Editing Exponential decay (section)

== Solution of the differential equation ==

The equation that describes exponential decay is
:<math>\frac{dN(t)}{dt} = -\lambda N(t)</math>
or, by rearranging (applying the technique called [[separation of variables]]),
:<math>\frac{dN(t)}{N(t)} = -\lambda dt.</math>

Integrating, we have
:<math>\ln N = -\lambda t + C \,</math>
where C is the [[constant of integration]], and hence
:<math>N(t) = e^C e^{-\lambda t} = N_0 e^{-\lambda t} \,</math>
where the final substitution, ''N''<sub>0</sub> = ''e''<sup>''C''</sup>, is obtained by evaluating the equation at ''t'' = 0, as ''N''<sub>0</sub> is defined as being the quantity at ''t'' = 0.

This is the form of the equation that is most commonly used to describe exponential decay.  Any one of decay constant, mean lifetime, or half-life is sufficient to characterise the decay.  The notation λ for the decay constant is a remnant of the usual notation for an [[eigenvalue]].  In this case, λ is the eigenvalue of the [[additive inverse|negative]] of the [[differential operator]] with ''N''(''t'') as the corresponding [[eigenfunction]].

=== Derivation of the mean lifetime ===
Given an assembly of elements, the number of which decreases ultimately to zero, the '''mean lifetime''', <math>\tau</math>, (also called simply the '''lifetime''') is the [[expected value]] of the amount of time before an object is removed from 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.

Starting from the population formula

:<math>N = N_0 e^{-\lambda t}, \,</math>

first let ''c'' be the normalizing factor to convert to a [[probability density function]]:

:<math>1 = \int_0^\infty c \cdot N_0 e^{-\lambda t}\, dt = c \cdot \frac{N_0}{\lambda}</math>

or, on rearranging,

:<math>c = \frac{\lambda}{N_0}.</math>

Exponential decay is a [[scalar multiplication|scalar multiple]] of the [[exponential distribution]] (i.e. the individual lifetime of each object is exponentially distributed), which has a [[Exponential distribution#Properties|well-known expected value]].  We can compute it here using [[integration by parts]].

:<math>\tau = \langle t \rangle = \int_0^\infty t \cdot c \cdot N_0 e^{-\lambda t}\, dt = \int_0^\infty \lambda t e^{-\lambda t}\, dt = \frac{1}{\lambda}.</math>

=== Decay by two or more processes ===<!-- This section is linked from [[Half-life]] -->
{{see also|Branching fraction}}
A quantity may decay via two or more different processes simultaneously. In general, these processes (often called "decay modes", "decay channels", "decay routes" etc.) have different probabilities of occurring, and thus occur at different rates with different half-lives, in parallel. The total decay rate of the quantity&nbsp;''N'' is given by the ''sum'' of the decay routes; thus, in the case of two processes:

:<math>-\frac{dN(t)}{dt} = N\lambda _1 + N\lambda _2 = (\lambda _1 + \lambda _2)N.</math>

The solution to this equation is given in the previous section, where the sum of <math>\lambda _1 + \lambda _2\,</math> is treated as a new total decay constant <math>\lambda _c</math>.

:<math>N(t) = N_0 e^{-(\lambda _1 + \lambda _2) t} = N_0 e^{-(\lambda _c) t}.</math>

'''Partial mean life''' associated with individual processes is by definition the [[multiplicative inverse]] of corresponding partial decay constant: <math>\tau = 1/\lambda</math>. A combined <math>\tau_c</math> can be given in terms of <math>\lambda</math>s:

:<math>\frac{1}{\tau_c} = \lambda_c = \lambda_1 + \lambda_2 = \frac{1}{\tau_1} + \frac{1}{\tau_2}</math>

:<math>\tau_c = \frac{\tau_1 \tau_2}{\tau_1 + \tau_2}. </math>

Since half-lives differ from mean life <math>\tau</math> by a constant factor, the same equation holds in terms of the two corresponding half-lives:

:<math>T_{1/2} = \frac{t_1 t_2}{t_1 + t_2} </math>

where <math>T _{1/2}</math> is the combined or total half-life for the process, <math>t_1</math> and <math>t_2</math> are so-named '''partial half-lives''' of corresponding processes. Terms "partial half-life" and "partial mean life" denote quantities derived from a decay constant as if the given decay mode were the only decay mode for the quantity. The term "partial half-life" is misleading, because it cannot be measured as a time interval for which a certain quantity is [[one half|halved]].

In terms of separate decay constants, the total half-life <math>T _{1/2}</math> can be shown to be

:<math>T_{1/2} = \frac{\ln 2}{\lambda _c} = \frac{\ln 2}{\lambda _1 + \lambda _2}.</math>

For a decay by three simultaneous exponential processes the total half-life can be computed as above:

:<math>T_{1/2} = \frac{\ln 2}{\lambda _c} = \frac{\ln 2}{\lambda_1 + \lambda_2 + \lambda_3} = \frac{t_1 t_2 t_3}{(t_1 t_2) + (t_1 t_3) + (t_2 t_3)}.</math>

=== Decay series / coupled decay ===

In [[nuclear science]] and [[pharmacokinetics]], the agent of interest might be situated in a decay chain, where the accumulation is governed by exponential decay of a source agent, while the agent of interest itself decays by means of an exponential process. 

These systems are solved using the [[Bateman equation]]. 

In the pharmacology setting, some ingested substances might be absorbed into the body by a process reasonably modeled as exponential decay, or might be deliberately [[modified-release dosage|formulated]] to have such a release profile.