Exponential growth: Meaning (information, definition, explanation, facts)

In mathematics, a quantity that grows exponentially is one that grows at a rate proportional to its size. Anything that grows by the same percentage every year (or every month, day, hour etc.) is growing exponentially. For example, if the average number of offspring of each individual (or couple) in a population remains constant, the growth is proportional to the number of individuals. Such an exponentially growing population grows three times as fast in individuals per year when there are six million individuals, as it does when there are two million. Bank accounts with fixed-rate compound interest grow exponentially provided there are no deposits, withdrawals or service charges.

Misnomer

The phrase exponential growth is also incorrectly used by persons not versed in quantitative matters to mean merely surprisingly fast growth (a potential malapropism). In fact, a population can grow exponentially but at a very slow rate (while the population is small, for instance), and can grow surprisingly fast without growing exponentially. Indeed, the logistic function grows approximately exponentially when it is growing very slowly, but nowhere near exponentially when it is growing fastest.

If we call x this quantity, the rate of change dx/dt obeys by definition the differential equation:

dx / dt = κx

where κ > 0 is the constant of proportionality (the average number of offspring in the case of the population). (See logistic function for a simple correction of this model growth where κ is not constant). The solution to this equation is the exponential function x(t) = Ce^κt, whence the name of the associated growth. C here is an arbitrary constant, determined by the initial size of the population.

In the long run, exponential growth of any kind will however overtake linear growth of any kind (the basis of the Malthusian catastrophe) as well as any polynomial growth, i.e., for all α:

There is a whole hierarchy of conceivable growth laws that are sub-exponential and also super-linear, and of course growth faster than exponential is also possible. The linear and exponential models are merely simple candidates but are those of greatest occurrence in nature.

• Investing. The effect of compound interest over many years has a substantial effect on savings and a person's ability to retire. See also rule of 72
• Biology.
  • Bacteria in a culture dish will grow exponentially until the available food is exhausted.
  • A new virus (SARS, West Nile, smallpox) of sufficient infectivity (κ > 0) will spread exponentially. Each infected person can infect multiple new people.
  • Human population.
• An atomic bomb. Each uranium atom that undergoes fission produces multiple neutrons, each of which can be absorbed by adjacent uranium atoms, causing them to fission in turn. If the probability of neutron absorption exceeds the probability of neutron escape (a function of the shape and mass of the uranium), κ > 0 and so the production rate of neutrons and induced uranium fissions increases exponentially.
• Engineering
  • Processing power of computers. See also Moore's law.
  • Charging and discharging of capacitors and changes in current in inductors are also exponential growth and decay phenomena. Engineers use a rule of five time constants to estimate when a steady state has been reached.