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:

-\frac{dN}{dt} = \lambda N.

The solution to this equation is [1]:

N(t) = N_0 e^{-\lambda t}. \,

Here N(t) is the quantity at time t, and $N_0 = N(0)$ is the (initital) 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 $N_0$ denotes the original quantity at $t=0$ . (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 $N(t)$ as the corresponding eigenfunction).

Plot-exponential-decay — Large decay constants make the quantity vanish almost immediately; smaller decay constants lead to almost-imperceptible decrease. This plot shows decay for decay constants of 25, 5, 1, 1/5, and 1/25. Plot generated using MAPLE.

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,

\tau = \frac{1}{\lambda}.

The mean lifetime (also called the exponential time constant) is thus seen to be a simple "scaling time":

N(t) = N_0 e^{-t/\tau}. \,

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 $t_{1/2}$ . The half-life can be written in terms of the decay constant, or the mean lifetime, as:

t_{1/2} = \frac{\ln 2}{\lambda} = \tau \ln 2.

When this expression is inserted for $\tau$ in the exponential equation above, and ln2 is absorbed into the base, this equation becomes:

N(t) = N_0 2^{-t/t_{1/2}}. \,

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.

Solution of the differential equation[]

The equation that describes exponential decay is

-\frac{dN(t)}{dt} = \lambda N(t)

\frac{dN(t)}{N(t)} = -\lambda dt.

Integrating, we have

\ln N(t) = -\lambda t + D \,

were D is the constant of integration

N(t) = Ce^{-\lambda t} \,

where $C = e^D$ . If we evaluate this equation at $t=0$ , we see that $e^D = C = N_0$ .

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:

-\frac{dN(t)}{dt} = N\lambda _1 + N\lambda _2 = (\lambda _1 + \lambda _2)N\,

The solution to this equation is given in the previous section, where the sum of $\lambda _1 + \lambda _2\,$ is treated as a new total decay constant $\lambda _c\,$ .

N(t) = N_0 e^{-(\lambda _1 + \lambda _2) t} = N_0 e^{-(\lambda _c) t}  \,

Since $\tau = 1/\lambda\,$ , a combined $\tau_c\,$ can be given in terms of $\lambda\,$ s:

\frac{1}{\tau_c} = \lambda_c = \lambda_1 + \lambda_2 = \frac{1}{\tau_1} + \frac{1}{\tau_2}\,

\tau_c = \frac{\tau_1 \tau_2}{\tau_1 + \tau_2} \,

Since half-lives differ from mean life $\tau \,$ by a constant factor, the same equation holds in terms of the two corresponding half-lives:

T_{1/2} = \frac{t _1 t _2}{t _1 + t_2} \,

where $T_{1/2}$ is the combined or total half-life for the process, and $t_1$ is the half-life of the first process, and $t_2$ is the half life of the second process.

In terms of separate decay constants, the total half-life $T_{1/2}$ can be shown to be:

T_{1/2} = \frac{\ln 2}{\lambda _c} = \frac{\ln 2}{\lambda _1 + \lambda _2} \,

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:

T_{1/2} = \frac{t _1 t _2 t _3}{(t _1 t_2) + (t _1 t_3) + (t _2 t _3)} = \frac{\ln 2}{\lambda _c} = \frac{\ln 2}{\lambda _1 + \lambda _2 + \lambda _3} \,

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 (in the limit of slow processes; equivalent to "good" heat conduction inside the object, so that its temperature remains relatively uniform throught its volume). 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 (or, equivalently, the potential) stored on a capacitor (capacitance C) decays 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. (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.)

Some vibrations may decay exponentially; this characteristic is often used in creating ADSR envelopes in synthesizers.

In pharmacology and toxicology, it is found that many administered substances are distributed and metabolized (see clearance) 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 (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).