What are the equations for the quadratic and power laws in unstable systems?

[Trollfaz]

http://www.nat.vu.nl/~tvisser/nonexponential.pdf
I always thought that an unstable system will decay exponentially but I recently learned that it obeys the quadratic law in the short time and power law in the long time. Can somebody tell me the equation that governs the quadratic law and power law?

 

[fresh_42]

Without reading the paper or knowing which part you refer to, I would interpret it as follows:
The general behavior is exponential, which means ##t \longmapsto e^t = 1 + t + \frac{1}{2}t^2 + \frac{1}{6}t^3 + \frac{1}{24}t^4 + \ldots##. This works in all cases. Now if we consider short term behavior, which means values of ##t## close to zero, it is sufficient to look at the leading terms, e.g. until the order of two which would be the quadratic approximation: ##t \longmapsto 1 + t + \frac{1}{2}t^2 +R## with a pretty small value ##R##. The long term behavior, i.e. values of ##t## greater than one, on the other hand requires to consider the entire series, which means an exponential behavior.
Of course one could as well consider other numbers ##a \neq e## as basis, or add a factor ##c\cdot e^t##, but this doesn't change the qualitative statement given. And in some cases (very small values for ##t \approx 0##), it might be sufficient to only look at the linear approximation ##t \longmapsto 1+t +R'##.

 

[jtbell]

And in some cases (very small values for ##t \approx 0##),

Or very small compared to the half-life, ##t \ll t_{1/2}##.

 

[mfb]

Not every transition between states follows an exponential law. Transitions between different atomic states in an external field are one example, neutral meson oscillations are another one. They can have sinusoidal oscillations between the states, and if you approximate this for very small time differences, it follows a quadratic law (with zero linear coefficient) as ##\cos(x) \approx 1 - \frac 1 2 x^2##. The Quantum Zeno effect uses such a system to suppress changes by repeated observations.

 

[Trollfaz]

So some transitions will purely follow the exponential law while others will deviate at short and long times?

 

[fresh_42]

So some transitions will purely follow the exponential law while others will deviate at short and long times?

No. In general they follow a single rule, not necessarily but most often. The point is that many processes, some listed above, can be approximated by linear or quadratic equations in certain ranges. E.g. ##t_{1/2} = 4,500,000,000 ## years for ##{}^{238}U##. If one has a sample in the laboratory and you measure ##\alpha -##decays, then the result will only depend on the amount of isotopes and only linearly on time, i.e. you have a linear behavior although the overall rule is not. You simply can't measure the terms of higher order in a short time. @mfb gave you an example of an oscillating process which can be approximated by a quadratic equation. It simply makes no sense to calculate with ##10## valid digits if the experiment can only measure two.

 

[Trollfaz]

The Quantum Zeno effect uses such a system to suppress changes by repeated observations

I suppose that only certain interactions is able to reset the decay. Say repeatedly shooting a photon
at an excited electron that is about to decay will keep it excited.

And fresh_42 does this mean that the probability that the atom decay at any given time changes?

 

[fresh_42]

And fresh_42 does this mean that the probability that the atom decay at any given time changes?

Not at any given time, but over a given time span. The longer you observe the sample, the more isotopes will have been decayed (radiocarbon dating). A single isotope doesn't know anything about the past or the future (Schrödinger's cat). You can only have statistical statements about many of them or about a time span, usually both.