Exponential decay
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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:
- <math>\frac{dN}{dt} = -\lambda N.</math>
The solution to this equation is
- <math>N = Ce^{-\lambda t}. \,</math>
This is the form of the equation that is most commonly used to describe exponential decay. The constant of integration <math>C</math> is often written <math>N_0</math> since it denotes the original quantity.
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Measuring rates of decay: half-life and average lifetime
An important characteristic of exponential decay 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 <math>t_{1/2}</math>. The equation describing half-life is
- <math>t_{1/2} = \frac{\ln 2}{\lambda}.</math>
Some forms of exponential decay have an alternative characterization. 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 is described by the equation
- <math>\tau = \frac{1}{\lambda}.</math>
The following table shows the reduction of the quantity in terms of the number of half-lives elapsed.
| Half-lives | Percent of quantity remaining |
|---|---|
| 0 | 100% |
| 1 | 50 |
| 2 | 25 |
| 3 | 12.5 |
| 4 | 6.25 |
| 5 | 3.125 |
| 6 | 1.5625 |
| 7 | 0.78125% |
Solution of the differential equation
The equation that describes exponential decay is
- <math>\frac{dN}{dt} = -\lambda N</math>
- <math>\frac{dN}{N} = -\lambda dt.</math>
Integrating, we have
- <math>\ln N = -\lambda t + D \,</math>
- <math>N = Ce^{-\lambda t} \,</math>
where <math>C = e^D</math>. If we evaluate this equation at <math>t=0</math>, we see that also <math>C = N_0</math>.
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. For instance, in the case of two simultaneous decay processes, the decay of the quantity N is given by:
- <math>N(t) = N_0 e^{-\lambda _1 t} e^{-\lambda _2 t} = N_0 e^{-(\lambda _1 + \lambda _2) t}</math>
The total half-life <math>T _{1/2}</math> can be shown to be:
- <math>T_{1/2} = \frac{\ln 2}{\lambda _1 + \lambda _2} \,</math>
or, in terms of the two half-lives:
- <math>T_{1/2} = \frac{t _1 t _2}{t _1 + t_2} \,</math>
where <math>t _1</math> is the half-life of the first process, and <math>t _2</math> is the half life of the second process.
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 science
- In a sample of radionuclides or other particles that undergo radioactive decay to a different state, the number of particles in the original state follows exponential decay, as long as there are many radionuclides remaining.
- 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, enzyme-catalyzed reactions behave this way.
- Atmospheric pressure decreases exponentially with increasing height above sea level, at a rate of about 12% per 1000m.
- The electric charge (or, equivalently, the potential) stored on a capacitor decays exponentially, if the capacitor experiences a constant external load. (Furthermore, the particular case of a capacitor discharging through several parallel resistors 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 decay exponentially; this characteristic is often used in creating ADSR envelopes in synthesizers.
- In pharmacology and toxicology, it is found that many substances spread through the body, and are metabolized (see clearance), according to exponential decay patterns. The "alpha half-life" and "beta half-life" of a substance measure how quickly a substance spreads and is eliminated.
Social science
- The popularity of fads, fashions and other cultural memes (for instance, attendance of popular films) 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 (the number of properties of the language that are still identical) 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 half-life of knowledge).