View source for Exponential decay

{{short description|Decrease in value at a rate proportional to the current value}}
[[Image:Plot-exponential-decay.svg|thumb|upright=1.5|A quantity undergoing exponential decay. Larger decay constants make the quantity vanish much more rapidly. This plot shows decay for decay constant ({{mvar|λ}}) of 25, 5, 1, 1/5, and 1/25 for {{mvar|x}} from 0 to 5.]]

A [[quantity]] is subject to '''exponential decay''' if it decreases at a rate [[Proportionality (mathematics)|proportional]] to its current value.  Symbolically, this process can be expressed by the following [[differential equation]], where {{mvar|N}} is the quantity and {{mvar|λ}} ([[lambda]]) is a positive rate called the '''exponential decay constant''', '''disintegration constant''',<ref>{{harvtxt|Serway|Moses|Moyer|1989|p=384}}</ref> '''rate constant''',<ref>{{harvtxt|Simmons|1972|p=15}}</ref> or '''transformation constant''':<ref>{{harvtxt|McGraw-Hill|2007}}</ref>

:<math>\frac{dN(t)}{dt} = -\lambda N(t).</math>

The solution to this equation (see [[#Solution_of_the_differential_equation|derivation]] below) is:

:<math>N(t) = N_0 e^{-\lambda t}, </math>

where {{math|''N''(''t'')}} is the quantity at time {{mvar|t}}, {{math|1=''N''<sub>0</sub> = ''N''(0)}} is the initial quantity, that is, the quantity at time {{math|1=''t'' = 0}}.

== Measuring rates of decay ==

=== Mean lifetime ===
If the decaying quantity, ''N''(''t''), is the number of discrete elements in a certain [[set (mathematics)|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]]''', <math>\tau</math>, relates to the decay rate constant, &lambda;, in the following way:
:<math>\tau = \frac{1}{\lambda}.</math>

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, <math>\tau</math>, instead of the decay constant, &lambda;:
:<math>N(t) = N_0 e^{-t/\tau}, </math>
and that <math>\tau</math> is the time at which the population of the assembly is reduced to {{Fraction|1|[[e (mathematical constant)|''e'']]}} ≈ 0.367879441 times its initial value. This is equivalent to <math>\log_{2}{e}</math> ≈ 1.442695 half-lives.

For example, if the initial population of the assembly, ''N''(0), is 1000, then the population at time <math>\tau</math>, <math>N(\tau)</math>, 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===
{{main|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''<sub>1/2</sub>.  The half-life can be written in terms of the decay constant, or the mean lifetime, as:

:<math>t_{1/2} = \frac{\ln (2)}{\lambda} = \tau \ln (2).</math>

When this expression is inserted for <math>\tau</math> in the exponential equation above, and [[Natural logarithm of 2|ln&nbsp;2]] is absorbed into the base, this equation becomes:

:<math>N(t) = N_0 2^{-t/t_{1/2}}. </math>

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

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

: <math>\tau = \frac{t_{1/2}}{\ln (2)} \approx 1.4427 \cdot t_{1/2}.</math>

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
:<math>\frac{dN(t)}{dt} = -\lambda N(t)</math>
or, by rearranging (applying the technique called [[separation of variables]]),
:<math>\frac{dN(t)}{N(t)} = -\lambda dt.</math>

Integrating, we have
:<math>\ln N = -\lambda t + C \,</math>
where C is the [[constant of integration]], and hence
:<math>N(t) = e^C e^{-\lambda t} = N_0 e^{-\lambda t} \,</math>
where the final substitution, ''N''<sub>0</sub> = ''e''<sup>''C''</sup>, is obtained by evaluating the equation at ''t'' = 0, as ''N''<sub>0</sub> 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 this case, λ is the eigenvalue of the [[additive inverse|negative]] of the [[differential operator]] with ''N''(''t'') as the corresponding [[eigenfunction]]. The units of the decay constant are s<sup>−1</sup>{{Citation needed|date=November 2016}}.

=== Derivation of the mean lifetime ===
Given an assembly of elements, the number of which decreases ultimately to zero, the '''mean lifetime''', <math>\tau</math>, (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

:<math>N = N_0 e^{-\lambda t}, \,</math>

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

:<math>1 = \int_0^\infty c \cdot N_0 e^{-\lambda t}\, dt = c \cdot \frac{N_0}{\lambda}</math>

or, on rearranging,

:<math>c = \frac{\lambda}{N_0}.</math>

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

:<math>\tau = \langle t \rangle = \int_0^\infty t \cdot c \cdot N_0 e^{-\lambda t}\, dt = \int_0^\infty \lambda t e^{-\lambda t}\, dt = \frac{1}{\lambda}.</math>

=== Decay by two or more processes ===<!-- This section is linked from [[Half-life]] -->
{{see also|Branching fraction}}
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&nbsp;''N'' is given by the ''sum'' of the decay routes; thus, in the case of two processes:

:<math>-\frac{dN(t)}{dt} = N\lambda _1 + N\lambda _2 = (\lambda _1 + \lambda _2)N.</math>

The solution to this equation is given in the previous section, where the sum of <math>\lambda _1 + \lambda _2\,</math> is treated as a new total decay constant <math>\lambda _c</math>.

:<math>N(t) = N_0 e^{-(\lambda _1 + \lambda _2) t} = N_0 e^{-(\lambda _c) t}.</math>

'''Partial mean life''' associated with individual processes is by definition the [[multiplicative inverse]] of corresponding partial decay constant: <math>\tau = 1/\lambda</math>. A combined <math>\tau_c</math> can be given in terms of <math>\lambda</math>s:

:<math>\frac{1}{\tau_c} = \lambda_c = \lambda_1 + \lambda_2 = \frac{1}{\tau_1} + \frac{1}{\tau_2}</math>

:<math>\tau_c = \frac{\tau_1 \tau_2}{\tau_1 + \tau_2}. </math>

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

:<math>T_{1/2} = \frac{t_1 t_2}{t_1 + t_2} </math>

where <math>T _{1/2}</math> is the combined or total half-life for the process, <math>t_1</math> and <math>t_2</math> 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 [[one half|halved]].

In terms of separate decay constants, the total half-life <math>T _{1/2}</math> can be shown to be

:<math>T_{1/2} = \frac{\ln 2}{\lambda _c} = \frac{\ln 2}{\lambda _1 + \lambda _2}.</math>

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

:<math>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)}.</math>

=== 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 [[modified-release dosage|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]]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]] holds.  For small samples, a more general analysis is necessary, accounting for a [[Poisson process]].

=== Natural sciences ===<!-- This section is linked from [[Methicillin-resistant Staphylococcus aureus]] -->
* '''[[Chemical reactions]]:''' The [[reaction rate|rate]]s of certain types of [[chemical reaction]]s depend on the concentration of one or another [[reactant]]. Reactions whose rate depends only on the concentration of one reactant (known as [[Rate equation#First-order reactions|first-order reactions]]) consequently follow exponential decay. For instance, many [[enzyme]]-[[catalysis|catalyzed]] reactions behave this way.
* '''[[Electrostatics]]:''' The [[electric charge]] (or, equivalently, the [[electric potential|potential]]) contained in a [[capacitor]] (capacitance ''C'') discharges with exponential decay (when the capacitor experiences a constant [[External electric load|external load]] of resistance ''R'') and similarly charges with the [[mirror image]] of exponential decay (when the capacitor is charged from a constant voltage source though a constant resistance). The exponential time-constant for the process is <math>\tau = R \, C ,</math> so the half-life is <math>R \, C \, \ln(2) .</math> The same equations can be applied to [[Duality (electrical circuits)|the dual]] of current in an inductor.
** Furthermore, the particular case of a capacitor or inductor changing through several [[Series and parallel circuits#Parallel circuits|parallel]] [[resistor]]s makes an interesting example of multiple decay processes, with each resistor representing a separate process. In fact, the expression for the [[resistor#Series and parallel circuits|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.{{citation needed|date=November 2017}}
* '''[[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).  See also [[Newton's law of cooling]].
* '''[[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 [[metabolism|metabolize]]d (see ''[[clearance (medicine)|clearance]]'') according to exponential decay patterns.  The [[biological half-life|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 [[Harmonic oscillator|damped mechanical oscillators]], and used in creating [[ADSR envelope]]s in [[Synthesizer#Sound basics|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 [[List of Ig Nobel Prize winners|Ig Nobel Prize]] for demonstrating that [[beer]] froth obeys the law of exponential decay.<ref>{{Cite journal| last1 = Leike | first1 = A.| title = Demonstration of the exponential decay law using beer froth| journal = European Journal of Physics| volume = 23| pages = 21–26| year = 2002| issue = 1| doi = 10.1088/0143-0807/23/1/304|bibcode = 2002EJPh...23...21L | citeseerx = 10.1.1.693.5948| s2cid = 250873501}}</ref>

=== 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 ===
{{see also|Exponential backoff}}
* The core '''[[Routing|routing protocol]]''' on the [[Internet]], [[BGP]], has to maintain a [[routing table]] in order to remember the paths a [[Packet (information technology)|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 (computing)|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 [[Random-access memory|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.

{{wide image|doubling_time_vs_half_life.svg|640px|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 [http://upload.wikimedia.org/wikipedia/commons/8/88/Doubling_time_vs_half_life.svg SVG version], hover over a graph to highlight it and its complement.}}

==See also==
* [[Exponential formula]]
* [[Exponential growth]]
* [[Radioactive decay]] for the mathematics of chains of exponential processes with differing constants

==Notes==
{{Reflist}}

==References==
* {{ cite encyclopedia | title = McGraw-Hill Encyclopedia of Science & Technology | edition = 10th | location = New York | publisher = [[McGraw-Hill]] | year = 2007 | isbn = 978-0-07-144143-8 | ref = {{harvid|McGraw-Hill|2007}} }}
* {{ citation | first1 = Raymond A. | last1 = Serway | first2 = Clement J. | last2 = Moses | first3 = Curt A. | last3 = Moyer | year = 1989 | isbn = 0-03-004844-3 | title = Modern Physics | publisher = [[Harcourt Brace Jovanovich]] | location = Fort Worth }}
* {{ citation | first1 = George F. | last1 = Simmons | author-link = George F. Simmons | year = 1972 | title = Differential Equations with Applications and Historical Notes | publisher = [[McGraw-Hill]] | location = New York | lccn = 75173716 }}

==External links==
* [https://www.fxsolver.com/browse/formulas/Exponential+decay Exponential decay calculator]
* [http://vam.anest.ufl.edu/simulations/stochasticonecompartment.php A stochastic simulation of exponential decay]
* [https://web.archive.org/web/20060617205436/http://www.facstaff.bucknell.edu/mastascu/elessonshtml/SysDyn/SysDyn3TCBasic.htm Tutorial on time constants]

[[Category:Exponentials]]