Exponential decay

A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where N is the quantity and λ (lambda) is a positive rate called the exponential decay constant:

The solution to this equation (see derivation below) is:

Here N(t) is the quantity at time t, and N₀ = N(0) is the initial quantity, i.e. the quantity at time t = 0.

Measuring rates of decay

Mean lifetime

If the decaying quantity, N(t), is the number of discrete elements in a certain set, it is possible to compute the average length of time that an element remains in the set. This is called the mean lifetime (or simply the lifetime or the exponential time constant), τ, and it can be shown that it relates to the decay rate, λ, in the following way:

The mean lifetime can be looked at as a "scaling time", because we can write the exponential decay equation in terms of the mean lifetime, τ, instead of the decay constant, λ: