Decrease in value at a rate proportional to the current value
A quantity undergoing exponential decay. Larger decay constants make the quantity vanish much more rapidly. This plot shows decay for decay constant (λ) of 25, 5, 1, 1/5, and 1/25 forx from 0 to 5.
Aquantity is subject toexponential decay if it decreases at a rateproportional to its current value. Symbolically, this process can be expressed by the followingdifferential equation, whereN is the quantity andλ (lambda) is a positive rate called theexponential decay constant,disintegration constant,[1]rate constant,[2] ortransformation constant:[3]
The solution to this equation (seederivation below) is:
whereN(t) is the quantity at timet,N0 =N(0) is the initial quantity, that is, the quantity at timet = 0.
If the decaying quantity,N(t), is the number of discrete elements in a certainset, it is possible to compute the average length of time that an element remains in the set. This is called themean lifetime (or simply thelifetime), where theexponentialtime constant,, relates to the decay rate constant, λ, in the following way:
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,, instead of the decay constant, λ:
and that is the time at which the population of the assembly is reduced to1⁄e ≈ 0.367879441 times its initial value. This is equivalent to ≈ 1.442695 half-lives.
For example, if the initial population of the assembly,N(0), is 1000, then the population at time,, is 368.
A very similar equation will be seen below, which arises when the base of the exponential is chosen to be 2, rather thane. In that case the scaling time is the "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. (IfN(t) is discrete, then this is the median life-time rather than the mean life-time.) This time is called thehalf-life, and often denoted by the symbolt1/2. The half-life can be written in terms of the decay constant, or the mean lifetime, as:
When this expression is inserted for in the exponential equation above, andln 2 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.
Therefore, the mean lifetime is equal to the half-life divided by the natural log of 2, or:
For example,polonium-210 has a half-life of 138 days, and a mean lifetime of 200 days.
where the final substitution,N0 =eC, is obtained by evaluating the equation att = 0, asN0 is defined as being the quantity att = 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 aneigenvalue. In this case, λ is the eigenvalue of thenegative of thedifferential operator withN(t) as the correspondingeigenfunction.
Given an assembly of elements, the number of which decreases ultimately to zero, themean lifetime,, (also called simply thelifetime) is theexpected value of the amount of time before an object is removed from the assembly. Specifically, if theindividual 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 thearithmetic mean of the individual lifetimes.
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 thesum of the decay routes; thus, in the case of two processes:
The solution to this equation is given in the previous section, where the sum of is treated as a new total decay constant.
Partial mean life associated with individual processes is by definition themultiplicative inverse of corresponding partial decay constant:. A combined can be given in terms ofs:
Since half-lives differ from mean life by a constant factor, the same equation holds in terms of the two corresponding half-lives:
where is the combined or total half-life for the process, and are so-namedpartial 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 ishalved.
In terms of separate decay constants, the total half-life can be shown to be
For a decay by three simultaneous exponential processes the total half-life can be computed as above:
Innuclear science andpharmacokinetics, 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.
In the pharmacology setting, some ingested substances might be absorbed into the body by a process reasonably modeled as exponential decay, or might be deliberatelyformulated to have such a release profile.
Exponential decay occurs in a wide variety of situations. Most of these fall into the domain of thenatural sciences.
Many decay processes that are often treated as exponential, are really only exponential so long as the sample is large and thelaw of large numbers holds. For small samples, a more general analysis is necessary, accounting for aPoisson process.
Electrostatics: In aRC circuit, theelectric charge (or, equivalently, thepotential) contained in acapacitor (capacitanceC) discharges through a constantexternal load (resistanceR) with exponential decay and similarly charges with themirror 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 so the half-life is The same equations can be applied tothe dual of current in an inductor.
Furthermore, the particular case of a capacitor or inductor changing through severalparallelresistors makes an interesting example of multiple decay processes, with each resistor representing a separate process. In fact, the expression for theequivalent resistance of two resistors in parallel mirrors the equation for the half-life with two decay processes.
Heat transfer: If an object at onetemperature 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 alsoNewton'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 andtoxicology: It is found that many administered substances are distributed andmetabolized (seeclearance) according to exponential decay patterns. Thebiological 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 ofelectromagnetic 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 theBeer-Lambert law.
Radioactivity: In a sample of aradionuclide that undergoesradioactive 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 aradiogenic nuclide.
Thermoelectricity: The decline in resistance of a Negative Temperature CoefficientThermistor as temperature is increased.
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 simpleglottochronology, 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 betweentwo languages requires additional assumptions, independent of exponential decay).
The corerouting protocol on theInternet,BGP, has to maintain arouting table in order to remember the paths apacket can be deviated to. When one of these paths repeatedly changes its state fromavailable tonot available (andvice versa), the BGProuter controlling that path has to repeatedly add and remove the path record from its routing table (flaps the path), thus spending local resources such asCPU andRAM and, even more, broadcasting useless information to peer routers. To prevent this undesired behavior, an algorithm namedroute 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.
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 theSVG version, hover over a graph to highlight it and its complement.
