Exponential decay
A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. Symbolically, this can be expressed as the following
differential equation, where
N is the quantity and λ is a positive number called the decay constant:
The solution to this equation is :
Here N is the quantity at time t, and is the quantity, at time t=0.
This is the form of the equation that is most commonly used to describe exponential decay. The constant of integration denotes the original quantity at .
Encyclopedia
A quantity is said to be subject to
exponential decay if it decreases at a rate proportional to its value. Symbolically, this can be expressed as the following
differential equation, where
N is the quantity and λ is a positive number called the
decay constant:
The solution to this equation is :
Here N is the quantity at time
t, and is the quantity, at time
t=0.
This is the form of the equation that is most commonly used to describe exponential decay. The constant of integration denotes the original quantity at . .
Measuring rates of decay
Mean lifetime
If the decaying quantity is the number of discrete elements of a
set, it is possible to compute the average length of time for which an element remains in the set. This is called the
mean lifetime, and it can be shown that it relates to the decay rate,
The mean lifetime is thus seen to be a simple "scaling time":
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-life, and often denoted by the symbol . 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 ln2 is absorbed into the base, this equation becomes:
Thus, the amount of material left is raised to the number of half-lives that have passed. Thus, after 3 half-lives there will be of the original material left.
Solution of the differential equation
The equation that describes exponential decay is
Integrating, we have
were D is the constant of integration
where . If we evaluate this equation at , we see that .
Decay by two or more processes
A quantity may decay via two or more different processes simultaneously. These processes may have different probabilities of occurring, and thus will occur at different rates with different half-lives, in parallel. In the case of two simultaneous decay processes, the total decay rate of the quantity
N is given by the
sum of the two decay routes:
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:
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 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 reciprocal of a similar sum of three reciprocals:
Applications and examples
Exponential decay occurs in a wide variety of situations. Most of these fall into the domain of the
natural sciences. 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.
Natural sciences
- 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.
- 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 . See also Newton's law of cooling
...
.
- The rates of certain types of chemical reactions depend on the concentration of one or another reactant. These reaction rates consequently follow exponential decay. For instance, many enzyme-catalyzed reactions behave this way.
- Atmospheric pressure decreases approximately exponentially with increasing height above sea level, at a rate of about 12% per 1000m.
- The electric charge stored on a capacitor decays exponentially, if the capacitor experiences a constant external load . The exponential time-constant t for the process is R C, and the half-life is therefore R C ln2.
- Some vibrations may decay exponentially; this characteristic is often used in creating ADSR envelope
- LFO [i]
...
s in
synthesizers.
- In pharmacology and toxicology, it is found that many administered substances are distributed and metabolized according to exponential decay patterns. The "alpha half-life" and "beta half-life" of a substance measure how quickly a substance is distributed and eliminated.
- The intensity of electromagnetic radiation such as light or X-rays or gamma rays in an absorbant medium, follows an exponential decrease with distance into the absorbing medium.
Social sciences
- The popularity of fads, fashions and other cultural memes often decays exponentially.
- The field of glottochronology attempts to determine the time elapsed since the divergence of two languages from a common root, using the assumption that linguistic changes are introduced at a steady rate; given this assumption, we expect the similarity between them to decrease exponentially.
- In history of science, some believe that the body of knowledge of any particular science is gradually disproven according to an exponential decay pattern .
See also
