Mean lifetime

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Given an assembly of elements, the number of which decreases ultimately to zero, the lifetime (also called the mean lifetime) is a certain number that characterizes the rate of reduction ("decay") of the assembly. Specifically, if the individual lifetime of an element of the assembly is the time elapsed between some reference time and the removal of that element from the assembly, the mean lifetime is the arithmetic mean of the individual lifetimes.

Typically, the notion of mean lifetime is used in connection with exponential decay. The remainder of this article confines itself to this particular decay pattern.

Mean lifetime in exponential decay

The mean lifetime τ of elements in an exponentially decaying assembly is equal to the reciprocal of the decay constant (cf. Exponential decay). Thus, it is the time needed for the assembly to be reduced by a factor of e. It is related to the half-life [t_] by

[\tau \cdot \ln 2 = t_ ].

Derivation

For purposes of this derivation, the following terminology and notation (some of it introduced above) will be useful:

A large (but decreasing) number of elements is grouped into an assembly.
The individual lifetime of an element is the time elapsed between some reference time and the removal of that element from the assembly.
The total lifetime (T) is the sum of the individual lifetimes.
The mean lifetime (τ) is the arithmetic mean of the individual lifetimes.
The population (N) is the number of elements in the assembly at any given time.
The initial population ([N_0]) is the population at some initial reference time.

In exponential decay, the population is governed by the following formula:

[N = N_0 e^ \,]

During a differential period of time dt, the population shrinks slightly. The (positive) number of elements leaving the assembly is equal to the negative of the change in population:

[-dN = N_0\lambda e^ dt \,]

If dt is taken at some time t, the individual lifetimes of the elements are simply equal to t. These elements' contribution to the total lifetime is thus given by:

[dT = t \cdot -dN = t\cdot N_0\lambda e^ dt \,]

[\tau = \frac \int_^ dT = \frac \int_^ N_0\lambda e^\, t\, dt]

[\tau = \lambda \int_^ t\, e^ dt]

Integration by parts yields:

[\tau = \frac]

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