Mean Lifetime
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A
quantity Quantity or amount is a property that can exist as a Counting, multitude or Magnitude (mathematics), magnitude, which illustrate discontinuity (mathematics), discontinuity and continuum (theory), continuity. Quantities can be compared in terms o ...
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 In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
, where is the quantity and (
lambda Lambda (}, ''lám(b)da'') is the 11th letter of the Greek alphabet, representing the voiced alveolar lateral approximant . In the system of Greek numerals, lambda has a value of 30. Lambda is derived from the Phoenician Lamed . Lambda gave rise ...
) is a positive rate called the exponential decay constant, disintegration constant, rate constant, or transformation constant: :\frac = -\lambda N. The solution to this equation (see
derivation Derivation may refer to: Language * Morphological derivation, a word-formation process * Parse tree or concrete syntax tree, representing a string's syntax in formal grammars Law * Derivative work, in copyright law * Derivation proceeding, a proc ...
below) is: :N(t) = N_0 e^, where is the quantity at time , is the initial quantity, that is, the quantity at time .


Measuring rates of decay


Mean lifetime

If the decaying quantity, ''N''(''t''), is the number of discrete elements in a certain
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
, 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 In physics and engineering, the time constant, usually denoted by the Greek letter (tau), is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system.Concretely, a first-order LTI system is a sy ...
, \tau, relates to the decay rate constant, λ, in the following way: :\tau = \frac. 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^, 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_ = \frac = \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^. 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 \approx 1.44 \cdot t_. For example,
polonium-210 Polonium-210 (210Po, Po-210, historically radium F) is an isotope of polonium. It undergoes alpha decay to stable 206Pb with a half-life of 138.376 days (about months), the longest half-life of all naturally occurring polonium isotopes. First i ...
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 = -\lambda N or, by rearranging (applying the technique called
separation of variables In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs ...
), :\frac = -\lambda dt. Integrating, we have :\ln N = -\lambda t + C \, where C is the
constant of integration In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connected ...
, and hence :N(t) = e^C e^ = N_0 e^ \, where the final substitution, ''N''0 = ''e''''C'', is obtained by evaluating the equation at ''t'' = 0, as ''N''0 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 characterise the decay. The notation λ for the decay constant is a remnant of the usual notation for an
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
. In this case, λ is the eigenvalue of the negative of the
differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
with ''N''(''t'') as the corresponding
eigenfunction In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
. 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 In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
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 In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The colle ...
of the individual lifetimes. Starting from the population formula :N = N_0 e^, \, first let ''c'' be the normalizing factor to convert to a
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
: :1 = \int_0^\infty c \cdot N_0 e^\, dt = c \cdot \frac or, on rearranging, :c = \frac. Exponential decay is a scalar multiple of the
exponential distribution In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...
(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 In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
. :\tau = \langle t \rangle = \int_0^\infty t \cdot c \cdot N_0 e^\, dt = \int_0^\infty \lambda t e^\, dt = \frac.


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 = 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^ = N_0 e^. Partial mean life associated with individual processes is by definition the
multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a rat ...
of corresponding partial decay constant: \tau = 1/\lambda. A combined \tau_c can be given in terms of \lambdas: :\frac = \lambda_c = \lambda_1 + \lambda_2 = \frac + \frac :\tau_c = \frac. 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_ = \frac where T _ 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 _ can be shown to be :T_ = \frac = \frac. For a decay by three simultaneous exponential processes the total half-life can be computed as above: :T_ = \frac = \frac = \frac.


Decay series / coupled decay

In
nuclear science Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter. Nuclear physics should not be confused with atomic physics, which studies the ...
and
pharmacokinetics Pharmacokinetics (from Ancient Greek ''pharmakon'' "drug" and ''kinetikos'' "moving, putting in motion"; see chemical kinetics), sometimes abbreviated as PK, is a branch of pharmacology dedicated to determining the fate of substances administered ...
, 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 science Natural science is one of the branches of science concerned with the description, understanding and prediction of natural phenomena, based on empirical evidence from observation and experimentation. Mechanisms such as peer review and repeatab ...
s. 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 In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials shou ...
holds. For small samples, a more general analysis is necessary, accounting for a
Poisson process In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...
.


Natural sciences

*
Chemical reactions A chemical reaction is a process that leads to the chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the positions of electrons in the forming and breaking ...
: The rates of certain types of
chemical reaction A chemical reaction is a process that leads to the IUPAC nomenclature for organic transformations, chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the pos ...
s depend on the concentration of one or another
reactant In chemistry, a reagent ( ) or analytical reagent is a substance or compound added to a system to cause a chemical reaction, or test if one occurs. The terms ''reactant'' and ''reagent'' are often used interchangeably, but reactant specifies a ...
. Reactions whose rate depends only on the concentration of one reactant (known as first-order reactions) consequently follow exponential decay. For instance, many
enzyme Enzymes () are proteins that act as biological catalysts by accelerating chemical reactions. The molecules upon which enzymes may act are called substrates, and the enzyme converts the substrates into different molecules known as products. A ...
-
catalyzed Catalysis () is the process of increasing the rate of a chemical reaction by adding a substance known as a catalyst (). Catalysts are not consumed in the reaction and remain unchanged after it. If the reaction is rapid and the catalyst recyc ...
reactions behave this way. *
Electrostatics Electrostatics is a branch of physics that studies electric charges at rest (static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amber ...
: The
electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
(or, equivalently, the
potential Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple re ...
) contained in a
capacitor A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of ...
(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 Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster of IBM ...
resistor A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active el ...
s 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 Geophysics () is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. The term ''geophysics'' som ...
:
Atmospheric pressure Atmospheric pressure, also known as barometric pressure (after the barometer), is the pressure within the atmosphere of Earth. The standard atmosphere (symbol: atm) is a unit of pressure defined as , which is equivalent to 1013.25 millibars, 7 ...
decreases approximately exponentially with increasing height above sea level, at a rate of about 12% per 1000m. *
Heat transfer Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
: If an object at one
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
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). See also
Newton's law of cooling In the study of heat transfer, Newton's law of cooling is a physical law which states that The rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its environment. The law is frequently q ...
. *
Luminescence Luminescence is spontaneous emission of light by a substance not resulting from heat; or "cold light". It is thus a form of cold-body radiation. It can be caused by chemical reactions, electrical energy, subatomic motions or stress on a cryst ...
: 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 Pharmacology is a branch of medicine, biology and pharmaceutical sciences concerned with drug or medication action, where a drug may be defined as any artificial, natural, or endogenous (from within the body) molecule which exerts a biochemica ...
and
toxicology Toxicology is a scientific discipline, overlapping with biology, chemistry, pharmacology, and medicine, that involves the study of the adverse effects of chemical substances on living organisms and the practice of diagnosing and treating expo ...
: It is found that many administered substances are distributed and
metabolize Metabolism (, from el, μεταβολή ''metabolē'', "change") is the set of life-sustaining chemical reactions in organisms. The three main functions of metabolism are: the conversion of the energy in food to energy available to run cell ...
d (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 In physics, physical optics, or wave optics, is the branch of optics that studies interference, diffraction, polarization, and other phenomena for which the ray approximation of geometric optics is not valid. This usage tends not to include effec ...
: The intensity of
electromagnetic radiation In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic field, electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, inf ...
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 A radionuclide (radioactive nuclide, radioisotope or radioactive isotope) is a nuclide that has excess nuclear energy, making it unstable. This excess energy can be used in one of three ways: emitted from the nucleus as gamma radiation; transfer ...
that undergoes
radioactive decay Radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is consid ...
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 A radiogenic nuclide is a nuclide that is produced by a process of radioactive decay. It may itself be radioactive (a radionuclide) or stable (a stable nuclide). Radiogenic nuclides (more commonly referred to as radiogenic isotopes) form some ...
nuclide. *
Thermoelectricity The thermoelectric effect is the direct conversion of temperature differences to electric voltage and vice versa via a thermocouple. A thermoelectric device creates a voltage when there is a different temperature on each side. Conversely, when ...
: The decline in resistance of a Negative Temperature Coefficient
Thermistor A thermistor is a type of resistor whose resistance is strongly dependent on temperature, more so than in standard resistors. The word thermistor is a portmanteau of ''thermal'' and ''resistor''. Thermistors are divided based on their conduction ...
as temperature is increased. *
Vibrations Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin ''vibrationem'' ("shaking, brandishing"). The oscillations may be periodic, such as the motion of a pendulum—or random, such ...
: Some vibrations may decay exponentially; this characteristic is often found in damped mechanical oscillators, and used in creating
ADSR envelope ADSR may refer to: * ADSR envelope (attack decay sustain release), a common type of music envelope * Accelerator-driven sub-critical reactor, a nuclear reactor using a particle accelerator to generate a fission reaction in a sub-critical assembly ...
s in
synthesizers A synthesizer (also spelled synthesiser) is an electronic musical instrument that generates audio signals. Synthesizers typically create sounds by generating waveforms through methods including subtractive synthesis, additive synthesis and ...
. An
overdamped Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples incl ...
system will simply return to equilibrium via an exponential decay. * Beer froth: Arnd Leike, of the
Ludwig Maximilian University of Munich The Ludwig Maximilian University of Munich (simply University of Munich or LMU; german: Ludwig-Maximilians-Universität München) is a public research university in Munich, Germany. It is Germany's sixth-oldest university in continuous operatio ...
, won an
Ig Nobel Prize The Ig Nobel Prize ( ) is a satiric prize awarded annually since 1991 to celebrate ten unusual or trivial achievements in scientific research. Its aim is to "honor achievements that first make people laugh, and then make them think." The name of ...
for demonstrating that
beer Beer is one of the oldest and the most widely consumed type of alcoholic drink in the world, and the third most popular drink overall after water and tea. It is produced by the brewing and fermentation of starches, mainly derived from ce ...
froth obeys the law of exponential decay.


Social sciences

*
Finance Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of fina ...
: 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 Glottochronology (from Attic Greek γλῶττα ''tongue, language'' and χρόνος ''time'') is the part of lexicostatistics which involves comparative linguistics and deals with the chronological relationship between languages.Sheila Embleton ( ...
, 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 A routing protocol specifies how routers communicate with each other to distribute information that enables them to select routes between nodes on a computer network. Routers perform the traffic directing functions on the Internet; data packets ...
on the
Internet The Internet (or internet) is the global system of interconnected computer networks that uses the Internet protocol suite (TCP/IP) to communicate between networks and devices. It is a '' network of networks'' that consists of private, pub ...
,
BGP Border Gateway Protocol (BGP) is a standardized exterior gateway protocol designed to exchange routing and reachability information among autonomous systems (AS) on the Internet. BGP is classified as a path-vector routing protocol, and it mak ...
, has to maintain a
routing table In computer networking, a routing table, or routing information base (RIB), is a data table stored in a router or a network host that lists the routes to particular network destinations, and in some cases, metrics (distances) associated with tho ...
in order to remember the paths a
packet Packet may refer to: * A small container or pouch ** Packet (container), a small single use container ** Cigarette packet ** Sugar packet * Network packet, a formatted unit of data carried by a packet-mode computer network * Packet radio, a form ...
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 Ram, ram, or RAM may refer to: Animals * A male sheep * Ram cichlid, a freshwater tropical fish People * Ram (given name) * Ram (surname) * Ram (director) (Ramsubramaniam), an Indian Tamil film director * RAM (musician) (born 1974), Dutch * ...
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.


See also

*
Exponential formula In combinatorial mathematics, the exponential formula (called the polymer expansion in physics) states that the exponential generating function for structures on finite sets is the exponential of the exponential generating function for connected str ...
*
Exponential 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 q ...
*
Radioactive decay Radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is consid ...
for the mathematics of chains of exponential processes with differing constants


Notes


References

* * * {{ citation , first1 = George F. , last1 = Simmons , author-link = George F. Simmons , year = 1972 , title = Differential Equations with Applications and Historical Notes , publisher =
McGraw-Hill McGraw Hill is an American educational publishing company and one of the "big three" educational publishers that publishes educational content, software, and services for pre-K through postgraduate education. The company also publishes referenc ...
, location = New York , lccn = 75173716


External links


Exponential decay calculator

A stochastic simulation of exponential decay


Exponentials