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:
where N(t) is the quantity at time t, and N0 = N(0) is the initial quantity, i.e. the quantity at time t = 0.
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 the exponential decay equation can be written in terms of the mean lifetime, τ, instead of the decay constant, λ:
and that τ is the time at which the population of the assembly is reduced to 1/e = 0.367879441 times its initial value.
For example, if the initial population of the assembly, N(0), is 1000, then the population at time τ, N(τ), is 368.
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".
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 t1/2. The half-life can be written in terms of the decay constant, or the mean lifetime, as:
When this expression is inserted for in the exponential equation above, and ln 2 is absorbed into the base, this equation becomes:
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