Exponential decay
Overview
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
, where N is the quantity and λ (lambda) is a positive number called the decay constant:
The solution to this equation (see derivation below) is:
Exponential rate of change
Here N(t) is the quantity at time t, and N_{0} = N(0) is the initial quantity, i.e.
Unanswered Questions
Encyclopedia
A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. Symbolically, this process can be expressed by the following differential equation
, where N is the quantity and λ (lambda) is a positive number called the decay constant:
The solution to this equation (see derivation below) is:
Exponential rate of change
Here N(t) is the quantity at time t, and N_{0} = N(0) is the initial quantity, i.e. the quantity at time t = 0.
The mean lifetime (also called the exponential time constant
) can be looked at as a "scaling time", because we can write the exponential decay equation in terms of the mean lifetime, , instead of the decay constant, :
We can see that is the time at which the population of the assembly is reduced to 1/e = 0.367879441 times its initial value. (E.g., if the initial population of the assembly, N(0), is 1000, then at time , the population, N(), 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".
, 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:
When this expression is inserted for in the exponential equation above, and ln 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 1/2^{3} = 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:
E.g. Polonium
-210 has a half-life of 138 days, and a mean lifetime of 200 days.
or, by rearranging,
Integrating, we have
where C is the constant of integration, and hence
where the final substitution, , is obtained by evaluating the equation at , as is defined as being the quantity at .
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 opposite
of the differentiation operator with as the corresponding eigenfunction
. The units of the decay constant are s^{−1}.
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
we firstly let c be the normalizing factor to convert to a probability space
:
or, on rearranging,
We see that 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
.
The solution to this equation is given in the previous section, where the sum of is treated as a new total decay constant .
Since , a combined can be given in terms of s:
In words: the mean life for combined decay channels is the harmonic mean
of the mean lives associated with the individual processes divided by the total number of processes.
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, is the half-life of the first process, and is the half-life of the second process.
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, as the harmonic mean of separate mean lives:
s. Any application of mathematics
to the social sciences or humanities
is risky and uncertain, because of the extraordinary complexity of human behavior. However, a few roughly exponential phenomena have been identified there as well.
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
.
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
, where N is the quantity and λ (lambda) is a positive number called the decay constant:
The solution to this equation (see derivation below) is:
Exponential rate of change
Here N(t) is the quantity at time t, and N_{0} = N(0) is the initial quantity, i.e. the quantity at time t = 0.
Mean lifetime
If the decaying quantity, N(t), is the number of discrete elements in a certain 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), , and it can be shown that it relates to the decay rate, , in the following way:The mean lifetime (also called the exponential time constant
Time constant
In physics and engineering, the time constant, usually denoted by the Greek letter \tau , is the risetime characterizing the response to a time-varying input of a first-order, linear time-invariant system.Concretely, a first-order LTI system is a system that can be modeled by a single first order...
) can be looked at as a "scaling time", because we can write the exponential decay equation in terms of the mean lifetime, , instead of the decay constant, :
We can see that is the time at which the population of the assembly is reduced to 1/e = 0.367879441 times its initial value. (E.g., if the initial population of the assembly, N(0), is 1000, then at time , the population, N(), 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. This time is called the half-lifeHalf-life
Half-life, abbreviated t½, is the period of time it takes for the amount of a substance undergoing decay to decrease by half. The name was originally used to describe a characteristic of unstable atoms , but it may apply to any quantity which follows a set-rate decay.The original term, dating to...
, 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:
When this expression is inserted for in the exponential equation above, and ln 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 1/2^{3} = 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:
E.g. Polonium
Polonium
Polonium is a chemical element with the symbol Po and atomic number 84, discovered in 1898 by Marie Skłodowska-Curie and Pierre Curie. A rare and highly radioactive element, polonium is chemically similar to bismuth and tellurium, and it occurs in uranium ores. Polonium has been studied for...
-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 isor, by rearranging,
Integrating, we have
where C is the constant of integration, and hence
where the final substitution, , is obtained by evaluating the equation at , as is defined as being the quantity at .
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 opposite
Additive inverse
In mathematics, the additive inverse, or opposite, of a number a is the number that, when added to a, yields zero.The additive inverse of a is denoted −a....
of the differentiation operator with as the corresponding eigenfunction
Eigenfunction
In mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has...
. 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, , (also called simply the lifetime) is the expected valueExpected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
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
Arithmetic mean
In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...
of the individual lifetimes.
Starting from the population formula
we firstly let c be the normalizing factor to convert to a probability space
Probability space
In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...
:
or, on rearranging,
We see that exponential decay is a scalar multiple
Scalar multiplication
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra . In an intuitive geometrical context, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction...
of the exponential distribution
Exponential distribution
In probability theory and statistics, the exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson process, i.e...
(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
Integration by parts
In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other integrals...
.
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:The solution to this equation is given in the previous section, where the sum of is treated as a new total decay constant .
Since , a combined can be given in terms of s:
In words: the mean life for combined decay channels is the harmonic mean
Harmonic mean
In mathematics, the harmonic mean is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired....
of the mean lives associated with the individual processes divided by the total number of processes.
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, is the half-life of the first process, and is the half-life of the second process.
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, as the harmonic mean of separate mean lives:
Applications and examples
Exponential decay occurs in a wide variety of situations. Most of these fall into the domain of the natural scienceNatural science
The natural sciences are branches of science that seek to elucidate the rules that govern the natural world by using empirical and scientific methods...
s. Any application of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
to the social sciences or humanities
Humanities
The humanities are academic disciplines that study the human condition, using methods that are primarily analytical, critical, or speculative, as distinguished from the mainly empirical approaches of the natural sciences....
is risky and uncertain, because of the extraordinary complexity of human behavior. However, a few roughly exponential phenomena have been identified there as well.
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
Law of large numbers
In probability theory, the law of large numbers is a theorem that describes the result of performing the same experiment a large number of times...
holds. For small samples, a more general analysis is necessary, accounting for a Poisson process
Poisson process
A Poisson process, named after the French mathematician Siméon-Denis Poisson , is a stochastic process in which events occur continuously and independently of one another...
.
Natural sciences
- Radioactivity: In a sample of a radionuclideRadionuclideA radionuclide is an atom with an unstable nucleus, which is a nucleus characterized by excess energy available to be imparted either to a newly created radiation particle within the nucleus or to an atomic electron. The radionuclide, in this process, undergoes radioactive decay, and emits gamma...
that undergoes radioactive decayRadioactive decayRadioactive decay is the process by which an atomic nucleus of an unstable atom loses energy by emitting ionizing particles . The emission is spontaneous, in that the atom decays without any physical interaction with another particle from outside the atom...
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.
- Heat transferHeat transferHeat transfer is a discipline of thermal engineering that concerns the exchange of thermal energy from one physical system to another. Heat transfer is classified into various mechanisms, such as heat conduction, convection, thermal radiation, and phase-change transfer...
: If an object at one temperatureTemperatureTemperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
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.
- Chemical reactions: The rateReaction rateThe reaction rate or speed of reaction for a reactant or product in a particular reaction is intuitively defined as how fast or slow a reaction takes place...
s of certain types of chemical reactionChemical reactionA chemical reaction is a process that leads to the transformation of one set of chemical substances to another. Chemical reactions can be either spontaneous, requiring no input of energy, or non-spontaneous, typically following the input of some type of energy, such as heat, light or electricity...
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 enzymeEnzymeEnzymes are proteins that catalyze chemical reactions. In enzymatic reactions, the molecules at the beginning of the process, called substrates, are converted into different molecules, called products. Almost all chemical reactions in a biological cell need enzymes in order to occur at rates...
-catalyzedCatalysisCatalysis is the change in rate of a chemical reaction due to the participation of a substance called a catalyst. Unlike other reagents that participate in the chemical reaction, a catalyst is not consumed by the reaction itself. A catalyst may participate in multiple chemical transformations....
reactions behave this way.
- Geophysics: Atmospheric pressureAtmospheric pressureAtmospheric pressure is the force per unit area exerted into a surface by the weight of air above that surface in the atmosphere of Earth . In most circumstances atmospheric pressure is closely approximated by the hydrostatic pressure caused by the weight of air above the measurement point...
decreases approximately exponentially with increasing height above sea level, at a rate of about 12% per 1000m.
- ElectrostaticsElectrostaticsElectrostatics is the branch of physics that deals with the phenomena and properties of stationary or slow-moving electric charges....
: The electric chargeElectric chargeElectric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
(or, equivalently, the potentialElectric potentialIn classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...
) stored on a capacitorCapacitorA capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...
(capacitance C) decays exponentially, if the capacitor experiences a constant external loadExternal electric loadIf an electric circuit has a well-defined output terminal, the circuit connected to this terminal is the load....
(resistance R). The exponential time-constant τ for the process is R C, and the half-life is therefore R C ln2. (Furthermore, the particular case of a capacitor discharging through several parallel resistorResistorA linear resistor is a linear, passive two-terminal electrical component that implements electrical resistance as a circuit element.The current through a resistor is in direct proportion to the voltage across the resistor's terminals. Thus, the ratio of the voltage applied across a resistor's...
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.)
- Vibrations: Some vibrations may decay exponentially; this characteristic is often found in damped mechanical oscillatorsDampingIn physics, damping is any effect that tends to reduce the amplitude of oscillations in an oscillatory system, particularly the harmonic oscillator.In mechanics, friction is one such damping effect...
, and used in creating ADSR envelopes in synthesizers.
- PharmacologyPharmacologyPharmacology is the branch of medicine and biology concerned with the study of drug action. More specifically, it is the study of the interactions that occur between a living organism and chemicals that affect normal or abnormal biochemical function...
and toxicologyToxicologyToxicology is a branch of biology, chemistry, and medicine concerned with the study of the adverse effects of chemicals on living organisms...
: It is found that many administered substances are distributed and metabolizeMetabolismMetabolism is the set of chemical reactions that happen in the cells of living organisms to sustain life. These processes allow organisms to grow and reproduce, maintain their structures, and respond to their environments. Metabolism is usually divided into two categories...
d (see clearanceClearance (medicine)In medicine, the clearance is a measurement of the renal excretion ability. Although clearance may also involve other organs than the kidney, it is almost synonymous with renal clearance or renal plasma clearance. Each substance has a specific clearance that depends on its filtration characteristics...
) according to exponential decay patterns. The biological half-livesBiological half-lifeThe biological half-life or elimination half-life of a substance is the time it takes for a substance to lose half of its pharmacologic, physiologic, or radiologic activity, as per the MeSH definition...
"alpha half-life" and "beta half-life" of a substance measure how quickly a substance is distributed and eliminated.
- Physical opticsPhysical opticsIn physics, physical optics, or wave optics, is the branch of optics which studies interference, diffraction, polarization, and other phenomena for which the ray approximation of geometric optics is not valid...
: The intensity of electromagnetic radiationElectromagnetic radiationElectromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...
such as light or X-rays or gamma rays in an absorbent medium, follows an exponential decrease with distance into the absorbing medium.
- Thermoelectricity: The decline in resistance of a Negative Temperature Coefficient ThermistorThermistorA thermistor is a type of resistor whose resistance varies significantly with temperature, more so than in standard resistors. The word is a portmanteau of thermal and resistor...
as temperature is increased.
Social sciences
- In simple glottochronologyGlottochronologyGlottochronology is that part of lexicostatistics dealing with the chronological relationship between languages....
, the (debatable) assumption of a constant decay rate in languages allows to estimate the age of single languages. (To compute the time of split between TWO languages requires additional assumptions, which have nothing to do with exponential decay). - In history of scienceHistory of scienceThe history of science is the study of the historical development of human understandings of the natural world and the domains of the social sciences....
, some believe that the body of knowledge of any particular science is gradually disproved according to an exponential decay pattern (see half-life of knowledgeHalf-life of knowledgeThe half-life of knowledge is the amount of time that has to elapse before half of the knowledge in a particular area is superseded or shown to be untrue. The concept is attributed to Fritz Machlup...
).
Computer science
- BGP, the core routing protocolRoutingRouting is the process of selecting paths in a network along which to send network traffic. Routing is performed for many kinds of networks, including the telephone network , electronic data networks , and transportation networks...
on the InternetInternetThe Internet is a global system of interconnected computer networks that use the standard Internet protocol suite to serve billions of users worldwide...
, has to maintain a routing tableRouting tableIn computer networking a routing table, or Routing Information Base , is a data table stored in a router or a networked computer that lists the routes to particular network destinations, and in some cases, metrics associated with those routes. The routing table contains information about the...
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 and RAMRam-Animals:*Ram, an uncastrated male sheep*Ram cichlid, a species of freshwater fish endemic to Colombia and Venezuela-Military:*Battering ram*Ramming, a military tactic in which one vehicle runs into another...
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 functionExponential functionIn mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
- Exponential growthExponential growthExponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value...
- Radioactive decayRadioactive decayRadioactive decay is the process by which an atomic nucleus of an unstable atom loses energy by emitting ionizing particles . The emission is spontaneous, in that the atom decays without any physical interaction with another particle from outside the atom...
for the mathematics of chains of exponential processes with differing constants