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[Phys-l] models of radioactivity (was: Poisson stat.)



On 04/17/2009 10:25 PM, Hugh Haskell wrote:

we know that half-life (T) can be
expressed as

T = (ln 2)/p (1)

What do you mean by "we", Kemosabe?

I don't know equation (1). I don't want to know it. It can't
possibly be true. Plug in p=.999 and "discover" that almost all
of the sample has disappeared in less than 1.5 half-lives, if you
believe equation (1). Not recommended.

There are some take-home messages here that can be stated quite
simply:
1) Plug and chug is a bad idea. Formulas such as equation (1) come
with provisos, and if you ignore the provisos you get nonsense.
2) Exponentials are nonlinear. Exponentials are transcendental.
You can tell at a glance that an algebraic, linearized expression
such as equation (1) cannot possibly be the whole story.

I agree that throwing dice to simulate radioactivity is a
pedagogically valuable tool. It is a good model of radioactivity.

OK.

Good, but not excellent.

If equation (1) is the basis for withholding the higher rating, I'm
not buying it. Equation (1) is a straw man, not a real critique.

If there is a real critique hiding somewhere, please explain.


=====================

The fact that dice-rolls are discrete in time whereas real radioactive
decay is considered continuous is also not a valid critique of the
dice model.

That's because if we look at the decay only at intervals separated
by Δt, then exp((i Δt)/τ) looks an awful lot like (exp(Δt/τ))^i
where the latter is just some number raised to the ith power, where
i is an integer. The smooth, ever-changing exponential curve goes
smack dab through all the points defined by the discrete, integer
exponential.

If you're looking for flaws in the model, discrete time is completely
the wrong place to look.