Are there any examples of common substances whose decay is not exponential?

We're used to thinking about radioactivity in terms of half-lives. This is a concept that makes sense only for a decay that is exponential. However, there are plenty of physics articles on the subject of non exponential decay. It seems to be theoretically ubiquitous. For example:

The decay of unstable quantum states is an ubiquitous process in virtually all fields of physics and energy ranges, from particle and nuclear physics to condensed matter, or atomic and molecular science. The exponential decay, by far the most common type, is surrounded by deviations at short and long times$^{1,2}$. The short-time deviations have been much discussed, in particular in connection with the Zeno effect$^{3,4,5}$ and the anti-Zeno effect$^{6,7,8,9}$. Experimental observations of short$^{10,11}$ and long$^{12}$ time deviations are very recent. A difficulty for the experimental verification of long-time deviations has been the weakness of the decaying signal$^{13}$, but also the measurement itself may be responsible, because of the suppression of the initial state reconstruction$^{2,14}$.

1) L. A. Khalfin, Zurn. Eksp. Teor. Fiz. 33, 1371 (1957), English translation: Sov. Phys. JETP 6 1053 (1958).
2) L. Fonda and G. C. Ghirardi, Il Nuovo Cimento 7A, 180 (1972).

10.1103/PhysRevA.74.062102, F. Delgado, J. G. Muga, G. Garcia-Calderon
Suppression of Zeno effect for distant detectors

So are there any examples of deviations from long time decay? If not, then why not? Is the theory wrong or simply impractical? And is there a simple, intuitive explanation for why long decays should not be exponential?


3 Answers 3


To be honest, the paper listed as reference 12 in the OP's quote (1) gives as good an answer to most of these questions as you can hope to get:

For example, taking $\omega(E)$ as a Lorentzian function for all E yields the well-known exponential decay at all times. However, in real physical systems, $\omega(E)$ must always have a lower limit, which is, for example, associated with the rest mass of scattering particles.

In other words: a system with a perfectly exponentially decaying probability of remaining in the initial state, of the form


must necessarily have an energy distribution of that initial state that is exactly a Lorentzian:

$\omega(E)=\frac{1}{\sqrt{2 \pi}}\frac{2/\tau}{(1/\tau)^2+(E-E_0)^2}$

But this is a function that is nonzero on the entire interval $(-\infty,\infty)$, while any real system necessarily has a minimum energy. Using some fancier Fourier analysis (2), one can prove that this means that in the $t \rightarrow \infty$ limit, any decay must go slower than exponential, and it generally turns out to be a power law. As rob mentions, this non-exponential behavior can also be seen as the result of the inverse decay process becoming non-negligible, and that point of view is also discussed in (2).

Most of the theory and review papers on this stuff that I've seen dates back to the 50s-80s, and don't even speculate on experimental prospects. However, (1) from above claims to give the first observation (in 2006) of this power-law decay. The authors note that in searches for this behavior using radioactive decay, one must look for deviations at the level of $10^{-60}$ of the initial signal (!), so it's not surprising that no one has been successful. Instead, they use the decay of the excited state of some organic molecule instead, which is very broad compared to the resonant energy and as a result crosses into this nonexponential regime much sooner.

As a final side note, the OP doesn't ask about short time deviations from exponential decay, but it's easy to see where those come from too. Using

$P(t)=|\langle \psi_0 | e^{-i H t} | \psi_0 \rangle|^2$, and Taylor expanding the time evolution exponential, one finds

$P(t)\approx 1-\sigma_E^2 t^2$,

where $\sigma_E$ is the energy spread that $\psi_0$ has among the states of $H$. This is a very general consideration that applies to pretty much any time evolution.

  • 1
    $\begingroup$ I agree with this (+1), but I am a little confused by "while any real system necessarily has a minimum energy" - isn't the issue that the function needs to be zero for $\omega < 0$? Also, I don't think the Fourier analysis needs to be that fancy - doesn't the Paley-Wiener theorem give you what you want here? $\endgroup$ Jun 17, 2016 at 8:43
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    $\begingroup$ @WetSavannaAnimalakaRodVance: Yes, the statement that any system has a minimum energy is equivalently that $\omega(E)$ must vanish below some minimum (which we can take as zero). $\endgroup$
    – Rococo
    Jun 17, 2016 at 21:56
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    $\begingroup$ And yes, Paley-Wiener is the relevant theorem here (as described in the linked article). Perhaps this is a standard result; I described it as such simply because I had not previously come across it, nor is it immediately obvious to me why it should be true. If these are not the case for you, so much the better :) $\endgroup$
    – Rococo
    Jun 17, 2016 at 21:59

For very long times, a decay process starts to compete with the inverse process. For instance, right now you are bathed in an ocean of matter and antimatter neutrinos with lots of different energies. For a given beta-decaying nucleus, some fraction of these background neutrinos will have enough energy to drive the inverse decay process, transforming the "daughter" nucleus into the "parent." Thus if you start off with a population of parent nuclei, you don't necessarily end up with zero parent nuclei and all daughter nuclei, as pure exponential decay would predict; instead you end up with a tiny fraction of the parent nuclei remaining in the sample. The size of this steady-state fraction depends on the local neutrino density and energy spectrum. You can make the same argument for other decay modes.


http://arxiv.org/abs/1304.6885 There are many papers on apparent sinusoidal modulation of overall beta-decay. One explanation is competing pathways with different kinetics whose rates add to beats. The obvious non-exponential decay case is electron-capture decay of a fully ionized atom.


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