A
quantity is subject to exponential decay if it decreases at a rate
proportional
Proportionality, proportion or proportional may refer to:
Mathematics
* Proportionality (mathematics), the property of two variables being in a multiplicative relation to a constant
* Ratio, of one quantity to another, especially of a part compare ...
to its current value. Symbolically, this process can be expressed by the following
differential equation, 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 ris ...
) is a positive rate called the exponential decay constant, disintegration constant, rate constant, or transformation constant:
:
The solution to this equation (see
derivation below) is:
:
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, 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,
, 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 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
,
, 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:
:
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:
:
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
:
or, by rearranging (applying the technique called
separation of variables),
:
Integrating, we have
:
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 connect ...
, and hence
:
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 denot ...
. 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,
, (also called simply the lifetime) is the
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
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 coll ...
of the individual lifetimes.
Starting from the population formula
:
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) c ...
:
:
or, on rearranging,
:
Exponential decay is a
scalar multiple
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector b ...
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 antiderivative. ...
.
:
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
.
:
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''/''b ...
of corresponding partial decay constant:
. A combined
can be given in terms of
s:
:
:
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-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
can be shown to be
:
For a decay by three simultaneous exponential processes the total half-life can be computed as above:
:
Decay series / coupled decay
In
nuclear science and
pharmacokinetics
Pharmacokinetics (from Ancient Greek ''pharmakon'' "drug" and ''kinetikos'' "moving, putting in motion"; see chemical kinetics), sometimes abbreviated as PK, is a branch of pharmacology dedicated to determining the fate of substances administered ...
, 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
In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials sho ...
holds. For small samples, a more general analysis is necessary, accounting for a
Poisson process.
Natural sciences
*
Chemical reactions: The
rate
Rate or rates may refer to:
Finance
* Rates (tax), a type of taxation system in the United Kingdom used to fund local government
* Exchange rate, rate at which one currency will be exchanged for another
Mathematics and science
* Rate (mathema ...
s 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 break ...
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 substrate (chemistry), substrates, and the enzyme converts the substrates into different molecule ...
-
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 respecti ...
(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 a ...
(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 resistor
A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias activ ...
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.)
*
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'' som ...
:
Atmospheric pressure
Atmospheric pressure, also known as barometric pressure (after the barometer), is the pressure within the atmosphere of Earth. The standard atmosphere (symbol: atm) is a unit of pressure defined as , which is equivalent to 1013.25 millibar ...
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 conduction ...
: 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 on ...
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
Pharmacology is a branch of medicine, biology and pharmaceutical sciences concerned with drug or medication action, where a drug may be defined as any artificial, natural, or endogenous (from within the body) molecule which exerts a biochemi ...
and
toxicology: 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, (visible ...
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
A radionuclide (radioactive nuclide, radioisotope or radioactive isotope) is a nuclide that has excess nuclear energy, making it unstable. This excess energy can be used in one of three ways: emitted from the nucleus as gamma radiation; transfer ...
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 thermoelectric effect is the direct conversion of temperature differences to electric voltage and vice versa via a thermocouple. A thermoelectric device creates a voltage when there is a different temperature on each side. Conversely, whe ...
: 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 envelopes in
synthesizers. An
overdamped system will simply return to equilibrium via an exponential decay.
* Beer froth: Arnd Leike, of the
Ludwig Maximilian University of Munich
The Ludwig Maximilian University of Munich (simply University of Munich or LMU; german: link=no, Ludwig-Maximilians-Universität München) is a public research university in Munich, Bavaria, Germany. Originally established as the University of ...
, 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 cer ...
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, 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 ''internetworking, network of networks'' that consists ...
,
BGP, has to maintain a
routing table
In computer networking, a routing table, or routing information base (RIB), is a data table stored in a router or a network host that lists the routes to particular network destinations, and in some cases, metrics (distances) associated with t ...
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, and ...
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 referen ...
, location = New York , lccn = 75173716
External links
Exponential decay calculatorA stochastic simulation of exponential decay
Exponentials