Editing Exponential decay (section)

== 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]]:''' In a [[RC circuit]], the [[electric charge]] (or, equivalently, the [[electric potential|potential]]) contained in a [[capacitor]] (capacitance ''C'') discharges through a constant [[External electric load|external load]] (resistance ''R'') with exponential decay 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.}}