Non-exponential growth and decay

All the quantities we have seen so far, such as populations and radioactive samples, grow or decay exponentially. That is, they are exponential functions of time \(t\). However, many quantities in the world grow or decay in a non-exponential fashion.

For instance, the electric field strength \(E\) of a charged particle decreases rapidly as you move away from the particle. At a distance of \(r\) from the particle, the field's strength is

\[ E = \dfrac{C}{r^2}, \]

where \(C\) is a constant. The electric field strength is said to obey an inverse-square law, rather than an exponential law; the 'decay' involves \(r^{-2}\) rather than \(e^{-r}\).

Many other quantities in the world — such as gravitational force, sound intensity and radio signal strength — obey inverse-square laws.

Another interesting example is Zipf's law, which concerns spoken language. Suppose we rank all the words in a language according to their frequency as spoken. Zipf's law states that the frequency \(f\) in spoken language of the word ranked \(r\) is inversely proportional to \(r\). That is,

\[ f = \dfrac{C}{r}, \]

for some constant \(C\). In other words, word frequency decreases in inverse proportion to rank. (See Wikipedia Open new window for more on Zipf's law.)

In addition to these examples, many quantities grow non-exponentially. For instance, if you travel in a straight line at constant velocity \(v\), then your displacement \(s\) grows linearly with time, \(s = vt\). If you travel in a straight line, initially stationary, with constant acceleration \(a\), then your velocity \(v\) grows linearly with time, \(v=at\), and your displacement grows quadratically with time, \(s = \dfrac{1}{2} at^2\). See the module Motion in a straight line for further details.

When two quantities \(x\) and \(y\) are related by an equality of the form \(y = C x^\alpha\), where \(C\) and \(\alpha\) are non-zero constants, we say they obey a power-law relationship. The examples from this section are all power-law relationships:

Exponential growth or decay is not a power-law relationship.

We will consider power-law relationships further in the section History and applications (Logarithmic plots).

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