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A
quantity Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a u ...
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, ...
, 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 ri ...
) 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 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, \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 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 connecte ...
, 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 denote ...
. 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 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 co ...
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) ca ...
: :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 (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 antiderivat ...
. :\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 multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/' ...
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 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 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 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 ...
s 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 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 ...
- catalyzed reactions behave this way. * Electrostatics: 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 res ...
(or, equivalently, the potential) 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 o ...
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 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'' so ...
: Atmospheric pressure 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 conducti ...
: 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. *
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 crys ...
: 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 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 e ...
: 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 In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
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 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. 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 i ...
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 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: 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 packet ...
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, p ...
,
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 makes ...
, 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 A central processing unit (CPU), also called a central processor, main processor or just processor, is the electronic circuitry that executes instructions comprising a computer program. The CPU performs basic arithmetic, logic, controlling, a ...
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.


See also

* Exponential formula *
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 ...
* Radioactive decay 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 refere ...
, location = New York , lccn = 75173716


External links


Exponential decay calculator

A stochastic simulation of exponential decay


Exponentials