Re: Simulating radioactive decay.

[brian whatcott <inet@INTELLISYS.NET>]

At 10:29 5/15/99 -0400, Robert Cohen wrote:
>  A Taylor series expansion of ln(1-p) gives...
>
>   ln(1-p) = - p - p*p/2 - p*p*p/3 - p*p*p*p/4 - ...
>
> Thus, replacing ln(1-p) with p is only valid when
>
>        p/2 + p*p/3 + p*p*p/4 + ... << 1


> ... one could then ask why we use   T =  ln(2) / p
> instead of the more accurate        T = -ln(2) / ln(1-p) ?
>
> The answer, I believe, is that the expression comes from:
>     dN/dt = - p N = - lambda N
>
> which assumes continuous changes in N, not discrete changes.
> Robert Cohen

Robert's excellent note here reminds me irresistibly of the
epitome of a physicist's fruitful mode of progression,
using appropriate expansions and approximations.

 Feynman's biographical book mentions a case where he needed
to remind a bombmaker of a case where the higher orders of
an expansion could not be ignored.

  The continuous expression that Robert cites is of course
due to Rutherford and Soddy in their 1902 paper. They modelled
this discrete radioactive decay process with a continuous
equation.

The numbers in their populations were high enough to provide
the desired level of accuracy.
brian whatcott <inet@intellisys.net>
Altus OK