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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dN}{dt} = -\lambda N.}

The solution to this equation is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N = Ce^{-\lambda t}. \,}

This is the form of the equation that is most commonly used to describe exponential decay. The constant of integration Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} is often written Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_0} since it denotes the original quantity.

Solution of the differential equation

The equation that describes exponential decay is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dN}{dt} = -\lambda N}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dN}{N} = -\lambda dt.}

Integrating, we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ln N = -\lambda t + D \,}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N = Ce^{-\lambda t} \,}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C = e^D.}

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_{1/2}} . The equation describing half-life is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_{1/2} = \frac{\ln 2}{\lambda}.}

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau = \frac{1}{\lambda}.}

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%

Applications

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 broadly exponential phenomena have been identified there as well.

  • 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.
  • 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.
  • In pharmacology and toxicology, it is found that the body metabolizes many administered substances in an exponential manner. The "half-life" of a drug is a measure of how quickly the drug is metabolized by the body.
  • 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.
  • Atmospheric pressure decreases exponentially with increasing height above sea level, at a rate of about 12% per 1000m.
  • The popularity of fads, fashions and other cultural memes (for instance, attendance of popular films) often decays exponentially.

See also