Ï„ (tau), characterizes the frequency response of a first-order, linear time-invariant (LTI) system. ...

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. 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: Half-Life For a quantity subject to exponential decay, the half-life is the time required for the quantity to fall to half of its initial value. ...

When this expression is inserted for τ in the exponential equation above, and ln2 is absorbed into the base, this equation becomes:

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.

Solution of the differential equation

The equation that describes exponential decay is

Integrating, we have

where D is the constant of integration.

where C = e^D. If we evaluate this equation at t = 0, we see that e^D = C = N₀.

Decay by two or more processes

A quantity may decay via two or more different processes simultaneously. These processes may have different probabilities of occurring, and thus will occur at different rates with different half-lives, in parallel. In the case of two simultaneous decay processes, the total decay rate of the quantity N is given by the sum of the two decay routes:

The solution to this equation is given in the previous section, where the sum of is treated as a new total decay constant .

Since , a combined can be given in terms of s:

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

where T_{1 / 2} is the combined or total half-life for the process, and t₁ is the half-life of the first process, and t₂ is the half life of the second process.

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

For a decay by three simultaneous exponential processes the total half-life can be computed, as above, as the reciprocal of a similar sum of three reciprocals:

Applications and examples

Exponential decay occurs in a wide variety of situations. Most of these fall into the domain of the natural sciences. Any application of mathematics to the social sciences or humanities is risky and uncertain, because of the extraordinary complexity of human behavior. However, a few roughly exponential phenomena have been identified there as well. The lunar farside as seen from Apollo 11 Natural science is the rational study of the universe via rules or laws of natural order. ... Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... The social sciences are groups of academic disciplines that study the human aspects of the world. ... The examples and perspective in this article may not represent a worldwide view. ...

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. The law of large numbers is a fundamental concept in statistics and probability that describes how the average of a randomly selected sample from a large population is likely to be close to the average of the whole population. ... A simple Poisson process, named after the French mathematician Siméon-Denis Poisson (1781 - 1840), is a stochastic process which is defined in terms of the occurrences of events. ...

Natural sciences

• 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.

• 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 throught its volume). See also Newton's law of cooling.

• The rates of certain types of chemical reactions depend on the concentration of one or another reactant. These reaction rates consequently follow exponential decay. For instance, many enzyme-catalyzed reactions behave this way.

• Atmospheric pressure decreases approximately exponentially with increasing height above sea level, at a rate of about 12% per 1000m.

• The electric charge (or, equivalently, the potential) stored on a capacitor (capacitance C) decays 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. (Furthermore, the particular case of a capacitor discharging 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.)

• Some vibrations may decay exponentially; this characteristic is often used in creating ADSR envelopes in synthesizers.

• In pharmacology and toxicology, it is found that many administered substances are distributed and metabolized (see clearance) according to exponential decay patterns. The "alpha half-life" and "beta half-life" of a substance measure how quickly a substance is distributed and eliminated.

• The intensity of electromagnetic radiation such as light or X-rays or gamma rays in an absorbant medium, follows an exponential decrease with distance into the absorbing medium.

A radionuclide is an atom with an unstable nucleus. ... Radioactive decay is the set of various processes by which unstable atomic nuclei emit subatomic particles (radiation). ... Fig. ... Heat flow along perfectly insulated wire Conduction is the transfer of heat or electric current from one substance to another by direct contact. ... The rusting of iron. ... A chemical reaction occurs when vapours of hydrogen chloride and ammonia meet to form a cloud of a new substance, ammonium chloride Chemical reaction is a process that results in the interconversion of chemical substances [1]. The substance or substances initially involved in a chemical reaction are called reactants. ... A reactant or reagent is any substance initially present in a chemical reaction. ... Ribbon diagram of the enzyme TIM, surrounded by the space-filling model of the protein. ... In chemistry and biology, catalysis is the acceleration (increase in rate) of a chemical reaction by means of a substance, called a catalyst, that is itself not consumed by the overall reaction. ... Diurnal (daily) rhythm of air pressure in northern Germany (black curve is air pressure) Atmospheric pressure is the pressure at any point in the Earths atmosphere. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interactions. ... Electric potential is the potential energy per unit of charge associated with a static (time-invariant) electric field, also called the electrostatic potential, typically measured in volts. ... Capacitors: SMD ceramic at top left; SMD tantalum at bottom left; through-hole tantalum at top right; through-hole electrolytic at bottom right. ... If an electric circuit has a well-defined output terminal, the circuit connected to this terminal (or its input impedance) is the load. ... Left: Series / Right: Parallel Arrows indicate direction of current. ... Resistor symbols (US and Japan) Resistor symbols (Europe, IEC) A pack of resistors A resistor is a two-terminal electrical or electronic component that resists an electric current by producing a voltage drop between its terminals in accordance with Ohms law. ... Resistor symbols (US and Japan) Resistor symbols (Europe, IEC) A pack of resistors A resistor is a two-terminal electrical or electronic component that resists an electric current by producing a voltage drop between its terminals in accordance with Ohms law. ... An ADSR envelope is a parameter used in synthesizers, including those that produce sound by subtractive synthesis, to control the sound produced. ... A synthesizer (or synthesiser) is an electronic musical instrument designed to produce electronically generated sound, using techniques such as additive, subtractive, FM, physical modelling synthesis, or phase distortion. ... Pharmacology (in Greek: pharmacon (φάρμακον) meaning drug, and logos (λόγος) meaning science) is the study of how substances interact with living organisms to produce a change in function. ... Toxicology (from the Greek words toxicos and logos [1]) is the study of the adverse effects of chemicals on living organisms [2]. It is the study of symptoms, mechanisms, treatments and detection of poisoning, especially the poisoning of people. ... It has been suggested that this article or section be merged with cell metabolism and carbohydrates. ... In medicine, the clearance, also renal clearance or renal plasma clearance (when referring to the function of the kidney), of a substance is the inverse of the time constant that describes its removal rate from the body divided by its volume of distribution (or total body water). ... Electromagnetic radiation can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. ...

Social sciences

• The popularity of fads, fashions and other cultural memes (for instance, attendance of popular films) often decays exponentially.

• The field of glottochronology attempts to determine the time elapsed since the divergence of two languages from a common root, using the assumption that linguistic changes are introduced at a steady rate; given this assumption, we expect the similarity between them (the number of properties of the language that are still identical) to decrease exponentially.

• In history of science, some believe that the body of knowledge of any particular science is gradually disproven according to an exponential decay pattern (see half-life of knowledge).

In biochemistry, flavin adenine dinucleotide (FAD) is the precursor molecule to FADH2. ... The term fashion usually applies to a prevailing mode of expression, but quite often applies to a personal mode of expression that may or may not adhere to prevailing ideals. ... It has been suggested that Memetic engineering be merged into this article or section. ... Film is a term that encompasses motion pictures as individual projects, as well as the field in general. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... The history of science investigates the historical record of human events that are pertinent to the cultural context and the secular development of what is currently called science, namely, a body of empirical and theoretical knowledge, produced by a global community of researchers, making use of specific techniques for the... The half-life of knowledge is the amount of time that has to elapse before half of the knowledge in a particular area is superseded or shown to be untrue. ...

Computer science

• BGP, the core routing protocol on the Internet, has to mantain a routing table in order to remember the paths a packet can be deviated to. When one of this 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 unuseful 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.

The border gateway protocol (BGP) is one of the core routing protocols in the Internet. ... In computer networking the term routing refers to selecting paths in a computer network along which to send data. ... This article describes routing in computer networks, a method of finding paths from origins to destinations, along which information can be passed. ... A packet is the fundamental unit of information carriage in all modern computer networks that use packet switching. ... A D-Link Wi-Fi NAT router, popular for home and small office networks A router is a computer networking device that forwards data packets across a network toward their destinations, through a process known as routing. ... CPU can stand for: in computing: Central processing unit in journalism: Commonwealth Press Union in law enforcement: Crime prevention unit in software: Critical patch update, a type of software patch distributed by Oracle Corporation in Macleans College is often known as Ash Lim. ... Look up RAM, Ram, ram in Wiktionary, the free dictionary. ...

See also

• Half-life
• Mean lifetime
• Elimination half-life
• Rate law, in chemistry
• Time constant, not only in electric engineering [2] and [3]
• Exponential growth, in mathematics
• Exponential distribution, in statistics