# Can someone please explain the "infrared catastrophe"?

In my readings I've run into this idea of an "infrared catastrophe" associated with 1/f noise. As far as I can tell it is because when you graph the periodogram of the 1/f signal you see the PSD goes to infinity as frequency goes to 0. Not sure what that means practically though. If we are talking about a sound wave, does that mean the sound becomes infinitely loud at low frequencies? What is the "catastrophe"?

• You might find this helpful. It discusses the theoretical infrared divergence problem and demonstrates how it is avoided in one particular application (stochastic processes with 1/f noise).
– user27578
Commented Mar 12, 2014 at 23:03

The infrared catastrophy seems to be named after a mimic of the ultraviolet catastrophy of the black body radiation .

In physics, an infrared divergence or infrared catastrophe is a situation in which an integral, for example a Feynman diagram, diverges because of contributions of objects with very small energy approaching zero, or, equivalently, because of physical phenomena at very long distances.

The infrared (IR) divergence only appears in theories with massless particles (such as photons). They represent a legitimate effect that a complete theory often implies. One way to deal with it is to impose an infrared cutoff and take the limit as the cutoff approaches zero and/or refine the question. Another way is to assign the massless particle a fictitious mass, and then take the limit as the fictitious mass vanishes.

The divergence is usually in terms of particle number and not empirically troubling, in that all measurable quantities remain finite

So it is a problem in calculating measurable quantities.

$1\over f$ noise does have finite energy. It does not have an infrared catastrophe. The infrared catastrophe was the (unrealistic) result of an attempt to theoretically explain blackbody radiation. The result implied that blackbody radiation were infinitely powerful in the infrared. This did not appear to be the case, so this result was regarded as a catastrophe to the model. It was.

• The so called infrared catastrophy is not in black body radiation, there it is an ultraviolet catastrophy and it is avoided by positing the quantum theory. see my answer. Commented Mar 12, 2014 at 21:06
• @annav: actually, you're both right. There were a few pre-Planck attempts at describing blackbody radiation, each unsatisfactory in some way. Wien's law worked well at short wavelengths but diverged at long wavelengths. A few years later, the Rayleigh-Jeans law was developed which worked well at long wavelengths but diverged at short wavelengths. Planck's law resolved both the infrared and ultraviolet catatrophes.
– user27578
Commented Mar 12, 2014 at 22:57
• @user7358, though you are correct about blackbody radiation, you are incorrect about 1/f noise. A true 1/f spectrum does indeed have an infrared divergence.
– user27578
Commented Mar 12, 2014 at 23:02